[{"id":"arxiv:0907.4207","type":"article-journal","author":[{"given":"Cédric","family":"Bény"}],"title":"Conditions for the approximate correction of algebras","issued":{"date-parts":[[2009,7,24]]},"note":"arxivid:0907.4207\narxiv_version_number:1"},{"id":"arxiv:0907.5391","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.104.120501","source":"Crossref","title":"General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels","volume":"104","author":[{"given":"Cédric","family":"Bény","sequence":"first","affiliation":[]},{"given":"Ognyan","family":"Oreshkov","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,3,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.104.120501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120501","note":"arxivid:0907.5391"},{"id":"arxiv:quant-ph/9806029","type":"article-journal","author":[{"given":"Dorit","family":"Aharonov"},{"given":"Alexei","family":"Kitaev"},{"given":"Noam","family":"Nisan"}],"title":"Quantum Circuits with Mixed States","issued":{"date-parts":[[1998,6,8]]},"note":"arxivid:quant-ph/9806029\narxiv_version_number:1"},{"id":"arxiv:1806.10324","type":"article-journal","author":[{"given":"Cédric","family":"Bény"},{"given":"Zoltán","family":"Zimborás"},{"given":"Fernando","family":"Pastawski"}],"title":"Approximate recovery with locality and symmetry constraints","issued":{"date-parts":[[2019,4,12]]},"note":"arxivid:1806.10324\narxiv_version_number:2"},{"id":"arxiv:0711.3438","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.78.032330","source":"Crossref","title":"Complementarity of private and correctable subsystems in quantum cryptography and error correction","volume":"78","author":[{"given":"Dennis","family":"Kretschmann","sequence":"first","affiliation":[]},{"given":"David W.","family":"Kribs","sequence":"additional","affiliation":[]},{"given":"Robert W.","family":"Spekkens","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,9,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.032330","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032330","note":"arxivid:0711.3438"},{"id":"arxiv:1411.7041","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/jhep04(2015)163","source":"Crossref","title":"Bulk locality and quantum error correction in AdS/CFT","volume":"2015","author":[{"given":"Ahmed","family":"Almheiri","sequence":"first","affiliation":[]},{"given":"Xi","family":"Dong","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Harlow","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2015,4]]},"URL":"http://dx.doi.org/10.1007/JHEP04(2015)163","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"163","note":"alternative-id:1201\narxivid:1411.7041"},{"id":"arxiv:1612.00017","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.7.021022","source":"Crossref","title":"Code Properties from Holographic Geometries","volume":"7","author":[{"given":"Fernando","family":"Pastawski","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2017,5,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.7.021022","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021022","note":"arxivid:1612.00017"},{"id":"arxiv:2012.14001","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>There are two parts to this work: first, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large <jats:italic>N</jats:italic> and a large gap in the spectrum of operators in the emergence of complementary recovery.</jats:p><jats:p>Second, we study the operator algebra exact quantum error correction for any von Neumann algebra. We show that similar to the finite dimensional case, for any error map in between von Neumann algebras the Petz dual of the error map is a recovery map if the inclusion of the correctable subalgebra of operators has finite index.</jats:p>","DOI":"10.1007/jhep01(2022)170","source":"Crossref","title":"Real-space RG, error correction and Petz map","volume":"2022","author":[{"given":"Keiichiro","family":"Furuya","sequence":"first","affiliation":[]},{"given":"Nima","family":"Lashkari","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4143-5276","authenticated-orcid":false,"given":"Shoy","family":"Ouseph","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,1]]},"URL":"http://dx.doi.org/10.1007/JHEP01(2022)170","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"170","note":"alternative-id:17662\narxivid:2012.14001"},{"id":"arxiv:2203.01379","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevd.106.126006","source":"Crossref","title":"Code properties of the holographic Sierpinski triangle","volume":"106","author":[{"given":"Ning","family":"Bao","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9732-5300","authenticated-orcid":true,"given":"Joydeep","family":"Naskar","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.106.126006","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"126006","note":"arxivid:2203.01379"},{"id":"doi:10.1007/978-3-642-18245-7","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-18245-7","source":"Crossref","title":"A Panoramic View of Riemannian Geometry","author":[{"given":"Marcel","family":"Berger","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2003]]},"ISBN":["9783540653172","9783642182457"],"URL":"http://dx.doi.org/10.1007/978-3-642-18245-7","note":"publisher-location:Berlin, Heidelberg"},{"id":"preset:Vilenkin78","type":"book","author":[{"family":"Vilenkin","given":"N.I."}],"title":"\\emph{Special Functions and the Theory of Group Representations}","volume":"22","publisher":"American Mathematical Society","issued":"1978"},{"id":"arxiv:1009.4375","type":"article-journal","author":[{"given":"Boaz","family":"Barak"},{"given":"Zeev","family":"Dvir"},{"given":"Avi","family":"Wigderson"},{"given":"Amir","family":"Yehudayoff"}],"title":"Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes","issued":{"date-parts":[[2011,3,10]]},"note":"arxivid:1009.4375\narxiv_version_number:2"},{"id":"doi:10.1109/TIT.1973.1054954","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1973.1054954","page":"92-95","source":"Crossref","title":"A class of codes for asymmetric channels and a problem from the additive theory of numbers","volume":"19","author":[{"given":"R.","family":"Varshamov","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1973,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1973.1054954","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ISIT.2011.6033693","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2011.6033693","page":"1061-1065","source":"Crossref","title":"On L&lt;inf&gt;1&lt;/inf&gt;-distance error control codes","author":[{"given":"Luca G.","family":"Tallini","sequence":"first","affiliation":[]},{"given":"Bella","family":"Bose","sequence":"additional","affiliation":[]}],"event":"2011 IEEE International Symposium on Information Theory - ISIT","container-title":"2011 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2011,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2011.6033693"},{"id":"manual:-D.-J.-C.-MacKay-J.-Sayir-and","type":"manuscript","author":[{"family":"MacKay","given":"D.J.C."},{"family":"Sayir","given":"J."},{"family":"Goldman","given":"N."}],"title":"Near-Capacity Codes for Fountain Channels with Insertions, Deletions, and Substitutions, with Applications to DNA Archives","note":"Unpublished manuscript, 2015"},{"id":"arxiv:1612.08837","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2017.2789292","page":"5156-5169","source":"Crossref","title":"Codes in the Space of Multisets—Coding for Permutation Channels With Impairments","volume":"64","author":[{"ORCID":"https://orcid.org/0000-0002-2395-7628","authenticated-orcid":false,"given":"Mladen","family":"Kovacevic","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5008-4527","authenticated-orcid":false,"given":"Vincent Y. F.","family":"Tan","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2017.2789292","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1612.08837"},{"id":"doi:10.1109/TIT.2018.2791990","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2018.2791990","page":"1595-1603","source":"Crossref","title":"Lattice Codes for Deletion and Repetition Channels","volume":"64","author":[{"given":"Lin","family":"Sok","sequence":"first","affiliation":[]},{"given":"Jean-Claude","family":"Belfiore","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4078-8301","authenticated-orcid":false,"given":"Patrick","family":"Sole","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5752-943X","authenticated-orcid":false,"given":"Aslan","family":"Tchamkerten","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2018.2791990","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.79947","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.79947","page":"778-788","source":"Crossref","title":"Generating functions and lower bounds on rates for limited error-correcting codes","volume":"37","author":[{"given":"V.D.","family":"Kolesnik","sequence":"first","affiliation":[{"name":"LIAP, Leningrad, USSR"}]},{"given":"V.Y.","family":"Krachkovsky","sequence":"additional","affiliation":[{"name":"LIAP, Leningrad, USSR"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1991,5]]},"URL":"http://dx.doi.org/10.1109/18.79947","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.256522","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.256522","page":"1089-1093","source":"Crossref","title":"A generalized Gilbert-Varshamov bound derived via analysis of a code-search algorithm","volume":"39","author":[{"given":"J.","family":"Gu","sequence":"first","affiliation":[]},{"given":"T.","family":"Fuja","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1993,5]]},"URL":"http://dx.doi.org/10.1109/18.256522","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.623158","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.623158","page":"1605-1606","source":"Crossref","title":"The generalized Gilbert-Varshamov bound is implied by Turan's theorem [code construction]","volume":"43","author":[{"given":"L.M.G.M.","family":"Tolhuizen","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1109/18.623158","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2402.14712","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2024.3483303","page":"244-262","source":"Crossref","title":"Gilbert–Varshamov Bound for Codes in <i>L</i>₁ Metric Using Multivariate Analytic Combinatorics","volume":"71","author":[{"ORCID":"https://orcid.org/0000-0003-3027-9789","authenticated-orcid":false,"given":"Keshav","family":"Goyal","sequence":"first","affiliation":[{"name":"School of Physical and Mathematical Sciences, Nanyang Technological University, Nanyang Avenue, Singapore"}]},{"ORCID":"https://orcid.org/0009-0006-6310-8141","authenticated-orcid":false,"given":"Duc","family":"Tu Dao","sequence":"additional","affiliation":[{"name":"Genome Institute of Singapore, Fusionopolis Way, Singapore"}]},{"ORCID":"https://orcid.org/0000-0002-2395-7628","authenticated-orcid":false,"given":"Mladen","family":"Kovačević","sequence":"additional","affiliation":[{"name":"Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia"}]},{"ORCID":"https://orcid.org/0000-0001-5611-0848","authenticated-orcid":false,"given":"Han","family":"Mao Kiah","sequence":"additional","affiliation":[{"name":"School of Physical and Mathematical Sciences, Nanyang Technological University, Nanyang Avenue, Singapore"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2025,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3483303","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2402.14712"},{"id":"doi:10.1007/BF02566968","type":"journal-article","publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"1","DOI":"10.1007/bf02566968","page":"141-147","source":"Crossref","title":"Theorems in the additive theory of numbers","volume":"37","author":[{"given":"R. C.","family":"Bose","sequence":"first","affiliation":[]},{"given":"S.","family":"Chowla","sequence":"additional","affiliation":[]}],"container-title":"Commentarii Mathematici Helvetici","original-title":[],"language":"en","issued":{"date-parts":[[1962,12]]},"URL":"http://dx.doi.org/10.1007/BF02566968","ISSN":["0010-2571","1420-8946"],"container-title-short":"Commentarii Mathematici Helvetici","note":"alternative-id:BF02566968"},{"id":"arxiv:2509.20545","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>\n                    We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces—composite permutation-invariant spaces of many qubits or qudits, composite constant-excitation Fock state spaces of many bosonic modes, and monolithic nuclear state spaces of atoms, ions, and molecules. By identifying all three spaces with discrete simplices and representations of the Lie group\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>S</a:mi>\n                      <a:mi>U</a:mi>\n                      <a:mo stretchy=\"false\">(</a:mo>\n                      <a:mi>q</a:mi>\n                      <a:mo stretchy=\"false\">)</a:mo>\n                    </a:math>\n                    , we prove that many codes and their gates in\n                    <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <e:mi>S</e:mi>\n                      <e:mi>U</e:mi>\n                      <e:mo stretchy=\"false\">(</e:mo>\n                      <e:mi>q</e:mi>\n                      <e:mo stretchy=\"false\">)</e:mo>\n                    </e:math>\n                    can be interconverted between the three state spaces. We construct code instances for all three spaces using classical\n                    <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <i:msub>\n                        <i:mi>ℓ</i:mi>\n                        <i:mn>1</i:mn>\n                      </i:msub>\n                    </i:math>\n                    codes and Tverberg’s theorem, a classic result from convex geometry. We obtain families of quantum codes with distance that scales almost linearly with the code length\n                    <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <k:mi>N</k:mi>\n                    </k:math>\n                    by constructing\n                    <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <m:msub>\n                        <m:mi>ℓ</m:mi>\n                        <m:mn>1</m:mn>\n                      </m:msub>\n                    </m:math>\n                    codes based on combinatorial patterns called Sidon sets and utilizing their Tverberg partitions. This compares favorably with the existing designs for all the state spaces. We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters, bosonic codes with exotic Gaussian gates, as well as examples of short codes with distance larger than the known constructions.\n                  </jats:p>","DOI":"10.1103/kx3b-4nrp","source":"Crossref","title":"Quantum Error Correction beyond\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <mml:mi>S</mml:mi>\n                      <mml:mi>U</mml:mi>\n                      <mml:mo stretchy=\"false\">(</mml:mo>\n                      <mml:mn>2</mml:mn>\n                      <mml:mo stretchy=\"false\">)</mml:mo>\n                    </mml:math>\n                    : Spin, Bosonic, and Permutation-Invariant Codes from Convex Geometry","volume":"7","author":[{"ORCID":"https://orcid.org/0009-0003-9960-1841","authenticated-orcid":true,"given":"Arda","family":"Aydin","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"College Park"}]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"}]},{"ORCID":"https://orcid.org/0000-0002-8972-4413","authenticated-orcid":true,"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"College Park"},{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,2,27]]},"URL":"http://dx.doi.org/10.1103/kx3b-4nrp","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010341","note":"arxivid:2509.20545"},{"id":"doi:10.1007/b100498","type":"book","publisher":"Springer US","DOI":"10.1007/b100498","source":"Crossref","title":"Coded Modulation Systems","author":[{"given":"John B.","family":"Anderson","sequence":"first","affiliation":[]},{"given":"Arne","family":"Svensson","sequence":"additional","affiliation":[]}],"container-title":"Information Technology: Transmission, Processing and Storage","original-title":[],"language":"en","editor":[{"given":"Jack Keil","family":"Wolf","sequence":"additional","affiliation":[]},{"given":"Robert J.","family":"McEliece","sequence":"additional","affiliation":[]},{"given":"John","family":"Proakis","sequence":"additional","affiliation":[]},{"given":"William H.","family":"Tranter","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2002]]},"ISBN":["9780306472794","9780306477928"],"URL":"http://dx.doi.org/10.1007/b100498","ISSN":["1389-6938"],"note":"publisher-location:Boston, MA"},{"id":"doi:10.1017/9781316822708","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Written in the intuitive yet rigorous style that readers of A Foundation in Digital Communication have come to expect, this second edition includes entirely new chapters on the radar problem (with Lyapunov's theorem) and intersymbol interference channels, new discussion of the baseband representation of passband noise, and a simpler, more geometric derivation of the optimal receiver for the additive white Gaussian noise channel. Other key topics covered include the definition of the power spectral density of nonstationary stochastic processes, the geometry of the space of energy-limited signals, the isometry properties of the Fourier transform, and complex sampling. Including over 500 homework problems and all the necessary mathematical background, this is the ideal text for one- or two-semester graduate courses on digital communications and courses on stochastic processes and detection theory. Solutions to problems and video lectures are available online.</jats:p>","DOI":"10.1017/9781316822708","source":"Crossref","title":"A Foundation in Digital Communication","author":[{"given":"Amos","family":"Lapidoth","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2017,2,8]]},"ISBN":["9781107177321","9781316822708"],"URL":"http://dx.doi.org/10.1017/9781316822708","note":"edition-number:2"},{"id":"manual:-J.-G.-Proakis-and-M.-Salehi.","type":"article-journal","author":[{"family":"Proakis","given":"J.G."},{"family":"Salehi","given":"M."}],"volume":"4","publisher":"McGraw-hill","container-title":"Digital communications","issued":"2001","publisher-place":"New York"},{"id":"doi:10.1016/B978-0-12-373580-5.X5033","type":"edited-book","publisher":"Elsevier","DOI":"10.1016/b978-0-12-373580-5.x5033-9","source":"Crossref","title":"Wireless Communications &amp; Networking","container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2007]]},"ISBN":["9780123735805"],"URL":"http://dx.doi.org/10.1016/B978-0-12-373580-5.X5033-9"},{"id":"doi:10.1109/49.29618","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/49.29618","page":"968-973","source":"Crossref","title":"A Leech lattice modem","volume":"7","author":[{"given":"G.R.","family":"Lang","sequence":"first","affiliation":[]},{"given":"F.M.","family":"Longstaff","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[1989]]},"URL":"http://dx.doi.org/10.1109/49.29618","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1109/ALLERTON.2010.5706965","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/allerton.2010.5706965","page":"620-628","source":"Crossref","title":"The impact of constellation cardinality on Gaussian channel capacity","author":[{"given":"Yihong","family":"Wu","sequence":"first","affiliation":[]},{"given":"Sergio","family":"Verdu","sequence":"additional","affiliation":[]}],"event":"2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)","container-title":"2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)","original-title":[],"issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1109/ALLERTON.2010.5706965"},{"id":"arxiv:1603.05970","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.95.062343","source":"Crossref","title":"Coherent-state constellations and polar codes for thermal Gaussian channels","volume":"95","author":[{"given":"Felipe","family":"Lacerda","sequence":"first","affiliation":[]},{"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[]},{"given":"Volkher B.","family":"Scholz","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,6,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.95.062343","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062343","note":"arxivid:1603.05970"},{"id":"doi:10.1109/ISIT.2016.7541749","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2016.7541749","page":"2499-2503","source":"Crossref","title":"Coherent state constellations for Bosonic Gaussian channels","author":[{"given":"Felipe","family":"Lacerda","sequence":"first","affiliation":[]},{"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[]},{"given":"Volkher B.","family":"Scholz","sequence":"additional","affiliation":[]}],"event":"2016 IEEE International Symposium on Information Theory (ISIT)","container-title":"2016 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2016,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2016.7541749"},{"id":"doi:10.1109/ITW.1998.706453","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.1998.706453","page":"98-99","source":"Crossref","title":"Improving the performance of asymmetric M-PAM signal constellations in Euclidean space by embedding them in hyperbolic space","author":[{"given":"E.B.","family":"Da Silva","sequence":"first","affiliation":[]},{"given":"R.","family":"Palazzo","sequence":"additional","affiliation":[]},{"given":"S.R.","family":"Costa","sequence":"additional","affiliation":[]}],"event":"1998 Information Theory Workshop","container-title":"1998 Information Theory Workshop (Cat. No.98EX131)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ITW.1998.706453"},{"id":"doi:10.1017/CBO9780511811401","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>At the heart of any modern communication system is the modem, connecting the data source to the communication channel. This first course in the mathematical theory of modem design introduces the theory of digital modulation and coding that underpins the design of digital telecommunications systems. A detailed treatment of core subjects is provided, including baseband and passband modulation and demodulation, equalization, and sequence estimation. The modulation waveforms for communication channels and digital recording channels are treated in a common setting and with unified terminology. A variety of more advanced topics is also covered, such as trellis codes, turbo codes, the Viterbi algorithm, block codes, maximum likelihood and maximum posterior probability, iterative demodulation, and jamming. Numerous end-of-chapter exercises are also included to test the reader's understanding throughout. This insightful book is ideal for senior undergraduate students studying digital communications and is also a useful reference for practising engineers.</jats:p>","DOI":"10.1017/cbo9780511811401","source":"Crossref","title":"Modem Theory","author":[{"given":"Richard E.","family":"Blahut","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2009,11,26]]},"ISBN":["9780521780148","9780511811401"],"URL":"http://dx.doi.org/10.1017/CBO9780511811401","note":"edition-number:1"},{"id":"doi:10.1109/JLT.2015.2510034","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/jlt.2015.2510034","page":"1599-1609","source":"Crossref","title":"Rate Adaptation and Reach Increase by Probabilistically Shaped 64-QAM: An Experimental Demonstration","volume":"34","author":[{"given":"Fred","family":"Buchali","sequence":"first","affiliation":[{"name":"Alcatel-Lucent Bell Labs, Stuttgart, Germany"}]},{"given":"Fabian","family":"Steiner","sequence":"additional","affiliation":[{"name":"Fachgebiet Methoden der Signalverarbeitung, Technical University of Munich, Munich, Germany"}]},{"given":"Georg","family":"Bocherer","sequence":"additional","affiliation":[{"name":"Institute for Communications Engineering, Technical University of Munich, Munich, Germany"}]},{"given":"Laurent","family":"Schmalen","sequence":"additional","affiliation":[{"name":"Alcatel-Lucent Bell Labs, Stuttgart, Germany"}]},{"given":"Patrick","family":"Schulte","sequence":"additional","affiliation":[{"name":"Institute for Communications Engineering, Technical University of Munich, Munich, Germany"}]},{"given":"Wilfried","family":"Idler","sequence":"additional","affiliation":[{"name":"Alcatel-Lucent Bell Labs, Stuttgart, Germany"}]}],"container-title":"Journal of Lightwave Technology","original-title":[],"issued":{"date-parts":[[2016,4,1]]},"URL":"http://dx.doi.org/10.1109/JLT.2015.2510034","ISSN":["0733-8724","1558-2213"],"container-title-short":"J. Lightwave Technol."},{"id":"manual:-International-Telecommunicat","type":"document","title":"International Telecommunication Union-T, Recommendation V.29: 9600 Bits Per Second Modem Standardized For Use on Point-to-Point 4-Wire Leased Telephone-Tpe Circuits","issued":"1993"},{"id":"manual:-E.-B.-Silva-R.-Palazzo-Jr.-a","type":"paper-conference","author":[{"family":"Silva","given":"E.B."},{"family":"Palazzo","given":"R.","suffix":"Jr."},{"family":"Firer","given":"M."}],"title":"Performance analysis of QAM-like constellations in hyperbolic space","container-title":"2000 International Symposium on Information Theory and its Applications","issued":"2000","publisher-place":"Honolulu, USA"},{"id":"doi:10.1109/icc.1994.368804","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.1994.368804","page":"645-649","source":"Crossref","title":"Turbo-codes and high spectral efficiency modulation","author":[{"given":"S.","family":"Le Goff","sequence":"first","affiliation":[]},{"given":"A.","family":"Glavieux","sequence":"additional","affiliation":[]},{"given":"C.","family":"Berrou","sequence":"additional","affiliation":[]}],"event":"ICC/SUPERCOMM'94 - 1994 International Conference on Communications","container-title":"Proceedings of ICC/SUPERCOMM'94 - 1994 International Conference on Communications","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/icc.1994.368804"},{"id":"arxiv:cond-mat/0607736","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.75.075103","source":"Crossref","title":"Exact topological quantum order in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math>and beyond: Branyons and brane-net condensates","volume":"75","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2007,2,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.75.075103","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"075103","note":"arxivid:cond-mat/0607736"},{"id":"doi:10.1109/TCOM.1983.1095820","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tcom.1983.1095820","page":"458-461","source":"Crossref","title":"Redundancy, the Discrete Fourier Transform, and Impulse Noise Cancellation","volume":"31","author":[{"given":"J.","family":"Wolf","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[1983,3]]},"URL":"http://dx.doi.org/10.1109/TCOM.1983.1095820","ISSN":["0096-2244"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/JSAC.1984.1146063","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/jsac.1984.1146063","page":"381-392","source":"Crossref","title":"Coding of Real-Number Sequences for Error Correction: A Digital Signal Processing Problem","volume":"2","author":[{"given":"T.","family":"Marshall","sequence":"first","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"language":"en","issued":{"date-parts":[[1984,3]]},"URL":"http://dx.doi.org/10.1109/JSAC.1984.1146063","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.4028/www.scientific.net/AMR.3","type":"journal-article","publisher":"Trans Tech Publications, Ltd.","abstract":"<jats:p>We proposed a novel decoding algorithm for Analog Codes (Reed-Solomon Codes over complex numbers), the <jats:italic>syndrome repairing</jats:italic> (SR) algorithm, for gross error correction in signal transmission. Simulations show that, if the number of gross errors is not too large and the amplitude of background noise is small enough (compared to the amplitude of gross errors), the SR algorithm recovers the original signal with nearly the same accuracy as if no gross errors occur upon transmission. In particular, if the transmission is background-noise-free, then the recovery is exact.</jats:p>","DOI":"10.4028/www.scientific.net/amr.341-342.514","page":"514-518","source":"Crossref","title":"Analog Codes for Gross Error Correction in Signal Transmission","volume":"341-342","author":[{"given":"Fang Ning","family":"Hu","sequence":"first","affiliation":[{"name":"Hangzhou Dianzi University"}]},{"given":"Werner","family":"Henkel","sequence":"additional","affiliation":[{"name":"Jacobs University Bremen"}]},{"given":"Ming Jie","family":"Zhao","sequence":"additional","affiliation":[{"name":"University of Manchester"}]}],"container-title":"Advanced Materials Research","original-title":[],"issued":{"date-parts":[[2011,9,27]]},"URL":"http://dx.doi.org/10.4028/www.scientific.net/AMR.341-342.514","ISSN":["1662-8985"],"container-title-short":"AMR"},{"id":"doi:10.1109/ICASSP.2008.4518494","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icassp.2008.4518494","page":"3853-3856","source":"Crossref","title":"Explicit measurements with almost optimal thresholds for compressed sensing","author":[{"given":"Farzad","family":"Parvaresh","sequence":"first","affiliation":[]},{"given":"Babak","family":"Hassibi","sequence":"additional","affiliation":[]}],"event":"ICASSP 2008 - 2008 IEEE International Conference on Acoustics, Speech and Signal Processing","container-title":"2008 IEEE International Conference on Acoustics, Speech and Signal Processing","original-title":[],"issued":{"date-parts":[[2008,3]]},"URL":"http://dx.doi.org/10.1109/ICASSP.2008.4518494","ISSN":["1520-6149"]},{"id":"manual:-M.-Mohamed-S.-Rizkalla-H.-Zo","type":"paper-conference","author":[{"family":"Mohamed","given":"M."},{"family":"Rizkalla","given":"S."},{"family":"Zoerlein","given":"H."},{"family":"Bossert","given":"M."}],"title":"Deterministic Compressed Sensing with Power Decoding for Complex Reed-Solomon Codes","container-title":"SCC 2015; 10th International ITG Conference on Systems, Communications and Coding","issued":"2015","page":"1–6","publisher-place":"Hamburg, Germany"},{"id":"manual:-W.-Henkel-(2000).-Analog-cod","type":"chapter","author":[{"family":"Henkel","given":"W."}],"title":"Analog codes for peak-to-average ratio reduction","container-title":"ITG FACHBERICHT","issued":"2000","page":"151–156"},{"id":"preset:MacSlo","type":"book","author":[{"family":"MacWilliams","given":"F.J."},{"family":"Sloane","given":"N.J.A."}],"title":"\\emph{The Theory of Error-Correcting Codes}","publisher":"Elsevier","issued":"1977"},{"id":"doi:10.1007/BF01232378","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01232378","page":"309-313","source":"Crossref","title":"The antipode construction for sphere packings","volume":"123","author":[{"given":"J. H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Inventiones Mathematicae","original-title":[],"language":"en","issued":{"date-parts":[[1996,12]]},"URL":"http://dx.doi.org/10.1007/BF01232378","ISSN":["0020-9910","1432-1297"],"container-title-short":"Invent Math","note":"alternative-id:BF01232378"},{"id":"doi:10.1007/978-1-4757-6568-7","type":"book","publisher":"Springer New York","DOI":"10.1007/978-1-4757-6568-7","source":"Crossref","title":"Sphere Packings, Lattices and Groups","author":[{"given":"J. H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Grundlehren der mathematischen Wissenschaften","original-title":[],"issued":{"date-parts":[[1999]]},"ISBN":["9781441931344","9781475765687"],"URL":"http://dx.doi.org/10.1007/978-1-4757-6568-7","ISSN":["0072-7830"],"note":"publisher-location:New York, NY"},{"id":"doi:10.4153/CJM-1967-017-0","type":"journal-article","publisher":"Canadian Mathematical Society","abstract":"<jats:p>These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.</jats:p><jats:p>The principal results of these notes are the following. New sphere packings are given in [2<jats:sup><jats:italic>m</jats:italic></jats:sup>], <jats:italic>m</jats:italic> ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2<jats:sup><jats:italic>m</jats:italic></jats:sup>], <jats:italic>m</jats:italic> ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.</jats:p>","DOI":"10.4153/cjm-1967-017-0","page":"251-267","source":"Crossref","title":"Notes on Sphere Packings","volume":"19","author":[{"given":"John","family":"Leech","sequence":"first","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1967]]},"URL":"http://dx.doi.org/10.4153/CJM-1967-017-0","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j. math.","note":"alternative-id:S0008414X00054250"},{"id":"preset:EricZin","type":"book","editor":[{"family":"Ericson","given":"T."},{"family":"Zinoviev","given":"V."}],"title":"\\emph{Codes on Euclidean Spheres}","publisher":"Elsevier","issued":"2001"},{"id":"doi:10.1137/0605031","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/0605031","page":"294-305","source":"Crossref","title":"On the Voronoi Regions of Certain Lattices","volume":"5","author":[{"given":"J. H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Algebraic Discrete Methods","original-title":[],"language":"en","issued":{"date-parts":[[1984,9]]},"URL":"http://dx.doi.org/10.1137/0605031","ISSN":["0196-5212","2168-345X"],"container-title-short":"SIAM. J. on Algebraic and Discrete Methods","note":"alternative-id:10.1137/0605031"},{"id":"doi:10.4153/CJM-1971-081-3","type":"journal-article","publisher":"Canadian Mathematical Society","issue":"4","abstract":"<jats:p>Error-correcting codes are used in several constructions for packings of equal spheres in <jats:italic>n</jats:italic>-dimensional Euclidean spaces <jats:italic>E<jats:sup>n</jats:sup>.</jats:italic> These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2<jats:sup><jats:italic>m</jats:italic></jats:sup> for <jats:italic>m</jats:italic> ≧ 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in <jats:italic>E<jats:sup>n</jats:sup></jats:italic> for <jats:italic>n</jats:italic> = <jats:italic>2<jats:sup>m</jats:sup></jats:italic> satisfies</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00053724_eqn1\"/></jats:disp-formula></jats:p><jats:p>In this paper we make systematic use of error-correcting codes to obtain sphere packings in <jats:italic>E<jats:sup>n</jats:sup>,</jats:italic> including several of the densest packings known and several new packings.</jats:p>","DOI":"10.4153/cjm-1971-081-3","page":"718-745","source":"Crossref","title":"Sphere Packings and Error-Correcting Codes","volume":"23","author":[{"given":"John","family":"Leech","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1971,8]]},"URL":"http://dx.doi.org/10.4153/CJM-1971-081-3","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j. math.","note":"alternative-id:S0008414X00053724"},{"id":"doi:10.1017/CBO9780511807077","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Fundamentals of Error Correcting Codes is an in-depth introduction to coding theory from both an engineering and mathematical viewpoint. As well as covering classical topics, there is much coverage of techniques which could only be found in specialist journals and book publications. Numerous exercises and examples and an accessible writing style make this a lucid and effective introduction to coding theory for advanced undergraduate and graduate students, researchers and engineers, whether approaching the subject from a mathematical, engineering or computer science background.</jats:p>","DOI":"10.1017/cbo9780511807077","source":"Crossref","title":"Fundamentals of Error-Correcting Codes","author":[{"given":"W. Cary","family":"Huffman","sequence":"first","affiliation":[]},{"given":"Vera","family":"Pless","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2003,6,26]]},"ISBN":["9780521782807","9780521131704","9780511807077"],"URL":"http://dx.doi.org/10.1017/CBO9780511807077","note":"edition-number:1"},{"id":"preset:Sole93","type":"chapter","author":[{"family":"Sole","given":"P."}],"title":"Generalized theta functions for lattice vector quantization","container-title":"Coding and Quantization, DIMACS Series in Dr,crete Mathenulies and Theoretical Computer Science","volume":"14","publisher":"American Math. Soc","issued":"1993","page":"27–32","publisher-place":"Providence, RH"},{"id":"doi:10.1109/18.370138","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.370138","page":"366-377","source":"Crossref","title":"Quaternary quadratic residue codes and unimodular lattices","volume":"41","author":[{"given":"A.","family":"Bonnecaze","sequence":"first","affiliation":[]},{"given":"P.","family":"Sole","sequence":"additional","affiliation":[]},{"given":"A.R.","family":"Calderbank","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1995,3]]},"URL":"http://dx.doi.org/10.1109/18.370138","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/3-540-57843-9_20","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-57843-9_20","page":"194-205","source":"Crossref","title":"Quaternary constructions of formally self-dual binary codes and unimodular lattices","author":[{"given":"Alexis","family":"Bonnecaze","sequence":"first","affiliation":[]},{"given":"Patrick","family":"Solé","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[1994]]},"ISBN":["9783540578437","9783540483571"],"URL":"http://dx.doi.org/10.1007/3-540-57843-9_20","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1016/S0012-365X(99)00032-1","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/s0012-365x(99)00032-1","page":"1-21","source":"Crossref","title":"Niemeier lattices and Type II codes over Z4","volume":"205","author":[{"given":"Alexis","family":"Bonnecaze","sequence":"first","affiliation":[]},{"given":"Philippe","family":"Gaborit","sequence":"additional","affiliation":[]},{"given":"Masaaki","family":"Harada","sequence":"additional","affiliation":[]},{"given":"Masaaki","family":"Kitazume","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Solé","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1999,7]]},"URL":"http://dx.doi.org/10.1016/S0012-365X(99)00032-1","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X99000321"},{"id":"doi:10.1006/eujc.1999.0360","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1006/eujc.1999.0360","page":"473-485","source":"Crossref","title":"Z4-Code Constructions for the Niemeier Lattices and their Embeddings in the Leech Lattice","volume":"21","author":[{"given":"Masaaki","family":"Harada","sequence":"first","affiliation":[]},{"given":"Masaaki","family":"Kitazume","sequence":"additional","affiliation":[]}],"container-title":"European Journal of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2000,5]]},"URL":"http://dx.doi.org/10.1006/eujc.1999.0360","ISSN":["0195-6698"],"container-title-short":"European Journal of Combinatorics","note":"alternative-id:S0195669899903608"},{"id":"preset:Bambah54","type":"paper-conference","author":[{"family":"Bambah","given":"R.P."},{"family":"Gupta","given":"H."}],"title":"On lattice coverings by spheres","container-title":"Proceedings of the National Institute of Sciences of India","volume":"20","publisher":"Indian National Science Academy","issued":"1954"},{"id":"manual:-B.-N.-Delaunay-and-S.-S.-Rys","type":"article-journal","author":[{"family":"Delaunay","given":"B.N."},{"family":"Ryskov","given":"S.S."}],"title":"Solution of the problem of least dense lattice covering of a four-dimensional space by equal spheres","volume":"4","container-title":"Soviet Mathematics Doklady","issued":"1963"},{"id":"manual:-S.-S.-Ryshkov-and-E.-P.-Bara","type":"article-journal","author":[{"family":"Ryshkov","given":"S.S."},{"family":"Baranovskii","given":"E.P."}],"title":"Solution of the problem of least dense lattice covering of five-dimensional space by equal spheres","volume":"222","container-title":"Doklady Akademii Nauk SSSR","issue":"1","issued":"1975","page":"39–42"},{"id":"manual:-S.-S.-Ryshkov-and-E.-P.-Bara","type":"article-journal","author":[{"family":"Ryshkov","given":"S.S."},{"family":"Baranovskii","given":"E.P."}],"title":"C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings)","volume":"137","container-title":"Proceedings of the Steklov Institute of Mathematics","issued":"1976","page":"3–131"},{"id":"doi:10.1137/0604005","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","DOI":"10.1137/0604005","page":"30-41","source":"Crossref","title":"The Optimal Lattice Quantizer in Three Dimensions","volume":"4","author":[{"given":"E. S.","family":"Barnes","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Algebraic Discrete Methods","original-title":[],"language":"en","issued":{"date-parts":[[1983,3]]},"URL":"http://dx.doi.org/10.1137/0604005","ISSN":["0196-5212","2168-345X"],"container-title-short":"SIAM. J. on Algebraic and Discrete Methods","note":"alternative-id:10.1137/0604005"},{"id":"arxiv:1311.0879","type":"article-journal","author":[{"given":"H.","family":"Bombin"}],"title":"Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes","issued":{"date-parts":[[2015,8,6]]},"note":"arxivid:1311.0879\narxiv_version_number:6"},{"id":"arxiv:math/0001038","type":"article-journal","author":[{"given":"Gabriele","family":"Nebe"},{"given":"E. M.","family":"Rains"},{"given":"N. J. A.","family":"Sloane"}],"title":"The invariants of the Clifford groups","issued":{"date-parts":[[2000,10,25]]},"note":"arxivid:math/0001038\narxiv_version_number:2"},{"id":"arxiv:0712.1939","type":"article-journal","author":[{"given":"Christine","family":"Bachoc"}],"title":"Designs, groups and lattices","issued":{"date-parts":[[2007,12,12]]},"note":"arxivid:0712.1939\narxiv_version_number:1"},{"id":"arxiv:2404.17677","type":"article-journal","author":[{"given":"Vadym","family":"Kliuchnikov"},{"given":"Sebastian","family":"Schönnenbeck"}],"title":"Stabilizer operators and Barnes-Wall lattices","issued":{"date-parts":[[2024,5,30]]},"note":"arxivid:2404.17677\narxiv_version_number:2"},{"id":"doi:10.1017/S1446788700025064","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"1","abstract":"<jats:p>Let <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_inline1\" /> be a positive definite quadratic form of determinant <jats:italic>D</jats:italic>, and let <jats:italic>M</jats:italic> be the minimum of <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>) for integral x ≠ 0. Then we set <jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_eqn1\" /></jats:disp-formula> and <jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1446788700025064_eqn2\" /></jats:disp-formula> the maximum being over all positive forms <jats:italic>f</jats:italic> in <jats:italic>n</jats:italic> variables. <jats:italic>f</jats:italic> is said to be extreme if <jats:italic>y</jats:italic> γ<jats:sub>n</jats:sub>(<jats:italic>f</jats:italic>) is a local maximum for varying <jats:italic>f</jats:italic>, <jats:italic>absolutely extreme</jats:italic> if <jats:italic>y</jats:italic> γ(<jats:italic>f</jats:italic>) is an absolute maximum, i.e. if <jats:italic>y</jats:italic> γ(f) = γ<jats:sub>n</jats:sub>.</jats:p>","DOI":"10.1017/s1446788700025064","page":"47-63","source":"Crossref","title":"Some extreme forms defined in terms of Abelian groups","volume":"1","author":[{"given":"E. S.","family":"Barnes","sequence":"first","affiliation":[]},{"given":"G. E.","family":"Wall","sequence":"additional","affiliation":[]}],"container-title":"Journal of the Australian Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1959,8]]},"URL":"http://dx.doi.org/10.1017/S1446788700025064","ISSN":["0004-9735"],"container-title-short":"J. Aust. Math. Soc.","note":"alternative-id:S1446788700025064"},{"id":"doi:10.24033/asens.1241","type":"journal-article","publisher":"Societe Mathematique de France","issue":"1","DOI":"10.24033/asens.1241","page":"17-51","source":"Crossref","title":"Une famille infinie de formes quadratiques entières; leurs groupes d'automorphismes","volume":"6","author":[{"given":"Michel","family":"Broué","sequence":"first","affiliation":[]},{"given":"Michel","family":"Enguehard","sequence":"first","affiliation":[]}],"container-title":"Annales scientifiques de l'École normale supérieure","original-title":[],"issued":{"date-parts":[[1973]]},"URL":"http://dx.doi.org/10.24033/asens.1241","ISSN":["0012-9593","1873-2151"],"container-title-short":"Ann. Sci. École Norm. Sup."},{"id":"doi:10.1049/el:19840043","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"2","DOI":"10.1049/el:19840043","page":"62-63","source":"Crossref","title":"Error control codes for QAM signalling","volume":"20","author":[{"given":"E.L.","family":"Cusack","sequence":"first","affiliation":[{"name":"British Telecom Research Laboratories, Martlesham Heath, Ipswich, Suffolk IP5 7RE, UK"}]}],"container-title":"Electronics Letters","original-title":[],"language":"en","issued":{"date-parts":[[1984,1,19]]},"URL":"http://dx.doi.org/10.1049/el:19840043","ISSN":["0013-5194","1350-911X"],"container-title-short":"Electron. Lett. (UK)","note":"alternative-id:10.1049/el:19840043"},{"id":"doi:10.1109/18.21245","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.21245","page":"1123-1151","source":"Crossref","title":"Coset codes. I. Introduction and geometrical classification","volume":"34","author":[{"given":"G.D.","family":"Forney","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1988,9]]},"URL":"http://dx.doi.org/10.1109/18.21245","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Sidelnikov97","type":"article-journal","author":[{"family":"Sidelnikov","given":"V.M."}],"title":"On a finite group of matrices and codes on the Euclidean sphere (in Russian","volume":"33","container-title":"Problemy Peredachi Informatsii","issued":"1997","page":"35–54"},{"id":"doi:10.1109/ISIT.1997.613373","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.1997.613373","page":"436","source":"Crossref","title":"On a finite group of matrices generating orbit codes on Euclidean sphere","author":[{"given":"V.M.","family":"Sidelnikov","sequence":"first","affiliation":[]}],"event":"IEEE International Symposium on Information Theory","container-title":"Proceedings of IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.1997.613373"},{"id":"doi:10.2307/2370513","type":"journal-article","publisher":"JSTOR","issue":"1","DOI":"10.2307/2370513","page":"1","source":"Crossref","title":"Determination of All Primitive Collineation Groups in More than Four Variables which Contain Homologies","volume":"36","author":[{"given":"Howard H.","family":"Mitchell","sequence":"first","affiliation":[]}],"container-title":"American Journal of Mathematics","original-title":[],"issued":{"date-parts":[[1914,1]]},"URL":"http://dx.doi.org/10.2307/2370513","ISSN":["0002-9327"],"container-title-short":"American Journal of Mathematics"},{"id":"doi:10.1017/S0305004100060746","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"3","abstract":"<jats:p>This paper studies the Coxeter–Todd lattice <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004100060746_inline1\" /> its automorphism group (which is Mitchell's reflection group 6·<jats:italic>PSU</jats:italic>(4, 3)·2), and the associated 12-dimensional real lattice <jats:italic>K</jats:italic><jats:sub>12</jats:sub>. We give several constructions for <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004100060746_inline1\" />, which is a <jats:italic>Z</jats:italic>[ω]-lattice where ω = <jats:italic>e</jats:italic><jats:sup>2πi/3</jats:sup>; enumerate the congruence classes of <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004100060746_inline2\" /> and <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0305004100060746_inline3\" /> where θ = ω − ω¯; prove the lattice is unique; determine its covering radius and deep holes; and study its connections with the lattice <jats:italic>E</jats:italic><jats:sub>6</jats:sub> and the Leech lattice. A number of new dense lattices in dimensions up to about 10<jats:sup>7</jats:sup> are constructed. We also give an explicit basis for the invariants of the Mitchell group. The paper concludes with an extensive bibliography.</jats:p>","DOI":"10.1017/s0305004100060746","page":"421-440","source":"Crossref","title":"The Coxeter–Todd lattice, the Mitchell group, and related sphere packings","volume":"93","author":[{"given":"J. H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Mathematical Proceedings of the Cambridge Philosophical Society","original-title":[],"language":"en","issued":{"date-parts":[[1983,5]]},"URL":"http://dx.doi.org/10.1017/S0305004100060746","ISSN":["0305-0041","1469-8064"],"container-title-short":"Math. Proc. Camb. Phil. Soc.","note":"alternative-id:S0305004100060746"},{"id":"doi:10.4153/CJM-1953-043-4","type":"journal-article","publisher":"Canadian Mathematical Society","abstract":"<jats:p>Let f(x<jats:sub>1</jats:sub>, … , x<jats:sub>n</jats:sub>) be a positive definite quadratic\n                        form of determinant Δ; let M be its minimum value for integers\n                        x<jats:sub>1</jats:sub>, … , x<jats:sub>n</jats:sub> not all zero; and let 2<jats:sub>s</jats:sub> be the\n                        number of times this minimum is attained, i.e., the number of solutions of\n                        the Diophantine equation</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00044072_eqn01\" /></jats:disp-formula></jats:p>","DOI":"10.4153/cjm-1953-043-4","page":"384-392","source":"Crossref","title":"An Extreme Duodenary Form","volume":"5","author":[{"given":"H. S. M.","family":"Coxeter","sequence":"first","affiliation":[]},{"given":"J. A.","family":"Todd","sequence":"additional","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1953]]},"URL":"http://dx.doi.org/10.4153/CJM-1953-043-4","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j.\n                        math.","note":"alternative-id:S0008414X00044072"},{"id":"doi:10.1007/BF01232233","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01232233","page":"27-56","source":"Crossref","title":"On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane)","volume":"117","author":[{"given":"P.","family":"Buser","sequence":"first","affiliation":[]},{"given":"P.","family":"Sarnak","sequence":"additional","affiliation":[]}],"container-title":"Inventiones Mathematicae","original-title":[],"language":"en","issued":{"date-parts":[[1994,12]]},"URL":"http://dx.doi.org/10.1007/BF01232233","ISSN":["0020-9910","1432-1297"],"container-title-short":"Invent Math","note":"alternative-id:BF01232233"},{"id":"preset:Gosset1900","type":"article-journal","author":[{"family":"Gosset","given":"T."}],"title":"On the regular and semi-regular figures in space of n dimensions","volume":"29","container-title":"Messenger of Mathematics","issued":"1900","page":"43–48"},{"id":"preset:Blichfeldt25","type":"article-journal","author":[{"family":"Blichfeldt","given":"H.F."}],"title":"On the minimum value of positive real quadratic forms in 6 variables","volume":"31","container-title":"Bulletin of American Math. Soc","issued":"1925","page":"386"},{"id":"doi:10.1007/BF01454863","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01454863","page":"605-608","source":"Crossref","title":"The minimum value of quadratic forms, and the closest packing of spheres","volume":"101","author":[{"given":"H. F.","family":"Blichfeldt","sequence":"first","affiliation":[]}],"container-title":"Mathematische Annalen","original-title":[],"language":"en","issued":{"date-parts":[[1929,12]]},"URL":"http://dx.doi.org/10.1007/BF01454863","ISSN":["0025-5831","1432-1807"],"container-title-short":"Math. Ann.","note":"alternative-id:BF01454863"},{"id":"doi:10.1007/BF01201341","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01201341","page":"1-15","source":"Crossref","title":"The minimum values of positive quadratic forms in six, seven and eight variables","volume":"39","author":[{"given":"H. F.","family":"Blichfeldt","sequence":"first","affiliation":[]}],"container-title":"Mathematische Zeitschrift","original-title":[],"language":"en","issued":{"date-parts":[[1935,12]]},"URL":"http://dx.doi.org/10.1007/BF01201341","ISSN":["0025-5874","1432-1823"],"container-title-short":"Math Z","note":"alternative-id:BF01201341"},{"id":"doi:10.1112/plms/s3-13.1.549","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1112/plms/s3-13.1.549","page":"549-576","source":"Crossref","title":"The Class-Number of a Positive Quadratic Form","volume":"s3-13","author":[{"given":"G. L.","family":"Watson","sequence":"first","affiliation":[{"name":"University College; London"}]}],"container-title":"Proceedings of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1963]]},"URL":"http://dx.doi.org/10.1112/plms/s3-13.1.549","ISSN":["0024-6115"],"container-title-short":"Proceedings of the London Mathematical Society"},{"id":"arxiv:1603.06518","type":"journal-article","publisher":"Annals of Mathematics","issue":"3","DOI":"10.4007/annals.2017.185.3.8","source":"Crossref","title":"The sphere packing problem in dimension $24$","volume":"185","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[{"name":"Microsoft Research New England, Cambridge, MA"}]},{"given":"Abhinav","family":"Kumar","sequence":"additional","affiliation":[{"name":"Stony Brook University Stony Brook, NY"}]},{"given":"Stephen","family":"Miller","sequence":"additional","affiliation":[{"name":"Rutgers University, Piscataway, NJ"}]},{"given":"Danylo","family":"Radchenko","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Mathematics, Bonn, Germany"}]},{"given":"Maryna","family":"Viazovska","sequence":"additional","affiliation":[{"name":"Berlin Mathematical School and Humboldt University of Berlin, Berlin, Germany"}]}],"container-title":"Annals of Mathematics","original-title":[],"issued":{"date-parts":[[2017,5,1]]},"URL":"http://dx.doi.org/10.4007/annals.2017.185.3.8","ISSN":["0003-486X"],"container-title-short":"Ann. of Math. (2)","note":"arxivid:1603.06518"},{"id":"manual:-N.-M.-Vetchinkin-Uniqueness-","type":"document","author":[{"family":"Vetchinkin","given":"N.M."}],"title":"Uniqueness of classes of positive quadratic and highest kissing number of 240 in eight dimensions"},{"id":"arxiv:1902.05438","type":"article-journal","author":[{"given":"Henry","family":"Cohn"},{"given":"Abhinav","family":"Kumar"},{"given":"Stephen D.","family":"Miller"},{"given":"Danylo","family":"Radchenko"},{"given":"Maryna","family":"Viazovska"}],"title":"Universal optimality of the $E_8$ and Leech lattices and interpolation formulas","issued":{"date-parts":[[2022,6,10]]},"note":"arxivid:1902.05438\narxiv_version_number:3"},{"id":"arxiv:math/0607446","type":"journal-article","publisher":"American Mathematical Society (AMS)","issue":"1","abstract":"<p>We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\n  <mml:semantics>\n    <mml:mi>m</mml:mi>\n    <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> distances between distinct points in it and it is a spherical <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 2 m minus 1 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mo stretchy=\"false\">(</mml:mo>\n      <mml:mn>2</mml:mn>\n      <mml:mi>m</mml:mi>\n      <mml:mo>−</mml:mo>\n      <mml:mn>1</mml:mn>\n      <mml:mo stretchy=\"false\">)</mml:mo>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">(2m-1)</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>-design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E 8\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi>E</mml:mi>\n      <mml:mn>8</mml:mn>\n    </mml:msub>\n    <mml:annotation encoding=\"application/x-tex\">E_8</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> and Leech lattices. We also prove the same result for the vertices of the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"600\">\n  <mml:semantics>\n    <mml:mn>600</mml:mn>\n    <mml:annotation encoding=\"application/x-tex\">600</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>-cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact two-point homogeneous spaces, and we conclude with an extension to Euclidean space.</p>","DOI":"10.1090/s0894-0347-06-00546-7","page":"99-148","source":"Crossref","title":"Universally optimal distribution of points on spheres","volume":"20","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[]},{"given":"Abhinav","family":"Kumar","sequence":"additional","affiliation":[]}],"container-title":"Journal of the American Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[2006,9,5]]},"URL":"http://dx.doi.org/10.1090/S0894-0347-06-00546-7","ISSN":["0894-0347","1088-6834"],"container-title-short":"J. Amer. Math. Soc.","note":"alternative-id:S0894-0347-06-00546-7\narxivid:math/0607446"},{"id":"arxiv:2508.04819","type":"article-journal","author":[{"given":"Sayan","family":"Chakraborty"},{"given":"Victor V.","family":"Albert"}],"title":"Hybrid Oscillator-Qudit Quantum Processors: stabilizer states, stabilizer codes, symplectic operations, and non-commutative geometry","issued":{"date-parts":[[2026,1,13]]},"note":"arxivid:2508.04819\narxiv_version_number:2"},{"id":"arxiv:1801.06998","type":"journal-article","publisher":"IOP Publishing","issue":"32","DOI":"10.1088/1751-8121/aac192","page":"325301","source":"Crossref","title":"A fermionic code related to the exceptional group <i>E</i> <sub>8</sub>","volume":"51","author":[{"ORCID":"https://orcid.org/0000-0002-3436-9952","authenticated-orcid":false,"given":"Péter","family":"Lévay","sequence":"first","affiliation":[]},{"given":"Frédéric","family":"Holweck","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2018,6,29]]},"URL":"http://dx.doi.org/10.1088/1751-8121/aac192","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1801.06998"},{"id":"arxiv:math/9906019","type":"article-journal","author":[{"given":"Noam D.","family":"Elkies"}],"title":"A characterization of the Z^n lattice","issued":{"date-parts":[[1999,6,2]]},"note":"arxivid:math/9906019\narxiv_version_number:1"},{"id":"arxiv:q-alg/9607013","type":"article-journal","author":[{"given":"C.","family":"Dong"},{"given":"H.","family":"Li"},{"given":"G.","family":"Mason"},{"given":"S. P.","family":"Norton"}],"title":"Associative subalgebras of the Griess algebra and related topics","issued":{"date-parts":[[1996,7,9]]},"note":"arxivid:q-alg/9607013\narxiv_version_number:1"},{"id":"doi:10.1016/S0012-365X(98)00358-6","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/s0012-365x(98)00358-6","page":"215-228","source":"Crossref","title":"Optimal self-dual codes over Z4","volume":"203","author":[{"given":"Eric","family":"Rains","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1999,5]]},"URL":"http://dx.doi.org/10.1016/S0012-365X(98)00358-6","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X98003586"},{"id":"arxiv:2309.02382","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    The Moonshine module is a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$c=24$$</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                          <mml:mrow>\n                            <mml:mi>c</mml:mi>\n                            <mml:mo>=</mml:mo>\n                            <mml:mn>24</mml:mn>\n                          </mml:mrow>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    conformal field theory (CFT) whose automorphism group is the Monster group. It was argued by Dixon, Ginsparg, and Harvey in Dixon et al. (Commun Math Phys 119:221–241, 1988.\n                    <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"10.1007/BF01217740\" ext-link-type=\"doi\">https://doi.org/10.1007/BF01217740</jats:ext-link>\n                    ) that there exists a spin lift of the Moonshine CFT with superconformal symmetry. Reference Dixon et al. (1988) did not provide an explicit construction of a superconformal current. The present paper provides an explicit construction of a supercurrent. In fact, we will construct several superconformal currents in a spin lift of the Moonshine CFT using techniques developed in Harvey and Moore (JHEP 05:146, 2020.\n                    <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"10.1007/JHEP05(2020)146\" ext-link-type=\"doi\">https://doi.org/10.1007/JHEP05(2020)146</jats:ext-link>\n                    .\n                    <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"http://arxiv.org/abs/2003.13700\" ext-link-type=\"uri\">arXiv:2003.13700</jats:ext-link>\n                    [hep-th]). In particular, our construction relies on error correcting codes.\n                  </jats:p>","DOI":"10.1007/s00220-025-05471-7","source":"Crossref","title":"Beauty and the Beast Part 2: Apprehending the Missing Supercurrent","volume":"406","author":[{"given":"Gregory W.","family":"Moore","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6385-704X","authenticated-orcid":false,"given":"Ranveer Kumar","family":"Singh","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,30]]},"URL":"http://dx.doi.org/10.1007/s00220-025-05471-7","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","page":"303","note":"alternative-id:5471\narxivid:2309.02382"},{"id":"doi:10.1098/rspa.1982.0071","type":"journal-article","publisher":"The Royal Society","issue":"1781","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>In a recent paper with R. A. Parker we classified the points at maximum distance from the Leech lattice (the ‘deepest holes’ in that lattice), and showed that there are 23 classes of such holes, the classes being in one: one correspondence with the 23 Niemeier lattices in 24 dimensions. We now present 23 constructions for the Leech lattice, one for each class of hole or Niemeier lattice. Two of these are the usual constructions of the Leech lattice from the Golay codes over GF(2) and GF(3).</jats:p>","DOI":"10.1098/rspa.1982.0071","page":"275-283","source":"Crossref","title":"Twenty-three constructions for the Leech lattice","volume":"381","author":[{"given":"John Horton","family":"Conway","sequence":"first","affiliation":[{"name":"Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB2 1SB, U. K."}]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[{"name":"Mathematics and Statistics Research Center, Bell Laboratories, Murray Hill, New Jersey 07974, U. S. A."}]}],"container-title":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1982,6,8]]},"URL":"http://dx.doi.org/10.1098/rspa.1982.0071","ISSN":["0080-4630","2053-9169"]},{"id":"doi:10.1109/18.391252","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/18.391252","page":"1097-1106","source":"Crossref","title":"The Leech lattice, the octacode, and decoding algorithms","volume":"41","author":[{"family":"Feng-Wen Sun","sequence":"first","affiliation":[]},{"given":"H.C.A.","family":"van Tilborg","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1995,7]]},"URL":"http://dx.doi.org/10.1109/18.391252","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:math/0207208","type":"article-journal","author":[{"given":"A. Roger","family":"Hammons"},{"given":"P. Vijay","family":"Kumar"},{"given":"A. R.","family":"Calderbank"},{"given":"N. J. A.","family":"Sloane"},{"given":"Patrick","family":"Solé"}],"title":"The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes","issued":{"date-parts":[[2002,7,23]]},"note":"arxivid:math/0207208\narxiv_version_number:1"},{"id":"doi:10.1016/0021-8693(83)90074-1","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0021-8693(83)90074-1","page":"151-188","source":"Crossref","title":"The complex leech lattice and maximal subgroups of the Suzuki group","volume":"84","author":[{"given":"Robert A","family":"Wilson","sequence":"first","affiliation":[]}],"container-title":"Journal of Algebra","original-title":[],"language":"en","issued":{"date-parts":[[1983,9]]},"URL":"http://dx.doi.org/10.1016/0021-8693(83)90074-1","ISSN":["0021-8693"],"container-title-short":"Journal of Algebra","note":"alternative-id:0021869383900741"},{"id":"doi:10.1007/s10623-023-01293-7","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s10623-023-01293-7","page":"771-785","source":"Crossref","title":"Classification of extremal type II $$\\mathbb {Z}_4$$-codes of length 24","volume":"92","author":[{"given":"Akihiro","family":"Munemasa","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6188-2893","authenticated-orcid":false,"given":"Rowena Alma L.","family":"Betty","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2023,9,10]]},"URL":"http://dx.doi.org/10.1007/s10623-023-01293-7","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:1293"},{"id":"doi:10.1016/0022-314X(73)90068-1","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0022-314x(73)90068-1","page":"142-178","source":"Crossref","title":"Definite quadratische formen der dimension 24 und diskriminante 1","volume":"5","author":[{"given":"Hans-Volker","family":"Niemeier","sequence":"first","affiliation":[]}],"container-title":"Journal of Number Theory","original-title":[],"language":"de","issued":{"date-parts":[[1973,4]]},"URL":"http://dx.doi.org/10.1016/0022-314X(73)90068-1","ISSN":["0022-314X"],"container-title-short":"Journal of Number Theory","note":"alternative-id:0022314X73900681"},{"id":"doi:10.1016/0097-3165(75)90042-4","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0097-3165(75)90042-4","page":"313-335","source":"Crossref","title":"On the classification and enumeration of self-dual codes","volume":"18","author":[{"given":"Vera","family":"Pless","sequence":"first","affiliation":[]},{"given":"N.J.A","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1975,5]]},"URL":"http://dx.doi.org/10.1016/0097-3165(75)90042-4","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316575900424"},{"id":"doi:10.1016/0012-365X(93)E0088-L","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(93)e0088-l","page":"193-223","source":"Crossref","title":"A new construction of lattices from codes over GF(3)","volume":"135","author":[{"given":"P.S.","family":"Montague","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1994,12]]},"URL":"http://dx.doi.org/10.1016/0012-365X(93)E0088-L","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X93E0088L"},{"id":"doi:10.1006/eujc.2002.0557","type":"journal-article","publisher":"Elsevier BV","issue":"5","DOI":"10.1006/eujc.2002.0557","page":"573-581","source":"Crossref","title":"Z6-Code Constructions of the Leech Lattice and the Niemeier Lattices","volume":"23","author":[{"given":"Masaaki","family":"Harada","sequence":"first","affiliation":[]},{"given":"Masaaki","family":"Kitazume","sequence":"additional","affiliation":[]}],"container-title":"European Journal of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2002,7]]},"URL":"http://dx.doi.org/10.1006/eujc.2002.0557","ISSN":["0195-6698"],"container-title-short":"European Journal of Combinatorics","note":"alternative-id:S0195669802905573"},{"id":"doi:10.1006/jnth.1998.2306","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1006/jnth.1998.2306","page":"359-389","source":"Crossref","title":"The Shadow Theory of Modular and Unimodular Lattices","volume":"73","author":[{"given":"E.M","family":"Rains","sequence":"first","affiliation":[]},{"given":"N.J.A","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Journal of Number Theory","original-title":[],"language":"en","issued":{"date-parts":[[1998,12]]},"URL":"http://dx.doi.org/10.1006/jnth.1998.2306","ISSN":["0022-314X"],"container-title-short":"Journal of Number Theory","note":"alternative-id:S0022314X9892306X"},{"id":"doi:10.1007/3-540-30731-1","type":"book","publisher":"Springer-Verlag","DOI":"10.1007/3-540-30731-1","source":"Crossref","title":"Self-Dual Codes and Invariant Theory","container-title":"Algorithms and Computation in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2006]]},"ISBN":["354030729X"],"URL":"http://dx.doi.org/10.1007/3-540-30731-1","note":"publisher-location:Berlin/Heidelberg"},{"id":"arxiv:2208.14457","type":"article-journal","author":[{"given":"Johan","family":"Henriksson"},{"given":"Brian","family":"McPeak"}],"title":"Averaging over codes and an $SU(2)$ modular bootstrap","issued":{"date-parts":[[2023,11,16]]},"note":"arxivid:2208.14457\narxiv_version_number:2"},{"id":"doi:10.1016/B978-0-444-87492-4.50023","type":"book-chapter","publisher":"Elsevier","DOI":"10.1016/b978-0-444-87492-4.50023-9","page":"246-251","source":"Crossref","title":"New Heterotic String Theories in Uncompactified Dimensions &lt; 10","author":[{"given":"K.S.","family":"NARAIN","sequence":"first","affiliation":[]}],"container-title":"Current Physics–Sources and Comments","original-title":[],"issued":{"date-parts":[[1989]]},"URL":"http://dx.doi.org/10.1016/B978-0-444-87492-4.50023-9","ISSN":["0922-503X"]},{"id":"doi:10.1016/0370-2693(90)90821-M","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0370-2693(90)90821-m","page":"165-172","source":"Crossref","title":"Conformal field theory, triality and the monster group","volume":"236","author":[{"given":"L.","family":"Dolan","sequence":"first","affiliation":[]},{"given":"P.","family":"Goddard","sequence":"additional","affiliation":[]},{"given":"P.","family":"Montague","sequence":"additional","affiliation":[]}],"container-title":"Physics Letters B","original-title":[],"language":"en","issued":{"date-parts":[[1990,2]]},"URL":"http://dx.doi.org/10.1016/0370-2693(90)90821-M","ISSN":["0370-2693"],"container-title-short":"Physics Letters B","note":"alternative-id:037026939090821M"},{"id":"arxiv:hep-th/9410029","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf02103716","page":"61-120","source":"Crossref","title":"Conformal field theories, representations and lattice constructions","volume":"179","author":[{"given":"L.","family":"Dolan","sequence":"first","affiliation":[]},{"given":"P.","family":"Goddard","sequence":"additional","affiliation":[]},{"given":"P.","family":"Montague","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1996,7]]},"URL":"http://dx.doi.org/10.1007/BF02103716","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF02103716\narxivid:hep-th/9410029"},{"id":"doi:10.1109/18.568705","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.568705","page":"969-976","source":"Crossref","title":"Type II codes over Z/sub 4/","volume":"43","author":[{"given":"A.","family":"Bonnecaze","sequence":"first","affiliation":[]},{"given":"P.","family":"Sole","sequence":"additional","affiliation":[]},{"given":"C.","family":"Bachoc","sequence":"additional","affiliation":[]},{"given":"B.","family":"Mourrain","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1997,5]]},"URL":"http://dx.doi.org/10.1109/18.568705","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.761269","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/18.761269","page":"1194-1205","source":"Crossref","title":"Type II codes, even unimodular lattices, and invariant rings","volume":"45","author":[{"given":"E.","family":"Bannai","sequence":"first","affiliation":[]},{"given":"S.T.","family":"Dougherty","sequence":"additional","affiliation":[]},{"given":"M.","family":"Harada","sequence":"additional","affiliation":[]},{"given":"M.","family":"Oura","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999,5]]},"URL":"http://dx.doi.org/10.1109/18.761269","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:math/0502313","type":"article-journal","author":[{"given":"Claude","family":"Pache"}],"title":"Shells of selfdual lattices viewed as spherical designs","issued":{"date-parts":[[2005,2,15]]},"note":"arxivid:math/0502313\narxiv_version_number:1"},{"id":"manual:-van-Emde-Boas-P.-Another-NP-","type":"article-journal","author":[{"family":"Emde","particle":"van"},{"family":"P","given":"Boas"}],"title":"Another NP-complete partition problem and the complexity of computing short vectors in lattices","container-title":"TR","issued":"1981"},{"id":"manual:-P.-L.-Zador-Development-and-","type":"thesis","author":[{"family":"Zador","given":"P.L."}],"title":"Development and evaluation of procedures for quantizing multivariate distributions","genre":"PhD thesis,","publisher":"Stanford Univ","issued":"1963"},{"id":"doi:10.1109/TIT.1982.1056490","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1982.1056490","page":"139-149","source":"Crossref","title":"Asymptotic quantization error of continuous signals and the quantization dimension","volume":"28","author":[{"given":"P.","family":"Zador","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1982,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1982.1056490","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-van-Emde-Boas-P.-(1981).-Ano","type":"report","author":[{"family":"Emde Boas","given":"P.","particle":"van"}],"title":"Another NP-complete problem and the complexity of computing short vectors in a lattice","genre":"Technical Report,","publisher":"Department of Mathematics, University of Amsterdam","issued":"1981"},{"id":"doi:10.1137/S0097539700373039","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","DOI":"10.1137/s0097539700373039","page":"2008-2035","source":"Crossref","title":"The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant","volume":"30","author":[{"given":"Daniele","family":"Micciancio","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[2001,1]]},"URL":"http://dx.doi.org/10.1137/S0097539700373039","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/S0097539700373039"},{"id":"doi:10.1007/BF01457454","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/bf01457454","page":"515-534","source":"Crossref","title":"Factoring polynomials with rational coefficients","volume":"261","author":[{"given":"A. K.","family":"Lenstra","sequence":"first","affiliation":[]},{"suffix":"Jr.","given":"H. W.","family":"Lenstra","sequence":"additional","affiliation":[]},{"given":"L.","family":"Lovász","sequence":"additional","affiliation":[]}],"container-title":"Mathematische Annalen","original-title":[],"language":"en","issued":{"date-parts":[[1982,12]]},"URL":"http://dx.doi.org/10.1007/BF01457454","ISSN":["0025-5831","1432-1807"],"container-title-short":"Math. Ann.","note":"alternative-id:BF01457454"},{"id":"doi:10.1007/978-0-387-77993-5","type":"book","publisher":"Springer New York","DOI":"10.1007/978-0-387-77993-5","source":"Crossref","title":"An Introduction to Mathematical Cryptography","container-title":"Undergraduate Texts in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2008]]},"ISBN":["9780387779935","9780387779942"],"URL":"http://dx.doi.org/10.1007/978-0-387-77993-5","ISSN":["0172-6056"],"note":"publisher-location:New York, NY"},{"id":"doi:10.1109/18.651040","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.651040","page":"273-278","source":"Crossref","title":"Lattice codes can achieve capacity on the AWGN channel","volume":"44","author":[{"given":"R.","family":"Urbanke","sequence":"first","affiliation":[]},{"given":"B.","family":"Rimoldi","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.651040","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1975.1055409","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1975.1055409","page":"441-445","source":"Crossref","title":"The upper error bound of a new near-optimal code","volume":"21","author":[{"given":"R.","family":"de Buda","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1975,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1975.1055409","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-R.-de-Buda-and-W.-Kassem-Abo","type":"chapter","author":[{"family":"Buda","given":"R.","particle":"de"},{"family":"Kassem","given":"W."}],"title":"About lattices and the random coding theorem","container-title":"Abstracts of Papers, IEEE Inter. Symp. Info. Theory 1981, IEEE Press","issued":"1981","page":"145","publisher-place":"NY"},{"id":"manual:-W.-Kassem-Optimal-Lattice-Co","type":"thesis","author":[{"family":"Kassem","given":"W."}],"title":"Optimal Lattice Codes for the Gaussian Channel","genre":"PhD thesis,","publisher":"McMaster Univ","issued":"1981,doi:10.1109/18.651040","publisher-place":"Hamilton, Ontario"},{"id":"doi:10.1090/S0025-5718-1985-0777278-","type":"journal-article","publisher":"American Mathematical Society (AMS)","issue":"170","abstract":"<p>The standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper Z Superscript m\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:msup>\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n            <mml:mi mathvariant=\"bold\">Z</mml:mi>\n          </mml:mrow>\n        </mml:mrow>\n        <mml:mi>m</mml:mi>\n      </mml:msup>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">{{\\mathbf {Z}}^m}</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> in a suitable box. However, it suffices to consider those <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold x element-of bold upper Z Superscript m\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n          <mml:mi mathvariant=\"bold\">x</mml:mi>\n        </mml:mrow>\n      </mml:mrow>\n      <mml:mo>∈<!-- ∈ --></mml:mo>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:msup>\n          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n              <mml:mi mathvariant=\"bold\">Z</mml:mi>\n            </mml:mrow>\n          </mml:mrow>\n          <mml:mi>m</mml:mi>\n        </mml:msup>\n      </mml:mrow>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">{\\mathbf {x}} \\in {{\\mathbf {Z}}^m}</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula> which lie in a suitable ellipsoid having a much smaller volume than the box. We show in this paper that searching through that ellipsoid is in many cases much more efficient. If combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions. Several randomly constructed numerical examples illustrate the superiority of our new method over the known ones.</p>","DOI":"10.1090/s0025-5718-1985-0777278-8","page":"463-471","source":"Crossref","title":"Improved methods for calculating vectors of short length in a lattice, including a complexity analysis","volume":"44","author":[{"given":"U.","family":"Fincke","sequence":"first","affiliation":[]},{"given":"M.","family":"Pohst","sequence":"additional","affiliation":[]}],"container-title":"Mathematics of Computation","original-title":[],"language":"en","issued":{"date-parts":[[1985]]},"URL":"http://dx.doi.org/10.1090/S0025-5718-1985-0777278-8","ISSN":["0025-5718","1088-6842"],"container-title-short":"Math. Comp.","note":"alternative-id:S0025-5718-1985-0777278-8"},{"id":"doi:10.1007/bf01581144","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1-3","DOI":"10.1007/bf01581144","page":"181-199","source":"Crossref","title":"Lattice basis reduction: Improved practical algorithms and solving subset sum problems","volume":"66","author":[{"given":"C. P.","family":"Schnorr","sequence":"first","affiliation":[]},{"given":"M.","family":"Euchner","sequence":"additional","affiliation":[]}],"container-title":"Mathematical Programming","original-title":[],"language":"en","issued":{"date-parts":[[1994,8]]},"URL":"http://dx.doi.org/10.1007/bf01581144","ISSN":["0025-5610","1436-4646"],"container-title-short":"Mathematical Programming","note":"alternative-id:BF01581144"},{"id":"doi:10.1007/b98975","type":"book","publisher":"Springer New York","DOI":"10.1007/b98975","source":"Crossref","title":"Sphere Packings","container-title":"Universitext","original-title":[],"language":"en","editor":[{"given":"John","family":"Talbot","sequence":"first","affiliation":[]}],"issued":{"date-parts":[[1999]]},"ISBN":["9780387987941","9780387227801"],"URL":"http://dx.doi.org/10.1007/b98975","note":"publisher-location:New York, NY"},{"id":"manual:-C.-A.-Rogers.-Packing-and-co","type":"book","author":[{"family":"Rogers","given":"C.A."}],"title":"Packing and covering","volume":"54","publisher":"University Press","issued":"1964","publisher-place":"Cambridge"},{"id":"preset:LMFDB","type":"document","title":"The LMFDB Collaboration, The L-functions and modular forms database","issued":"2024","URL":"https://www.lmfdb.org"},{"id":"manual:-G.-Nebe-and-N.-J.-A.-Sloane.","type":"document","author":[{"family":"Nebe","given":"G."},{"family":"Sloane","given":"N.J.A."}],"title":"Catalogue of Lattices","URL":"https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html","note":"Available at \\url{https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html}"},{"id":"manual:-H.-Cohn.-Kissing-numbers.-ur","type":"document","author":[{"family":"Cohn","given":"H."}],"title":"Kissing numbers","URL":"https://cohn.mit.edu/kissing-numbers","note":"Available at \\url{https://cohn.mit.edu/kissing-numbers}"},{"id":"manual:-J.-Cannon-W.-Bosma-C.-Fieker","type":"document","author":[{"family":"Cannon","given":"J."},{"family":"Bosma","given":"W."},{"family":"Fieker","given":"C."},{"family":"Steel","given":"A."}],"title":"HANDBOOK OF MAGMA FUNCTIONS","issued":"2008"},{"id":"doi:10.1145/190347.190362","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/190347.190362","page":"52-57","source":"Crossref","title":"Programming with algebraic structures","author":[{"given":"Wieb","family":"Bosma","sequence":"first","affiliation":[]},{"given":"John","family":"Cannon","sequence":"additional","affiliation":[]},{"given":"Graham","family":"Matthews","sequence":"additional","affiliation":[]}],"event":"the international symposium","container-title":"Proceedings of the international symposium on Symbolic and algebraic computation  - ISSAC '94","original-title":[],"issued":{"date-parts":[[1994]]},"URL":"http://dx.doi.org/10.1145/190347.190362","note":"publisher-location:New York, New York, USA\nproceedings-subject:Symbolic and algebraic computation"},{"id":"doi:10.1006/jsco.1996.0125","type":"journal-article","publisher":"Elsevier BV","issue":"3-4","DOI":"10.1006/jsco.1996.0125","page":"235-265","source":"Crossref","title":"The Magma Algebra System I: The User Language","volume":"24","author":[{"given":"WIEB","family":"BOSMA","sequence":"first","affiliation":[]},{"given":"JOHN","family":"CANNON","sequence":"additional","affiliation":[]},{"given":"CATHERINE","family":"PLAYOUST","sequence":"additional","affiliation":[]}],"container-title":"Journal of Symbolic Computation","original-title":[],"language":"en","issued":{"date-parts":[[1997,9]]},"URL":"http://dx.doi.org/10.1006/jsco.1996.0125","ISSN":["0747-7171"],"container-title-short":"Journal of Symbolic Computation","note":"alternative-id:S074771719690125X"},{"id":"preset:Levenshtein78","type":"paper-conference","author":[{"family":"Levenshtein","given":"V.I."}],"title":"On choosing polynomials to obtain bounds in packing problems","container-title":"Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission","issued":"1978","publisher-place":"Part II, Moscow, Vilnius"},{"id":"preset:Levenshtein79","type":"article-journal","author":[{"family":"Levenshtein","given":"V.I."}],"title":"On bounds for packings in n-dimensional Euclidean space","volume":"245","container-title":"Doklady Akademii Nauk SSSR","issue":"6","issued":"1979","page":"1299–1303"},{"id":"preset:Levenshtein83","type":"article-journal","author":[{"family":"Levenshtein","given":"V.I."}],"title":"Bounds for packings of metric spaces and some of their applications","volume":"40","container-title":"Problemy Kibernetiki","issued":"1983","page":"43–110"},{"id":"doi:10.1007/BF00053379","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1-2","DOI":"10.1007/bf00053379","page":"1-82","source":"Crossref","title":"Designs as maximum codes in polynomial metric spaces","volume":"29","author":[{"given":"V. I.","family":"Levenshtein","sequence":"first","affiliation":[]}],"container-title":"Acta Applicandae Mathematicae","original-title":[],"language":"en","issued":{"date-parts":[[1992,11]]},"URL":"http://dx.doi.org/10.1007/BF00053379","ISSN":["0167-8019","1572-9036"],"container-title-short":"Acta Appl Math","note":"alternative-id:BF00053379"},{"id":"preset:HPLevBounds","type":"chapter","author":[{"family":"Levenshtein","given":"V.I."}],"title":"Universal bounds for codes and designs","container-title":"\\emph{Handbook of Coding Theory}","volume":"I, Part 1","editor":[{"family":"V. S. Pless","particle":"eds."},{"family":"Huffman","given":"W.C."}],"publisher":"Elsevier","issued":"1998","page":"499–648"},{"id":"arxiv:math/0110009","type":"journal-article","publisher":"Annals of Mathematics","issue":"2","DOI":"10.4007/annals.2003.157.689","page":"689-714","source":"Crossref","title":"New upper bounds on sphere packings I","volume":"157","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[]},{"given":"Noam","family":"Elkies","sequence":"additional","affiliation":[]}],"container-title":"Annals of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2003,3,1]]},"URL":"http://dx.doi.org/10.4007/annals.2003.157.689","ISSN":["0003-486X"],"container-title-short":"Ann. Math.","note":"arxivid:math/0110009"},{"id":"arxiv:math/0608426","type":"journal-article","publisher":"American Mathematical Society (AMS)","issue":"3","abstract":"<p>Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions. In particular our computations give the (known) values for the cases <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n equals 3 comma 4 comma 8 comma 24\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi>n</mml:mi>\n      <mml:mo>=</mml:mo>\n      <mml:mn>3</mml:mn>\n      <mml:mo>,</mml:mo>\n      <mml:mn>4</mml:mn>\n      <mml:mo>,</mml:mo>\n      <mml:mn>8</mml:mn>\n      <mml:mo>,</mml:mo>\n      <mml:mn>24</mml:mn>\n    </mml:mrow>\n    <mml:annotation encoding=\"application/x-tex\">n = 3, 4, 8, 24</mml:annotation>\n  </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","DOI":"10.1090/s0894-0347-07-00589-9","page":"909-924","source":"Crossref","title":"New upper bounds for kissing numbers from semidefinite programming","volume":"21","author":[{"given":"Christine","family":"Bachoc","sequence":"first","affiliation":[]},{"given":"Frank","family":"Vallentin","sequence":"additional","affiliation":[]}],"container-title":"Journal of the American Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[2007,11,29]]},"URL":"http://dx.doi.org/10.1090/S0894-0347-07-00589-9","ISSN":["0894-0347","1088-6834"],"container-title-short":"J. Amer. Math. Soc.","note":"alternative-id:S0894-0347-07-00589-9\narxivid:math/0608426"},{"id":"arxiv:math/0005266","type":"article-journal","author":[{"given":"Gerald","family":"Höhn"}],"title":"Self-dual Codes over the Kleinian Four Group","issued":{"date-parts":[[2000,5,26]]},"note":"arxivid:math/0005266\narxiv_version_number:1"},{"id":"doi:10.1016/0012-365X(95)00354-Y","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(95)00354-y","page":"29-39","source":"Crossref","title":"On sublattices of the hexagonal lattice","volume":"170","author":[{"given":"M.","family":"Bernstein","sequence":"first","affiliation":[]},{"given":"N.J.A.","family":"Sloane","sequence":"additional","affiliation":[]},{"given":"Paul E.","family":"Wright","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1997,6]]},"URL":"http://dx.doi.org/10.1016/0012-365X(95)00354-Y","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9500354Y"},{"id":"doi:10.1515/zna-1995-0802","type":"journal-article","publisher":"Walter de Gruyter GmbH","issue":"8","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p> The coincidence problem is analyzed in an illustrative fashion for some lattices and modules in two and three dimensions which are important for crystals and quasicrystals. We give a complete description of the groups of coincidence rotations with their associated indices and encapsulate their statistics by means of generating functions.</jats:p>","DOI":"10.1515/zna-1995-0802","page":"711-717","source":"Crossref","title":"Algebraic Solution of the Coincidence Problem in Two and Three Dimensions","volume":"50","author":[{"given":"Michael","family":"Baake","sequence":"first","affiliation":[{"name":"Institut für Theoretische Physik, Universität Tübingen, D-72076 Tübingen, Germany"}]},{"given":"Peter A. B.","family":"Pleasants","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia"}]}],"container-title":"Zeitschrift für Naturforschung A","original-title":[],"language":"en","issued":{"date-parts":[[1995,8,1]]},"URL":"http://dx.doi.org/10.1515/zna-1995-0802","ISSN":["1865-7109","0932-0784"],"note":"alternative-id:10.1515/zna-1995-0802"},{"id":"doi:10.2307/2371320","type":"journal-article","publisher":"JSTOR","issue":"3","DOI":"10.2307/2371320","page":"665","source":"Crossref","title":"The Number of Circles Covering a Set","volume":"61","author":[{"given":"Richard","family":"Kershner","sequence":"first","affiliation":[]}],"container-title":"American Journal of Mathematics","original-title":[],"issued":{"date-parts":[[1939,7]]},"URL":"http://dx.doi.org/10.2307/2371320","ISSN":["0002-9327"],"container-title-short":"American Journal of Mathematics"},{"id":"manual:-L.-Fejes-Toth-Sur-Ia-represe","type":"article-journal","author":[{"family":"Toth","given":"L.Fejes"}],"title":"Sur Ia representation d' une population infinie par une nombre fini d'elements","volume":"10","container-title":"AMAH","issued":"1959","page":"299–304"},{"id":"doi:10.1007/978-3-642-65234-9","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-65234-9","source":"Crossref","title":"Lagerungen in der Ebene auf der Kugel und im Raum","author":[{"given":"László Fejes","family":"Tóth","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1972]]},"ISBN":["9783642652356","9783642652349"],"URL":"http://dx.doi.org/10.1007/978-3-642-65234-9","note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/TIT.1979.1056067","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1979.1056067","page":"373-380","source":"Crossref","title":"Asymptotically optimal block quantization","volume":"25","author":[{"given":"A.","family":"Gersho","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1979,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1979.1056067","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1982.1056492","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1982.1056492","page":"137-139","source":"Crossref","title":"The hexagon theorem","volume":"28","author":[{"given":"D.","family":"Newman","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1982,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1982.1056492","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TCOM.1962.1088634","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tcom.1962.1088634","page":"90-95","source":"Crossref","title":"A Coherent Digital Amplitude and Phase Modulation Scheme","volume":"10","author":[{"given":"C.","family":"Campopiano","sequence":"first","affiliation":[]},{"given":"B.","family":"Glazer","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[1962,3]]},"URL":"http://dx.doi.org/10.1109/TCOM.1962.1088634","ISSN":["0096-2244"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/TCOM.1974.1092061","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tcom.1974.1092061","page":"28-38","source":"Crossref","title":"Optimization of Two-Dimensional Signal Constellations in the Presence of Gaussian Noise","volume":"22","author":[{"given":"G.","family":"Foschini","sequence":"first","affiliation":[]},{"given":"R.","family":"Gitlin","sequence":"additional","affiliation":[]},{"given":"S.","family":"Weinstein","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[1974,1]]},"URL":"http://dx.doi.org/10.1109/TCOM.1974.1092061","ISSN":["0096-2244"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1017/S0017089500007047","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"1","abstract":"<jats:p>Let <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_inline1\"/> be a positive definite binary quadratic form with real coefficients and discriminant <jats:italic>b</jats:italic><jats:sub>2</jats:sub> − 4<jats:italic>ac</jats:italic> = −1.</jats:p><jats:p>Among such forms, let <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_inline2\"/>. The Epstein zeta function of <jats:italic>f</jats:italic> is denned to be</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_eqnU1\"/></jats:disp-formula></jats:p><jats:p>Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s &gt; 0,</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_eqn1\"/></jats:disp-formula></jats:p><jats:p>We prove a corresponding result for theta functions. For real α &gt; 0, let</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_eqnU2\"/></jats:disp-formula></jats:p><jats:p>This function satisfies the functional equation</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0017089500007047_eqn2\"/></jats:disp-formula></jats:p><jats:p>(This may be proved by using the formula (4) below, and then twice applying the identity (8).)</jats:p>","DOI":"10.1017/s0017089500007047","page":"75-85","source":"Crossref","title":"Minimal theta functions","volume":"30","author":[{"given":"Hugh L.","family":"Montgomery","sequence":"first","affiliation":[]}],"container-title":"Glasgow Mathematical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1988,1]]},"URL":"http://dx.doi.org/10.1017/S0017089500007047","ISSN":["0017-0895","1469-509X"],"container-title-short":"Glasgow Math. J.","note":"alternative-id:S0017089500007047"},{"id":"preset:coxeter","type":"book","author":[{"family":"Coxeter","given":"H.S.M."}],"title":"\\emph{Regular Polytopes}","publisher":"Courier Corporation","issued":"1973"},{"id":"doi:10.1007/BF01442912","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/bf01442912","page":"581-583","source":"Crossref","title":"Sur les formes quadratiques positives quaternaires","volume":"5","author":[{"given":"A.","family":"Korkine","sequence":"first","affiliation":[]},{"given":"G.","family":"Zolotareff","sequence":"additional","affiliation":[]}],"container-title":"Mathematische Annalen","original-title":[],"language":"fr","issued":{"date-parts":[[1872,12]]},"URL":"http://dx.doi.org/10.1007/BF01442912","ISSN":["0025-5831","1432-1807"],"container-title-short":"Math. Ann.","note":"alternative-id:BF01442912"},{"id":"arxiv:2010.02238","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevresearch.3.013118","source":"Crossref","title":"Four-dimensional toric code with non-Clifford transversal gates","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-8265-4774","authenticated-orcid":true,"given":"Tomas","family":"Jochym-O'Connor","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9614-2836","authenticated-orcid":true,"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.3.013118","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013118","note":"arxivid:2010.02238"},{"id":"manual:-Sur-les-formes-quadratiques-","type":"document","title":"Sur les formes quadratiques positives","note":"Zus. Mit S. Zolotareff) Korkine in: Mathematische Annalen"},{"id":"manual:-J.-Kepler.-The-six-cornered-","type":"book","author":[{"family":"Kepler","given":"J."}],"title":"The six-cornered snowflake","publisher":"Paul Dry Books","issued":"2010"},{"id":"manual:-C.-F.-Gauss-Besprechung-des-","type":"article-journal","author":[{"family":"Gauss","given":"C.F."},{"family":"L. A. Seeber","given":"Besprechung","particle":"des Buchs von"}],"title":"Untersuchungen uber die Eigenschaften der positiven terndren quadratischen Formen usw","volume":"II","container-title":"Gottingsche Gelehrte Anzeigen","issued":"1831-07-09","page":"18 8–196,"},{"id":"arxiv:math/9811071","type":"article-journal","author":[{"given":"Thomas C.","family":"Hales"}],"title":"An overview of the Kepler conjecture","issued":{"date-parts":[[2002,5,20]]},"note":"arxivid:math/9811071\narxiv_version_number:2"},{"id":"arxiv:1501.02155","type":"article-journal","author":[{"given":"Thomas","family":"Hales"},{"given":"Mark","family":"Adams"},{"given":"Gertrud","family":"Bauer"},{"given":"Dat Tat","family":"Dang"},{"given":"John","family":"Harrison"},{"given":"Truong","family":"Le Hoang"},{"given":"Cezary","family":"Kaliszyk"},{"given":"Victor","family":"Magron"},{"given":"Sean","family":"McLaughlin"},{"given":"Thang Tat","family":"Nguyen"},{"given":"Truong Quang","family":"Nguyen"},{"given":"Tobias","family":"Nipkow"},{"given":"Steven","family":"Obua"},{"given":"Joseph","family":"Pleso"},{"given":"Jason","family":"Rute"},{"given":"Alexey","family":"Solovyev"},{"given":"An Hoai Thi","family":"Ta"},{"given":"Trung Nam","family":"Tran"},{"given":"Diep Thi","family":"Trieu"},{"given":"Josef","family":"Urban"},{"given":"Ky Khac","family":"Vu"},{"given":"Roland","family":"Zumkeller"}],"title":"A formal proof of the Kepler conjecture","issued":{"date-parts":[[2015,1,9]]},"note":"arxivid:1501.02155\narxiv_version_number:1"},{"id":"manual:-N.-M.-Vetchinkin-Uniqueness-","type":"document","author":[{"family":"Vetchinkin","given":"N.M."}],"title":"Uniqueness of classes of positive quadratic and highest known kissing number in seven dimensions"},{"id":"manual:-N.-M.-Vetchinkin-Uniqueness-","type":"article-journal","author":[{"family":"Vetchinkin","given":"N.M."}],"title":"Uniqueness of classes of positive quadratic forms, on which values of Hermite constants are reached for 6≤n≤8","volume":"152","container-title":"Trudy Matematicheskogo Instituta imeni VA Steklova","issued":"1980","page":"34–86"},{"id":"arxiv:math/0507118","type":"article-journal","author":[{"given":"Laurent","family":"Manivel"}],"title":"Configurations of lines and models of Lie algebras","issued":{"date-parts":[[2005,7,6]]},"note":"arxivid:math/0507118\narxiv_version_number:1"},{"id":"manual:-V.-A.-Kotelnikov-The-theory-","type":"thesis","author":[{"family":"Kotelnikov","given":"V.A."}],"title":"The theory of optimum noise immunity","genre":"PhD thesis,","publisher":"Molotov Energy Institute","issued":"1947-01","publisher-place":"Moscow"},{"id":"doi:10.1109/JRPROC.1949.232969","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/jrproc.1949.232969","page":"10-21","source":"Crossref","title":"Communication in the Presence of Noise","volume":"37","author":[{"given":"C.E.","family":"Shannon","sequence":"first","affiliation":[]}],"container-title":"Proceedings of the IRE","original-title":[],"issued":{"date-parts":[[1949,1]]},"URL":"http://dx.doi.org/10.1109/JRPROC.1949.232969","ISSN":["0096-8390"],"container-title-short":"Proc. IRE"},{"id":"preset:KLbounds","type":"article-journal","author":[{"family":"Kabatiansky","given":"G.A."},{"family":"Levenshtein","given":"V.I."}],"title":"On Bounds for Packings on a Sphere and in Space","volume":"14","container-title":"Probl. Peredachi Inf","issue":"1","issued":"1978","page":"3–25"},{"id":"doi:10.1112/plms/s3-8.4.609","type":"journal-article","publisher":"Wiley","issue":"4","DOI":"10.1112/plms/s3-8.4.609","page":"609-620","source":"Crossref","title":"The Packing of Equal Spheres","volume":"s3-8","author":[{"given":"C. A.","family":"Rogers","sequence":"first","affiliation":[{"name":"The University; Birmingham, 15"}]}],"container-title":"Proceedings of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1958,10]]},"URL":"http://dx.doi.org/10.1112/plms/s3-8.4.609","ISSN":["0024-6115"],"container-title-short":"Proceedings of the London Mathematical Society"},{"id":"doi:10.1070/SM1974v024n01ABEH001911","type":"journal-article","publisher":"Steklov Mathematical Institute","issue":"1","DOI":"10.1070/sm1974v024n01abeh001911","page":"147-157","source":"Crossref","title":"NEW BOUNDS FOR DENSEST PACKING OF SPHERES IN<i>n</i>-DIMENSIONAL EUCLIDEAN SPACE","volume":"24","author":[{"given":"V M","family":"Sidel'nikov","sequence":"first","affiliation":[]}],"container-title":"Mathematics of the USSR-Sbornik","original-title":[],"issued":{"date-parts":[[1974,2,28]]},"URL":"http://dx.doi.org/10.1070/SM1974v024n01ABEH001911","ISSN":["0025-5734"],"container-title-short":"Math. USSR Sb."},{"id":"arxiv:2312.10026","type":"article-journal","author":[{"given":"Marcelo","family":"Campos"},{"given":"Matthew","family":"Jenssen"},{"given":"Marcus","family":"Michelen"},{"given":"Julian","family":"Sahasrabudhe"}],"title":"A new lower bound for sphere packing","issued":{"date-parts":[[2023,12,15]]},"note":"arxivid:2312.10026\narxiv_version_number:1"},{"id":"arxiv:2504.05042","type":"article-journal","author":[{"given":"Boaz","family":"Klartag"}],"title":"Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid","issued":{"date-parts":[[2026,1,25]]},"note":"arxivid:2504.05042\narxiv_version_number:2"},{"id":"doi:10.1112/S0025579300001030","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1112/s0025579300001030","page":"1-6","source":"Crossref","title":"A note on coverings","volume":"4","author":[{"given":"C. A.","family":"Rogers","sequence":"first","affiliation":[{"name":"The UniversityBirmingham 15"}]}],"container-title":"Mathematika","original-title":[],"language":"en","issued":{"date-parts":[[1957,6]]},"URL":"http://dx.doi.org/10.1112/S0025579300001030","ISSN":["0025-5793","2041-7942"],"container-title-short":"Mathematika","note":"alternative-id:10.1112/S0025579300001030"},{"id":"doi:10.1112/S0025579300002059","type":"journal-article","publisher":"Wiley","issue":"2","DOI":"10.1112/s0025579300002059","page":"147-157","source":"Crossref","title":"Covering space with equal spheres","volume":"6","author":[{"given":"H. S. M.","family":"Coxeter","sequence":"first","affiliation":[{"name":"The University of Toronto. University CollegeLondon"}]},{"given":"L.","family":"Few","sequence":"additional","affiliation":[{"name":"The University of Toronto. University CollegeLondon"}]},{"given":"C. A.","family":"Rogers","sequence":"additional","affiliation":[{"name":"The University of Toronto. University CollegeLondon"}]}],"container-title":"Mathematika","original-title":[],"language":"en","issued":{"date-parts":[[1959,12]]},"URL":"http://dx.doi.org/10.1112/S0025579300002059","ISSN":["0025-5793","2041-7942"],"container-title-short":"Mathematika","note":"alternative-id:10.1112/S0025579300002059"},{"id":"doi:10.1002/j.1538-7305.1959.tb03905","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1002/j.1538-7305.1959.tb03905.x","page":"611-656","source":"Crossref","title":"Probability of Error for Optimal Codes in a Gaussian Channel","volume":"38","author":[{"given":"Claude E.","family":"Shannon","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1959,5]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1959.tb03905.x","ISSN":["0005-8580"]},{"id":"doi:10.1002/j.1538-7305.1963.tb04013","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1002/j.1538-7305.1963.tb04013.x","page":"681-707","source":"Crossref","title":"Bounds on Communication","volume":"42","author":[{"given":"David","family":"Slepian","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1963,5]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1963.tb04013.x","ISSN":["0005-8580"]},{"id":"manual:-E.-Agrell-Database-of-sphere","type":"document","author":[{"family":"Agrell","given":"E."}],"title":"Database of sphere packings, 2019","publisher":"\\url{https://codes.se/packings}","note":"Accessed (2015","publisher-place":"Online"},{"id":"doi:10.1109/18.720549","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.720549","page":"2561-2595","source":"Crossref","title":"The art of signaling: fifty years of coding theory","volume":"44","author":[{"given":"A.R.","family":"Calderbank","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.720549","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:cs/0611112","type":"article-journal","author":[{"given":"Daniel J.","family":"Costello"},{"given":"G. David","family":"Forney"}],"title":"Channel Coding: The Road to Channel Capacity","issued":{"date-parts":[[2006,11,22]]},"note":"arxivid:cs/0611112\narxiv_version_number:1"},{"id":"arxiv:1905.01319","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>We establish a precise relation between the modular bootstrap, used to con- strain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiral algebra U(1)<jats:sup>\n              <jats:italic>c</jats:italic>\n            </jats:sup> maps exactly to the Cohn-Elkies linear programming bound on the sphere packing density in <jats:italic>d</jats:italic> = 2<jats:italic>c</jats:italic> dimensions. We also show that the analytic functionals developed earlier for the correlator conformal bootstrap can be adapted to this context. For <jats:italic>c</jats:italic> = 4 and <jats:italic>c</jats:italic> = 12, these functionals exactly repro- duce the “magic functions” used recently by Viazovska [1] and Cohn et al. [2] to solve the sphere packing problem in dimensions 8 and 24. The same functionals are also applied to general 2D CFTs, with only Virasoro symmetry. In the limit of large central charge, we relate sphere packing to bounds on the black hole spectrum in 3D quantum gravity, and prove analytically that any such theory must have a nontrivial primary state of dimension <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$ {\\Delta}_0\\underset{\\sim }{&lt;}c/\\mathrm{8.503.} $$</jats:tex-math>\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mi>Δ</mml:mi>\n                    <mml:mn>0</mml:mn>\n                  </mml:msub>\n                  <mml:munder>\n                    <mml:mo>&lt;</mml:mo>\n                    <mml:mo>˜</mml:mo>\n                  </mml:munder>\n                  <mml:mi>c</mml:mi>\n                  <mml:mo>/</mml:mo>\n                  <mml:mn>8.503.</mml:mn>\n                </mml:math>\n              </jats:alternatives>\n            </jats:inline-formula>\n          </jats:p>","DOI":"10.1007/jhep12(2019)048","source":"Crossref","title":"Sphere packing and quantum gravity","volume":"2019","author":[{"given":"Thomas","family":"Hartman","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2613-0906","authenticated-orcid":false,"given":"Dalimil","family":"Mazáč","sequence":"additional","affiliation":[]},{"given":"Leonardo","family":"Rastelli","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,12]]},"URL":"http://dx.doi.org/10.1007/JHEP12(2019)048","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"48","note":"alternative-id:11943\narxivid:1905.01319"},{"id":"arxiv:2006.02560","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra U(1)<jats:sup><jats:italic>c</jats:italic></jats:sup><jats:italic>×</jats:italic> U(1)<jats:sup><jats:italic>c</jats:italic></jats:sup>, or equivalently the linear programming bound for sphere packing in 2<jats:italic>c</jats:italic> dimensions. We give a more detailed picture of the behavior for finite <jats:italic>c</jats:italic> than was previously available, and we extrapolate as <jats:italic>c →</jats:italic> ∞. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimen- sions. Furthermore, we study when these bounds can be tight. Besides the known cases <jats:italic>c</jats:italic> = 1<jats:italic>/</jats:italic>2, 4, and 12 and the conjectured case <jats:italic>c</jats:italic> = 1, our calculations numerically rule out sharp bounds for all other <jats:italic>c &lt;</jats:italic> 90, by combining the modular bootstrap with linear programming bounds for spherical codes.</jats:p>","DOI":"10.1007/jhep12(2020)066","source":"Crossref","title":"High-dimensional sphere packing and the modular bootstrap","volume":"2020","author":[{"given":"Nima","family":"Afkhami-Jeddi","sequence":"first","affiliation":[]},{"given":"Henry","family":"Cohn","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Hartman","sequence":"additional","affiliation":[]},{"given":"David","family":"de Laat","sequence":"additional","affiliation":[]},{"given":"Amirhossein","family":"Tajdini","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,12]]},"URL":"http://dx.doi.org/10.1007/JHEP12(2020)066","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"66","note":"alternative-id:14355\narxivid:2006.02560"},{"id":"arxiv:2202.11012","type":"article-journal","author":[{"given":"Thomas","family":"Hartman"},{"given":"Dalimil","family":"Mazac"},{"given":"David","family":"Simmons-Duffin"},{"given":"Alexander","family":"Zhiboedov"}],"title":"Snowmass White Paper: The Analytic Conformal Bootstrap","issued":{"date-parts":[[2022,2,22]]},"note":"arxivid:2202.11012\narxiv_version_number:1"},{"id":"doi:10.1201/9781315371993","type":"book","publisher":"Chapman and Hall/CRC","DOI":"10.1201/9781315371993","source":"Crossref","title":"Introduction to Coding Theory","author":[{"given":"Jürgen","family":"Bierbrauer","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2016,10,14]]},"ISBN":["9781315371993"],"URL":"http://dx.doi.org/10.1201/9781315371993","note":"publisher-location:Second edition. | Boca Raton : Taylor & Francis, 2017. | Series: Discrete mathematics and its applications | “A CRC title.”\nedition-number:2"},{"id":"doi:10.1016/S0012-365X(99)00183-1","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/s0012-365x(99)00183-1","page":"211-244","source":"Crossref","title":"Linear codes over finite fields and finite projective geometries","volume":"213","author":[{"given":"Ivan N.","family":"Landjev","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2000,2]]},"URL":"http://dx.doi.org/10.1016/S0012-365X(99)00183-1","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X99001831"},{"id":"doi:10.1049/el:19700293","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"13","DOI":"10.1049/el:19700293","page":"419-421","source":"Crossref","title":"Linear binary anticodes","volume":"6","author":[{"given":"P.G.","family":"Farrell","sequence":"first","affiliation":[{"name":"Electronics Laboratories, University of Kent at Canterbury, Canterbury, Kent, UK"}]}],"container-title":"Electronics Letters","original-title":[],"language":"en","issued":{"date-parts":[[1970,6,25]]},"URL":"http://dx.doi.org/10.1049/el:19700293","ISSN":["0013-5194","1350-911X"],"container-title-short":"Electron. Lett. (UK)","note":"alternative-id:10.1049/el:19700293"},{"id":"doi:10.1049/el:19740269","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"16","DOI":"10.1049/el:19740269","page":"340-341","source":"Crossref","title":"Further properties of linear binary anticodes","volume":"10","author":[{"given":"P.G.","family":"Farrell","sequence":"first","affiliation":[{"name":"Electronics Laboratories, University of Kent at Canterbury, Canterbury, Kent CT2 7NT, UK"}]},{"given":"A.","family":"Farrag","sequence":"additional","affiliation":[{"name":"Electronics Laboratories, University of Kent at Canterbury, Canterbury, Kent CT2 7NT, UK"}]}],"container-title":"Electronics Letters","original-title":[],"language":"en","issued":{"date-parts":[[1974,8,8]]},"URL":"http://dx.doi.org/10.1049/el:19740269","ISSN":["0013-5194","1350-911X"],"container-title-short":"Electron. Lett. (UK)","note":"alternative-id:10.1049/el:19740269"},{"id":"manual:-B.-I.-Belov-V.-N.-Logachev-a","type":"article-journal","author":[{"family":"Belov","given":"B.I."},{"family":"Logachev","given":"V.N."},{"family":"Sandimirov","given":"V.P."}],"title":"Construction of a Class of Linear Binary Codes Achieving the Varshamov-Griesmer Bound","volume":"10","container-title":"Problemy Peredachi Informatsii","issue":"3","issued":"1974","page":"36–44"},{"id":"manual:-R.-Hill-Optimal-Linear-Codes","type":"chapter","author":[{"family":"Hill","given":"R."}],"title":"Optimal Linear Codes in","editor":[{"family":"Mitchell","given":"C."}],"container-title":"Cryptography and Coding.\"","issued":"1992","page":"75–104"},{"id":"manual:-B.-I.-Belov-A-conjecture-on-","type":"article-journal","author":[{"family":"Belov","given":"B.I."}],"title":"A conjecture on the Griesmer bound","volume":"182","container-title":"Optimization Methods and Their Applications,(Russian), Sibirsk. Energet. Inst. Sibirsk. Otdel. Akad. Nauk SSSR, Irkutsk","issued":"1974","page":"100–106"},{"id":"doi:10.1002/j.1538-7305.1956.tb02379","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1002/j.1538-7305.1956.tb02379.x","page":"203-234","source":"Crossref","title":"A Class of Binary Signaling Alphabets","volume":"35","author":[{"given":"David","family":"Slepian","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1956,1]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1956.tb02379.x","ISSN":["0005-8580"]},{"id":"doi:10.1002/j.1538-7305.1960.tb03958","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1002/j.1538-7305.1960.tb03958.x","page":"1219-1252","source":"Crossref","title":"Some Further Theory of Group Codes","volume":"39","author":[{"given":"David","family":"Slepian","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1960,9]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1960.tb03958.x","ISSN":["0005-8580"]},{"id":"preset:HKSclass","type":"chapter","author":[{"family":"Östergård","given":"P.R.J."}],"title":"Construction and Classification of Codes","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1109/18.641542","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.641542","page":"1757-1766","source":"Crossref","title":"The intractability of computing the minimum distance of a code","volume":"43","author":[{"given":"A.","family":"Vardy","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1109/18.641542","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:cs/0304044","type":"article-journal","author":[{"given":"M. N.","family":"Vyalyi"}],"title":"Hardness of approximating the weight enumerator of a binary linear code","issued":{"date-parts":[[2003,4,30]]},"note":"arxivid:cs/0304044\narxiv_version_number:1"},{"id":"preset:Varshamov65","type":"article-journal","author":[{"family":"Varshamov","given":"R.R."}],"title":"Some features of linear codes that correct asymmetric errors","volume":"9","container-title":"Soviet Physics Doklady","issued":"1965"},{"id":"arxiv:1310.7536","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2015.2401567","page":"1879-1886","source":"Crossref","title":"New Constructions of Codes for Asymmetric Channels via Concatenation","volume":"61","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]},{"given":"Graeme","family":"Smith","sequence":"additional","affiliation":[]},{"given":"John","family":"Smolin","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2015,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2015.2401567","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1310.7536"},{"id":"arxiv:0909.5200","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.104.050503","source":"Crossref","title":"Tradeoffs for Reliable Quantum Information Storage in 2D Systems","volume":"104","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]},{"given":"Barbara","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,2,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.104.050503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050503","note":"arxivid:0909.5200"},{"id":"arxiv:2309.16411","type":"article-journal","author":[{"given":"Nouédyn","family":"Baspin"}],"title":"On combinatorial structures in linear codes","issued":{"date-parts":[[2023,9,28]]},"note":"arxivid:2309.16411\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.1978.1055873","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1978.1055873","page":"384-386","source":"Crossref","title":"On the inherent intractability of certain coding problems (Corresp.)","volume":"24","author":[{"given":"E.","family":"Berlekamp","sequence":"first","affiliation":[]},{"given":"R.","family":"McEliece","sequence":"additional","affiliation":[]},{"given":"H.","family":"van Tilborg","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1978,5]]},"URL":"http://dx.doi.org/10.1109/TIT.1978.1055873","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-G.-S.-Evseev-Complexity-of-d","type":"article-journal","author":[{"family":"Evseev","given":"G.S."}],"title":"Complexity of decoding for linear codes","volume":"19","container-title":"Problems of Information Transmission","issue":"1","issued":"1983","page":"1–6"},{"id":"arxiv:1408.1310","type":"article-journal","author":[{"given":"Yunghsiang S.","family":"Han"},{"given":"Hung-Ta","family":"Pai"},{"given":"Po-Ning","family":"Chen"},{"given":"Ting-Yi","family":"Wu"}],"title":"Maximum-likelihood Soft-decision Decoding for Binary Linear Block Codes Based on Their Supercodes","issued":{"date-parts":[[2014,8,6]]},"note":"arxivid:1408.1310\narxiv_version_number:1"},{"id":"arxiv:1607.04793","type":"article-journal","author":[{"given":"Eliya","family":"Nachmani"},{"given":"Yair","family":"Beery"},{"given":"David","family":"Burshtein"}],"title":"Learning to Decode Linear Codes Using Deep Learning","issued":{"date-parts":[[2016,9,30]]},"note":"arxivid:1607.04793\narxiv_version_number:2"},{"id":"arxiv:2203.14966","type":"article-journal","author":[{"given":"Yoni","family":"Choukroun"},{"given":"Lior","family":"Wolf"}],"title":"Error Correction Code Transformer","issued":{"date-parts":[[2022,3,27]]},"note":"arxivid:2203.14966\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.1972.1054746","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1972.1054746","page":"170-182","source":"Crossref","title":"Class of algorithms for decoding block codes with channel measurement information","volume":"18","author":[{"given":"D.","family":"Chase","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1972,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1972.1054746","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:HPBounds","type":"chapter","author":[{"family":"Brouwer","given":"A.E."}],"title":"Bounds on the size of linear codes","container-title":"\\emph{Handbook of Coding Theory}","volume":"I, Part 1","editor":[{"family":"V. S. Pless","particle":"eds."},{"family":"Huffman","given":"W.C."}],"publisher":"Elsevier","issued":"1998","page":"295–461"},{"id":"doi:10.1109/SFCS.2005.16","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/sfcs.2005.16","page":"317-326","source":"Crossref","title":"Almost Orthogonal Linear Codes are Locally Testable","author":[{"given":"T.","family":"Kaufman","sequence":"first","affiliation":[]},{"given":"S.","family":"Litsyn","sequence":"additional","affiliation":[]}],"event":"46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)","container-title":"46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.2005.16"},{"id":"doi:10.1109/FOCS.2007.8","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/focs.2007.8","page":"590-600","source":"Crossref","title":"Sparse Random Linear Codes are Locally Decodable and Testable","author":[{"given":"Tali","family":"Kaufman","sequence":"first","affiliation":[]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[]}],"event":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","container-title":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","original-title":[],"issued":{"date-parts":[[2007,10]]},"URL":"http://dx.doi.org/10.1109/FOCS.2007.8"},{"id":"preset:Bonisoli84","type":"article-journal","author":[{"family":"Bonisoli","given":"A."}],"title":"Every equidistant linear code is a sequence of dual Hamming codes","volume":"18","container-title":"Ars Combinatoria","issued":"1984","page":"181–186"},{"id":"doi:10.1016/S0019-9958(63)80010-8","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/s0019-9958(63)80010-8","page":"315-330","source":"Crossref","title":"Error-correcting codes: An axiomatic approach","volume":"6","author":[{"suffix":"Jr.","given":"Assmus","family":"E.F.","sequence":"first","affiliation":[]},{"given":"H.F.","family":"Mattson","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1963,12]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(63)80010-8","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995863800108"},{"id":"doi:10.1137/0114009","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","DOI":"10.1137/0114009","page":"106-111","source":"Crossref","title":"Linear Codes of Constant Weight","volume":"14","author":[{"given":"E.","family":"Weiss","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1966,1]]},"URL":"http://dx.doi.org/10.1137/0114009","ISSN":["0036-1399","1095-712X"],"container-title-short":"SIAM J. Appl. Math.","note":"alternative-id:10.1137/0114009"},{"id":"doi:10.1109/TIT.1986.1057187","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1986.1057187","page":"337-348","source":"Crossref","title":"Lexicographic codes: Error-correcting codes from game theory","volume":"32","author":[{"given":"J.","family":"Conway","sequence":"first","affiliation":[]},{"given":"N.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1986,5]]},"URL":"http://dx.doi.org/10.1109/TIT.1986.1057187","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:HKSrings","type":"chapter","author":[{"family":"Dougherty","given":"S.T."}],"title":"Codes over rings","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"arxiv:quant-ph/0608027","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2014.2313559","page":"3073-3089","source":"Crossref","title":"Catalytic Quantum Error Correction","volume":"60","author":[{"given":"Todd A.","family":"Brun","sequence":"first","affiliation":[]},{"given":"Igor","family":"Devetak","sequence":"additional","affiliation":[]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2313559","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0608027"},{"id":"arxiv:quant-ph/0610092","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"5798","abstract":"<jats:p>We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard quantum error–correcting codes, thus allowing us to “quantize” all of classical linear coding theory. In particular, efficient modern classical codes that attain the Shannon capacity can be made into entanglement-assisted quantum codes attaining the hashing bound (closely related to the quantum capacity). For systems without large amounts of shared entanglement, these codes can also be used as catalytic codes, in which a small amount of initial entanglement enables quantum communication.</jats:p>","DOI":"10.1126/science.1131563","page":"436-439","source":"Crossref","title":"Correcting Quantum Errors with Entanglement","volume":"314","author":[{"given":"Todd","family":"Brun","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089, USA."}]},{"given":"Igor","family":"Devetak","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089, USA."}]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2006,10,20]]},"URL":"http://dx.doi.org/10.1126/science.1131563","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.1131563\narxivid:quant-ph/0610092"},{"id":"doi:10.1007/s10623-014-9997-6","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-014-9997-6","page":"193-202","source":"Crossref","title":"Entanglement-assisted quantum codes from arbitrary binary linear codes","volume":"77","author":[{"given":"Jianfa","family":"Qian","sequence":"first","affiliation":[]},{"given":"Lina","family":"Zhang","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2014,7,26]]},"URL":"http://dx.doi.org/10.1007/s10623-014-9997-6","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9997"},{"id":"arxiv:1004.3791","type":"journal-article","publisher":"IOP Publishing","issue":"8","DOI":"10.1088/1367-2630/12/8/083039","page":"083039","source":"Crossref","title":"Majorana fermion codes","volume":"12","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Barbara M","family":"Terhal","sequence":"additional","affiliation":[]},{"given":"Bernhard","family":"Leemhuis","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2010,8,17]]},"URL":"http://dx.doi.org/10.1088/1367-2630/12/8/083039","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1004.3791"},{"id":"arxiv:1703.00459","type":"article-journal","author":[{"given":"Sagar","family":"Vijay"},{"given":"Liang","family":"Fu"}],"title":"Quantum Error Correction for Complex and Majorana Fermion Qubits","issued":{"date-parts":[[2017,3,1]]},"note":"arxivid:1703.00459\narxiv_version_number:1"},{"id":"arxiv:2312.17518","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","abstract":"<jats:title>Abstract</jats:title><jats:p>CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes <jats:inline-formula><jats:alternatives><jats:tex-math>$$(C_1, C_2)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(</mml:mo>\n                    <mml:msub>\n                      <mml:mi>C</mml:mi>\n                      <mml:mn>1</mml:mn>\n                    </mml:msub>\n                    <mml:mo>,</mml:mo>\n                    <mml:msub>\n                      <mml:mi>C</mml:mi>\n                      <mml:mn>2</mml:mn>\n                    </mml:msub>\n                    <mml:mo>)</mml:mo>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:tex-math>$$C_1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mi>C</mml:mi>\n                    <mml:mn>1</mml:mn>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula> contains <jats:inline-formula><jats:alternatives><jats:tex-math>$$C_2$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mi>C</mml:mi>\n                    <mml:mn>2</mml:mn>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$C_2$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mi>C</mml:mi>\n                    <mml:mn>2</mml:mn>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula> is even, and the shortening of the dual of <jats:inline-formula><jats:alternatives><jats:tex-math>$$C_1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mi>C</mml:mi>\n                    <mml:mn>1</mml:mn>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula> with respect to the support of each codeword of <jats:inline-formula><jats:alternatives><jats:tex-math>$$C_2$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mi>C</mml:mi>\n                    <mml:mn>2</mml:mn>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula> is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes <jats:inline-formula><jats:alternatives><jats:tex-math>$$(C_1, C_2)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(</mml:mo>\n                    <mml:msub>\n                      <mml:mi>C</mml:mi>\n                      <mml:mn>1</mml:mn>\n                    </mml:msub>\n                    <mml:mo>,</mml:mo>\n                    <mml:msub>\n                      <mml:mi>C</mml:mi>\n                      <mml:mn>2</mml:mn>\n                    </mml:msub>\n                    <mml:mo>)</mml:mo>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula> is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.\n</jats:p>","DOI":"10.1007/s11128-024-04427-5","source":"Crossref","title":"An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance","volume":"23","author":[{"ORCID":"https://orcid.org/0000-0001-9662-8051","authenticated-orcid":false,"given":"Eduardo","family":"Camps-Moreno","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9832-7145","authenticated-orcid":false,"given":"Hiram H.","family":"López","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8977-8171","authenticated-orcid":false,"given":"Gretchen L.","family":"Matthews","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7304-0087","authenticated-orcid":false,"given":"Diego","family":"Ruano","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0944-7584","authenticated-orcid":false,"given":"Rodrigo","family":"San-José","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0122-7699","authenticated-orcid":false,"given":"Ivan","family":"Soprunov","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,6]]},"URL":"http://dx.doi.org/10.1007/s11128-024-04427-5","ISSN":["1573-1332"],"container-title-short":"Quantum Inf Process","page":"230","note":"alternative-id:4427\narxivid:2312.17518"},{"id":"arxiv:2011.09746","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We study a three-fold variant of the hypergraph product code construction, differing from the standard homological product of three classical codes. When instantiated with 3 classical LDPC codes, this \"XYZ product\" yields a non CSS quantum LDPC code which might display a large minimum distance. The simplest instance of this construction, corresponding to the product of 3 repetition codes, is a non CSS variant of the 3-dimensional toric code known as the Chamon code. The general construction was introduced in Denise Maurice's PhD thesis, but has remained poorly understood so far. The reason is that while hypergraph product codes can be analyzed with combinatorial tools, the XYZ product codes also depend crucially on the algebraic properties of the parity-check matrices of the three classical codes, making their analysis much more involved.Our main motivation for studying XYZ product codes is that the natural representatives of logical operators are two-dimensional objects. This contrasts with standard hypergraph product codes in 3 dimensions which always admit one-dimensional logical operators. In particular, specific instances of XYZ product codes with constant rate might display a minimum distance as large as <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x0398;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. While we do not prove this result here, we obtain the dimension of a large class of XYZ product codes, and when restricting to codes with dimension 1, we reduce the problem of computing the minimum distance to a more elementary combinatorial problem involving binary 3-tensors. We also discuss in detail some families of XYZ product codes that can be embedded in three dimensions with local interaction. Some of these codes seem to share properties with Haah's cubic codes and might be interesting candidates for self-correcting quantum memories with a logarithmic energy barrier.</jats:p>","DOI":"10.22331/q-2022-07-14-766","page":"766","source":"Crossref","title":"Quantum XYZ Product Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[{"name":"Inria, France"}]},{"ORCID":"https://orcid.org/0000-0003-3823-6804","authenticated-orcid":false,"given":"Simon","family":"Apers","sequence":"additional","affiliation":[{"name":"CNRS, IRIF, Université Paris Cité"}]},{"ORCID":"https://orcid.org/0000-0002-3445-0179","authenticated-orcid":false,"given":"Christophe","family":"Vuillot","sequence":"additional","affiliation":[{"name":"Inria, France"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,14]]},"URL":"http://dx.doi.org/10.22331/q-2022-07-14-766","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2011.09746"},{"id":"arxiv:quant-ph/0610088","type":"article-journal","author":[{"given":"Dave","family":"Bacon"},{"given":"Andrea","family":"Casaccino"}],"title":"Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes","issued":{"date-parts":[[2006,10,17]]},"note":"arxivid:quant-ph/0610088\narxiv_version_number:2"},{"id":"arxiv:2507.05408","type":"article-journal","author":[{"given":"Victor","family":"Barizien"},{"given":"Hugo","family":"Jacinto"},{"given":"Nicolas","family":"Sangouard"}],"title":"Accessible Quantum Gates on Classical Stabilizer Codes","issued":{"date-parts":[[2025,7,7]]},"note":"arxivid:2507.05408\narxiv_version_number:1"},{"id":"arxiv:2404.16129","type":"article-journal","author":[{"given":"Edward","family":"Farhi"},{"given":"Stephen P.","family":"Jordan"}],"title":"Efficiently constructing a quantum uniform superposition over bit strings near a binary linear code","issued":{"date-parts":[[2024,4,24]]},"note":"arxivid:2404.16129\narxiv_version_number:1"},{"id":"arxiv:2408.06299","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Entanglement is essential for quantum information processing, but is limited by noise. We address this by developing high-yield entanglement distillation protocols with several advancements. (1) We extend the 2-to-1 recurrence entanglement distillation protocol to higher-rate <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mi>n</a:mi></a:math>-to-<d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><d:mo stretchy=\"false\">(</d:mo><d:mi>n</d:mi><d:mo>−</d:mo><d:mn>1</d:mn><d:mo stretchy=\"false\">)</d:mo></d:math> protocols that can correct any single-qubit errors. These protocols are evaluated through numerical simulations focusing on fidelity and yield. We also outline a method to adapt any classical error-correcting code for entanglement distillation, where the code can correct both bit-flip and phase-flip errors by incorporating Hadamard gates. (2) We propose a constant-depth decoder for stabilizer codes that transforms logical states into physical ones using single-qubit measurements. This decoder is applied to entanglement distillation protocols, reducing circuit depth and enabling protocols derived from high-performance quantum error-correcting codes. We demonstrate this by evaluating the circuit complexity for entanglement distillation protocols based on surface codes and quantum convolutional codes. (3) Our stabilizer entanglement distillation techniques advance quantum computing. We propose a fault-tolerant protocol for constant-depth encoding and decoding of arbitrary states in surface codes, with potential extensions to more general quantum low-density parity-check codes. This protocol is feasible with state-of-the-art reconfigurable atom arrays and surpasses the limits of conventional logarithmic depth encoders. Overall, our study integrates stabilizer formalism, measurement-based quantum computing, and entanglement distillation, advancing both quantum communication and computing.</jats:p>","DOI":"10.1103/prxquantum.6.010339","source":"Crossref","title":"Stabilizer Entanglement Distillation and Efficient Fault-Tolerant Encoders","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-5259-3483","authenticated-orcid":true,"given":"Yu","family":"Shi","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03m2x1q45","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Arizona"},{"id":[{"id":"https://ror.org/03m2x1q45","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Arizona"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0002-5265-0609","authenticated-orcid":true,"given":"Ashlesha","family":"Patil","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03m2x1q45","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Arizona"},{"id":[{"id":"https://ror.org/03m2x1q45","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Arizona"}]},{"ORCID":"https://orcid.org/0000-0002-2581-4380","authenticated-orcid":true,"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03m2x1q45","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Arizona"},{"id":[{"id":"https://ror.org/03m2x1q45","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Arizona"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,4]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.010339","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010339","note":"arxivid:2408.06299"},{"id":"doi:10.1016/0095-8956(91)90066-S","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0095-8956(91)90066-s","page":"250-258","source":"Crossref","title":"Constructing self-dual codes using graphs","volume":"52","author":[{"given":"Haluk","family":"Oral","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series B","original-title":[],"language":"en","issued":{"date-parts":[[1991,7]]},"URL":"http://dx.doi.org/10.1016/0095-8956(91)90066-S","ISSN":["0095-8956"],"container-title-short":"Journal of Combinatorial Theory, Series B","note":"alternative-id:009589569190066S"},{"id":"arxiv:1411.7046","type":"journal-article","publisher":"IOP Publishing","issue":"1","DOI":"10.1088/1367-2630/18/1/013050","page":"013050","source":"Crossref","title":"A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)","volume":"18","author":[{"given":"Courtney G","family":"Brell","sequence":"first","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2016,1,25]]},"URL":"http://dx.doi.org/10.1088/1367-2630/18/1/013050","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1411.7046"},{"id":"preset:HKSbasics","type":"chapter","author":[{"family":"Huffman","given":"W.C."},{"family":"Kim","given":"J.-L."},{"family":"Solé","given":"P."}],"title":"Basics of coding theory","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1007/3-540-28991-7","type":"book","publisher":"Springer-Verlag","DOI":"10.1007/3-540-28991-7","source":"Crossref","title":"Classification Algorithms for Codes and Designs","container-title":"Algorithms and Computation in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2006]]},"ISBN":["3540289909"],"URL":"http://dx.doi.org/10.1007/3-540-28991-7","note":"publisher-location:Berlin/Heidelberg"},{"id":"doi:10.1109/TIT.1977.1055688","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1977.1055688","page":"157-166","source":"Crossref","title":"New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities","volume":"23","author":[{"given":"R.","family":"McEliece","sequence":"first","affiliation":[]},{"given":"E.","family":"Rodemich","sequence":"additional","affiliation":[]},{"given":"H.","family":"Rumsey","sequence":"additional","affiliation":[]},{"given":"L.","family":"Welch","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1977,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1977.1055688","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s00454-008-9128-0","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00454-008-9128-0","page":"199-207","source":"Crossref","title":"Linear Programming Bounds for Codes via a Covering Argument","volume":"41","author":[{"given":"Michael","family":"Navon","sequence":"first","affiliation":[]},{"given":"Alex","family":"Samorodnitsky","sequence":"additional","affiliation":[]}],"container-title":"Discrete &amp; Computational Geometry","original-title":[],"language":"en","issued":{"date-parts":[[2008,12,10]]},"URL":"http://dx.doi.org/10.1007/s00454-008-9128-0","ISSN":["0179-5376","1432-0444"],"container-title-short":"Discrete Comput Geom","note":"alternative-id:9128"},{"id":"doi:10.1137/S0895480102408353","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/s0895480102408353","page":"700-718","source":"Crossref","title":"Generalized Alon--Boppana Theorems and Error-Correcting Codes","volume":"19","author":[{"given":"Joel","family":"Friedman","sequence":"first","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2005,1]]},"URL":"http://dx.doi.org/10.1137/S0895480102408353","ISSN":["0895-4801","1095-7146"],"container-title-short":"SIAM J. Discrete Math.","note":"alternative-id:10.1137/S0895480102408353"},{"id":"doi:10.1142/9789812832245_0002","type":"proceedings-article","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/9789812832245_0002","page":"15-24","source":"Crossref","title":"A Functional View of Upper Bounds on Codes","author":[{"given":"Alexander","family":"Barg","sequence":"first","affiliation":[{"name":"Department of ECE/ISR, University of Maryland, College Park MD 20742, USA"},{"name":"Dobrushin Math. Laboratory, Institute for Information Transmission Problems, Bol'shoj Karetnyj 19, Moscow 101447, Russia"}]},{"given":"Dmitry","family":"Nogin","sequence":"additional","affiliation":[{"name":"Dobrushin Math. Laboratory, Institute for Information Transmission Problems, Bol'shoj Karetnyj 19, Moscow 101447, Russia"}]}],"event":"Proceedings of the First International Workshop","container-title":"Coding and Cryptology","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1142/9789812832245_0002","note":"alternative-id:10.1142/9789812832245_0002\n10.1142/6915"},{"id":"doi:10.1134/S0032946006020025","type":"journal-article","publisher":"Pleiades Publishing Ltd","issue":"2","DOI":"10.1134/s0032946006020025","page":"77-89","source":"Crossref","title":"Spectral approach to linear programming bounds on codes","volume":"42","author":[{"given":"A. M.","family":"Barg","sequence":"first","affiliation":[]},{"given":"D. Yu.","family":"Nogin","sequence":"additional","affiliation":[]}],"container-title":"Problems of Information Transmission","original-title":[],"language":"en","issued":{"date-parts":[[2006,6]]},"URL":"http://dx.doi.org/10.1134/S0032946006020025","ISSN":["0032-9460","1608-3253"],"container-title-short":"Probl Inf Transm","note":"alternative-id:2002"},{"id":"arxiv:2104.14587","type":"article-journal","author":[{"given":"Alex","family":"Samorodnitsky"}],"title":"One more proof of the first linear programming bound for binary codes and two conjectures","issued":{"date-parts":[[2021,4,29]]},"note":"arxivid:2104.14587\narxiv_version_number:1"},{"id":"arxiv:2303.16619","type":"article-journal","author":[{"given":"Nati","family":"Linial"},{"given":"Elyassaf","family":"Loyfer"}],"title":"An Elementary Proof of the First LP Bound on the Rate of Binary Codes","issued":{"date-parts":[[2023,3,29]]},"note":"arxivid:2303.16619\narxiv_version_number:1"},{"id":"arxiv:1504.05155","type":"article-journal","author":[{"given":"Scott","family":"Aaronson"},{"given":"Daniel","family":"Grier"},{"given":"Luke","family":"Schaeffer"}],"title":"The Classification of Reversible Bit Operations","issued":{"date-parts":[[2015,4,20]]},"note":"arxivid:1504.05155\narxiv_version_number:1"},{"id":"doi:10.1007/978-3-030-01177-2_67","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-030-01177-2_67","page":"898-919","source":"Crossref","title":"Ternary Computing to Strengthen Cybersecurity","author":[{"given":"Bertrand","family":"Cambou","sequence":"first","affiliation":[]},{"given":"Donald","family":"Telesca","sequence":"additional","affiliation":[]}],"container-title":"Advances in Intelligent Systems and Computing","original-title":[],"issued":{"date-parts":[[2018,11,2]]},"ISBN":["9783030011765","9783030011772"],"URL":"http://dx.doi.org/10.1007/978-3-030-01177-2_67","ISSN":["2194-5357","2194-5365"],"note":"publisher-location:Cham"},{"id":"doi:10.1109/CCWC.2019.8666511","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/ccwc.2019.8666511","source":"Crossref","title":"Key Exchange using Ternary system to Enhance Security","author":[{"given":"Sareh","family":"Assiri","sequence":"first","affiliation":[]},{"given":"Bertrand","family":"Cambou","sequence":"additional","affiliation":[]},{"given":"D. Duane","family":"Booher","sequence":"additional","affiliation":[]},{"given":"Dina","family":"Ghanai Miandoab","sequence":"additional","affiliation":[]},{"given":"Mohammad","family":"Mohammadinodoushan","sequence":"additional","affiliation":[]}],"event":"2019 IEEE 9th Annual Computing and Communication Workshop and Conference (CCWC)","container-title":"2019 IEEE 9th Annual Computing and Communication Workshop and Conference (CCWC)","original-title":[],"issued":{"date-parts":[[2019,1]]},"URL":"http://dx.doi.org/10.1109/CCWC.2019.8666511"},{"id":"preset:Delsarte72","type":"article-journal","author":[{"family":"Delsarte","given":"P."}],"title":"Bounds for unrestricted codes, by linear programming","volume":"27","container-title":"Philips Research Reports","issued":"1972","page":"272–289"},{"id":"arxiv:1710.03109","type":"article-journal","author":[{"given":"Umberto","family":"Martínez-Peñas"}],"title":"Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring","issued":{"date-parts":[[2018,2,8]]},"note":"arxivid:1710.03109\narxiv_version_number:2"},{"id":"arxiv:1407.3413","type":"article-journal","author":[{"given":"Bei","family":"Zeng"},{"given":"Duan-Lu","family":"Zhou"}],"title":"Topological and Error-Correcting Properties for Symmetry-Protected Topological Order","issued":{"date-parts":[[2014,7,12]]},"note":"arxivid:1407.3413\narxiv_version_number:1"},{"id":"doi:10.3390/math11244950","type":"journal-article","publisher":"MDPI AG","issue":"24","abstract":"<jats:p>Binary codes have a special place in coding theory since they are one of the most commonly used in practice. There are classes of codes specific only to the binary case. One such class is self-complementary codes. Self-complementary linear codes are binary codes that, together with any vector, contain its complement as well. This paper is about binary linear self-complementary codes. A natural goal in coding theory is to find a linear code with a given length n and dimension k such that the minimum distance d is maximal. Codes with these properties are called optimal. Another important issue is classifying the optimal codes, i.e., finding exactly one representative of each equivalence class. In some sense, the classification problem is more general than the minimum distance bounds problem. In this work, we summarize the classification results for self-complementary codes with the maximum possible minimum distance and a length of up to 20. For the classification, we developed a new algorithm that is much more efficient compared to existing ones in some cases.</jats:p>","DOI":"10.3390/math11244950","page":"4950","source":"Crossref","title":"Some Results on Self-Complementary Linear Codes","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0003-3489-7744","authenticated-orcid":false,"given":"Maria","family":"Pashinska-Gadzheva","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 5000 Veliko Tarnovo, Bulgaria"}]},{"ORCID":"https://orcid.org/0000-0002-6730-1129","authenticated-orcid":false,"given":"Iliya","family":"Bouyukliev","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 5000 Veliko Tarnovo, Bulgaria"}]},{"ORCID":"https://orcid.org/0000-0003-2503-5325","authenticated-orcid":false,"given":"Valentin","family":"Bakoev","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Informatics, St. Cyril and St. Methodius University of Veliko Tarnovo, 5000 Veliko Tarnovo, Bulgaria"}]}],"container-title":"Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,14]]},"URL":"http://dx.doi.org/10.3390/math11244950","ISSN":["2227-7390"],"container-title-short":"Mathematics","note":"alternative-id:math11244950"},{"id":"arxiv:quant-ph/0701065","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.99.130505","source":"Crossref","title":"Simple Family of Nonadditive Quantum Codes","volume":"99","author":[{"given":"John A.","family":"Smolin","sequence":"first","affiliation":[]},{"given":"Graeme","family":"Smith","sequence":"additional","affiliation":[]},{"given":"Stephanie","family":"Wehner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,9,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.99.130505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130505","note":"arxivid:quant-ph/0701065"},{"id":"doi:10.1109/TIT.1962.1057721","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1962.1057721","page":"200-202","source":"Crossref","title":"Some bounds for error-correcting codes","volume":"8","author":[{"given":"L.","family":"Grey","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1962,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1962.1057721","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/978-3-642-41970-6_33","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-41970-6_33","page":"288-304","source":"Crossref","title":"Über Steinersche Systeme","author":[{"given":"Ernst","family":"Witt","sequence":"first","affiliation":[]}],"container-title":"Springer Collected Works in Mathematics","original-title":[],"issued":{"date-parts":[[1998]]},"ISBN":["9783642416316","9783642419706"],"URL":"http://dx.doi.org/10.1007/978-3-642-41970-6_33","ISSN":["2194-9875","2194-9883"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1017/CBO9780511549533","type":"monograph","publisher":"Cambridge University Press","DOI":"10.1017/cbo9780511549533","source":"Crossref","title":"Design Theory","author":[{"given":"Thomas","family":"Beth","sequence":"first","affiliation":[]},{"given":"Deiter","family":"Jungnickel","sequence":"additional","affiliation":[]},{"given":"Hanfried","family":"Lenz","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1999,11,18]]},"ISBN":["9780521444323","9780511549533"],"URL":"http://dx.doi.org/10.1017/CBO9780511549533","note":"edition-number:2"},{"id":"doi:10.1007/BF01191366","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/bf01191366","page":"376-378","source":"Crossref","title":"On an infinite class of Steiner systems constructed from affine spaces","volume":"47","author":[{"given":"J. D.","family":"Key","sequence":"first","affiliation":[]},{"given":"A.","family":"Wagner","sequence":"additional","affiliation":[]}],"container-title":"Archiv der Mathematik","original-title":[],"language":"en","issued":{"date-parts":[[1986,10]]},"URL":"http://dx.doi.org/10.1007/BF01191366","ISSN":["0003-889X","1420-8938"],"container-title-short":"Arch. Math","note":"alternative-id:BF01191366"},{"id":"arxiv:1401.3665","type":"article-journal","author":[{"given":"Peter","family":"Keevash"}],"title":"The existence of designs","issued":{"date-parts":[[2024,11,27]]},"note":"arxivid:1401.3665\narxiv_version_number:4"},{"id":"doi:10.1016/0012-365X(87)90061-6","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0012-365x(87)90061-6","page":"301-311","source":"Crossref","title":"Non-trivial t-designs without repeated blocks exist for all t","volume":"65","author":[{"given":"Luc","family":"Teirlinck","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1987,7]]},"URL":"http://dx.doi.org/10.1016/0012-365X(87)90061-6","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X87900616"},{"id":"arxiv:1611.06827","type":"article-journal","author":[{"given":"Stefan","family":"Glock"},{"given":"Daniela","family":"Kühn"},{"given":"Allan","family":"Lo"},{"given":"Deryk","family":"Osthus"}],"title":"The existence of designs via iterative absorption: hypergraph $F$-designs for arbitrary $F$","issued":{"date-parts":[[2020,2,28]]},"note":"arxivid:1611.06827\narxiv_version_number:3"},{"id":"arxiv:1802.05900","type":"article-journal","author":[{"given":"Peter","family":"Keevash"}],"title":"The existence of designs II","issued":{"date-parts":[[2018,2,16]]},"note":"arxivid:1802.05900\narxiv_version_number:1"},{"id":"arxiv:2411.18291","type":"article-journal","author":[{"given":"Peter","family":"Keevash"}],"title":"A short proof of the existence of designs","issued":{"date-parts":[[2024,11,27]]},"note":"arxivid:2411.18291\narxiv_version_number:1"},{"id":"doi:10.1006/jcta.1997.2765","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1006/jcta.1997.2765","page":"280-291","source":"Crossref","title":"Quasi-Symmetric Designs and Codes Meeting the Grey–Rankin Bound","volume":"78","author":[{"given":"Gary","family":"McGuire","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1997,5]]},"URL":"http://dx.doi.org/10.1006/jcta.1997.2765","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S0097316597927655"},{"id":"doi:10.1090/surv/175","type":"monograph","publisher":"American Mathematical Society","DOI":"10.1090/surv/175","source":"Crossref","title":"Algebraic Design Theory","author":[{"given":"Warwick","family":"de Launey","sequence":"first","affiliation":[]},{"given":"Dane","family":"Flannery","sequence":"additional","affiliation":[]}],"container-title":"Mathematical Surveys and Monographs","original-title":[],"issued":{"date-parts":[[2011,7,27]]},"ISBN":["9780821844960","9781470414023"],"URL":"http://dx.doi.org/10.1090/surv/175","ISSN":["0076-5376","2331-7159"],"note":"publisher-location:Providence, Rhode Island"},{"id":"preset:HKSdesigns","type":"chapter","author":[{"family":"Tonchev","given":"V.D."}],"title":"Codes and designs","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"arxiv:0811.1254","type":"book-chapter","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/9789812837172_0004","page":"121-158","source":"Crossref","title":"CODING THEORY AND ALGEBRAIC COMBINATORICS","author":[{"given":"MICHAEL","family":"HUBER","sequence":"first","affiliation":[{"name":"Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany"}]}],"container-title":"Series on Coding Theory and Cryptology","original-title":[],"issued":{"date-parts":[[2010,2]]},"URL":"http://dx.doi.org/10.1142/9789812837172_0004","ISSN":["1793-2238"],"note":"alternative-id:10.1142/9789812837172_0004\n10.1142/SCTC\n10.1142/7116\narxivid:0811.1254"},{"id":"doi:10.1201/9781420010541","type":"book","publisher":"Chapman and Hall/CRC","DOI":"10.1201/9781420010541","source":"Crossref","title":"Handbook of Combinatorial Designs","container-title":[],"original-title":[],"language":"en","editor":[{"given":"Charles J.","family":"Colbourn","sequence":"first","affiliation":[]},{"given":"Jeffrey H.","family":"Dinitz","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2006,11,2]]},"ISBN":["9780429138485"],"URL":"http://dx.doi.org/10.1201/9781420010541","note":"edition-number:0"},{"id":"doi:10.1017/CBO9780511623714","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Although graph theory, design theory, and coding theory had their origins in various areas of applied mathematics, today they are to be found under the umbrella of discrete mathematics. Here the authors have considerably reworked and expanded their earlier successful books on graphs, codes and designs, into an invaluable textbook. They do not seek to consider each of these three topics individually, but rather to stress the many and varied connections between them. The discrete mathematics needed is developed in the text, making this book accessible to any student with a background of undergraduate algebra. Many exercises and useful hints are included througout, and a large number of references are given.</jats:p>","DOI":"10.1017/cbo9780511623714","source":"Crossref","title":"Designs, Graphs, Codes and their Links","author":[{"given":"P. J.","family":"Cameron","sequence":"first","affiliation":[]},{"given":"J. H. van","family":"Lint","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1991,9,19]]},"ISBN":["9780521413251","9780521423854","9780511623714"],"URL":"http://dx.doi.org/10.1017/CBO9780511623714","note":"edition-number:1"},{"id":"doi:10.1007/b97564","type":"book","publisher":"Springer-Verlag","DOI":"10.1007/b97564","source":"Crossref","title":"Combinatorial Designs","container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2004]]},"ISBN":["0387954872"],"URL":"http://dx.doi.org/10.1007/b97564","note":"publisher-location:New York"},{"id":"doi:10.1137/1016056","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/1016056","page":"349-388","source":"Crossref","title":"Coding and Combinatorics","volume":"16","author":[{"given":"E. F.","family":"Assmus, Jr.","sequence":"first","affiliation":[]},{"given":"H. F.","family":"Mattson, Jr.","sequence":"additional","affiliation":[]}],"container-title":"SIAM Review","original-title":[],"language":"en","issued":{"date-parts":[[1974,7]]},"URL":"http://dx.doi.org/10.1137/1016056","ISSN":["0036-1445","1095-7200"],"container-title-short":"SIAM Rev.","note":"alternative-id:10.1137/1016056"},{"id":"doi:10.1016/S0021-9800(69)80115-8","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0021-9800(69)80115-8","page":"122-151","source":"Crossref","title":"New 5-designs","volume":"6","author":[{"suffix":"Jr.","given":"E.F.","family":"Assmus","sequence":"first","affiliation":[]},{"suffix":"Jr.","given":"H.F.","family":"Mattson","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory","original-title":[],"language":"en","issued":{"date-parts":[[1969,3]]},"URL":"http://dx.doi.org/10.1016/S0021-9800(69)80115-8","ISSN":["0021-9800"],"container-title-short":"Journal of Combinatorial Theory","note":"alternative-id:S0021980069801158"},{"id":"arxiv:2310.14281","type":"article-journal","author":[{"given":"Madoka","family":"Awada"}],"title":"Infinite series of $3$-designs in the extended quadratic residue code","issued":{"date-parts":[[2024,6,5]]},"note":"arxivid:2310.14281\narxiv_version_number:4"},{"id":"doi:10.1111/j.1749-6632.1970.tb56485","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1111/j.1749-6632.1970.tb56485.x","page":"310-313","source":"Crossref","title":"The Weight of the Symmetry Code for p=29 and the 5‐Designs Contained Therein","volume":"175","author":[{"given":"Vera","family":"Pless","sequence":"first","affiliation":[]}],"container-title":"Annals of the New York Academy of Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1970,9]]},"URL":"http://dx.doi.org/10.1111/j.1749-6632.1970.tb56485.x","ISSN":["0077-8923","1749-6632"],"container-title-short":"Annals of the New York Academy of Sciences","note":"alternative-id:10.1111/j.1749-6632.1970.tb56485.x"},{"id":"doi:10.1016/0097-3165(72)90088-X","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(72)90088-x","page":"119-142","source":"Crossref","title":"Symmetry codes over GF(3) and new five-designs","volume":"12","author":[{"given":"Vera","family":"Pless","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1972,1]]},"URL":"http://dx.doi.org/10.1016/0097-3165(72)90088-X","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:009731657290088X"},{"id":"doi:10.4153/CJM-1957-003-8","type":"journal-article","publisher":"Canadian Mathematical Society","abstract":"<jats:p>The principal result of this paper is the representation of the Mathieu\n                        group <jats:italic>M</jats:italic><jats:sub>23</jats:sub> as a group of 11 × 11 matrices over the Galois Field GF(2).\n                        This is a new representation of <jats:italic>M</jats:italic><jats:sub>23</jats:sub> and in §5 an indication of how the techniques of this result\n                        might be extended to the Mathieu group <jats:italic>M</jats:italic><jats:sub>11</jats:sub> is given.</jats:p>","DOI":"10.4153/cjm-1957-003-8","page":"15-18","source":"Crossref","title":"A Note on the Mathieu Groups","volume":"9","author":[{"given":"Lowell J.","family":"Paige","sequence":"first","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1957]]},"URL":"http://dx.doi.org/10.4153/CJM-1957-003-8","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j.\n                        math.","note":"alternative-id:S0008414X00044321"},{"id":"doi:10.1016/S1571-0653(04)00398-1","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/s1571-0653(04)00398-1","page":"220-234","source":"Crossref","title":"Combinatorial designs and perfect codes","volume":"10","author":[{"given":"Kevin T.","family":"Phelps","sequence":"first","affiliation":[]}],"container-title":"Electronic Notes in Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2001,11]]},"URL":"http://dx.doi.org/10.1016/S1571-0653(04)00398-1","ISSN":["1571-0653"],"container-title-short":"Electronic Notes in Discrete Mathematics","note":"special_numbering:C\nalternative-id:S1571065304003981"},{"id":"doi:10.1109/18.133244","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.133244","page":"1261-1268","source":"Crossref","title":"A strengthening of the Assmus-Mattson theorem","volume":"37","author":[{"given":"A.R.","family":"Calderbank","sequence":"first","affiliation":[]},{"given":"P.","family":"IDelsarte","sequence":"additional","affiliation":[]},{"given":"N.J.A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1991]]},"URL":"http://dx.doi.org/10.1109/18.133244","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1201/9781003040897","type":"book","publisher":"CRC Press","DOI":"10.1201/9781003040897","source":"Crossref","title":"CRC Handbook of Combinatorial Designs","container-title":[],"original-title":[],"language":"en","editor":[{"given":"Charles J.","family":"Colbourn","sequence":"first","affiliation":[]}],"issued":{"date-parts":[[2010,12,12]]},"ISBN":["9781003040897"],"URL":"http://dx.doi.org/10.1201/9781003040897","note":"edition-number:0"},{"id":"doi:10.1007/978-1-4613-0165-3_6","type":"book-chapter","publisher":"Springer New York","DOI":"10.1007/978-1-4613-0165-3_6","page":"113-130","source":"Crossref","title":"Evaluation of Gallager Codes for Short Block Length and High Rate Applications","author":[{"given":"David J. C.","family":"MacKay","sequence":"first","affiliation":[]},{"given":"Matthew C.","family":"Davey","sequence":"additional","affiliation":[]}],"container-title":"The IMA Volumes in Mathematics and its Applications","original-title":[],"language":"en","issued":{"date-parts":[[2001]]},"ISBN":["9780387951737","9781461301653"],"URL":"http://dx.doi.org/10.1007/978-1-4613-0165-3_6","ISSN":["0940-6573"],"note":"publisher-location:New York, NY"},{"id":"doi:10.1109/ITW.2001.955146","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.2001.955146","page":"90-92","source":"Crossref","title":"Regular low-density parity-check codes from combinatorial designs","author":[{"given":"S.J.","family":"Johnson","sequence":"first","affiliation":[]},{"given":"S.R.","family":"Weller","sequence":"additional","affiliation":[]}],"event":"2001 IEEE Information Theory Workshop","container-title":"Proceedings 2001 IEEE Information Theory Workshop (Cat. No.01EX494)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ITW.2001.955146"},{"id":"doi:10.1109/GLOCOM.2001.965562","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/glocom.2001.965562","page":"970-974","source":"Crossref","title":"Construction of low-density parity-check codes from Kirkman triple systems","volume":"2","author":[{"given":"S.J.","family":"Johnson","sequence":"first","affiliation":[]},{"given":"S.R.","family":"Weller","sequence":"additional","affiliation":[]}],"event":"GLOBECOM '01. IEEE Global Telecommunications Conference","container-title":"GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/GLOCOM.2001.965562"},{"id":"doi:10.1109/TCOMM.2003.816946","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tcomm.2003.816946","page":"1413-1419","source":"Crossref","title":"Resolvable 2-designs for regular low-density parity-check codes","volume":"51","author":[{"given":"S.J.","family":"Johnson","sequence":"first","affiliation":[]},{"given":"S.R.","family":"Weller","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[2003,9]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2003.816946","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"manual:-J.-Seberry-M.-Yamada-(1992).","type":"article-journal","author":[{"family":"Seberry","given":"J."},{"family":"Yamada","given":"M."}],"title":"Hadamard matrices, Sequences, and Block Designs","container-title":"University of Wollongong. Journal contribution","issued":"1992","URL":"https://hdl.handle.net/10779/uow.27846666.v1","note":"Available at \\url{https://hdl.handle.net/10779/uow.27846666.v1}"},{"id":"doi:10.1002/sapm1933121321","type":"journal-article","publisher":"Wiley","issue":"1-4","DOI":"10.1002/sapm1933121321","page":"321-333","source":"Crossref","title":"A Combinatorial Problem","volume":"12","author":[{"given":"J. A.","family":"Todd","sequence":"first","affiliation":[]}],"container-title":"Journal of Mathematics and Physics","original-title":[],"language":"en","issued":{"date-parts":[[1933,4]]},"URL":"http://dx.doi.org/10.1002/sapm1933121321","ISSN":["0097-1421"],"container-title-short":"Journal of Mathematics and Physics","note":"alternative-id:10.1002/sapm1933121321"},{"id":"arxiv:1910.08265","type":"article-journal","author":[{"given":"Cunsheng","family":"Ding"},{"given":"Chunming","family":"Tang"}],"title":"Infinite families of near MDS codes holding $t$-designs","issued":{"date-parts":[[2019,12,15]]},"note":"arxivid:1910.08265\narxiv_version_number:2"},{"id":"arxiv:2001.00158","type":"article-journal","author":[{"given":"Chunming","family":"Tang"},{"given":"Cunsheng","family":"Ding"}],"title":"An infinite family of linear codes supporting 4-designs","issued":{"date-parts":[[2020,1,1]]},"note":"arxivid:2001.00158\narxiv_version_number:1"},{"id":"preset:HKSselfdual","type":"chapter","author":[{"family":"Bouyuklieva","given":"S."}],"title":"Self-dual codes","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1007/s00026-005-0250-x","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00026-005-0250-x","page":"189-198","source":"Crossref","title":"A Characterization of Designs Related to an Extremal Doubly-Even Self-Dual Code of Length 48","volume":"9","author":[{"given":"Masaaki","family":"Harada","sequence":"first","affiliation":[]},{"given":"Akihiro","family":"Munemasa","sequence":"additional","affiliation":[]},{"given":"Vladimir D.","family":"Tonchev","sequence":"additional","affiliation":[]}],"container-title":"Annals of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2005,7]]},"URL":"http://dx.doi.org/10.1007/s00026-005-0250-x","ISSN":["0218-0006","0219-3094"],"container-title-short":"Ann. Comb.","note":"alternative-id:250"},{"id":"doi:10.1109/18.568714","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.568714","page":"1021-1025","source":"Crossref","title":"Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs","volume":"43","author":[{"given":"V.D.","family":"Tonchev","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1997,5]]},"URL":"http://dx.doi.org/10.1109/18.568714","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1023/A:1008340128973","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1023/a:1008340128973","page":"271-289","source":"Crossref","title":"Designs in Product Association Schemes","volume":"16","author":[{"given":"William J.","family":"Martin","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1999,5]]},"URL":"http://dx.doi.org/10.1023/A:1008340128973","ISSN":["0925-1022","1573-7586"],"container-title-short":"Designs, Codes and Cryptography","note":"alternative-id:199821"},{"id":"doi:10.1007/BF01390767","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01390767","page":"5-20","source":"Crossref","title":"On the construction of perfect deletion-correcting codes using design theory","volume":"6","author":[{"given":"Patrick A. H.","family":"Bours","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1995,7]]},"URL":"http://dx.doi.org/10.1007/BF01390767","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:BF01390767"},{"id":"doi:10.1023/A:1008212622423","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1023/a:1008212622423","page":"81-87","source":"Crossref","title":"Existence of Perfect 3-Deletion-Correcting Codes","volume":"14","author":[{"given":"A.","family":"Mahmoodi","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1998,4]]},"URL":"http://dx.doi.org/10.1023/A:1008212622423","ISSN":["0925-1022","1573-7586"],"container-title-short":"Designs, Codes and Cryptography","note":"alternative-id:153104"},{"id":"doi:10.1016/0012-365X(95)00333-R","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(95)00333-r","page":"55-82","source":"Crossref","title":"Optimal constant weight codes over Zk and generalized designs","volume":"169","author":[{"given":"Tuvi","family":"Etzion","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1997,5]]},"URL":"http://dx.doi.org/10.1016/0012-365X(95)00333-R","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9500333R"},{"id":"doi:10.4153/CJM-1963-069-5","type":"journal-article","publisher":"Canadian Mathematical Society","abstract":"<jats:p>Given a set <jats:italic>E</jats:italic> of <jats:italic>v</jats:italic> elements, and given positive integers <jats:italic>k</jats:italic>, <jats:italic>l</jats:italic> (<jats:italic>l</jats:italic> ≤ <jats:italic>k</jats:italic> ≤ <jats:italic>v</jats:italic>), and λ, we understand by a <jats:italic>tactical configuration</jats:italic><jats:italic>C</jats:italic>[<jats:italic>k, l, λ, v</jats:italic>] (briefly, <jats:italic>configuration</jats:italic>) a system of subsets of <jats:italic>E</jats:italic>, having <jats:italic>k</jats:italic> elements each, such that every subset of <jats:italic>E</jats:italic> having <jats:italic>l</jats:italic> elements is contained in exactly λ sets of the system.</jats:p><jats:p>A necessary condition for the existence of a configuration <jats:italic>C</jats:italic>[<jats:italic>k, l, λ, v</jats:italic>] is known (6) to be</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0008414X00029862_eqn001\" /></jats:disp-formula></jats:p>","DOI":"10.4153/cjm-1963-069-5","page":"702-722","source":"Crossref","title":"On Some Tactical Configurations","volume":"15","author":[{"given":"Haim","family":"Hanani","sequence":"first","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1963]]},"URL":"http://dx.doi.org/10.4153/CJM-1963-069-5","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j. math.","note":"alternative-id:S0008414X00029862"},{"id":"doi:10.1007/s10623-007-9130-1","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s10623-007-9130-1","page":"125-140","source":"Crossref","title":"Undetected error probability of q-ary constant weight codes","volume":"48","author":[{"given":"Shu-Tao","family":"Xia","sequence":"first","affiliation":[]},{"given":"Fang-Wei","family":"Fu","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2007,11,29]]},"URL":"http://dx.doi.org/10.1007/s10623-007-9130-1","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9130"},{"id":"doi:10.1023/A:1025484821641","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1023/a:1025484821641","page":"187-199","source":"Crossref","title":"Designs in Additive Codes over GF(4)","volume":"30","author":[{"given":"Jon-Lark","family":"Kim","sequence":"first","affiliation":[]},{"given":"Vera","family":"Pless","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2003,9]]},"URL":"http://dx.doi.org/10.1023/A:1025484821641","ISSN":["0925-1022","1573-7586"],"container-title-short":"Designs, Codes and Cryptography","note":"alternative-id:5142561"},{"id":"arxiv:2010.09448","type":"article-journal","author":[{"given":"Cunsheng","family":"Ding"},{"given":"Chunming","family":"Tang"},{"given":"Vladimir D.","family":"Tonchev"}],"title":"The Projective General Linear Group $\\mathrm{PGL}_2(\\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$","issued":{"date-parts":[[2020,10,19]]},"note":"arxivid:2010.09448\narxiv_version_number:1"},{"id":"doi:10.1016/S0012-365X(98)00035-1","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/s0012-365x(98)00035-1","page":"129-137","source":"Crossref","title":"Extremal double circulant Type II codes over Z4 and construction of 5-(24, 10, 36) designs","volume":"194","author":[{"given":"T.Aaron","family":"Gulliver","sequence":"first","affiliation":[]},{"given":"Masaaki","family":"Harada","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1999,1]]},"URL":"http://dx.doi.org/10.1016/S0012-365X(98)00035-1","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X98000351"},{"id":"doi:10.1002/(SICI)1520-6610(1998)6:3","type":"journal-article","publisher":"Wiley","issue":"3","DOI":"10.1002/(sici)1520-6610(1998)6:3<225::aid-jcd4>3.0.co;2-h","page":"225-229","source":"Crossref","title":"New 5-designs constructed from the lifted Golay code over ?4","volume":"6","author":[{"given":"Masaaki","family":"Harada","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Designs","original-title":[],"language":"en","issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1002/(SICI)1520-6610(1998)6:3<225::AID-JCD4>3.0.CO;2-H","ISSN":["1063-8539","1520-6610"],"container-title-short":"J. Combin. Designs"},{"id":"doi:10.1016/S0378-3758(99)00117-2","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0378-3758(99)00117-2","page":"349-368","source":"Crossref","title":"3-Colored 5-Designs and Z4-Codes","volume":"86","author":[{"given":"A.","family":"Bonnecaze","sequence":"first","affiliation":[]},{"given":"E.","family":"Rains","sequence":"additional","affiliation":[]},{"given":"P.","family":"Solé","sequence":"additional","affiliation":[]}],"container-title":"Journal of Statistical Planning and Inference","original-title":[],"language":"en","issued":{"date-parts":[[2000,5]]},"URL":"http://dx.doi.org/10.1016/S0378-3758(99)00117-2","ISSN":["0378-3758"],"container-title-short":"Journal of Statistical Planning and Inference","note":"alternative-id:S0378375899001172"},{"id":"doi:10.1134/S0081543815010149","type":"journal-article","publisher":"Pleiades Publishing Ltd","issue":"1","DOI":"10.1134/s0081543815010149","page":"189-202","source":"Crossref","title":"A survey on tight Euclidean t-designs and tight relative t-designs in certain association schemes","volume":"288","author":[{"given":"Eiichi","family":"Bannai","sequence":"first","affiliation":[]},{"given":"Etsuko","family":"Bannai","sequence":"additional","affiliation":[]},{"given":"Yan","family":"Zhu","sequence":"additional","affiliation":[]}],"container-title":"Proceedings of the Steklov Institute of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2015,1]]},"URL":"http://dx.doi.org/10.1134/S0081543815010149","ISSN":["0081-5438","1531-8605"],"container-title-short":"Proc. Steklov Inst. Math.","note":"alternative-id:6885"},{"id":"arxiv:1506.08857","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.92.032316","source":"Crossref","title":"Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices","volume":"92","author":[{"given":"Dardo","family":"Goyeneche","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Alsina","sequence":"additional","affiliation":[]},{"given":"José I.","family":"Latorre","sequence":"additional","affiliation":[]},{"given":"Arnau","family":"Riera","sequence":"additional","affiliation":[]},{"given":"Karol","family":"Życzkowski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,9,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.92.032316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032316","note":"arxivid:1506.08857"},{"id":"arxiv:1708.05946","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.97.062326","source":"Crossref","title":"Entanglement and quantum combinatorial designs","volume":"97","author":[{"given":"Dardo","family":"Goyeneche","sequence":"first","affiliation":[]},{"given":"Zahra","family":"Raissi","sequence":"additional","affiliation":[]},{"given":"Sara","family":"Di Martino","sequence":"additional","affiliation":[]},{"given":"Karol","family":"Życzkowski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,6,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.062326","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062326","note":"arxivid:1708.05946"},{"id":"arxiv:2111.04055","type":"journal-article","publisher":"IOP Publishing","issue":"50","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    Goyeneche\n                    <jats:italic>et al</jats:italic>\n                    [2018\n                    <jats:italic>Phys. Rev. A</jats:italic>\n                    <jats:bold>97</jats:bold>\n                    062326] introduced several classes of quantum combinatorial designs, namely quantum Latin squares, quantum Latin cubes, and the notion of orthogonality on them. They also showed that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Moreover, they established a relationship between quantum combinatorial designs and a remarkable class of entangled states called\n                    <jats:italic>k</jats:italic>\n                    -uniform states, i.e. multipartite pure states such that every reduction to\n                    <jats:italic>k</jats:italic>\n                    parties is maximally mixed. In this article, we put forward the notions of incomplete quantum Latin squares and orthogonality on them and present construction methods for mutually orthogonal quantum Latin squares and mutually orthogonal quantum Latin cubes. Furthermore, we introduce the notions of generalized mutually orthogonal quantum Latin squares and generalized mutually orthogonal quantum Latin cubes, which are equivalent to quantum orthogonal arrays of size\n                    <jats:italic>d</jats:italic>\n                    <jats:sup>2</jats:sup>\n                    and\n                    <jats:italic>d</jats:italic>\n                    <jats:sup>3</jats:sup>\n                    , respectively, and thus naturally provide two- and three-uniform states.\n                  </jats:p>","DOI":"10.1088/1751-8121/ac3705","page":"505204","source":"Crossref","title":"Quantum combinatorial designs and\n                    <i>k</i>\n                    -uniform states","volume":"54","author":[{"ORCID":"https://orcid.org/0000-0001-7763-4632","authenticated-orcid":false,"given":"Yajuan","family":"Zang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9152-6515","authenticated-orcid":false,"given":"Paolo","family":"Facchi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8503-5151","authenticated-orcid":false,"given":"Zihong","family":"Tian","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2021,11,26]]},"URL":"http://dx.doi.org/10.1088/1751-8121/ac3705","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:2111.04055"},{"id":"arxiv:1008.4747","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.82.042338","source":"Crossref","title":"Entanglement-assisted quantum low-density parity-check codes","volume":"82","author":[{"given":"Yuichiro","family":"Fujiwara","sequence":"first","affiliation":[]},{"given":"David","family":"Clark","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Vandendriessche","sequence":"additional","affiliation":[]},{"given":"Maarten","family":"De Boeck","sequence":"additional","affiliation":[]},{"given":"Vladimir D.","family":"Tonchev","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,10,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.82.042338","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042338","note":"arxivid:1008.4747"},{"id":"arxiv:quant-ph/0208140","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.68.012316","source":"Crossref","title":"Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation","volume":"68","author":[{"given":"G.","family":"Alber","sequence":"first","affiliation":[]},{"given":"Th.","family":"Beth","sequence":"additional","affiliation":[]},{"given":"Ch.","family":"Charnes","sequence":"additional","affiliation":[]},{"given":"A.","family":"Delgado","sequence":"additional","affiliation":[]},{"given":"M.","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"M.","family":"Mussinger","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,7,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.012316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012316","note":"arxivid:quant-ph/0208140"},{"id":"doi:10.1023/A:1024188005329","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1/3","DOI":"10.1023/a:1024188005329","page":"51-70","source":"Crossref","title":[],"volume":"29","author":[{"given":"Thomas","family":"Beth","sequence":"first","affiliation":[]},{"given":"Christopher","family":"Charnes","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"Gernot","family":"Alber","sequence":"additional","affiliation":[]},{"given":"Aldo","family":"Delgado","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Mussinger","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"issued":{"date-parts":[[2003]]},"URL":"http://dx.doi.org/10.1023/A:1024188005329","ISSN":["0925-1022"],"note":"alternative-id:5122113"},{"id":"doi:10.1007/s10623-013-9829-0","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s10623-013-9829-0","page":"805-823","source":"Crossref","title":"Extremal properties of t-SEEDs and recursive constructions","volume":"73","author":[{"given":"Yiling","family":"Lin","sequence":"first","affiliation":[]},{"given":"Masakazu","family":"Jimbo","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2013,6,15]]},"URL":"http://dx.doi.org/10.1007/s10623-013-9829-0","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9829"},{"id":"preset:Delsarte73","type":"article-journal","author":[{"family":"Delsarte","given":"P."}],"title":"An algebraic approach to the association schemes of coding theory","volume":"10","container-title":"Philips Research Reports Suppl","issued":"1973","page":"–97"},{"id":"doi:10.1016/0097-3165(76)90017-0","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0097-3165(76)90017-0","page":"230-243","source":"Crossref","title":"Association schemes and t-designs in regular semilattices","volume":"20","author":[{"given":"Philippe","family":"Delsarte","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1976,3]]},"URL":"http://dx.doi.org/10.1016/0097-3165(76)90017-0","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316576900170"},{"id":"doi:10.1137/0134012","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","DOI":"10.1137/0134012","page":"157-166","source":"Crossref","title":"Hahn Polynomials, Discrete Harmonics, and<i>t</i>-Designs","volume":"34","author":[{"given":"Ph.","family":"Delsarte","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1978,1]]},"URL":"http://dx.doi.org/10.1137/0134012","ISSN":["0036-1399","1095-712X"],"container-title-short":"SIAM J. Appl. Math.","note":"alternative-id:10.1137/0134012"},{"id":"arxiv:0909.4767","type":"article-journal","author":[{"given":"Christine","family":"Bachoc"}],"title":"Semidefinite programming, harmonic analysis and coding theory","issued":{"date-parts":[[2010,9,8]]},"note":"arxivid:0909.4767\narxiv_version_number:2"},{"id":"arxiv:1007.2905","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-1-4614-0769-0_9","page":"219-269","source":"Crossref","title":"Invariant Semidefinite Programs","author":[{"given":"Christine","family":"Bachoc","sequence":"first","affiliation":[]},{"given":"Dion C.","family":"Gijswijt","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Schrijver","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Vallentin","sequence":"additional","affiliation":[]}],"container-title":"International Series in Operations Research &amp; Management Science","original-title":[],"language":"en","issued":{"date-parts":[[2011,9,26]]},"ISBN":["9781461407683","9781461407690"],"URL":"http://dx.doi.org/10.1007/978-1-4614-0769-0_9","ISSN":["0884-8289","2214-7934"],"note":"publisher-location:New York, NY\narxivid:1007.2905"},{"id":"doi:10.1109/18.887851","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/18.887851","page":"2373-2395","source":"Crossref","title":"Upper bounds for constant-weight codes","volume":"46","author":[{"given":"K.","family":"Zeger","sequence":"first","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]},{"given":"E.","family":"Agrell","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000]]},"URL":"http://dx.doi.org/10.1109/18.887851","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.37236/1162","type":"journal-article","publisher":"The Electronic Journal of Combinatorics","issue":"1","abstract":"<jats:p>Existing tables of constant weight codes are mainly confined to codes of length $n\\leq 28$.  This paper presents tables of codes of lengths $29 \\leq n \\leq 63$. The motivation for creating these tables was their application to the generation of good sets of frequency hopping lists in radio networks.  The complete generation of all relevant cases by a small number of algorithms is augmented in individual cases by miscellaneous constructions. These sometimes give a larger number of codewords than the algorithms.</jats:p>","DOI":"10.37236/1162","source":"Crossref","title":"A New Table of Constant Weight Codes of Length Greater than 28","volume":"13","author":[{"given":"D. H.","family":"Smith","sequence":"first","affiliation":[]},{"given":"L. A.","family":"Hughes","sequence":"additional","affiliation":[]},{"given":"S.","family":"Perkins","sequence":"additional","affiliation":[]}],"container-title":"The Electronic Journal of Combinatorics","original-title":[],"issued":{"date-parts":[[2006,5,12]]},"URL":"http://dx.doi.org/10.37236/1162","ISSN":["1077-8926"],"container-title-short":"Electron. J. Combin.","page":"A2"},{"id":"doi:10.1109/18.59932","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.59932","page":"1334-1380","source":"Crossref","title":"A new table of constant weight codes","volume":"36","author":[{"given":"A.E.","family":"Brouwer","sequence":"first","affiliation":[]},{"given":"J.B.","family":"Shearer","sequence":"additional","affiliation":[]},{"given":"N.J.A.","family":"Sloane","sequence":"additional","affiliation":[]},{"given":"W.D.","family":"Smith","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1990]]},"URL":"http://dx.doi.org/10.1109/18.59932","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Hamming78","type":"article-journal","author":[{"family":"Hamming","given":"R.W."}],"container-title":"Letter","issued":"1978-04-05"},{"id":"doi:10.1145/1457838.1457879","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/1457838.1457879","page":"222-231","source":"Crossref","title":"IBM 7070 data processing system","author":[{"given":"J.","family":"Svigals","sequence":"first","affiliation":[]}],"event":"Papers presented at the the March 3-5, 1959, western joint computer conference","container-title":"Papers presented at the the March 3-5, 1959, western joint computer conference on XX - IRE-AIEE-ACM '59 (Western)","original-title":[],"issued":{"date-parts":[[1959]]},"URL":"http://dx.doi.org/10.1145/1457838.1457879","note":"publisher-location:New York, New York, USA\nproceedings-subject:XX"},{"id":"doi:10.5948/UPO9781614440215","type":"monograph","publisher":"American Mathematical Society","DOI":"10.5948/upo9781614440215","source":"Crossref","title":"From Error-Correcting Codes Through Sphere Packings to Simple Groups","author":[{"given":"Thomas","family":"Thompson","sequence":"first","affiliation":[]}],"container-title":"Carus Mathematical Monographs","original-title":[],"language":"en","issued":{"date-parts":[[1983]]},"ISBN":["9780883850374","9781614440215"],"URL":"http://dx.doi.org/10.5948/UPO9781614440215","ISSN":["0069-0813","2637-7535"],"note":"publisher-location:Providence, Rhode Island"},{"id":"doi:10.1109/TIT.1962.1057714","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1962.1057714","page":"203-207","source":"Crossref","title":"A new upper bound for error-correcting codes","volume":"8","author":[{"given":"S.","family":"Johnson","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1962,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1962.1057714","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-K.-Lindstr-m.-The-nonexisten","type":"thesis","author":[{"family":"Lindström","given":"K."}],"title":"The nonexistence of unknown nearly perfect binary codes","genre":"PhD thesis,","publisher":"Turun yliopisto","issued":"1975"},{"id":"manual:-K.-Lindstrom-and-M.-J.-Aalto","type":"article-journal","author":[{"family":"Lindstrom","given":"K."},{"family":"Aaltonen","given":"M.J."}],"title":"The nonexistence of nearly perfect nonbinary codes for 1 =< e =< 10","volume":null,"container-title":"Ann. Univ. Turku, Ser. A I","issue":"172","issued":"1976"},{"id":"doi:10.1016/S0019-9958(77)90519-8","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0019-9958(77)90519-8","page":"40-47","source":"Crossref","title":"All nearly perfect codes are known","volume":"35","author":[{"given":"Kauko","family":"Lindström","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1977,9]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(77)90519-8","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995877905198"},{"id":"doi:10.1016/0012-365X(72)90025-8","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(72)90025-8","page":"65-88","source":"Crossref","title":"Nearly perfect binary codes","volume":"3","author":[{"given":"J.M.","family":"Goethals","sequence":"first","affiliation":[]},{"given":"S.L.","family":"Snover","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1972]]},"URL":"http://dx.doi.org/10.1016/0012-365X(72)90025-8","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X72900258"},{"id":"preset:Semakov71","type":"article-journal","author":[{"family":"Semakov","given":"N.V."},{"family":"Zinoviev","given":"V.A."},{"family":"Zaitsev","given":"G.V."}],"title":"Uniformly Packed Codes","volume":"7","container-title":"Problemy Peredachi Informatsii","issue":"1","issued":"1971","page":"38–50"},{"id":"preset:Vasilyev62","type":"article-journal","author":[{"family":"Vasilyev","given":"J.L."}],"title":"On nongroup close-packed codes (in Russian","volume":"8","note":"translated in Probleme der Kibernetik 8","container-title":"Problemy Kibernetiki","issued":"1962","page":"337–339,"},{"id":"preset:Krotov00","type":"article-journal","author":[{"family":"Krotov","given":"D.S."}],"title":"Lower bounds for the number of m-quasigroups of order four and of the number of perfect binary codes","volume":"7","container-title":"Diskretnyi Analiz i Issledovanie Operatsii, Ser","issue":"2","issued":"2000","page":"47–53"},{"id":"doi:10.1137/0604040","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/0604040","page":"398-403","source":"Crossref","title":"A Combinatorial Construction of Perfect Codes","volume":"4","author":[{"given":"K. T.","family":"Phelps","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Algebraic Discrete Methods","original-title":[],"language":"en","issued":{"date-parts":[[1983,9]]},"URL":"http://dx.doi.org/10.1137/0604040","ISSN":["0196-5212","2168-345X"],"container-title-short":"SIAM. J. on Algebraic and Discrete Methods","note":"alternative-id:10.1137/0604040"},{"id":"doi:10.1109/18.335887","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.335887","page":"754-763","source":"Crossref","title":"Perfect binary codes: constructions, properties, and enumeration","volume":"40","author":[{"given":"T.","family":"Etzion","sequence":"first","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1994,5]]},"URL":"http://dx.doi.org/10.1109/18.335887","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:math/0608278","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2008.917692","page":"1760-1765","source":"Crossref","title":"On the Number of $1$-Perfect Binary Codes: A Lower Bound","volume":"54","author":[{"given":"Denis S.","family":"Krotov","sequence":"first","affiliation":[]},{"given":"Sergey V.","family":"Avgustinovich","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.917692","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:math/0608278"},{"id":"arxiv:0806.2513","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2009.2027525","page":"4657-4660","source":"Crossref","title":"The Perfect Binary One-Error-Correcting Codes of Length &lt;formula formulatype=\"inline\"&gt;&lt;tex Notation=\"TeX\"&gt;$15$&lt;/tex&gt;&lt;/formula&gt;: Part I—Classification","volume":"55","author":[{"given":"P.","family":"Ostergard","sequence":"first","affiliation":[]},{"given":"O.","family":"Pottonen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2009,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2009.2027525","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0806.2513"},{"id":"doi:10.1016/0097-3165(86)90021-X","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(86)90021-x","page":"45-51","source":"Crossref","title":"Every finite group is the automorphism group of some perfect code","volume":"43","author":[{"given":"K.T","family":"Phelps","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1986,9]]},"URL":"http://dx.doi.org/10.1016/0097-3165(86)90021-X","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:009731658690021X"},{"id":"doi:10.1109/ITW.1998.706460","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.1998.706460","page":"114-115","source":"Crossref","title":"Perfect binary codes with trivial automorphism group","author":[{"given":"S.V.","family":"Avgustinovich","sequence":"first","affiliation":[]},{"given":"F.I.","family":"Solov'eva","sequence":"additional","affiliation":[]}],"event":"1998 Information Theory Workshop","container-title":"1998 Information Theory Workshop (Cat. No.98EX131)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ITW.1998.706460"},{"id":"doi:10.3934/amc.2012.6.121","type":"journal-article","publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"2","DOI":"10.3934/amc.2012.6.121","page":"121-130","source":"Crossref","title":"On the symmetry group of extended perfect binary codes of length $n+1$ and rank $n-\\log(n+1)+2$","volume":"6","author":[{"given":"Olof","family":"Heden","sequence":"first","affiliation":[]},{"name":",Department of Mathematics, KTH, S-100 44 Stockholm","sequence":"first","affiliation":[]},{"given":"Fabio","family":"Pasticci","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Westerbäck","sequence":"additional","affiliation":[]},{"name":",Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1, I-06123 Perugia","sequence":"additional","affiliation":[]}],"container-title":"Advances in Mathematics of Communications","original-title":[],"language":"en","issued":{"date-parts":[[2012]]},"URL":"http://dx.doi.org/10.3934/amc.2012.6.121","ISSN":["1930-5338"]},{"id":"manual:-F.-I.-Solov-eva-and-S.-T.-To","type":"article-journal","author":[{"family":"Solov'eva","given":"F.I."},{"family":"Topalova","given":"S.T."}],"title":"On Automorphism Groups of Perfect Binary Codes and Steiner Triple Systems","volume":"36","container-title":"Problemy Peredachi Informatsii","issue":"4","issued":"2000","page":"53–58"},{"id":"manual:-F.-I.-Solov-eva-and-S.-T.-To","type":"article-journal","author":[{"family":"Solov'eva","given":"F.I."},{"family":"Topalova","given":"S.T."}],"title":"Perfect binary codes and Steiner triple systems with maximum orders of automorphism groups","volume":"7","container-title":"Diskretnyi Analiz i Issledovanie Operatsii, Ser","issue":"4","issued":"2000","page":"101–110"},{"id":"manual:-S.-A.-Malyugin-On-the-order-","type":"article-journal","author":[{"family":"Malyugin","given":"S.A."}],"title":"On the order of the automorphism group of perfect binary codes","volume":"7","container-title":"Diskretnyi Analiz i Issledovanie Operatsii, Ser","issue":"4","issued":"2000","page":"91–100"},{"id":"doi:10.1016/j.disc.2011.05.002","type":"journal-article","publisher":"Elsevier BV","issue":"17","DOI":"10.1016/j.disc.2011.05.002","page":"1879-1885","source":"Crossref","title":"On the size of the symmetry group of a perfect code","volume":"311","author":[{"given":"Olof","family":"Heden","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2011,9]]},"URL":"http://dx.doi.org/10.1016/j.disc.2011.05.002","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X11002093"},{"id":"doi:10.1016/S0019-9958(73)80007-5","type":"journal-article","publisher":"Elsevier BV","issue":"5","DOI":"10.1016/s0019-9958(73)80007-5","page":"407-438","source":"Crossref","title":"Four fundamental parameters of a code and their combinatorial significance","volume":"23","author":[{"given":"Philippe","family":"Delsarte","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1973,12]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(73)80007-5","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995873800075"},{"id":"arxiv:0903.2749","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2010.2046197","page":"2571-2582","source":"Crossref","title":"The Perfect Binary One-Error-Correcting Codes of Length 15: Part II—Properties","volume":"56","author":[{"given":"Patric R. J.","family":"Ostergard","sequence":"first","affiliation":[]},{"given":"Olli","family":"Pottonen","sequence":"additional","affiliation":[]},{"given":"Kevin T.","family":"Phelps","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2046197","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0903.2749"},{"id":"preset:HKSbounds","type":"chapter","author":[{"family":"Boyvalenkov","given":"P."},{"family":"Danev","given":"D."}],"title":"Linear programming bounds","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"arxiv:quant-ph/0605094","type":"journal-article","publisher":"AIP Publishing","issue":"5","abstract":"<jats:p>We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for nonorientable surfaces it is impossible to construct homological codes based on qudits of dimension D&amp;gt;2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension D. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor’s 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.</jats:p>","DOI":"10.1063/1.2731356","source":"Crossref","title":"Homological error correction: Classical and quantum codes","volume":"48","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[{"name":"Universidad Complutense Departamento de Física Teórica I, , 28040 Madrid, Spain"}]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[{"name":"Universidad Complutense Departamento de Física Teórica I, , 28040 Madrid, Spain"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2007,5,1]]},"URL":"http://dx.doi.org/10.1063/1.2731356","ISSN":["0022-2488","1089-7658"],"page":"052105","note":"arxivid:quant-ph/0605094"},{"id":"doi:10.1016/s0019-9958(60)90287-4","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0019-9958(60)90287-4","page":"68-79","source":"Crossref","title":"On a class of error correcting binary group codes","volume":"3","author":[{"given":"R.C.","family":"Bose","sequence":"first","affiliation":[]},{"given":"D.K.","family":"Ray-Chaudhuri","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1960,3]]},"URL":"http://dx.doi.org/10.1016/s0019-9958(60)90287-4","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995860902874"},{"id":"doi:10.1016/S0019-9958(60)90870-6","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s0019-9958(60)90870-6","page":"279-290","source":"Crossref","title":"Further results on error correcting binary group codes","volume":"3","author":[{"given":"R.C.","family":"Bose","sequence":"first","affiliation":[]},{"given":"D.K.","family":"Ray-Chaudhuri","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1960,9]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(60)90870-6","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995860908706"},{"id":"manual:-A.-Hocquenghem-Codes-correct","type":"article-journal","author":[{"family":"Hocquenghem","given":"A."}],"title":"Codes correcteurs d'Erreurs","volume":"2","container-title":"Chiffres (Paris","issued":"1959","page":"147–156,"},{"id":"preset:HKScyclic","type":"chapter","author":[{"family":"Ding","given":"C."}],"title":"Cyclic Codes over Finite Fields","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1109/TIT.1960.1057586","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1960.1057586","page":"459-470","source":"Crossref","title":"Encoding and error-correction procedures for the Bose-Chaudhuri codes","volume":"6","author":[{"given":"W.","family":"Peterson","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1960,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1960.1057586","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-S.-Arimoto-Encoding-and-deco","type":"chapter","author":[{"family":"Arimoto","given":"S."}],"title":"Encoding and decoding of p-ary group codes and the correction system","container-title":"Information Processing in Japan (in Japanese","volume":"2","issued":"1961-11","page":"320–325,"},{"id":"preset:Blahut","type":"book","author":[{"family":"Blahut","given":"R.E."}],"title":"\\emph{Theory and Practice of Error-Control Codes}","publisher":"Addison-Wesley","issued":"1983"},{"id":"doi:10.1109/TIT.1969.1054260","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1969.1054260","page":"122-127","source":"Crossref","title":"Shift-register synthesis and BCH decoding","volume":"15","author":[{"given":"J.","family":"Massey","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1969,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1969.1054260","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Berlekamp","type":"book","author":[{"family":"Berlekamp","given":"E.R."}],"title":"\\emph{Algebraic Coding Theory}","publisher":"McGraw-Hill","issued":"1968"},{"id":"doi:10.1109/TIT.1971.1054655","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1971.1054655","page":"464-466","source":"Crossref","title":"Inversionless decoding of binary BCH codes","volume":"17","author":[{"given":"H.","family":"Burton","sequence":"first","affiliation":[{"name":"NY Univ., NY, NY, USA"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1971,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1971.1054655","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:PetersonWeldon","type":"book","author":[{"family":"Peterson","given":"W.W."},{"family":"Weldon","given":"E.J."}],"title":"\\emph{Error-Correcting Codes}","publisher":"MIT Press","issued":"1972"},{"id":"doi:10.1007/978-3-7091-2945-6","type":"book","publisher":"Springer Vienna","DOI":"10.1007/978-3-7091-2945-6","source":"Crossref","title":"Information Theory and Reliable Communication","author":[{"given":"Robert","family":"Gallager","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1972]]},"ISBN":["9783211811450","9783709129456"],"URL":"http://dx.doi.org/10.1007/978-3-7091-2945-6","note":"publisher-location:Vienna"},{"id":"doi:10.1016/S0019-9958(75)90090-X","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0019-9958(75)90090-x","page":"87-99","source":"Crossref","title":"A method for solving key equation for decoding goppa codes","volume":"27","author":[{"given":"Yasuo","family":"Sugiyama","sequence":"first","affiliation":[]},{"given":"Masao","family":"Kasahara","sequence":"additional","affiliation":[]},{"given":"Shigeichi","family":"Hirasawa","sequence":"additional","affiliation":[]},{"given":"Toshihiko","family":"Namekawa","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1975,1]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(75)90090-X","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S001999587590090X"},{"id":"doi:10.1017/CBO9780511606267","type":"monograph","publisher":"Cambridge University Press","DOI":"10.1017/cbo9780511606267","source":"Crossref","title":"The Theory of Information and Coding","author":[{"given":"Robert","family":"McEliece","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2002,4,18]]},"ISBN":["9780521000956","9780511606267"],"URL":"http://dx.doi.org/10.1017/CBO9780511606267","note":"edition-number:2"},{"id":"doi:10.1109/18.782097","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.782097","page":"1757-1767","source":"Crossref","title":"Improved decoding of Reed-Solomon and algebraic-geometry codes","volume":"45","author":[{"given":"V.","family":"Guruswami","sequence":"first","affiliation":[]},{"given":"M.","family":"Sudan","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999]]},"URL":"http://dx.doi.org/10.1109/18.782097","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/SFCS.1998.743426","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/sfcs.1998.743426","page":"28-37","source":"Crossref","title":"Improved decoding of Reed-Solomon and algebraic-geometric codes","author":[{"given":"V.","family":"Guruswami","sequence":"first","affiliation":[]},{"given":"M.","family":"Sudan","sequence":"additional","affiliation":[]}],"event":"39th Annual Symposium on Foundations of Computer Science","container-title":"Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.1998.743426"},{"id":"manual:-K.-M.-Cheung-and-F.-Pollara-","type":"article-journal","author":[{"family":"Cheung","given":"K.-M."},{"family":"Pollara","given":"F."}],"title":"Phobos lander coding system: Software and analysis","container-title":"The Telecommunications and Data Acquisition Report","issued":"1988"},{"id":"preset:LinCostello","type":"document","author":[{"family":"Lin","given":"S."},{"family":"Costello","given":"D.J."}],"title":"\\emph{Error Control Coding}","edition":"2nd","note":"Prentice-Hall, 2004"},{"id":"preset:KLdatabase","type":"book","author":[{"family":"Helmling","given":"M."},{"family":"Scholl","given":"S."},{"family":"Gensheimer","given":"F."},{"family":"Dietz","given":"T."},{"family":"Kraft","given":"K."},{"family":"Ruzika","given":"S."},{"family":"Wehn","given":"N."}],"title":"\\emph{Database of Channel Codes and ML Simulation Results}","publisher":"\\href{http://www.uni-kl.de/channel-codes}{URL}","issued":"2022"},{"id":"preset:Schurer15b","type":"document","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Cyclic Codes (BCH-Bound). From MinT—the database of optimal net, code, OA, and OOA parameters","note":"Version: 2015-09-03.","URL":"https://web.archive.org/web/20240420202309/https://mint.sbg.ac.at/desc_CCyclic-BCHBound.html"},{"id":"doi:10.1016/S0019-9958(60)90877-9","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s0019-9958(60)90877-9","page":"291-294","source":"Crossref","title":"Two-error correcting Bose-Chaudhuri codes are quasi-perfect","volume":"3","author":[{"given":"Daniel","family":"Gorenstein","sequence":"first","affiliation":[]},{"given":"W. Wesley","family":"Peterson","sequence":"additional","affiliation":[]},{"given":"Neal","family":"Zierler","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1960,9]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(60)90877-9","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995860908779"},{"id":"doi:10.1109/TIT.1979.1056043","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1979.1056043","page":"361-362","source":"Crossref","title":"No primitive binaryt-error-correcting BCH code witht &amp;gt; 2is quasi-perfect (Corresp.)","volume":"25","author":[{"given":"T.","family":"Helleseth","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1979,5]]},"URL":"http://dx.doi.org/10.1109/TIT.1979.1056043","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Hill","type":"book","author":[{"family":"Hill","given":"R."}],"title":"\\emph{A First Course in Coding Theory}","publisher":"Oxford University Press","issued":"1988"},{"id":"doi:10.1016/S0019-9958(68)90874-7","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/s0019-9958(68)90874-7","page":"378-400","source":"Crossref","title":"A class of optimum nonlinear double-error-correcting codes","volume":"13","author":[{"given":"Franco P.","family":"Preparata","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1968,10]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(68)90874-7","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995868908747"},{"id":"arxiv:cs/9501101","type":"article-journal","author":[{"given":"T. G.","family":"Dietterich"},{"given":"G.","family":"Bakiri"}],"title":"Solving Multiclass Learning Problems via Error-Correcting Output Codes","issued":{"date-parts":[[1995,1,1]]},"note":"arxivid:cs/9501101\narxiv_version_number:1"},{"id":"arxiv:1206.0260","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.87.022344","source":"Crossref","title":"Block synchronization for quantum information","volume":"87","author":[{"given":"Yuichiro","family":"Fujiwara","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,2,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.022344","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022344","note":"arxivid:1206.0260"},{"id":"arxiv:1904.02703","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We study the performance of medium-length quantum LDPC (QLDPC) codes in the depolarizing channel. Only degenerate codes with the maximal stabilizer weight much smaller than their minimum distance are considered. It is shown that with the help of OSD-like post-processing the performance of the standard belief propagation (BP) decoder on many QLDPC codes can be improved by several orders of magnitude. Using this new BP-OSD decoder we study the performance of several known classes of degenerate QLDPC codes including hypergraph product codes, hyperbicycle codes, homological product codes, and Haah's cubic codes. We also construct several interesting examples of short generalized bicycle codes. Some of them have an additional property that their syndromes are protected by small BCH codes, which may be useful for the fault-tolerant syndrome measurement. We also propose a new large family of QLDPC codes that contains the class of hypergraph product codes, where one of the used parity-check matrices is square. It is shown that in some cases such codes have better performance than hypergraph product codes. Finally, we demonstrate that the performance of the proposed BP-OSD decoder for some of the constructed codes is better than for a relatively large surface code decoded by a near-optimal decoder.</jats:p>","DOI":"10.22331/q-2021-11-22-585","page":"585","source":"Crossref","title":"Degenerate Quantum LDPC Codes With Good Finite Length Performance","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-6772-7867","authenticated-orcid":false,"given":"Pavel","family":"Panteleev","sequence":"first","affiliation":[{"name":"Faculty of Mechanics and Mathematics, Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation"}]},{"ORCID":"https://orcid.org/0000-0003-2695-3179","authenticated-orcid":false,"given":"Gleb","family":"Kalachev","sequence":"additional","affiliation":[{"name":"Faculty of Mechanics and Mathematics, Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,11,22]]},"URL":"http://dx.doi.org/10.22331/q-2021-11-22-585","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1904.02703"},{"id":"doi:10.1109/TIT.1986.1057134","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1986.1057134","page":"23-40","source":"Crossref","title":"On the minimum distance of cyclic codes","volume":"32","author":[{"given":"J.","family":"van Lint","sequence":"first","affiliation":[]},{"given":"R.","family":"Wilson","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1986,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1986.1057134","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1961.1057659","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1961.1057659","page":"234-244","source":"Crossref","title":"Error correcting codes and their implementation for data transmission systems","volume":"7","author":[{"given":"J.","family":"Meggitt","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1961,10]]},"URL":"http://dx.doi.org/10.1109/TIT.1961.1057659","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1962.1057777","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1962.1057777","page":"5-9","source":"Crossref","title":"The use of information sets in decoding cyclic codes","volume":"8","author":[{"given":"E.","family":"Prange","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1962,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1962.1057777","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1002/j.1538-7305.1964.tb04075","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1002/j.1538-7305.1964.tb04075.x","page":"485-505","source":"Crossref","title":"Permutation Decoding of Systematic Codes","volume":"43","author":[{"given":"Jessie","family":"Macwilliams","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1964,1]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1964.tb04075.x","ISSN":["0005-8580"]},{"id":"doi:10.1109/18.119714","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.119714","page":"516-518","source":"Crossref","title":"On binary cyclic codes of odd lengths from 101 to 127","volume":"38","author":[{"given":"D.","family":"Schomaker","sequence":"first","affiliation":[]},{"given":"M.","family":"Wirtz","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1992,3]]},"URL":"http://dx.doi.org/10.1109/18.119714","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2005.851735","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2005.851735","page":"2849-2858","source":"Crossref","title":"Locally Testable Cyclic Codes","volume":"51","author":[{"given":"L.","family":"Babai","sequence":"first","affiliation":[]},{"given":"A.","family":"Shpilka","sequence":"additional","affiliation":[]},{"given":"D.","family":"Stefankovic","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2005,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.851735","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/978-3-642-16367-8_12","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-16367-8_12","page":"211-227","source":"Crossref","title":"Invariance in Property Testing","author":[{"given":"Madhu","family":"Sudan","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2010]]},"ISBN":["9783642163661","9783642163678"],"URL":"http://dx.doi.org/10.1007/978-3-642-16367-8_12","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"preset:TVNAlgCodes","type":"article-journal","author":[{"family":"Tsfasman","given":"M."},{"family":"Vlǎduţ","given":"S."},{"family":"Nogin","given":"D."}],"title":"\\emph{Algebraic Geometric Codes","volume":"139","publisher":"American Mathematical Society","container-title":"Basic Notions}","issued":"2007"},{"id":"preset:TVNAlgCodesAdv","type":"article-journal","author":[{"family":"Tsfasman","given":"M."},{"family":"Vlǎduţ","given":"S."},{"family":"Nogin","given":"D."}],"title":"\\emph{Algebraic Geometric Codes","volume":"238","publisher":"American Mathematical Society","container-title":"Advanced Chapters}","issued":"2019"},{"id":"doi:10.1007/978-94-011-3810-9","type":"book","publisher":"Springer Netherlands","DOI":"10.1007/978-94-011-3810-9","source":"Crossref","title":"Algebraic-Geometric Codes","author":[{"given":"M. A.","family":"Tsfasman","sequence":"first","affiliation":[]},{"given":"S. G.","family":"Vlăduţ","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1991]]},"ISBN":["9781402003356","9789401138109"],"URL":"http://dx.doi.org/10.1007/978-94-011-3810-9","note":"publisher-location:Dordrecht"},{"id":"arxiv:1512.07081","type":"article-journal","author":[{"given":"Benjamin","family":"Audoux"},{"given":"Alain","family":"Couvreur"}],"title":"On tensor products of CSS Codes","issued":{"date-parts":[[2018,9,25]]},"note":"arxivid:1512.07081\narxiv_version_number:3"},{"id":"arxiv:1202.0928","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2012.6284206","page":"348-352","source":"Crossref","title":"Improved quantum hypergraph-product LDPC codes","author":[{"given":"Alexey A.","family":"Kovalev","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"event":"2012 IEEE International Symposium on Information Theory - ISIT","container-title":"2012 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2012,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2012.6284206","note":"arxivid:1202.0928"},{"id":"doi:10.1007/978-3-7091-2786-5_1","type":"book-chapter","publisher":"Springer Vienna","DOI":"10.1007/978-3-7091-2786-5_1","page":"3-15","source":"Crossref","title":"Duadic Codes and Generalizations","author":[{"given":"V.","family":"Pless","sequence":"first","affiliation":[]}],"container-title":"Eurocode ’92","original-title":[],"issued":{"date-parts":[[1993]]},"ISBN":["9783211825198","9783709127865"],"URL":"http://dx.doi.org/10.1007/978-3-7091-2786-5_1","note":"publisher-location:Vienna"},{"id":"arxiv:2408.01906","type":"article-journal","author":[{"given":"Xianhong","family":"Xie"},{"given":"Yaxin","family":"Zhao"},{"given":"Zhonghua","family":"Sun"},{"given":"Xiaobo","family":"Zhou"}],"title":"Binary $[n,(n\\pm1)/2]$ cyclic codes with good minimum distances from sequences","issued":{"date-parts":[[2024,8,4]]},"note":"arxivid:2408.01906\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.1984.1056944","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1984.1056944","page":"709-714","source":"Crossref","title":"Duadic Codes","volume":"30","author":[{"given":"J.","family":"Leon","sequence":"first","affiliation":[]},{"given":"J.","family":"Masley","sequence":"additional","affiliation":[]},{"given":"V.","family":"Pless","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1984,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1984.1056944","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2301.06446","type":"article-journal","author":[{"given":"Hai","family":"Liu"},{"given":"Chengju","family":"Li"},{"given":"Cunsheng","family":"Ding"}],"title":"Five infinite families of binary cyclic codes and their related codes with good parameters","issued":{"date-parts":[[2023,1,16]]},"note":"arxivid:2301.06446\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.1967.1054048","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1967.1054048","page":"619-621","source":"Crossref","title":"Optimal binary sequences for spread spectrum multiplexing (Corresp.)","volume":"13","author":[{"given":"R.","family":"Gold","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1967,10]]},"URL":"http://dx.doi.org/10.1109/TIT.1967.1054048","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s11265-006-7278-y","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s11265-006-7278-y","page":"25-42","source":"Crossref","title":"An Iterative Algorithm and Low Complexity Hardware Architecture for Fast Acquisition of Long PN Codes in UWB Systems","volume":"43","author":[{"given":"On Wa","family":"Yeung","sequence":"first","affiliation":[]},{"given":"Keith M.","family":"Chugg","sequence":"additional","affiliation":[]}],"container-title":"Journal of VLSI signal processing systems for signal, image and video technology","original-title":[],"language":"en","issued":{"date-parts":[[2006,4]]},"URL":"http://dx.doi.org/10.1007/s11265-006-7278-y","ISSN":["0922-5773"],"container-title-short":"J VLSI Sign Process Syst Sign Image Video Technol","note":"alternative-id:7278"},{"id":"manual:-M.-Hamid-and-A.-Miller-Gold-","type":"document","author":[{"family":"Hamid","given":"M."},{"family":"Miller","given":"A."}],"title":"Gold Code Generators in Virtex Devices","issued":"2000"},{"id":"doi:10.1002/j.2161-4296.1978.tb01325","type":"journal-article","publisher":"Institute of Navigation","issue":"2","DOI":"10.1002/j.2161-4296.1978.tb01325.x","page":"121-146","source":"Crossref","title":"GPS Signal Structure and Performance Characteristics","volume":"25","author":[{"suffix":"Jr.","given":"J. J.","family":"SPILKER","sequence":"first","affiliation":[]}],"container-title":"Navigation","original-title":[],"language":"en","issued":{"date-parts":[[1978,6]]},"URL":"http://dx.doi.org/10.1002/j.2161-4296.1978.tb01325.x","ISSN":["0028-1522"]},{"id":"doi:10.1109/ICC.2015.7249089","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.2015.7249089","page":"4840-4845","source":"Crossref","title":"Iterative decoding of Gold sequences","author":[{"given":"Mathieu","family":"des Noes","sequence":"first","affiliation":[]},{"given":"Valentin","family":"Savin","sequence":"additional","affiliation":[]},{"given":"Jean Marc","family":"Brossier","sequence":"additional","affiliation":[]},{"given":"Laurent","family":"Ros","sequence":"additional","affiliation":[]}],"event":"2015 IEEE International Conference on Signal Processing for Communications (ICC)","container-title":"2015 IEEE International Conference on Communications (ICC)","original-title":[],"issued":{"date-parts":[[2015,6]]},"URL":"http://dx.doi.org/10.1109/ICC.2015.7249089"},{"id":"doi:10.1109/PROC.1980.11697","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/proc.1980.11697","page":"593-619","source":"Crossref","title":"Crosscorrelation properties of pseudorandom and related sequences","volume":"68","author":[{"given":"D.V.","family":"Sarwate","sequence":"first","affiliation":[]},{"given":"M.B.","family":"Pursley","sequence":"additional","affiliation":[]}],"container-title":"Proceedings of the IEEE","original-title":[],"issued":{"date-parts":[[1980]]},"URL":"http://dx.doi.org/10.1109/PROC.1980.11697","ISSN":["0018-9219"],"container-title-short":"Proc. IEEE"},{"id":"preset:HKSpseudo","type":"chapter","author":[{"family":"Helleseth","given":"T."},{"family":"Li","given":"C."}],"title":"Pseudo-Noise Sequences","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1006/ffta.2001.0333","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1006/ffta.2001.0333","page":"171-183","source":"Crossref","title":"Type II Codes over F4","volume":"8","author":[{"given":"Philippe","family":"Gaborit","sequence":"first","affiliation":[]},{"given":"Vera","family":"Pless","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Solé","sequence":"additional","affiliation":[]},{"given":"Oliver","family":"Atkin","sequence":"additional","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2002,4]]},"URL":"http://dx.doi.org/10.1006/ffta.2001.0333","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S1071579701903338"},{"id":"manual:-Karlin-M.-MacWilliams-F.-J.-","type":"article-journal","author":[{"family":"M.","given":"Karlin"},{"family":"J","given":"MacWilliams F."}],"title":"Quadratic residue codes over GF(4) and their binary images","note":"Asilomar, CA 1972","container-title":"InIEEE Int. Symp. on Information Theory"},{"id":"doi:10.1109/TIT.1969.1054261","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1969.1054261","page":"81-92","source":"Crossref","title":"New binary coding results by circulants","volume":"15","author":[{"given":"M.","family":"Karlin","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1969,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1969.1054261","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/S0019-9958(78)90158-4","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0019-9958(78)90158-4","page":"148-153","source":"Crossref","title":"A note on extended quaternary quadratic residue codes and their binary images","volume":"38","author":[{"given":"M.","family":"Karlin","sequence":"first","affiliation":[]},{"given":"V.K.","family":"Bhargava","sequence":"additional","affiliation":[]},{"given":"S.E.","family":"Tavares","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1978,8]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(78)90158-4","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995878901584"},{"id":"doi:10.1109/18.945245","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.945245","page":"2242-2248","source":"Crossref","title":"On Type II codes over F/sub 4/","volume":"47","author":[{"given":"K.","family":"Betsumiya","sequence":"first","affiliation":[]},{"given":"T.A.","family":"Gulliver","sequence":"additional","affiliation":[]},{"given":"M.","family":"Harada","sequence":"additional","affiliation":[]},{"given":"A.","family":"Munemasa","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.945245","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-T.-Kasami.-Weight-distributi","type":"book","author":[{"family":"Kasami","given":"T."}],"title":"Weight distribution formula for some class of cyclic codes","publisher":"University of Illinois","issued":"1966","publisher-place":"Champaign, IL, USA"},{"id":"doi:10.1016/0012-365X(94)00030-M","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(94)00030-m","page":"133-143","source":"Crossref","title":"The weight hierarchy of the Kasami codes","volume":"145","author":[{"given":"Tor","family":"Helleseth","sequence":"first","affiliation":[]},{"given":"P","family":"Vijay Kumar","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1995,10]]},"URL":"http://dx.doi.org/10.1016/0012-365X(94)00030-M","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9400030M"},{"id":"doi:10.1007/s12095-015-0135-8","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s12095-015-0135-8","page":"7-18","source":"Crossref","title":"On the lifted Melas code","volume":"8","author":[{"given":"Adel","family":"Alahmadi","sequence":"first","affiliation":[]},{"given":"Hussain","family":"Alhazmi","sequence":"additional","affiliation":[]},{"given":"Tor","family":"Helleseth","sequence":"additional","affiliation":[]},{"given":"Rola","family":"Hijazi","sequence":"additional","affiliation":[]},{"given":"Najat","family":"Muthana","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Solé","sequence":"additional","affiliation":[]}],"container-title":"Cryptography and Communications","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,24]]},"URL":"http://dx.doi.org/10.1007/s12095-015-0135-8","ISSN":["1936-2447","1936-2455"],"container-title-short":"Cryptogr. Commun.","note":"alternative-id:135"},{"id":"doi:10.1147/rd.43.0364","type":"journal-article","publisher":"IBM","issue":"3","DOI":"10.1147/rd.43.0364","page":"364-366","source":"Crossref","title":"A Cyclic Code for Double Error Correction [Letter to the Editor]","volume":"4","author":[{"given":"C. M.","family":"Melas","sequence":"first","affiliation":[]}],"container-title":"IBM Journal of Research and Development","original-title":[],"issued":{"date-parts":[[1960,7]]},"URL":"http://dx.doi.org/10.1147/rd.43.0364","ISSN":["0018-8646","0018-8646"],"container-title-short":"IBM J. Res. &amp; Dev."},{"id":"doi:10.1137/S0895480193243365","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","DOI":"10.1137/s0895480193243365","page":"554-559","source":"Crossref","title":"Generalized Hamming Weights of Melas Codes and Dual Melas Codes","volume":"7","author":[{"given":"G.","family":"van der Geer","sequence":"first","affiliation":[]},{"given":"M.","family":"van der Vlugt","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1994,11]]},"URL":"http://dx.doi.org/10.1137/S0895480193243365","ISSN":["0895-4801","1095-7146"],"container-title-short":"SIAM J. Discrete Math.","note":"alternative-id:10.1137/S0895480193243365"},{"id":"manual:-Y.-H.-Chen-T.-K.-Truong-Y.-C","type":"article-journal","author":[{"family":"Chen","given":"Y.H."},{"family":"Truong","given":"T.K."},{"family":"Chang","given":"Y."},{"family":"Lee","given":"C.D."},{"family":"Chen","given":"S.H."}],"title":"Algebraic decoding of quadratic residue codes using Berlekamp-Massey algorithm","volume":"23","container-title":"Journal of information science and engineering","issue":"1","issued":"2007","page":"127–145"},{"id":"manual:-G.-J.-M.-Beenker-On-double-c","type":"document","author":[{"family":"Beenker","given":"G.J.M."}],"title":"On double circulant codes","issued":"1980"},{"id":"doi:10.1109/TIT.1970.1054540","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1970.1054540","page":"717-719","source":"Crossref","title":"New family of single-error correcting codes","volume":"16","author":[{"given":"N.","family":"Sloane","sequence":"first","affiliation":[]},{"given":"D.","family":"Whitehead","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1970,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1970.1054540","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1403.6192","type":"article-journal","author":[{"given":"Yixuan","family":"Xie"},{"given":"Jinhong","family":"Yuan"},{"given":"Yuichiro","family":"Fujiwara"}],"title":"Quantum Synchronizable Codes From Quadratic Residue Codes and Their Supercodes","issued":{"date-parts":[[2014,8,18]]},"note":"arxivid:1403.6192\narxiv_version_number:3"},{"id":"doi:10.1016/0012-365X(75)90047-3","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0012-365x(75)90047-3","page":"211-224","source":"Crossref","title":"Unrestricted codes with the golay parameters are unique","volume":"12","author":[{"given":"P.","family":"Delsarte","sequence":"first","affiliation":[]},{"given":"J.M.","family":"Goethals","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1975]]},"URL":"http://dx.doi.org/10.1016/0012-365X(75)90047-3","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X75900473"},{"id":"preset:Golay49","type":"paper-conference","author":[{"family":"Golay","given":"M.J.E."}],"title":"Notes on digital coding","container-title":"Proceedings of the IEEE","volume":"37","issued":"1949","page":"657"},{"id":"doi:10.1016/0097-3165(71)90043-4","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0097-3165(71)90043-4","page":"178-186","source":"Crossref","title":"On the Golay perfect binary code","volume":"11","author":[{"given":"J.-M","family":"Goethals","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1971,9]]},"URL":"http://dx.doi.org/10.1016/0097-3165(71)90043-4","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316571900434"},{"id":"doi:10.1109/TIT.1986.1057197","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1986.1057197","page":"561-567","source":"Crossref","title":"Decoding the Golay codes","volume":"32","author":[{"given":"V.","family":"Pless","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1986,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1986.1057197","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1142/9789814287517_0004","type":"book-chapter","publisher":"Co-Published with Indian Institute of Science (IISc), Bangalore, India","DOI":"10.1142/9789814287517_0004","page":"41-50","source":"Crossref","title":"Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm","author":[{"given":"ANDREW J.","family":"VITERBI","sequence":"first","affiliation":[{"name":"Department of Engineering, University of California, Los Angeles, Calif., USA"}]}],"container-title":"The Foundations of the Digital Wireless World","original-title":[],"language":"en","issued":{"date-parts":[[2009,9]]},"ISBN":["9789814287500","9789814287517"],"URL":"http://dx.doi.org/10.1142/9789814287517_0004","note":"alternative-id:10.1142/9789814287517_0004\n10.1142/7464"},{"id":"doi:10.1049/el:19931456","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"25","DOI":"10.1049/el:19931456","page":"2170-2171","source":"Crossref","title":"New simple encoder and trellis decoder for Golay codes","volume":"29","author":[{"given":"B.","family":"Honary","sequence":"first","affiliation":[{"name":"Hull-Lancaster Communication Research Group, Engineering Department, Lancaster University, Lancaster, WLancs. LA1 4YR, UK"}]},{"given":"G.","family":"Markarian","sequence":"additional","affiliation":[{"name":"Hull-Lancaster Communication Research Group, Engineering Department, Lancaster University, Lancaster, WLancs. LA1 4YR, UK"}]}],"container-title":"Electronics Letters","original-title":[],"language":"en","issued":{"date-parts":[[1993,12,9]]},"URL":"http://dx.doi.org/10.1049/el:19931456","ISSN":["0013-5194","1350-911X"],"container-title-short":"Electron. Lett. (UK)","note":"alternative-id:10.1049/el:19931456"},{"id":"doi:10.1029/JA092iA13p14873","type":"journal-article","publisher":"American Geophysical Union (AGU)","issue":"A13","abstract":"<jats:p>The Voyager Uranus/Interstellar Mission is the continuation of the NASA program of exploration of the outer solar system. The first phase of the Voyager program included encounters with Jupiter and Saturn as summarized in the <jats:italic>Journal of Geophysical Research</jats:italic> (volume 86, pages 8123–8841, 1981, and volume 88, pages 8639–9018, 1983). With the successful completion of this first phase a second phase was undertaken with the objectives of exploring the Uranus system and investigating the interplanetary and interstellar media. Additional objectives included preserving the capability for extending the investigations to include an encounter with the Neptune system and a search for the heliopause.</jats:p>","DOI":"10.1029/ja092ia13p14873","page":"14873-14876","source":"Crossref","title":"The Voyager 2 encounter with Uranus","volume":"92","author":[{"given":"E. C.","family":"Stone","sequence":"first","affiliation":[]}],"container-title":"Journal of Geophysical Research: Space Physics","original-title":[],"language":"en","issued":{"date-parts":[[1987,12,30]]},"URL":"http://dx.doi.org/10.1029/JA092iA13p14873","ISSN":["0148-0227"],"container-title-short":"J. Geophys. Res.","note":"alternative-id:10.1029/JA092iA13p14873"},{"id":"manual:-E.-E.-Johnson.-An-Efficient-","type":"book","author":[{"family":"Johnson","given":"E.E."}],"title":"An Efficient Golay Codec For MIL-STD-188-141A and FED-STD-1045","publisher":"Department of Electrical and Computer Engineering, New Mexico State University","issued":"1991"},{"id":"doi:10.1109/18.720545","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.720545","page":"2477-2504","source":"Crossref","title":"Association schemes and coding theory","volume":"44","author":[{"given":"P.","family":"Delsarte","sequence":"first","affiliation":[]},{"given":"V.I.","family":"Levenshtein","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.720545","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/0097-3165(90)90069-9","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0097-3165(90)90069-9","page":"235-246","source":"Crossref","title":"The extended golay codes considered as ideals","volume":"55","author":[{"given":"Frank","family":"Bernhardt","sequence":"first","affiliation":[]},{"given":"Peter","family":"Landrock","sequence":"additional","affiliation":[]},{"given":"Olaf","family":"Manz","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1990,11]]},"URL":"http://dx.doi.org/10.1016/0097-3165(90)90069-9","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316590900699"},{"id":"arxiv:1604.07863","type":"article-journal","author":[{"given":"Steven T.","family":"Dougherty"},{"given":"Joe","family":"Gildea"},{"given":"Rhian","family":"Taylor"},{"given":"Alexander","family":"Tylyshchak"}],"title":"Constructions of Self-Dual and Formally Self-Dual Codes from Group Rings","issued":{"date-parts":[[2016,4,26]]},"note":"arxivid:1604.07863\narxiv_version_number:1"},{"id":"arxiv:1912.09167","type":"article-journal","author":[{"given":"Martino","family":"Borello"},{"given":"Wolfgang","family":"Willems"}],"title":"On the algebraic structure of quasi group codes","issued":{"date-parts":[[2021,11,10]]},"note":"arxivid:1912.09167\narxiv_version_number:2"},{"id":"manual:-M.-J.-T.-Guy-unpublished-","type":"manuscript","author":[{"family":"Guy","given":"M.J.T."}],"note":"unpublished"},{"id":"doi:10.1109/TIT.2008.928260","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2008.928260","page":"4381-4383","source":"Crossref","title":"A Group Ring Construction of the Extended Binary Golay Code","volume":"54","author":[{"given":"Ian","family":"McLoughlin","sequence":"first","affiliation":[]},{"given":"Ted","family":"Hurley","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.928260","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:HKSalgebra","type":"chapter","author":[{"family":"Willems","given":"W."}],"title":"Codes in Group Algebras","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"arxiv:math/0311319","type":"article-journal","author":[{"given":"A. R.","family":"Calderbank"},{"given":"N. J. A.","family":"Sloane"}],"title":"Modular and p-adic cyclic codes","issued":{"date-parts":[[2003,11,18]]},"note":"arxivid:math/0311319\narxiv_version_number:1"},{"id":"preset:Feit","type":"manuscript","author":[{"family":"Feit","given":"W."}],"title":"Some remarks on weight functions of spaces over GF(2","note":"unpublished (1972"},{"id":"doi:10.1016/0012-365X(74)90085-5","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/0012-365x(74)90085-5","page":"391-400","source":"Crossref","title":"Weight enumerators of self-orthogonal codes","volume":"9","author":[{"given":"C.L.","family":"Mallows","sequence":"first","affiliation":[]},{"given":"N.J.A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1974,10]]},"URL":"http://dx.doi.org/10.1016/0012-365X(74)90085-5","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X74900855"},{"id":"doi:10.1109/18.57203","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.57203","page":"1038-1050","source":"Crossref","title":"Orbit and coset analysis of the Golay and related codes","volume":"36","author":[{"given":"J.H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"N.J.A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1990]]},"URL":"http://dx.doi.org/10.1109/18.57203","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1005.4959","type":"article-journal","author":[{"given":"Dion C.","family":"Gijswijt"},{"given":"Hans D.","family":"Mittelmann"},{"given":"Alexander","family":"Schrijver"}],"title":"Semidefinite code bounds based on quadruple distances","issued":{"date-parts":[[2010,5,26]]},"note":"arxivid:1005.4959\narxiv_version_number:1"},{"id":"doi:10.1109/18.412695","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.412695","page":"1495-1499","source":"Crossref","title":"Even more efficient bounded-distance decoding of the hexacode, the Golay code, and the Leech lattice","volume":"41","author":[{"given":"A.","family":"Vardy","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1995]]},"URL":"http://dx.doi.org/10.1109/18.412695","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2206.04209","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.106.062421","source":"Crossref","title":"Golay codes and quantum contextuality","volume":"106","author":[{"given":"Mordecai","family":"Waegell","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0308-2959","authenticated-orcid":true,"given":"P. K.","family":"Aravind","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.106.062421","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062421","note":"arxivid:2206.04209"},{"id":"arxiv:1212.1913","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2014.2359201","page":"7442-7450","source":"Crossref","title":"Energy-Minimizing Error-Correcting Codes","volume":"60","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[]},{"given":"Yufei","family":"Zhao","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2359201","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1212.1913"},{"id":"doi:10.1007/BF03187604","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf03187604","page":"363-388","source":"Crossref","title":"Spherical codes and designs","volume":"6","author":[{"given":"P.","family":"Delsarte","sequence":"first","affiliation":[]},{"given":"J. M.","family":"Goethals","sequence":"additional","affiliation":[]},{"given":"J. J.","family":"Seidel","sequence":"additional","affiliation":[]}],"container-title":"Geometriae Dedicata","original-title":[],"language":"en","issued":{"date-parts":[[1977,9]]},"URL":"http://dx.doi.org/10.1007/BF03187604","ISSN":["0046-5755","1572-9168"],"container-title-short":"Geom Dedicata","note":"alternative-id:BF03187604"},{"id":"arxiv:2003.13700","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>Special conformal field theories can have symmetry groups which are interesting sporadic finite simple groups. Famous examples include the Monster symmetry group of a <jats:italic>c</jats:italic> = 24 two-dimensional conformal field theory (CFT) constructed by Frenkel, Lepowsky and Meurman, and the Conway symmetry group of a <jats:italic>c</jats:italic> = 12 CFT explored in detail by Duncan and Mack-Crane. The Mathieu moonshine connection between the K3 elliptic genus and the Mathieu group <jats:italic>M</jats:italic>\n            <jats:sub>24</jats:sub> has led to the study of K3 sigma models with large symmetry groups. A particular K3 CFT with a maximal symmetry group preserving (4<jats:italic>,</jats:italic> 4) superconformal symmetry was studied in beautiful work by Gaberdiel, Taormina, Volpato, and Wendland [41]. The present paper shows that in both the GTVW and <jats:italic>c</jats:italic> = 12 theories the construction of superconformal generators can be understood via the theory of quantum error correcting codes. The automorphism groups of these codes lift to symmetry groups in the CFT preserving the superconformal generators. In the case of the <jats:italic>N</jats:italic> = 1 supercurrent of the GTVW model our result, combined with a result of T. Johnson-Freyd implies the symmetry group is the maximal subgroup of <jats:italic>M</jats:italic>\n            <jats:sub>24</jats:sub> known as the sextet group. (The sextet group is also known as the holomorph of the hexacode.) Building on [41] the Ramond-Ramond sector of the GTVW model is related to the Miracle Octad Generator which in turn leads to a role for the Golay code as a group of symmetries of RR states. Moreover, (4<jats:italic>,</jats:italic> 1) superconformal symmetry suffices to define and decompose the elliptic genus of a K3 sigma model into characters of the <jats:italic>N</jats:italic> = 4 superconformal algebra. The symmetry group preserving (4<jats:italic>,</jats:italic> 1) is larger than that preserving (4<jats:italic>,</jats:italic> 4).</jats:p>","DOI":"10.1007/jhep05(2020)146","source":"Crossref","title":"Moonshine, superconformal symmetry, and quantum error correction","volume":"2020","author":[{"ORCID":"https://orcid.org/0000-0001-5860-274X","authenticated-orcid":false,"given":"Jeffrey A.","family":"Harvey","sequence":"first","affiliation":[]},{"given":"Gregory W.","family":"Moore","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,5,27]]},"URL":"http://dx.doi.org/10.1007/JHEP05(2020)146","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"146","note":"alternative-id:13084\narxivid:2003.13700"},{"id":"arxiv:2211.00092","type":"article-journal","author":[{"given":"Peter","family":"Boyvalenkov"},{"given":"Peter","family":"Dragnev"},{"given":"Douglas","family":"Hardin"},{"given":"Edward","family":"Saff"},{"given":"Maya","family":"Stoyanova"}],"title":"Universal minima of discrete potentials for sharp spherical codes","issued":{"date-parts":[[2023,9,12]]},"note":"arxivid:2211.00092\narxiv_version_number:2"},{"id":"arxiv:2202.06647","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We generalize the concept of folding from surface codes to CSS codes by considering certain dualities within them. In particular, this gives a general method to implement logical operations in suitable LDPC quantum codes using transversal gates and qubit permutations only.To demonstrate our approach, we specifically consider a [[30, 8, 3]] hyperbolic quantum code called Bring's code. Further, we show that by restricting the logical subspace of Bring's code to four qubits, we can obtain the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>f</mml:mi><mml:mi>u</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi></mml:math> Clifford group on that subspace.</jats:p>","DOI":"10.22331/q-2024-06-13-1372","page":"1372","source":"Crossref","title":"Fold-Transversal Clifford Gates for Quantum Codes","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-7211-5515","authenticated-orcid":false,"given":"Nikolas P.","family":"Breuckmann","sequence":"first","affiliation":[{"name":"Department of Computer Science, University College London, WC1E 6BT London, United Kingdom"}]},{"ORCID":"https://orcid.org/0000-0002-8932-3492","authenticated-orcid":false,"given":"Simon","family":"Burton","sequence":"additional","affiliation":[{"name":"Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,13]]},"URL":"http://dx.doi.org/10.22331/q-2024-06-13-1372","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2202.06647"},{"id":"doi:10.1109/TIT.2002.806146","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2002.806146","page":"53-59","source":"Crossref","title":"The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code","volume":"49","author":[{"given":"S.K.","family":"Houghten","sequence":"first","affiliation":[]},{"given":"C.W.H.","family":"Lam","sequence":"additional","affiliation":[]},{"given":"L.H.","family":"Thiel","sequence":"additional","affiliation":[]},{"given":"J.A.","family":"Parker","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2003,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.806146","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s10623-011-9530-0","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-011-9530-0","page":"29-41","source":"Crossref","title":"A group ring construction of the [48,24,12] type II linear block code","volume":"63","author":[{"given":"Ian","family":"McLoughlin","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2011,6,14]]},"URL":"http://dx.doi.org/10.1007/s10623-011-9530-0","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9530"},{"id":"arxiv:2408.12752","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/jsait.2025.3570832","page":"127-137","source":"Crossref","title":"Transversal Clifford and <i>T</i>-Gate Codes of Short Length and High Distance","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-3409-1126","authenticated-orcid":false,"given":"Shubham P.","family":"Jain","sequence":"first","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA"}]},{"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, USA"}]}],"container-title":"IEEE Journal on Selected Areas in Information Theory","original-title":[],"issued":{"date-parts":[[2025]]},"URL":"http://dx.doi.org/10.1109/JSAIT.2025.3570832","ISSN":["2641-8770"],"container-title-short":"IEEE J. Sel. Areas Inf. Theory","note":"arxivid:2408.12752"},{"id":"doi:10.1109/TIT.1962.1057682","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1962.1057682","page":"13-20","source":"Crossref","title":"Cyclic codes from irreducible polynomials for correction of multiple errors","volume":"8","author":[{"given":"L.-H.","family":"Zetterberg","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1962,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1962.1057682","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-P.-Kallquist-Decoding-of-Zet","type":"paper-conference","author":[{"family":"Kallquist","given":"P."}],"title":"Decoding of Zetterberg codes","container-title":"Proc. Fourth Joint Swedish-Soviet Workshop on Inform. Theory","issued":"Aug","page":"27–","publisher-place":"Gotland, Sweden"},{"id":"doi:10.1109/18.149509","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.149509","page":"1570-1573","source":"Crossref","title":"Algebraic decoding of the Zetterberg codes","volume":"38","author":[{"given":"S.M.","family":"Dodunekov","sequence":"first","affiliation":[]},{"given":"J.E.M.","family":"Nilsson","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1992]]},"URL":"http://dx.doi.org/10.1109/18.149509","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/LCOMM.2010.07.100784","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/lcomm.2010.07.100784","page":"662-663","source":"Crossref","title":"A Result on Zetterberg Codes","volume":"14","author":[{"given":"Ming-Haw","family":"Jing","sequence":"first","affiliation":[]},{"given":"Yaotsu","family":"Chang","sequence":"additional","affiliation":[]},{"given":"Chong-Dao","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Jian-Hong","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Zih-Heng","family":"Chen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Letters","original-title":[],"issued":{"date-parts":[[2010,7]]},"URL":"http://dx.doi.org/10.1109/LCOMM.2010.07.100784","ISSN":["1089-7798"],"container-title-short":"IEEE Commun. Lett."},{"id":"doi:10.1109/JRPROC.1961.287814","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/jrproc.1961.287814","page":"228-235","source":"Crossref","title":"Cyclic Codes for Error Detection","volume":"49","author":[{"given":"W.","family":"Peterson","sequence":"first","affiliation":[]},{"given":"D.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Proceedings of the IRE","original-title":[],"issued":{"date-parts":[[1961,1]]},"URL":"http://dx.doi.org/10.1109/JRPROC.1961.287814","ISSN":["0096-8390"],"container-title-short":"Proc. IRE"},{"id":"arxiv:2104.13663","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc42927.2021.9500279","source":"Crossref","title":"CRC Codes as Error Correction Codes","author":[{"given":"Wei","family":"An","sequence":"first","affiliation":[]},{"given":"Muriel","family":"Medard","sequence":"additional","affiliation":[]},{"given":"Ken R.","family":"Duffy","sequence":"additional","affiliation":[]}],"event":"ICC 2021 - IEEE International Conference on Communications","container-title":"ICC 2021 - IEEE International Conference on Communications","original-title":[],"issued":{"date-parts":[[2021,6]]},"URL":"http://dx.doi.org/10.1109/ICC42927.2021.9500279","note":"arxivid:2104.13663"},{"id":"manual:-C.-Borrelli-IEEE-802.3-cycli","type":"chapter","author":[{"family":"Borrelli","given":"C."}],"title":"IEEE 802.3 cyclic redundancy check","container-title":"application note: Virtex Series and Virtex-II Family","volume":"XAPP209 (v1. 0","issued":"2001"},{"id":"doi:10.1007/978-0-387-68192-4_2","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-0-387-68192-4_2","page":"19-65","source":"Crossref","title":"Basics of Cellular Communication","author":[{"given":"Mustafa","family":"Ergen","sequence":"first","affiliation":[]}],"container-title":"Mobile Broadband","original-title":[],"language":"en","issued":{"date-parts":[[2008,1,21]]},"ISBN":["9780387681894","9780387681924"],"URL":"http://dx.doi.org/10.1007/978-0-387-68192-4_2","note":"publisher-location:Boston, MA"},{"id":"doi:10.1109/CICN.2015.108","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/cicn.2015.108","page":"529-531","source":"Crossref","title":"Implementation of Cyclic Redundancy Check in Data Communication","author":[{"given":"Sang","family":"Sheng-Ju","sequence":"first","affiliation":[]}],"event":"2015 International Conference on Computational Intelligence and Communication Networks (CICN)","container-title":"2015 International Conference on Computational Intelligence and Communication Networks (CICN)","original-title":[],"issued":{"date-parts":[[2015,12]]},"URL":"http://dx.doi.org/10.1109/CICN.2015.108"},{"id":"manual:-R.-B.-Wells.-Applied-coding-","type":"book","author":[{"family":"Wells","given":"R.B."}],"title":"Applied coding and information theory for engineers","publisher":"Prentice-Hall, Inc","issued":"1998"},{"id":"doi:10.1109/DSN.2002.1028931","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/dsn.2002.1028931","page":"459-468","source":"Crossref","title":"32-bit cyclic redundancy codes for Internet applications","author":[{"given":"P.","family":"Koopman","sequence":"first","affiliation":[]}],"event":"International Conference on Dependable Systems and Networks","container-title":"Proceedings International Conference on Dependable Systems and Networks","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/DSN.2002.1028931"},{"id":"doi:10.1109/DSN.2004.1311885","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/dsn.2004.1311885","page":"145-154","source":"Crossref","title":"Cyclic redundancy code (CRC) polynomial selection for embedded networks","author":[{"given":"P.","family":"Koopman","sequence":"first","affiliation":[]},{"given":"T.","family":"Chakravarty","sequence":"additional","affiliation":[]}],"event":"International Conference on Dependable Systems and Networks, 2004","container-title":"International Conference on Dependable Systems and Networks, 2004","original-title":[],"issued":{"date-parts":[[2004]]},"URL":"http://dx.doi.org/10.1109/DSN.2004.1311885"},{"id":"doi:10.1109/LCOMM.2012.111612.121898","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/lcomm.2012.111612.121898","page":"2044-2047","source":"Crossref","title":"An Adaptive Successive Cancellation List Decoder for Polar Codes with Cyclic Redundancy Check","volume":"16","author":[{"given":"Bin","family":"Li","sequence":"first","affiliation":[]},{"given":"Hui","family":"Shen","sequence":"additional","affiliation":[]},{"given":"David","family":"Tse","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Letters","original-title":[],"issued":{"date-parts":[[2012,12]]},"URL":"http://dx.doi.org/10.1109/LCOMM.2012.111612.121898","ISSN":["1089-7798"],"container-title-short":"IEEE Commun. Lett."},{"id":"doi:10.1109/TIT.2015.2410251","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.2015.2410251","page":"2213-2226","source":"Crossref","title":"List Decoding of Polar Codes","volume":"61","author":[{"given":"Ido","family":"Tal","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Vardy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2015,5]]},"URL":"http://dx.doi.org/10.1109/TIT.2015.2410251","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ACCESS.2019.2901596","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/access.2019.2901596","page":"28559-28573","source":"Crossref","title":"CRC-Aided Logarithmic Stack Decoding of Polar Codes for Ultra Reliable Low Latency Communication in 3GPP New Radio","volume":"7","author":[{"given":"Luping","family":"Xiang","sequence":"first","affiliation":[]},{"given":"Zeynep B.","family":"Kaykac Egilmez","sequence":"additional","affiliation":[]},{"given":"Robert G.","family":"Maunder","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2636-5214","authenticated-orcid":false,"given":"Lajos","family":"Hanzo","sequence":"additional","affiliation":[]}],"container-title":"IEEE Access","original-title":[],"issued":{"date-parts":[[2019]]},"URL":"http://dx.doi.org/10.1109/ACCESS.2019.2901596","ISSN":["2169-3536"],"container-title-short":"IEEE Access"},{"id":"doi:10.1109/TIT.1954.1057466","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1954.1057466","page":"50-63","source":"Crossref","title":"Coding for constant-data-rate systems","volume":"4","author":[{"given":"R.","family":"Silverman","sequence":"first","affiliation":[]},{"given":"M.","family":"Balser","sequence":"additional","affiliation":[]}],"container-title":"Transactions of the IRE Professional Group on Information Theory","original-title":[],"issued":{"date-parts":[[1954,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1954.1057466","ISSN":["2168-2690","2168-2704"],"container-title-short":"Trans. IRE Prof. Group Inf. Theory"},{"id":"doi:10.1201/9781315115870","type":"book","publisher":"CRC Press","DOI":"10.1201/9781315115870","source":"Crossref","title":"Encyclopedia of Computer Science and Technology, Second Edition Volume I","container-title":[],"original-title":[],"issued":{"date-parts":[[2017,3,21]]},"ISBN":["9781315115870"],"URL":"http://dx.doi.org/10.1201/9781315115870"},{"id":"manual:-W.-W.-Peterson-Error-Correct","type":"article-journal","author":[{"family":"Peterson","given":"W.W."}],"title":"Error-Correcting Codes","volume":"206","publisher":"JSTOR","note":"Available at \\url{http://www.jstor.org/stable/24937230}.","container-title":"Scientific American","issue":"2","issued":"1962","page":"96–111","URL":"http://www.jstor.org/stable/24937230"},{"id":"doi:10.1007/978-1-4612-4072-3","type":"book","publisher":"Springer New York","DOI":"10.1007/978-1-4612-4072-3","source":"Crossref","title":"The Book of Numbers","author":[{"given":"John H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"Richard K.","family":"Guy","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[1996]]},"ISBN":["9781461284888","9781461240723"],"URL":"http://dx.doi.org/10.1007/978-1-4612-4072-3","note":"publisher-location:New York, NY"},{"id":"doi:10.1109/20.917609","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/20.917609","page":"737-741","source":"Crossref","title":"A low-density generator matrix interpretation of parallel concatenated single bit parity codes","volume":"37","author":[{"given":"T.R.","family":"Oenning","sequence":"first","affiliation":[]},{"family":"Jaekyun Moon","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Magnetics","original-title":[],"issued":{"date-parts":[[2001,3]]},"URL":"http://dx.doi.org/10.1109/20.917609","ISSN":["0018-9464"],"container-title-short":"IEEE Trans. Magn."},{"id":"doi:10.1109/18.796375","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/18.796375","page":"2345-2360","source":"Crossref","title":"Type IV self-dual codes over rings","volume":"45","author":[{"given":"S.T.","family":"Dougherty","sequence":"first","affiliation":[]},{"given":"P.","family":"Gaborit","sequence":"additional","affiliation":[]},{"given":"M.","family":"Harada","sequence":"additional","affiliation":[]},{"given":"A.","family":"Munemasa","sequence":"additional","affiliation":[]},{"given":"P.","family":"Sole","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999]]},"URL":"http://dx.doi.org/10.1109/18.796375","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1803.06987","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2018.8437652","page":"791-795","source":"Crossref","title":"Synthesis of Logical Clifford Operators via Symplectic Geometry","author":[{"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[]},{"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]},{"given":"Swanand","family":"Kadhe","sequence":"additional","affiliation":[]}],"event":"2018 IEEE International Symposium on Information Theory (ISIT)","container-title":"2018 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2018,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2018.8437652","note":"arxivid:1803.06987"},{"id":"doi:10.1109/TIT.1967.1053987","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1967.1053987","page":"349-350","source":"Crossref","title":"Optimum(n,2)codes for small values of channel error probability (Corresp.)","volume":"13","author":[{"given":"J.","family":"Cordaro","sequence":"first","affiliation":[]},{"given":"T.","family":"Wagner","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1967,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1967.1053987","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-R.-W.-Nobrega.-Komm.-url-htt","type":"document","author":[{"family":"Komm","given":"R.W.Nobrega"}],"issued":"2025","URL":"https://github.com/rwnobrega/komm","note":"Available at \\url{https://github.com/rwnobrega/komm},"},{"id":"preset:Nilsson65","type":"book","author":[{"family":"Nilsson","given":"N.J."}],"title":"\\emph{Learning Machines}","publisher":"McGraw-Hill","issued":"1965"},{"id":"doi:10.1109/43.41505","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/43.41505","page":"1206-1217","source":"Crossref","title":"Decomposition and factorization of sequential finite state machines","volume":"8","author":[{"given":"S.","family":"Devadas","sequence":"first","affiliation":[]},{"given":"A.R.","family":"Newton","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems","original-title":[],"issued":{"date-parts":[[1989]]},"URL":"http://dx.doi.org/10.1109/43.41505","ISSN":["0278-0070"],"container-title-short":"IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst."},{"id":"manual:-K.-Potdar-T.-S.-Pardawala-an","type":"article-journal","author":[{"family":"Potdar","given":"K."},{"family":"Pardawala","given":"T.S."},{"family":"Pai","given":"C.D."}],"title":"A comparative study of categorical variable encoding techniques for neural network classifiers","volume":"175","container-title":"International journal of computer applications","issue":"4","issued":"2017","page":"7–9"},{"id":"doi:10.1142/7238","type":"monograph","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/7238","source":"Crossref","title":"Pattern Classification Using Ensemble Methods","author":[{"given":"Lior","family":"Rokach","sequence":"first","affiliation":[{"name":"Ben-Gurion University of the NegevIsrael"}]}],"container-title":"Series in Machine Perception and Artificial Intelligence","original-title":[],"language":"en","issued":{"date-parts":[[2009,11]]},"ISBN":["9789814271066","9789814271073"],"URL":"http://dx.doi.org/10.1142/7238","ISSN":["1793-0839"],"note":"aliases:10.1142/9789814271073\nalternative-id:10.1142/7238\n10.1142/SMPAI"},{"id":"arxiv:0711.2914","type":"article-journal","author":[{"given":"Gidudu","family":"Anthony"},{"given":"Hulley","family":"Gregg"},{"given":"Marwala","family":"Tshilidzi"}],"title":"Image Classification Using SVMs: One-against-One Vs One-against-All","issued":{"date-parts":[[2007,11,19]]},"note":"arxivid:0711.2914\narxiv_version_number:1"},{"id":"doi:10.5120/ijca2017915495","type":"journal-article","publisher":"Foundation of Computer Science","issue":"4","DOI":"10.5120/ijca2017915495","page":"7-9","source":"Crossref","title":"A Comparative Study of Categorical Variable Encoding Techniques for Neural Network Classifiers","volume":"175","author":[{"given":"Kedar","family":"Potdar","sequence":"first","affiliation":[]},{"given":"Taher","family":"S.","sequence":"first","affiliation":[]},{"given":"Chinmay","family":"D.","sequence":"first","affiliation":[]}],"container-title":"International Journal of Computer Applications","original-title":[],"issued":{"date-parts":[[2017,10,17]]},"URL":"http://dx.doi.org/10.5120/ijca2017915495","ISSN":["0975-8887"],"container-title-short":"IJCA"},{"id":"doi:10.1162/08997660151134334","type":"journal-article","publisher":"MIT Press","issue":"5","abstract":"<jats:p>The best-known decomposition schemes of multiclass learning problems are one per class coding (OPC) and error-correcting output coding (ECOC). Both methods perform a prior decomposition, that is, before training of the classifier takes place. The impact of output codes on the inferred decision rules can be experienced only after learning. Therefore, we present a novel algorithm for the code design of multiclass learning problems. This algorithm applies a maximum-likelihood objective function in conjunction with the expectation-maximization (EM) algorithm. Minimizing the augmented objective function yields the optimal decomposition of the multiclass learning problem in two-class problems. Experimental results show the potential gain of the optimized output codes over OPC or ECOC methods.</jats:p>","DOI":"10.1162/08997660151134334","page":"1065-1102","source":"Crossref","title":"Stochastic Organization of Output Codes in Multiclass Learning Problems","volume":"13","author":[{"given":"Wolfgang","family":"Utschick","sequence":"first","affiliation":[{"name":"Institute for Network Theory and Signal Processing, Munich University of Technology, D-80290 Munich, Germany"}]},{"given":"Werner","family":"Weichselberger","sequence":"additional","affiliation":[{"name":"Institute for Network Theory and Signal Processing, Munich University of Technology, D-80290 Munich, Germany"}]}],"container-title":"Neural Computation","original-title":[],"language":"en","issued":{"date-parts":[[2001,5,1]]},"URL":"http://dx.doi.org/10.1162/08997660151134334","ISSN":["0899-7667","1530-888X"],"container-title-short":"Neural Computation","note":"alternative-id:10.1162/08997660151134334"},{"id":"arxiv:1009.4495","type":"article-journal","author":[{"given":"Subhash","family":"Kak"}],"title":"Unary Coding for Neural Network Learning","issued":{"date-parts":[[2010,9,22]]},"note":"arxivid:1009.4495\narxiv_version_number:1"},{"id":"manual:-I.-R.-Fiete-and-H.-S.-Seung-","type":"chapter","author":[{"family":"Fiete","given":"I.R."},{"family":"Seung","given":"H.S."}],"title":"Neural network models of birdsong production, learning, and coding","container-title":"New Encyclopedia of Neuroscience","editor":[{"family":"Squire","given":"Eds L."},{"family":"Albright","given":"T."},{"family":"Bloom","given":"F."},{"family":"Gage","given":"F."},{"given":"N."}],"publisher":"Spitzer. Elsevier","issued":"2007"},{"id":"arxiv:1802.07419","type":"document","categories":["quantum pcps","local hamiltonians","error-correcting codes"],"language":"en","author":[{"family":"Nirkhe","given":"Chinmay"},{"family":"Vazirani","given":"Umesh"},{"family":"Yuen","given":"Henry"}],"contributor":[{"family":"Chatzigiannakis","given":"Ioannis"},{"family":"Kaklamanis","given":"Christos"},{"family":"Marx","given":"Dániel"},{"family":"Sannella","given":"Donald"}],"editor":[{"family":"Chatzigiannakis","given":"Ioannis"},{"family":"Kaklamanis","given":"Christos"},{"family":"Marx","given":"Dániel"},{"family":"Sannella","given":"Donald"}],"issued":{"date-parts":[[2018]]},"abstract":"The No Low-Energy Trivial States (NLTS) conjecture of Freedman and Hastings (Quantum Information and Computation 2014), which asserts the existence of local Hamiltonians whose low-energy states cannot be generated by constant-depth quantum circuits, identifies a fundamental obstacle to resolving the quantum PCP conjecture. Progress towards the NLTS conjecture was made by Eldar and Harrow (Foundations of Computer Science 2017), who proved a closely related theorem called No Low-Error Trivial States (NLETS). In this paper, we give a much simpler proof of the NLETS theorem and use the same technique to establish superpolynomial circuit size lower bounds for noisy ground states of local Hamiltonians (assuming QCMA != QMA), resolving an open question of Eldar and Harrow. We discuss the new light our results cast on the relationship between NLTS and NLETS.\nFinally, our techniques imply the existence of approximate quantum low-weight check (qLWC) codes with linear rate, linear distance, and constant weight checks. These codes are similar to quantum LDPC codes except (1) each particle may participate in a large number of checks, and (2) errors only need to be corrected up to fidelity 1 - 1/poly(n). This stands in contrast to the best-known stabilizer LDPC codes due to Freedman, Meyer, and Luo which achieve a distance of O(sqrt{n log n}).\nThe principal technique used in our results is to leverage the Feynman-Kitaev clock construction to approximately embed a subspace of states defined by a circuit as the ground space of a local Hamiltonian.","container-title":"LIPIcs, Volume 107, ICALP 2018","DOI":"10.4230/LIPICS.ICALP.2018.91","volume":"107","number":"91","page":"91:1-91:11","page-first":"91:1","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"Approximate Low-Weight Check Codes and Circuit Lower Bounds for Noisy Ground States","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.91","note":"copyright:Creative Commons Attribution 3.0 Unported license\narxivid:1802.07419"},{"id":"doi:10.1090/gsm/047","type":"monograph","publisher":"American Mathematical Society","DOI":"10.1090/gsm/047","source":"Crossref","title":"Classical and Quantum Computation","author":[{"given":"A.","family":"Kitaev","sequence":"first","affiliation":[]},{"given":"A.","family":"Shen","sequence":"additional","affiliation":[]},{"given":"M.","family":"Vyalyi","sequence":"additional","affiliation":[]}],"container-title":"Graduate Studies in Mathematics","original-title":[],"issued":{"date-parts":[[2002,5,31]]},"ISBN":["9780821832295","9781470418007"],"URL":"http://dx.doi.org/10.1090/gsm/047","ISSN":["1065-7339"],"note":"publisher-location:Providence, Rhode Island"},{"id":"doi:10.1145/285243.285258","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"4","abstract":"<jats:p>The proliferation of applications that must reliably distribute bulk data to a large number of autonomous clients motivates the design of new multicast and broadcast protocols. We describe an ideal, fully scalable protocol for these applications that we call a digital fountain. A digital fountain allows any number of heterogeneous clients to acquire bulk data with optimal efficiency at times of their choosing. Moreover, no feedback channels are needed to ensure reliable delivery, even in the face of high loss rates.We develop a protocol that closely approximates a digital fountain using a new class of erasure codes that for large block sizes are orders of magnitude faster than standard erasure codes. We provide performance measurements that demonstrate the feasibility of our approach and discuss the design, implementation and performance of an experimental system.</jats:p>","DOI":"10.1145/285243.285258","page":"56-67","source":"Crossref","title":"A digital fountain approach to reliable distribution of bulk data","volume":"28","author":[{"given":"John W.","family":"Byers","sequence":"first","affiliation":[{"name":"UC Berkeley and International Computer Science Institute, Berkeley, California"}]},{"given":"Michael","family":"Luby","sequence":"additional","affiliation":[{"name":"International Computer Science Institute, Berkeley, California"}]},{"given":"Michael","family":"Mitzenmacher","sequence":"additional","affiliation":[{"name":"Digital Systems Research Center, Palo Alto, California"}]},{"given":"Ashutosh","family":"Rege","sequence":"additional","affiliation":[{"name":"International Computer Science Institute, Berkeley, California"}]}],"container-title":"ACM SIGCOMM Computer Communication Review","original-title":[],"language":"en","issued":{"date-parts":[[1998,10]]},"URL":"http://dx.doi.org/10.1145/285243.285258","ISSN":["0146-4833"],"container-title-short":"SIGCOMM Comput. Commun. Rev.","note":"alternative-id:10.1145/285243.285258"},{"id":"doi:10.1109/TIT.2006.874390","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2006.874390","page":"2551-2567","source":"Crossref","title":"Raptor codes","volume":"52","author":[{"given":"A.","family":"Shokrollahi","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2006,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2006.874390","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/MASS.2012.6502540","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/mass.2012.6502540","page":"398-406","source":"Crossref","title":"Cross-layer coordination for efficient contents delivery in LTE eMBMS traffic","author":[{"given":"Eilwoo","family":"Baik","sequence":"first","affiliation":[{"name":"Department of Computer Science Univerisity of California, Davis"}]},{"given":"Amit","family":"Pande","sequence":"additional","affiliation":[{"name":"Department of Computer Science Univerisity of California, Davis"}]},{"given":"Prasant","family":"Mohapatra","sequence":"additional","affiliation":[{"name":"Department of Computer Science Univerisity of California, Davis"}]}],"event":"2012 IEEE 9th International Conference on Mobile Ad-Hoc and Sensor Systems (MASS)","container-title":"2012 IEEE 9th International Conference on Mobile Ad-Hoc and Sensor Systems (MASS 2012)","original-title":[],"issued":{"date-parts":[[2012,10]]},"URL":"http://dx.doi.org/10.1109/MASS.2012.6502540"},{"id":"doi:10.1126/science.aaj2038","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6328","abstract":"<jats:title>A reliable and efficient DNA storage architecture</jats:title>\n                  <jats:p>DNA has the potential to provide large-capacity information storage. However, current methods have only been able to use a fraction of the theoretical maximum. Erlich and Zielinski present a method, DNA Fountain, which approaches the theoretical maximum for information stored per nucleotide. They demonstrated efficient encoding of information—including a full computer operating system—into DNA that could be retrieved at scale after multiple rounds of polymerase chain reaction.</jats:p>\n                  <jats:p>\n                    <jats:italic>Science</jats:italic>\n                    , this issue p.\n                    <jats:related-article xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"doi\" issue=\"6328\" page=\"950\" related-article-type=\"in-this-issue\" vol=\"355\" xlink:href=\"10.1126/science.aaj2038\">950</jats:related-article>\n                  </jats:p>","DOI":"10.1126/science.aaj2038","page":"950-954","source":"Crossref","title":"DNA Fountain enables a robust and efficient storage architecture","volume":"355","author":[{"ORCID":"https://orcid.org/0000-0003-3257-8387","authenticated-orcid":false,"given":"Yaniv","family":"Erlich","sequence":"first","affiliation":[{"name":"New York Genome Center, New York, NY 10013, USA."},{"name":"Department of Computer Science, Fu Foundation School of Engineering, Columbia University, New York, NY 10027, USA."},{"name":"Center for Computational Biology and Bioinformatics (C2B2), Department of Systems Biology, Columbia University, New York, NY 10027, USA."}]},{"ORCID":"https://orcid.org/0000-0003-1357-4233","authenticated-orcid":false,"given":"Dina","family":"Zielinski","sequence":"additional","affiliation":[{"name":"New York Genome Center, New York, NY 10013, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2017,3,3]]},"URL":"http://dx.doi.org/10.1126/science.aaj2038","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.aaj2038"},{"id":"doi:10.1049/ip-com:20050237","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"6","DOI":"10.1049/ip-com:20050237","page":"1062","source":"Crossref","title":"Fountain codes","volume":"152","author":[{"given":"D.J.C.","family":"MacKay","sequence":"first","affiliation":[]}],"container-title":"IEE Proceedings - Communications","original-title":[],"language":"en","issued":{"date-parts":[[2005]]},"URL":"http://dx.doi.org/10.1049/ip-com:20050237","ISSN":["1350-2425"],"container-title-short":"IEE Proc., Commun."},{"id":"preset:Joshi10","type":"paper-conference","author":[{"family":"Joshi","given":"G."},{"family":"Rhim","given":"J.B."},{"family":"Sun","given":"J."},{"family":"Wang","given":"D."}],"title":"Fountain codes","container-title":"Global telecommunications conference (GLOBECOM 2010","publisher":"IEEE","issued":"2010","page":"7–12"},{"id":"doi:10.1017/9781009283403","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Critical coding techniques have developed over the past few decades for data storage, retrieval and transmission systems, significantly mitigating costs for governments and corporations that maintain server systems containing large amounts of data. This book surveys the basic ideas of these coding techniques, which tend not to be covered in the graduate curricula, including pointers to further reading. Written in an informal style, it avoids detailed coverage of proofs, making it an ideal refresher or brief introduction for students and researchers in academia and industry who may not have the time to commit to understanding them deeply. Topics covered include fountain codes designed for large file downloads; LDPC and polar codes for error correction; network, rank metric, and subspace codes for the transmission of data through networks; post-quantum computing; and quantum error correction. Readers are assumed to have taken basic courses on algebraic coding and information theory.</jats:p>","DOI":"10.1017/9781009283403","source":"Crossref","title":"Essays on Coding Theory","author":[{"given":"Ian F.","family":"Blake","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2024,3,8]]},"ISBN":["9781009283403","9781009283373"],"URL":"http://dx.doi.org/10.1017/9781009283403","note":"edition-number:1"},{"id":"arxiv:1401.0734","type":"article-journal","author":[{"given":"Megasthenis","family":"Asteris"},{"given":"Alexandros G.","family":"Dimakis"}],"title":"Repairable Fountain Codes","issued":{"date-parts":[[2014,1,3]]},"note":"arxivid:1401.0734\narxiv_version_number:1"},{"id":"arxiv:1705.07983","type":"article-journal","author":[{"given":"Michael G.","family":"Luby"},{"given":"Roberto","family":"Padovani"},{"given":"Thomas J.","family":"Richardson"},{"given":"Lorenz","family":"Minder"},{"given":"Pooja","family":"Aggarwal"}],"title":"Liquid Cloud Storage","issued":{"date-parts":[[2017,5,22]]},"note":"arxivid:1705.07983\narxiv_version_number:1"},{"id":"doi:10.1109/SFCS.2002.1181950","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/sfcs.2002.1181950","source":"Crossref","title":"LT codes","author":[{"given":"M.","family":"Luby","sequence":"first","affiliation":[]}],"event":"43rd Annual IEEE Symposium on Foundations of Computer Science","container-title":"The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.2002.1181950"},{"id":"doi:10.1017/CBO9780511791338","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Having trouble deciding which coding scheme to employ, how to design a new scheme, or how to improve an existing system? This summary of the state-of-the-art in iterative coding makes this decision more straightforward. With emphasis on the underlying theory, techniques to analyse and design practical iterative coding systems are presented. Using Gallager's original ensemble of LDPC codes, the basic concepts are extended for several general codes, including the practically important class of turbo codes. The simplicity of the binary erasure channel is exploited to develop analytical techniques and intuition, which are then applied to general channel models. A chapter on factor graphs helps to unify the important topics of information theory, coding and communication theory. Covering the most recent advances, this text is ideal for graduate students in electrical engineering and computer science, and practitioners. Additional resources, including instructor's solutions and figures, available online: www.cambridge.org/9780521852296.</jats:p>","DOI":"10.1017/cbo9780511791338","source":"Crossref","title":"Modern Coding Theory","author":[{"given":"Tom","family":"Richardson","sequence":"first","affiliation":[]},{"given":"Rüdiger","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2008,3,17]]},"ISBN":["9780521852296","9780511791338"],"URL":"http://dx.doi.org/10.1017/CBO9780511791338","note":"edition-number:1"},{"id":"manual:-D.-J.-C.-MacKay.-2002.-Infor","type":"book","author":[{"family":"MacKay","given":"D.J.C."}],"title":"Information Theory, Inference & Learning Algorithms","publisher":"Cambridge University Press","issued":"2002","publisher-place":"USA"},{"id":"doi:10.1145/3501714.3501727","type":"book-chapter","publisher":"ACM","DOI":"10.1145/3501714.3501727","page":"129-138","source":"Crossref","title":"Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach","author":[{"given":"Judea","family":"Pearl","sequence":"first","affiliation":[]}],"container-title":"Probabilistic and Causal Inference","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,28]]},"ISBN":["9781450395861"],"URL":"http://dx.doi.org/10.1145/3501714.3501727","note":"publisher-location:New York, NY, USA\nedition-number:1\nalternative-id:10.1145/3501714.3501727\n10.1145/3501714"},{"id":"manual:-P.-Maymounkov-Online-codes-T","type":"report","author":[{"family":"Maymounkov","given":"P."}],"title":"Online codes","genre":"Technical report,","publisher":"New York University","issued":"2002"},{"id":"arxiv:1706.05814","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/tcomm.2017.2715805","page":"1-1","source":"Crossref","title":"Inactivation Decoding of LT and Raptor Codes: Analysis and Code Design","author":[{"given":"Francisco","family":"Lazaro","sequence":"first","affiliation":[]},{"given":"Gianluigi","family":"Liva","sequence":"additional","affiliation":[]},{"given":"Gerhard","family":"Bauch","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2017]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2017.2715805","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun.","note":"arxivid:1706.05814"},{"id":"doi:10.1109/18.910575","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910575","page":"569-584","source":"Crossref","title":"Efficient erasure correcting codes","volume":"47","author":[{"given":"M.G.","family":"Luby","sequence":"first","affiliation":[]},{"given":"M.","family":"Mitzenmacher","sequence":"additional","affiliation":[]},{"given":"M.A.","family":"Shokrollahi","sequence":"additional","affiliation":[]},{"given":"D.A.","family":"Spielman","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910575","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:HKSfountain","type":"chapter","author":[{"family":"Blake","given":"I.F."}],"title":"Coding for Erasures and Fountain Codes","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1145/258533.258573","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/258533.258573","page":"150-159","source":"Crossref","title":"Practical loss-resilient codes","author":[{"given":"Michael G.","family":"Luby","sequence":"first","affiliation":[]},{"given":"Michael","family":"Mitzenmacher","sequence":"additional","affiliation":[]},{"given":"M. Amin","family":"Shokrollahi","sequence":"additional","affiliation":[]},{"given":"Daniel A.","family":"Spielman","sequence":"additional","affiliation":[]},{"given":"Volker","family":"Stemann","sequence":"additional","affiliation":[]}],"event":"the twenty-ninth annual ACM symposium","container-title":"Proceedings of the twenty-ninth annual ACM symposium on Theory of computing  - STOC '97","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1145/258533.258573","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"doi:10.1016/S0012-365X(02)00513-7","type":"journal-article","publisher":"Elsevier BV","issue":"2-3","DOI":"10.1016/s0012-365x(02)00513-7","page":"549-557","source":"Crossref","title":"Error-correcting codes from graphs","volume":"257","author":[{"given":"Vladimir D.","family":"Tonchev","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2002,11]]},"URL":"http://dx.doi.org/10.1016/S0012-365X(02)00513-7","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X02005137"},{"id":"manual:-W.-H.-Haemers-C.-Parker-V.-P","type":"article-journal","author":[{"family":"Haemers","given":"W.H."},{"family":"Parker","given":"C."},{"family":"Pless","given":"V."},{"family":"Tonchev","given":"V.D."}],"title":"A design and a code invariant under the simple group \\(Co_3\\","volume":"A 62","container-title":"J. Comb. Theory","issued":"1993","page":"225–233,"},{"id":"doi:10.1007/978-3-642-74341-2","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-74341-2","source":"Crossref","title":"Distance-Regular Graphs","author":[{"given":"Andries E.","family":"Brouwer","sequence":"first","affiliation":[]},{"given":"Arjeh M.","family":"Cohen","sequence":"additional","affiliation":[]},{"given":"Arnold","family":"Neumaier","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1989]]},"ISBN":["9783642743436","9783642743412"],"URL":"http://dx.doi.org/10.1007/978-3-642-74341-2","note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1147/rd.45.0497","type":"journal-article","publisher":"IBM","issue":"5","DOI":"10.1147/rd.45.0497","page":"497-504","source":"Crossref","title":"On Moore Graphs with Diameters 2 and 3","volume":"4","author":[{"given":"A. J.","family":"Hoffman","sequence":"first","affiliation":[{"name":"General Electric Company, New Yolk, USA"}]},{"given":"R. R.","family":"Singleton","sequence":"additional","affiliation":[{"name":"General Electric Company, New Yolk, USA"}]}],"container-title":"IBM Journal of Research and Development","original-title":[],"issued":{"date-parts":[[1960,11]]},"URL":"http://dx.doi.org/10.1147/rd.45.0497","ISSN":["0018-8646","0018-8646"],"container-title-short":"IBM J. Res. &amp; Dev."},{"id":"manual:-T.-Kasami-A-topological-appr","type":"article-journal","author":[{"family":"Kasami","given":"T."}],"title":"A topological approach to construction of group codes","volume":"44","container-title":"J. Inst. Elec. Commun. Engrs.(Japan","issued":"1961","page":"1316–1321"},{"id":"manual:-D.-A.-Huffman-A-graph-theore","type":"chapter","author":[{"family":"Huffman","given":"D.A."}],"title":"A graph-theoretic formulation of binary group codes","container-title":"Summaries of papers presented at ICMCI","volume":"part 3","issued":"1964","page":"29–30"},{"id":"manual:-W.-D.-Frazer-A-graph-theoret","type":"paper-conference","author":[{"family":"Frazer","given":"W.D."}],"title":"A graph theoretic approach to linear codes","container-title":"Proc. Second Annual Allerton Conf. On Circuit & System Theory","issued":"1964"},{"id":"doi:10.1109/TIT.1965.1053789","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1965.1053789","page":"457-458","source":"Crossref","title":"Cut-set matrices and linear codes (Corresp.)","volume":"11","author":[{"given":"S.","family":"Hakimi","sequence":"first","affiliation":[]},{"given":"H.","family":"Frank","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1965,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1965.1053789","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1968.1054190","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1968.1054190","page":"584-591","source":"Crossref","title":"Graph theoretic error-correcting codes","volume":"14","author":[{"given":"S.","family":"Hakimi","sequence":"first","affiliation":[]},{"given":"J.","family":"Bredeson","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1968,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1968.1054190","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s10623-011-9594-x","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1-3","DOI":"10.1007/s10623-011-9594-x","page":"373-393","source":"Crossref","title":"Codes from incidence matrices of graphs","volume":"68","author":[{"given":"P.","family":"Dankelmann","sequence":"first","affiliation":[]},{"given":"J. D.","family":"Key","sequence":"additional","affiliation":[]},{"given":"B. G.","family":"Rodrigues","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2012,1,3]]},"URL":"http://dx.doi.org/10.1007/s10623-011-9594-x","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9594"},{"id":"doi:10.1007/978-3-642-01877-0_21","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-01877-0_21","page":"259-273","source":"Crossref","title":"On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction","author":[{"given":"Gilles","family":"Zémor","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2009]]},"ISBN":["9783642018138","9783642018770"],"URL":"http://dx.doi.org/10.1007/978-3-642-01877-0_21","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:1703.07847","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present an infinite family of protocols to distill magic states for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gate depth of the circuit scale linearly in the logarithm of the target error <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>δ</mml:mi></mml:math> (up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>δ</mml:mi></mml:math>). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the \"inner code\". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the \"outer code\". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.</jats:p>","DOI":"10.22331/q-2017-10-03-31","page":"31","source":"Crossref","title":"Magic state distillation with low space overhead and optimal asymptotic input count","volume":"1","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA"},{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"D.","family":"Poulin","sequence":"additional","affiliation":[{"name":"Département de Physique & Institut Quantique, Université de Sherbrooke, Quebec, Canada"}]},{"given":"D.","family":"Wecker","sequence":"additional","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,3]]},"URL":"http://dx.doi.org/10.22331/q-2017-10-03-31","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1703.07847"},{"id":"arxiv:1709.02789","type":"journal-issue","publisher":"Rinton Press","issue":"1&2","DOI":"10.26421/qic18.1-2","source":"Crossref","title":[],"volume":"18","container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2018,2]]},"URL":"http://dx.doi.org/10.26421/QIC18.1-2","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC","note":"arxivid:1709.02789"},{"id":"doi:10.1007/978-3-540-72780-4","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-540-72780-4","source":"Crossref","title":"Graphs, Networks and Algorithms","author":[{"given":"Dieter","family":"Jungnickel","sequence":"first","affiliation":[]}],"container-title":"Algorithms and Computation in Mathematics","original-title":[],"issued":{"date-parts":[[2008]]},"ISBN":["9783540727798","9783540727804"],"URL":"http://dx.doi.org/10.1007/978-3-540-72780-4","ISSN":["1431-1550"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:2312.08462","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>We explore a deep connection between fracton order and product codes. In particular, we propose and analyze conditions on classical seed codes which lead to fracton order in the resulting quantum product codes. Depending on the properties of the input codes, product codes can realize either Type-I or Type-II fracton models, in both nonlocal and local constructions. For the nonlocal case, we show that a recently proposed model of lineons on nonlocal graphs can be obtained as a hypergraph product code. Interestingly, constrained mobility in this model arises only from energy barriers associated with the graph. For the local case, we introduce a novel type of classical LDPC code defined on a planar aperiodic tiling. By considering the specific example of the pinwheel tiling, we demonstrate the systematic construction of local Type-I and Type-II fracton models as product codes. Our work establishes product codes as a natural setting for exploring fracton order.</jats:p>","DOI":"10.1103/f48m-rlh3","source":"Crossref","title":"Fracton models from product codes","volume":"7","author":[{"ORCID":"https://orcid.org/0009-0009-4192-6137","authenticated-orcid":true,"given":"Yi","family":"Tan","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-3107-1878","authenticated-orcid":true,"given":"Brenden","family":"Roberts","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"given":"Beni","family":"Yoshida","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"}]},{"ORCID":"https://orcid.org/0000-0003-0194-7266","authenticated-orcid":true,"given":"Norman Y.","family":"Yao","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,22]]},"URL":"http://dx.doi.org/10.1103/f48m-rlh3","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"L032062","note":"arxivid:2312.08462"},{"id":"doi:10.1109/TIT.1973.1055068","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1973.1055068","page":"711-713","source":"Crossref","title":"Justesen's construction--The low-rate case (Corresp.)","volume":"19","author":[{"given":"E.J.","family":"Weldon","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, University of Hawaii, Honolulu, Hawaii 96822, and ADTECH Incorporated, Honolulu, Hawaii"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1973,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1973.1055068","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/S0019-9958(74)91016-X","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/s0019-9958(74)91016-x","page":"341-350","source":"Crossref","title":"A modification of the constructive asymptotically good codes of Justesen for low rates","volume":"25","author":[{"given":"Yasuo","family":"Sugiyama","sequence":"first","affiliation":[]},{"given":"Masao","family":"Kasahara","sequence":"additional","affiliation":[]},{"given":"Shigeichi","family":"Hirasawa","sequence":"additional","affiliation":[]},{"given":"Toshihiko","family":"Namekawa","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1974,8]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(74)91016-X","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S001999587491016X"},{"id":"doi:10.1109/TIT.1978.1055861","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1978.1055861","page":"198-204","source":"Crossref","title":"A new class of asymptotically good codes beyond the Zyablov bound","volume":"24","author":[{"given":"Y.","family":"Sugiyama","sequence":"first","affiliation":[]},{"given":"M.","family":"Kasahara","sequence":"additional","affiliation":[]},{"given":"S.","family":"Hirasawa","sequence":"additional","affiliation":[]},{"given":"T.","family":"Namekawa","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1978,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1978.1055861","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1972.1054893","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1972.1054893","page":"652-656","source":"Crossref","title":"Class of constructive asymptotically good algebraic codes","volume":"18","author":[{"given":"J.","family":"Justesen","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1972,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1972.1054893","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1145/100216.100244","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/100216.100244","page":"213-223","source":"Crossref","title":"Small-bias probability spaces: efficient constructions and applications","author":[{"given":"J.","family":"Naor","sequence":"first","affiliation":[]},{"given":"M.","family":"Naor","sequence":"additional","affiliation":[]}],"event":"the twenty-second annual ACM symposium","container-title":"Proceedings of the twenty-second annual ACM symposium on Theory of computing  - STOC '90","original-title":[],"issued":{"date-parts":[[1990]]},"URL":"http://dx.doi.org/10.1145/100216.100244","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"arxiv:2012.01453","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Implementing robust quantum error correction (QEC) is imperative for harnessing the promise of quantum technologies. We introduce a framework that takes <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>y</mml:mi></mml:math> classical code and explicitly constructs the corresponding QEC code. Our framework can be seen to generalize the CSS codes, and goes beyond the stabilizer formalism (Fig. 1). A concrete advantage is that the desirable properties of a classical code are automatically incorporated in the design of the resulting quantum code. We reify the theory by various illustrations some of which outperform the best previous constructions. We then introduce a local quantum spin-chain Hamiltonian whose ground space we analytically completely characterize. We utilize our framework to demonstrate that the ground space contains explicit quantum codes with linear distance. This side-steps the Bravyi-Terhal no-go theorem.</jats:p>","DOI":"10.22331/q-2024-11-27-1541","page":"1541","source":"Crossref","title":"Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians","volume":"8","author":[{"given":"Ramis","family":"Movassagh","sequence":"first","affiliation":[{"name":"IBM Research, MIT-IBM Watson AI lab, Cambridge MA, 02142, USA"},{"name":"Google Quantum AI, Los Angeles, CA, 90291, USA"}]},{"given":"Yingkai","family":"Ouyang","sequence":"additional","affiliation":[{"name":"School of Mathematical and Physical Sciences, University of Sheffield, Sheffield, UK"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,27]]},"URL":"http://dx.doi.org/10.22331/q-2024-11-27-1541","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2012.01453"},{"id":"arxiv:1911.03069","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>p</mml:mi></mml:math>-faces of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube (for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mi>p</mml:mi></mml:math>) and stabilizer constraints with faces of dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x00B1;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mrow><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mfrac linethickness=\"0\"><mml:mi>n</mml:mi><mml:mi>p</mml:mi></mml:mfrac><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:math> physical qubits and displays local testability with a soundness of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x03A9;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math> beating the current state-of-the-art of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mi>log</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors.We then extend this code family by considering the quotient of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube by arbitrary linear classical codes of length <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.</jats:p>","DOI":"10.22331/q-2022-02-24-661","page":"661","source":"Crossref","title":"Towards local testability for quantum coding","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[{"name":"Inria, France"}]},{"given":"Vivien","family":"Londe","sequence":"additional","affiliation":[{"name":"Microsoft, France"}]},{"ORCID":"https://orcid.org/0000-0002-6041-9554","authenticated-orcid":false,"given":"Gilles","family":"Zémor","sequence":"additional","affiliation":[{"name":"Institut de Mathématiques de Bordeaux, UMR 5251, France"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,24]]},"URL":"http://dx.doi.org/10.22331/q-2022-02-24-661","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1911.03069"},{"id":"doi:10.1145/100216.100225","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/100216.100225","page":"73-83","source":"Crossref","title":"Self-testing/correcting with applications to numerical problems","author":[{"given":"M.","family":"Blum","sequence":"first","affiliation":[]},{"given":"M.","family":"Luby","sequence":"additional","affiliation":[]},{"given":"R.","family":"Rubinfeld","sequence":"additional","affiliation":[]}],"event":"the twenty-second annual ACM symposium","container-title":"Proceedings of the twenty-second annual ACM symposium on Theory of computing  - STOC '90","original-title":[],"issued":{"date-parts":[[1990]]},"URL":"http://dx.doi.org/10.1145/100216.100225","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"doi:10.1016/0022-0000(93)90044-W","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0022-0000(93)90044-w","page":"549-595","source":"Crossref","title":"Self-testing/correcting with applications to numerical problems","volume":"47","author":[{"given":"Manuel","family":"Blum","sequence":"first","affiliation":[]},{"given":"Michael","family":"Luby","sequence":"additional","affiliation":[]},{"given":"Ronitt","family":"Rubinfeld","sequence":"additional","affiliation":[]}],"container-title":"Journal of Computer and System Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1993,12]]},"URL":"http://dx.doi.org/10.1016/0022-0000(93)90044-W","ISSN":["0022-0000"],"container-title-short":"Journal of Computer and System Sciences","note":"alternative-id:002200009390044W"},{"id":"doi:10.1109/TIT.2005.856958","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2005.856958","page":"4032-4039","source":"Crossref","title":"Testing Reed–Muller Codes","volume":"51","author":[{"given":"N.","family":"Alon","sequence":"first","affiliation":[]},{"given":"T.","family":"Kaufman","sequence":"additional","affiliation":[]},{"given":"M.","family":"Krivelevich","sequence":"additional","affiliation":[]},{"given":"S.","family":"Litsyn","sequence":"additional","affiliation":[]},{"given":"D.","family":"Ron","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2005,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.856958","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1137/S0097539704445615","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/s0097539704445615","page":"779-802","source":"Crossref","title":"Testing Polynomials over General Fields","volume":"36","author":[{"given":"Tali","family":"Kaufman","sequence":"first","affiliation":[]},{"given":"Dana","family":"Ron","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[2006,1]]},"URL":"http://dx.doi.org/10.1137/S0097539704445615","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/S0097539704445615"},{"id":"arxiv:math/0604353","type":"article-journal","author":[{"given":"Alex","family":"Samorodnitsky"}],"title":"Low-degree tests at large distances","issued":{"date-parts":[[2006,4,16]]},"note":"arxivid:math/0604353\narxiv_version_number:1"},{"id":"arxiv:0910.0641","type":"article-journal","author":[{"given":"Arnab","family":"Bhattacharyya"},{"given":"Swastik","family":"Kopparty"},{"given":"Grant","family":"Schoenebeck"},{"given":"Madhu","family":"Sudan"},{"given":"David","family":"Zuckerman"}],"title":"Optimal Testing of Reed-Muller Codes","issued":{"date-parts":[[2010,4,9]]},"note":"arxivid:0910.0641\narxiv_version_number:2"},{"id":"doi:10.1145/1007352.1007361","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/1007352.1007361","page":"1-10","source":"Crossref","title":"Robust pcps of proximity, shorter pcps and applications to coding","author":[{"given":"Eli","family":"Ben-Sasson","sequence":"first","affiliation":[{"name":"Radcliffe Institute for Advanced Study, Cambridge, MA"}]},{"given":"Oded","family":"Goldreich","sequence":"additional","affiliation":[{"name":"Weizmann Institute of Science, Rehovot, ISRAEL"}]},{"given":"Prahladh","family":"Harsha","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]},{"given":"Salil","family":"Vadhan","sequence":"additional","affiliation":[{"name":"Harvard University, Cambridge, MA"}]}],"event":"STOC04: Symposium of Theory of Computing 2004","container-title":"Proceedings of the thirty-sixth annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2004,6,13]]},"URL":"http://dx.doi.org/10.1145/1007352.1007361","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/1007352.1007361\n10.1145/1007352"},{"id":"doi:10.1145/780542.780631","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/780542.780631","page":"612-621","source":"Crossref","title":"Randomness-efficient low degree tests and short PCPs via epsilon-biased sets","author":[{"given":"Eli","family":"Ben-Sasson","sequence":"first","affiliation":[{"name":"Harvard University and MIT, Cambridge, MA"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachussettes Institute of Technology, Cambridge, MA"}]},{"given":"Salil","family":"Vadhan","sequence":"additional","affiliation":[{"name":"Harvard University, Cambridge, MA"}]},{"given":"Avi","family":"Wigderson","sequence":"additional","affiliation":[{"name":"Princeton, Princeton, NJ and the Hebrew University, Jerusalem"}]}],"event":"STOC03: The 35th Annual ACM Symposium on Theory of Computing","container-title":"Proceedings of the thirty-fifth annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2003,6,9]]},"URL":"http://dx.doi.org/10.1145/780542.780631","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/780542.780631\n10.1145/780542"},{"id":"arxiv:cs/0408066","type":"article-journal","author":[{"given":"Eli","family":"Ben-Sasson"},{"given":"Madhu","family":"Sudan"}],"title":"Robust Locally Testable Codes and Products of Codes","issued":{"date-parts":[[2004,8,30]]},"note":"arxivid:cs/0408066\narxiv_version_number:1"},{"id":"doi:10.1145/1236457.1236459","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"3","abstract":"<jats:p>The PCP theorem [Arora and Safra 1998; Arora et. al. 1998] says that every language in NP has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof. The celebrated equivalence of this theorem and inapproximability of certain optimization problems, due to Feige et al. [1996], has placed the PCP theorem at the heart of the area of inapproximability.</jats:p>\n          <jats:p>In this work, we present a new proof of the PCP theorem that draws on this equivalence. We give a combinatorial proof for the NP-hardness of approximating a certain constraint satisfaction problem, which can then be reinterpreted to yield the PCP theorem.</jats:p>\n          <jats:p>\n            Our approach is to consider the\n            <jats:italic>unsat value</jats:italic>\n            of a constraint system, which is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the underlying variables. We describe a new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system. The amplification step causes an increase in alphabet-size that is corrected by a (standard) PCP composition step. Iterative application of these two steps yields a proof for the PCP theorem.\n          </jats:p>\n          <jats:p>The amplification lemma relies on a new notion of “graph powering” that can be applied to systems of binary constraints. This powering amplifies the unsat-value of a constraint system provided that the underlying graph structure is an expander.</jats:p>\n          <jats:p>\n            We also extend our amplification lemma towards construction of assignment testers (alternatively, PCPs of Proximity) which are slightly stronger objects than PCPs. We then construct PCPs and locally-testable codes whose length is linear up to a\n            <jats:italic>polylog</jats:italic>\n            factor, and whose correctness can be probabilistically verified by making a\n            <jats:italic>constant</jats:italic>\n            number of queries. Namely, we prove\n            <jats:italic>SAT</jats:italic>\n            ∈\n            <jats:italic>PCP</jats:italic>\n            <jats:sub>1/2,1</jats:sub>\n            [log\n            <jats:sub>2</jats:sub>\n            (\n            <jats:italic>n</jats:italic>\n            ⋅poly log\n            <jats:italic>n</jats:italic>\n            ),\n            <jats:italic>O</jats:italic>\n            (1)].\n          </jats:p>","DOI":"10.1145/1236457.1236459","page":"12","source":"Crossref","title":"The PCP theorem by gap amplification","volume":"54","author":[{"given":"Irit","family":"Dinur","sequence":"first","affiliation":[{"name":"The Hebrew University, Jerusalem, Israel"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[2007,6]]},"URL":"http://dx.doi.org/10.1145/1236457.1236459","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/1236457.1236459"},{"id":"doi:10.1145/1162349.1162351","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"4","abstract":"<jats:p>\n            We initiate a systematic study of locally testable codes; that is, error-correcting codes that admit very efficient membership tests. Specifically, these are codes accompanied with tests that make a constant number of (random) queries into any given word and reject non-codewords with probability proportional to their distance from the code.Locally testable codes are believed to be the combinatorial core of PCPs. However, the relation is less immediate than commonly believed. Nevertheless, we show that certain PCP systems can be modified to yield locally testable codes. On the other hand, we adapt techniques that we develop for the construction of the latter to yield new PCPs.Our main results are locally testable codes and PCPs of almost-linear length. Specifically, we prove the existence of the following constructs:---Locally testable binary (linear) codes in which\n            <jats:italic>k</jats:italic>\n            information bits are encoded by a codeword of length\n            <jats:italic>k</jats:italic>\n            ⋅ exp(Õ(√(log\n            <jats:italic>k</jats:italic>\n            ))). This improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet.---PCP systems of almost-linear length for SAT. The length of the proof is\n            <jats:italic>n</jats:italic>\n            ⋅ exp(Õ(√(log\n            <jats:italic>n</jats:italic>\n            ))) and verification in performed by a constant number (i.e., 19) of queries, as opposed to previous results that used proof length\n            <jats:italic>n</jats:italic>\n            <jats:sup>\n              (1 +\n              <jats:italic>O</jats:italic>\n              (1/\n              <jats:italic>q</jats:italic>\n              ))\n            </jats:sup>\n            for verification by\n            <jats:italic>q</jats:italic>\n            queries.The novel techniques in use include a random projection of certain codewords and PCP-oracles that preserves local-testability, an adaptation of PCP constructions to obtain “linear PCP-oracles” for proving conjunctions of linear conditions, and design of PCPs with some new soundness properties---a direct construction of locally testable (linear) codes of subexponential length.\n          </jats:p>","DOI":"10.1145/1162349.1162351","page":"558-655","source":"Crossref","title":"Locally testable codes and PCPs of almost-linear length","volume":"53","author":[{"given":"Oded","family":"Goldreich","sequence":"first","affiliation":[{"name":"Weizmann Institute of Science, Rehovot, Israel"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[2006,7]]},"URL":"http://dx.doi.org/10.1145/1162349.1162351","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/1162349.1162351"},{"id":"doi:10.1109/TIT.2018.2809788","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2018.2809788","page":"5813-5831","source":"Crossref","title":"Locally Testable and Locally Correctable Codes approaching the Gilbert-Varshamov Bound","volume":"64","author":[{"given":"Sivakanth","family":"Gopi","sequence":"first","affiliation":[]},{"given":"Swastik","family":"Kopparty","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8917-8689","authenticated-orcid":false,"given":"Rafael","family":"Oliveira","sequence":"additional","affiliation":[]},{"given":"Noga","family":"Ron-Zewi","sequence":"additional","affiliation":[]},{"given":"Shubhangi","family":"Saraf","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2018.2809788","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1145/3051093","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"2","abstract":"<jats:p>\n            Locally correctable codes (LCCs) and locally testable codes (LTCs) are error-correcting codes that admit\n            <jats:italic>local</jats:italic>\n            algorithms for correction and detection of errors. Those algorithms are local in the sense that they only query a small number of entries of the corrupted codeword. The fundamental question about LCCs and LTCs is to determine the optimal tradeoff among their rate, distance, and query complexity.\n          </jats:p>\n          <jats:p>\n            In this work, we construct the first LCCs and LTCs with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist LCCs and LTCs with block length\n            <jats:italic>n</jats:italic>\n            , constant rate (which can even be taken arbitrarily close to 1), and constant relative distance, whose query complexity is exp(Õ(√log\n            <jats:italic>n</jats:italic>\n            )) (for LCCs) and (log\n            <jats:italic>n</jats:italic>\n            )\n            <jats:sup>\n              <jats:italic>O</jats:italic>\n              (log log\n              <jats:italic>n</jats:italic>\n              )\n            </jats:sup>\n            (for LTCs).\n          </jats:p>\n          <jats:p>\n            In addition to having small query complexity, our codes also achieve better tradeoffs between the rate and the relative distance than were previously known to be achievable by LCCs or LTCs. Specifically, over large (but constant size) alphabet, our codes approach the Singleton bound, that is, they have almost the best-possible relationship between their rate and distance. Over the binary alphabet, our codes meet the Zyablov bound. Such tradeoffs between the rate and the relative distance were previously not known for any\n            <jats:italic>o</jats:italic>\n            (\n            <jats:italic>n</jats:italic>\n            ) query complexity. Our results on LCCs also immediately give locally decodable codes with the same parameters.\n          </jats:p>","DOI":"10.1145/3051093","page":"1-42","source":"Crossref","title":"High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity","volume":"64","author":[{"given":"Swastik","family":"Kopparty","sequence":"first","affiliation":[{"name":"Department of Mathematics 8 Department of Computer Science, Rutgers University, Piscataway NJ, USA"}]},{"given":"Or","family":"Meir","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Haifa University, Haifa, Israel"}]},{"given":"Noga","family":"Ron-Zewi","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Ben-Gurion University, Be’er Sheva, Israel"}]},{"given":"Shubhangi","family":"Saraf","sequence":"additional","affiliation":[{"name":"Department of Mathematics 8 Department of Computer Science, Rutgers University, Piscataway NJ, USA"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[2017,4,30]]},"URL":"http://dx.doi.org/10.1145/3051093","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/3051093"},{"id":"arxiv:1002.3864","type":"article-journal","author":[{"given":"Prahladh","family":"Harsha"},{"given":"Moses","family":"Charikar"},{"given":"Matthew","family":"Andrews"},{"given":"Sanjeev","family":"Arora"},{"given":"Subhash","family":"Khot"},{"given":"Dana","family":"Moshkovitz"},{"given":"Lisa","family":"Zhang"},{"given":"Ashkan","family":"Aazami"},{"given":"Dev","family":"Desai"},{"given":"Igor","family":"Gorodezky"},{"given":"Geetha","family":"Jagannathan"},{"given":"Alexander S.","family":"Kulikov"},{"given":"Darakhshan J.","family":"Mir"},{"given":"Alantha","family":"Newman"},{"given":"Aleksandar","family":"Nikolov"},{"given":"David","family":"Pritchard"},{"given":"Gwen","family":"Spencer"}],"title":"Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)","issued":{"date-parts":[[2010,2,20]]},"note":"arxivid:1002.3864\narxiv_version_number:1"},{"id":"doi:10.1109/SFCS.1995.492573","type":"proceedings-article","publisher":"IEEE Comput. Soc. Press","DOI":"10.1109/sfcs.1995.492573","page":"422-431","source":"Crossref","title":"Free bits, PCPs and non-approximability-towards tight results","author":[{"given":"M.","family":"Bellare","sequence":"first","affiliation":[]},{"given":"O.","family":"Goldreich","sequence":"additional","affiliation":[]},{"given":"M.","family":"Sudan","sequence":"additional","affiliation":[]}],"event":"IEEE 36th Annual Foundations of Computer Science","container-title":"Proceedings of IEEE 36th Annual Foundations of Computer Science","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.1995.492573"},{"id":"doi:10.1145/502090.502098","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"4","abstract":"<jats:p>\n            We prove optimal, up to an arbitrary ε &gt; 0, inapproximability results for Max-E\n            <jats:italic>k</jats:italic>\n            -Sat for\n            <jats:italic>k</jats:italic>\n            ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime\n            <jats:italic>p</jats:italic>\n            and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.\n          </jats:p>","DOI":"10.1145/502090.502098","page":"798-859","source":"Crossref","title":"Some optimal inapproximability results","volume":"48","author":[{"given":"Johan","family":"Håstad","sequence":"first","affiliation":[{"name":"Royal Institute of Technology, Stockholm, Sweden"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[2001,7]]},"URL":"http://dx.doi.org/10.1145/502090.502098","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/502090.502098"},{"id":"doi:10.1137/S0097539796302531","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/s0097539796302531","page":"804-915","source":"Crossref","title":"Free Bits, PCPs, and Nonapproximability---Towards Tight Results","volume":"27","author":[{"given":"Mihir","family":"Bellare","sequence":"first","affiliation":[]},{"given":"Oded","family":"Goldreich","sequence":"additional","affiliation":[]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[1998,6]]},"URL":"http://dx.doi.org/10.1137/S0097539796302531","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/S0097539796302531"},{"id":"doi:10.1090/S0273-0979-06-01126-8","type":"journal-article","publisher":"American Mathematical Society (AMS)","issue":"04","DOI":"10.1090/s0273-0979-06-01126-8","page":"439-562","source":"Crossref","title":"Expander graphs and their applications","volume":"43","author":[{"given":"Shlomo","family":"Hoory","sequence":"first","affiliation":[]},{"given":"Nathan","family":"Linial","sequence":"additional","affiliation":[]},{"given":"Avi","family":"Wigderson","sequence":"additional","affiliation":[]}],"container-title":"Bulletin of the American Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[2006,8,7]]},"URL":"http://dx.doi.org/10.1090/S0273-0979-06-01126-8","ISSN":["0273-0979"],"container-title-short":"Bull. Amer. Math. Soc.","note":"alternative-id:S0273097906011268"},{"id":"arxiv:2111.04808","type":"article-journal","author":[{"given":"Irit","family":"Dinur"},{"given":"Shai","family":"Evra"},{"given":"Ron","family":"Livne"},{"given":"Alexander","family":"Lubotzky"},{"given":"Shahar","family":"Mozes"}],"title":"Locally Testable Codes with constant rate, distance, and locality","issued":{"date-parts":[[2021,12,15]]},"note":"arxivid:2111.04808\narxiv_version_number:2"},{"id":"doi:10.1145/780542.780594","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/780542.780594","page":"345-354","source":"Crossref","title":"Some 3CNF properties are hard to test","author":[{"given":"Eli","family":"Ben-Sasson","sequence":"first","affiliation":[{"name":"Harvard &amp; MIT, Cambridge, MA"}]},{"given":"Prahladh","family":"Harsha","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]},{"given":"Sofya","family":"Raskhodnikova","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]}],"event":"STOC03: The 35th Annual ACM Symposium on Theory of Computing","container-title":"Proceedings of the thirty-fifth annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2003,6,9]]},"URL":"http://dx.doi.org/10.1145/780542.780594","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/780542.780594\n10.1145/780542"},{"id":"arxiv:2202.13641","type":"article-journal","author":[{"given":"Anthony","family":"Leverrier"},{"given":"Gilles","family":"Zémor"}],"title":"Quantum Tanner codes","issued":{"date-parts":[[2022,9,16]]},"note":"arxivid:2202.13641\narxiv_version_number:3"},{"id":"doi:10.1111/j.1749-6632.1970.tb56492","type":"journal-article","publisher":"Wiley","issue":"1","abstract":"<jats:p>Conference matrices are used to construct a new infinite family of nonlinear error correcting codes with high minimum distance.</jats:p>","DOI":"10.1111/j.1749-6632.1970.tb56492.x","page":"363-365","source":"Crossref","title":"A NEW FAMILY OF NONLINEAR CODES OBTAINED FROM CONFERENCE MATRICES","volume":"175","author":[{"given":"N.J.A.","family":"Sloane","sequence":"first","affiliation":[]},{"given":"J.J.","family":"Seidel","sequence":"additional","affiliation":[]}],"container-title":"Annals of the New York Academy of Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1970,9]]},"URL":"http://dx.doi.org/10.1111/j.1749-6632.1970.tb56492.x","ISSN":["0077-8923","1749-6632"],"container-title-short":"Annals of the New York Academy of Sciences","note":"alternative-id:10.1111/j.1749-6632.1970.tb56492.x"},{"id":"preset:Klove81","type":"book","author":[{"family":"Kløve","given":"T."}],"title":"Error correcting codes for the asymmetric channel","publisher":"Department of Pure Mathematics, University of Bergen","issued":"1981"},{"id":"doi:10.1016/S0019-9958(79)90329-2","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0019-9958(79)90329-2","page":"20-36","source":"Crossref","title":"On the theory of binary asymmetric error correcting codes","volume":"40","author":[{"given":"Serban D.","family":"Constantin","sequence":"first","affiliation":[]},{"given":"T.R.N.","family":"Rao","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1979,1]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(79)90329-2","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995879903292"},{"id":"doi:10.1109/ITW.2009.5351208","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.2009.5351208","page":"135-138","source":"Crossref","title":"A comparison between single asymmetric and single insertion/deletion correcting codes using group-theory","author":[{"given":"Filip","family":"Paluncic","sequence":"first","affiliation":[]},{"given":"Hendrik C.","family":"Ferreira","sequence":"additional","affiliation":[]},{"given":"Willem A.","family":"Clarke","sequence":"additional","affiliation":[]}],"event":"2009 IEEE Information Theory Workshop (ITW 2009)","container-title":"2009 IEEE Information Theory Workshop","original-title":[],"issued":{"date-parts":[[2009,10]]},"URL":"http://dx.doi.org/10.1109/ITW.2009.5351208"},{"id":"doi:10.1016/S0019-9958(80)90082-0","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0019-9958(80)90082-0","page":"187-196","source":"Crossref","title":"The constantinrao construction for binary asymmetric error-correcting codes","volume":"44","author":[{"given":"Robert J.","family":"McEliece","sequence":"first","affiliation":[]},{"given":"Eugene R.","family":"Rodemich","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1980,2]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(80)90082-0","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995880900820"},{"id":"doi:10.1109/18.651063","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.651063","page":"340-345","source":"Crossref","title":"Systematic encoding of the Varshamov-Tenengol'ts codes and the Constantin-Rao codes","volume":"44","author":[{"given":"K.A.S.","family":"Abdel-Ghaffar","sequence":"first","affiliation":[]},{"given":"H.C.","family":"Ferreira","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.651063","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:0907.5149","type":"article-journal","author":[{"given":"Peter W.","family":"Shor"},{"given":"Graeme","family":"Smith"},{"given":"John A.","family":"Smolin"},{"given":"Bei","family":"Zeng"}],"title":"High performance single-error-correcting quantum codes for amplitude damping","issued":{"date-parts":[[2009,7,29]]},"note":"arxivid:0907.5149\narxiv_version_number:1"},{"id":"doi:10.1049/el:19740375","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"23","DOI":"10.1049/el:19740375","page":"471-472","source":"Crossref","title":"Two dual families of nonlinear binary codes","volume":"10","author":[{"given":"J.M.","family":"Goethals","sequence":"first","affiliation":[{"name":"MBLE Research Laboratory, 2 Avenue Van Becelaere, Brussels, B-1170, Belgium"}]}],"container-title":"Electronics Letters","original-title":[],"language":"en","issued":{"date-parts":[[1974,11,14]]},"URL":"http://dx.doi.org/10.1049/el:19740375","ISSN":["0013-5194","1350-911X"],"container-title-short":"Electron. Lett. (UK)","note":"alternative-id:10.1049/el:19740375"},{"id":"doi:10.1016/0097-3165(75)90090-4","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(75)90090-4","page":"26-50","source":"Crossref","title":"Alternating bilinear forms over GF(q)","volume":"19","author":[{"given":"P","family":"Delsarte","sequence":"first","affiliation":[]},{"given":"J.M","family":"Goethals","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1975,7]]},"URL":"http://dx.doi.org/10.1016/0097-3165(75)90090-4","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316575900904"},{"id":"doi:10.1016/0012-365X(90)90010-F","type":"journal-article","publisher":"Elsevier BV","issue":"2-3","DOI":"10.1016/0012-365x(90)90010-f","page":"249-263","source":"Crossref","title":"On the delsarte-goethals codes and their formal duals","volume":"83","author":[{"given":"Ferdinand B.","family":"Hergert","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1990,8]]},"URL":"http://dx.doi.org/10.1016/0012-365X(90)90010-F","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9090010F"},{"id":"arxiv:0801.2144","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.2008.4578694","page":"396-400","source":"Crossref","title":"Non-additive quantum codes from Goethals and Preparata codes","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Martin","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"2008 IEEE Information Theory Workshop (ITW)","container-title":"2008 IEEE Information Theory Workshop","original-title":[],"issued":{"date-parts":[[2008,5]]},"URL":"http://dx.doi.org/10.1109/ITW.2008.4578694","note":"arxivid:0801.2144"},{"id":"arxiv:0801.2150","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2008.4594996","source":"Crossref","title":"Quantum Goethals-Preparata codes","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Martin","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"2008 IEEE International Symposium on Information Theory - ISIT","container-title":"2008 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2008.4594996","note":"arxivid:0801.2150"},{"id":"manual:-S.-Ling-and-P.-Sole.-2008.-N","type":"document","author":[{"family":"Ling","given":"S."},{"family":"Sole","given":"P."}],"title":"Nonadditive quantum codes from Z4-codes","issued":"2008","URL":"https://hal.archives-ouvertes.fr/hal-00338309/fr/","note":"Available at \\url{https://hal.archives-ouvertes.fr/hal-00338309/fr/}"},{"id":"doi:10.1109/18.476333","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.476333","page":"2040-2048","source":"Crossref","title":"The algebraic decoding of the Z/sub 4/-linear Goethals code","volume":"41","author":[{"given":"T.","family":"Helleseth","sequence":"first","affiliation":[]},{"given":"P.V.","family":"Kumar","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1995]]},"URL":"http://dx.doi.org/10.1109/18.476333","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1142/3603","type":"monograph","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/3603","source":"Crossref","title":"Quaternary Codes","author":[{"given":"Z X","family":"Wan","sequence":"first","affiliation":[{"name":"Academia SinicaChina"}]}],"container-title":"Series on Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1997,11]]},"ISBN":["9789810232740","9789812798121"],"URL":"http://dx.doi.org/10.1142/3603","ISSN":["1793-0928"],"note":"aliases:10.1142/9789812798121\nalternative-id:10.1142/3603\n10.1142/SAM"},{"id":"doi:10.1137/0603015","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","DOI":"10.1137/0603015","page":"151-165","source":"Crossref","title":"Spreads, Translation Planes and Kerdock Sets. I","volume":"3","author":[{"given":"William M.","family":"Kantor","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Algebraic Discrete Methods","original-title":[],"language":"en","issued":{"date-parts":[[1982,6]]},"URL":"http://dx.doi.org/10.1137/0603015","ISSN":["0196-5212","2168-345X"],"container-title-short":"SIAM. J. on Algebraic and Discrete Methods","note":"alternative-id:10.1137/0603015"},{"id":"doi:10.1137/0603032","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/0603032","page":"308-318","source":"Crossref","title":"Spreads, Translation Planes and Kerdock Sets. II","volume":"3","author":[{"given":"W. M.","family":"Kantor","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Algebraic Discrete Methods","original-title":[],"language":"en","issued":{"date-parts":[[1982,9]]},"URL":"http://dx.doi.org/10.1137/0603032","ISSN":["0196-5212","2168-345X"],"container-title-short":"SIAM. J. on Algebraic and Discrete Methods","note":"alternative-id:10.1137/0603032"},{"id":"doi:10.1109/TIT.1983.1056676","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1983.1056676","page":"345-348","source":"Crossref","title":"On the inequivalence of generalized Preparata codes","volume":"29","author":[{"given":"W.","family":"Kantor","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1983,5]]},"URL":"http://dx.doi.org/10.1109/TIT.1983.1056676","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-R.-Sch-rer-and-W.-Ch.-Schmid","type":"article-journal","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Preparata Codes. From MinT—the database of optimal net, code, OA, and OOA parameters","container-title":"Version","issued":"2015-09-03","URL":"https://web.archive.org/web/20240420202309/https://mint.sbg.ac.at/desc_CPreparata.html","note":"Available at \\url{https://web.archive.org/web/20240420202309/https://mint.sbg.ac.at/desc_CPreparata.html}"},{"id":"manual:-G.-V.-Zaitsev-V.-A.-Zinoviev","type":"paper-conference","author":[{"family":"Zaitsev","given":"G.V."},{"family":"Zinoviev","given":"V.A."},{"family":"Semakov","given":"N.V."}],"title":"Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error-correcting codes","container-title":"Proc. 2nd International Symp. Inform. Theory","issued":"1973","page":"257–263"},{"id":"arxiv:math/9310227","type":"article-journal","author":[{"given":"A. R.","family":"Calderbank"},{"given":"A. Roger","family":"Hammons"},{"given":"P. Vijay","family":"Kumar"},{"given":"N. J. A.","family":"Sloane"},{"given":"Patrick","family":"Solé"}],"title":"A linear construction for certain Kerdock and Preparata codes","issued":{"date-parts":[[1993,10,1]]},"note":"arxivid:math/9310227\narxiv_version_number:1"},{"id":"manual:-F.-Gray-PULSE-CODE-COMMUNICA","type":"article-journal","author":[{"family":"Gray","given":"F."}],"title":"PULSE CODE COMMUNICATION","volume":"2632058","container-title":"United States Patent Number","issued":"1953"},{"id":"doi:10.1002/j.1538-7305.1958.tb03887","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1002/j.1538-7305.1958.tb03887.x","page":"815-826","source":"Crossref","title":"Gray Codes and Paths on the n-Cube","volume":"37","author":[{"given":"E. N.","family":"Gilbert","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1958,5]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1958.tb03887.x","ISSN":["0005-8580"]},{"id":"doi:10.1137/0209013","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","DOI":"10.1137/0209013","page":"130-141","source":"Crossref","title":"Combinatorial Gray Codes","volume":"9","author":[{"given":"J. T.","family":"Joichi","sequence":"first","affiliation":[]},{"given":"Dennis E.","family":"White","sequence":"additional","affiliation":[]},{"given":"S. G.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[1980,2]]},"URL":"http://dx.doi.org/10.1137/0209013","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/0209013"},{"id":"doi:10.1109/TSMCB.2012.2210706","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tsmcb.2012.2210706","page":"515-529","source":"Crossref","title":"(n, k, p)-Gray code for image systems","volume":"43","author":[{"family":"Yicong Zhou","sequence":"first","affiliation":[]},{"given":"K.","family":"Panetta","sequence":"additional","affiliation":[]},{"given":"S.","family":"Agaian","sequence":"additional","affiliation":[]},{"given":"C. L. P.","family":"Chen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Cybernetics","original-title":[],"issued":{"date-parts":[[2013,4]]},"URL":"http://dx.doi.org/10.1109/TSMCB.2012.2210706","ISSN":["2168-2267","2168-2275"],"container-title-short":"IEEE Trans. Cybern."},{"id":"doi:10.1145/360336.360343","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"9","abstract":"<jats:p>Algorithms are presented to generate the n-bit binary reflected Gray code and codewords of fixed weight in that code. Both algorithms are efficient in that the time required to generate the next element from the current one is constant. Applications to the generation of the combinations of n things taken k at a time, the compositions of integers, and the permutations of a multiset are discussed.</jats:p>","DOI":"10.1145/360336.360343","page":"517-521","source":"Crossref","title":"Efficient generation of the binary reflected gray code and its applications","volume":"19","author":[{"given":"James R.","family":"Bitner","sequence":"first","affiliation":[{"name":"Univ. of Texas, Austin"}]},{"given":"Gideon","family":"Ehrlich","sequence":"additional","affiliation":[{"name":"Bar-Ilan Univ., Ramat-Gan, Israel"}]},{"given":"Edward M.","family":"Reingold","sequence":"additional","affiliation":[{"name":"Univ. of Illinois, Urbana"}]}],"container-title":"Communications of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[1976,9]]},"URL":"http://dx.doi.org/10.1145/360336.360343","ISSN":["0001-0782","1557-7317"],"container-title-short":"Commun. ACM","note":"alternative-id:10.1145/360336.360343"},{"id":"doi:10.1364/AO.36.004463","type":"journal-article","publisher":"Optica Publishing Group","issue":"19","DOI":"10.1364/ao.36.004463","page":"4463","source":"Crossref","title":"Three-dimensional imaging based on Gray-code light projection: characterization of the measuring algorithm and development of a measuring system for industrial applications","volume":"36","author":[{"given":"G.","family":"Sansoni","sequence":"first","affiliation":[]},{"given":"S.","family":"Corini","sequence":"additional","affiliation":[]},{"given":"S.","family":"Lazzari","sequence":"additional","affiliation":[]},{"given":"R.","family":"Rodella","sequence":"additional","affiliation":[]},{"given":"F.","family":"Docchio","sequence":"additional","affiliation":[]}],"container-title":"Applied Optics","original-title":[],"language":"en","issued":{"date-parts":[[1997,7,1]]},"URL":"http://dx.doi.org/10.1364/AO.36.004463","ISSN":["0003-6935","1539-4522"],"container-title-short":"Appl. Opt."},{"id":"doi:10.1109/12.29465","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/12.29465","page":"1249-1268","source":"Crossref","title":"Optimum broadcasting and personalized communication in hypercubes","volume":"38","author":[{"given":"S.L.","family":"Johnsson","sequence":"first","affiliation":[]},{"given":"C.-T.","family":"Ho","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Computers","original-title":[],"issued":{"date-parts":[[1989]]},"URL":"http://dx.doi.org/10.1109/12.29465","ISSN":["0018-9340"],"container-title-short":"IEEE Trans. Comput."},{"id":"doi:10.1137/S0036144595295272","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","DOI":"10.1137/s0036144595295272","page":"605-629","source":"Crossref","title":"A Survey of Combinatorial Gray Codes","volume":"39","author":[{"given":"Carla","family":"Savage","sequence":"first","affiliation":[]}],"container-title":"SIAM Review","original-title":[],"language":"en","issued":{"date-parts":[[1997,1]]},"URL":"http://dx.doi.org/10.1137/S0036144595295272","ISSN":["0036-1445","1095-7200"],"container-title-short":"SIAM Rev.","note":"alternative-id:10.1137/S0036144595295272"},{"id":"doi:10.1201/9780429332760-9","type":"book-chapter","publisher":"CRC Press","DOI":"10.1201/9780429332760-9","page":"203-216","source":"Crossref","title":"Generalized Gray Codes and Their Properties","author":[{"given":"Linda S.","family":"Barasch","sequence":"first","affiliation":[]},{"given":"S.","family":"Lakshmivarahan","sequence":"additional","affiliation":[]},{"given":"Sudarshan K.","family":"Dhall","sequence":"additional","affiliation":[]}],"container-title":"Mathematics for Large Scale Computing","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,29]]},"ISBN":["9780429332760"],"URL":"http://dx.doi.org/10.1201/9780429332760-9","note":"edition-number:1"},{"id":"arxiv:2202.01280","type":"article-journal","author":[{"given":"Torsten","family":"Mütze"}],"title":"Combinatorial Gray codes-an updated survey","issued":{"date-parts":[[2024,7,30]]},"note":"arxivid:2202.01280\narxiv_version_number:4"},{"id":"arxiv:quant-ph/0312218","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.92.177902","source":"Crossref","title":"Efficient Decomposition of Quantum Gates","volume":"92","author":[{"given":"Juha J.","family":"Vartiainen","sequence":"first","affiliation":[]},{"given":"Mikko","family":"Möttönen","sequence":"additional","affiliation":[]},{"given":"Martti M.","family":"Salomaa","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2004,4,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.92.177902","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"177902","note":"arxivid:quant-ph/0312218"},{"id":"arxiv:1909.12847","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Simulation of quantum systems is expected to be one of the most important applications of quantum computing, with much of the theoretical work so far having focused on fermionic and spin-<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\frac{1}{2}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mfrac>\n                    <mml:mrow>\n                      <mml:mn>1</mml:mn>\n                    </mml:mrow>\n                    <mml:mrow>\n                      <mml:mn>2</mml:mn>\n                    </mml:mrow>\n                  </mml:mfrac>\n                </mml:math></jats:alternatives></jats:inline-formula> systems. Here, we instead consider encodings of <jats:italic>d</jats:italic>-level (i.e., qudit) quantum operators into multi-qubit operators, studying resource requirements for approximating operator exponentials by Trotterization. We primarily focus on spin-<jats:italic>s</jats:italic> and truncated bosonic operators in second quantization, observing desirable properties for approaches based on the Gray code, which to our knowledge has not been used in this context previously. After outlining a methodology for implementing an arbitrary encoding, we investigate the interplay between Hamming distances, sparsity patterns, bosonic truncation, and other properties of local operators. Finally, we obtain resource counts for five common Hamiltonian classes used in physics and chemistry, while modeling the possibility of converting between encodings within a Trotter step. The most efficient encoding choice is heavily dependent on the application and highly sensitive to <jats:italic>d</jats:italic>, although clear trends are present. These operation count reductions are relevant for running algorithms on near-term quantum hardware because the savings effectively decrease the required circuit depth. Results and procedures outlined in this work may be useful for simulating a broad class of Hamiltonians on qubit-based digital quantum computers.</jats:p>","DOI":"10.1038/s41534-020-0278-0","source":"Crossref","title":"Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-8510-8480","authenticated-orcid":false,"given":"Nicolas P. D.","family":"Sawaya","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7205-752X","authenticated-orcid":false,"given":"Tim","family":"Menke","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3557-2709","authenticated-orcid":false,"given":"Thi Ha","family":"Kyaw","sequence":"additional","affiliation":[]},{"given":"Sonika","family":"Johri","sequence":"additional","affiliation":[]},{"given":"Alán","family":"Aspuru-Guzik","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5579-451X","authenticated-orcid":false,"given":"Gian Giacomo","family":"Guerreschi","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,2]]},"URL":"http://dx.doi.org/10.1038/s41534-020-0278-0","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"49","note":"alternative-id:278\narxivid:1909.12847"},{"id":"doi:10.1515/dma.1991.1.4.365","type":"journal-article","publisher":"Walter de Gruyter GmbH","issue":"4","DOI":"10.1515/dma.1991.1.4.365","source":"Crossref","title":"Kerdock code in a cyclic form","volume":"1","author":[{"given":"A. A.","family":"NECHAEV","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics and Applications","original-title":[],"issued":{"date-parts":[[1991]]},"URL":"http://dx.doi.org/10.1515/dma.1991.1.4.365","ISSN":["0924-9265","1569-3929"]},{"id":"doi:10.1007/BF01388558","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01388558","page":"31-42","source":"Crossref","title":"Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others","volume":"4","author":[{"given":"J. H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1994,1]]},"URL":"http://dx.doi.org/10.1007/BF01388558","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:BF01388558"},{"id":"doi:10.1109/TIT.2009.2018336","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2009.2018336","page":"2659-2673","source":"Crossref","title":"Rank Modulation for Flash Memories","volume":"55","author":[{"family":"Anxiao Jiang","sequence":"first","affiliation":[]},{"given":"R.","family":"Mateescu","sequence":"additional","affiliation":[]},{"given":"M.","family":"Schwartz","sequence":"additional","affiliation":[]},{"given":"J.","family":"Bruck","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2009,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2009.2018336","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/BF01388485","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf01388485","page":"237-249","source":"Crossref","title":"The automorphism groups of the Delsarte-Goethals codes","volume":"3","author":[{"given":"Claude","family":"Carlet","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1993,7]]},"URL":"http://dx.doi.org/10.1007/BF01388485","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:BF01388485"},{"id":"doi:10.1109/ICC.2004.1312536","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.2004.1312536","page":"483-487 Vol.1","source":"Crossref","title":"Space-time signaling based on Kerdock and Delsafte-Goethals codes","author":[{"given":"A.R.","family":"Calderbank","sequence":"first","affiliation":[]},{"given":"S.N.","family":"Diggavi","sequence":"additional","affiliation":[]},{"given":"N.","family":"Al-Dhahir","sequence":"additional","affiliation":[]}],"event":"2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577)","container-title":"2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577)","original-title":[],"issued":{"date-parts":[[2004]]},"URL":"http://dx.doi.org/10.1109/ICC.2004.1312536"},{"id":"arxiv:1801.04537","type":"article-journal","author":[{"given":"Andrew","family":"Thompson"},{"given":"Robert","family":"Calderbank"}],"title":"Compressed Neighbour Discovery using Sparse Kerdock Matrices","issued":{"date-parts":[[2018,1,14]]},"note":"arxivid:1801.04537\narxiv_version_number:1"},{"id":"doi:10.1080/02522667.1991.10699078","type":"journal-article","publisher":"Taru Publications","issue":"3","DOI":"10.1080/02522667.1991.10699078","page":"387-400","source":"Crossref","title":"The Automorphism Groups of the Kerdock Codes","volume":"12","author":[{"given":"Claude","family":"Carlet","sequence":"first","affiliation":[]}],"container-title":"Journal of Information and Optimization Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1991,9]]},"URL":"http://dx.doi.org/10.1080/02522667.1991.10699078","ISSN":["0252-2667","2169-0103"],"container-title-short":"Journal of Information and Optimization Sciences","note":"alternative-id:10.1080/02522667.1991.10699078"},{"id":"doi:10.1016/S0019-9958(72)90376-2","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0019-9958(72)90376-2","page":"182-187","source":"Crossref","title":"A class of low-rate nonlinear binary codes","volume":"20","author":[{"given":"A.M.","family":"Kerdock","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1972,3]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(72)90376-2","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995872903762"},{"id":"manual:-R.-Sch-rer-and-W.-Ch.-Schmid","type":"article-journal","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Kerdock Codes. From MinT—the database of optimal net, code, OA, and OOA parameters","container-title":"Version","issued":"2015-09-03","URL":"https://web.archive.org/web/20240525071439/http://mint.sbg.ac.at/desc_CKerdock.html","note":"Available at \\url{https://web.archive.org/web/20240525071439/http://mint.sbg.ac.at/desc_CKerdock.html}"},{"id":"doi:10.1112/S0024611597000403","type":"journal-article","publisher":"Wiley","issue":"2","DOI":"10.1112/s0024611597000403","page":"436-480","source":"Crossref","title":"Z<sub>4</sub>\n-Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line-Sets","volume":"75","author":[{"given":"AR","family":"Calderbank","sequence":"first","affiliation":[{"name":"AT&T Bell Laboratories; Murray Hill NJ 07974 USA"}]},{"given":"PJ","family":"Cameron","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences; Queen Mary and Westfield College; London E1 4NS UK"}]},{"given":"WM","family":"Kantor","sequence":"additional","affiliation":[{"name":"Department of Mathematics; University of Oregon; Eugene Oregon 97403 USA"}]},{"given":"JJ","family":"Seidel","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computing Science; Eindhoven University of Technology; 5600 MB Eindhoven The Netherlands"}]}],"container-title":"Proceedings of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1997,9]]},"URL":"http://dx.doi.org/10.1112/S0024611597000403","ISSN":["0024-6115"],"container-title-short":"Proceedings of the London Mathematical Society"},{"id":"arxiv:1904.07842","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2020.3015683","page":"6104-6120","source":"Crossref","title":"Kerdock Codes Determine Unitary 2-Designs","volume":"66","author":[{"given":"Trung","family":"Can","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2084-9717","authenticated-orcid":false,"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5521-4397","authenticated-orcid":false,"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2020,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2020.3015683","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1904.07842"},{"id":"arxiv:2105.05879","type":"article-journal","author":[{"given":"Tim","family":"Fuchs"},{"given":"David","family":"Gross"},{"given":"Felix","family":"Krahmer"},{"given":"Richard","family":"Kueng"},{"given":"Dustin G.","family":"Mixon"}],"title":"Sketching with Kerdock's crayons: Fast sparsifying transforms for arbitrary linear maps","issued":{"date-parts":[[2021,5,12]]},"note":"arxivid:2105.05879\narxiv_version_number:1"},{"id":"preset:Levenshtein82","type":"article-journal","author":[{"family":"Levenshtein","given":"V.I."}],"title":"Bounds on the maximal cardinality of a code with bounded modulus of the inner product","volume":"25","container-title":"Soviet Mathematics Doklady","issue":"2","issued":"1982","page":"526–531"},{"id":"arxiv:1308.3188","type":"journal-article","publisher":"Mathematical Sciences Publishers","issue":"3","DOI":"10.2140/gt.2016.20.1289","page":"1289-1357","source":"Crossref","title":"Optimal simplices and codes in projective spaces","volume":"20","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[]},{"given":"Abhinav","family":"Kumar","sequence":"additional","affiliation":[]},{"given":"Gregory","family":"Minton","sequence":"additional","affiliation":[]}],"container-title":"Geometry &amp; Topology","original-title":[],"language":"en","issued":{"date-parts":[[2016,7,4]]},"URL":"http://dx.doi.org/10.2140/gt.2016.20.1289","ISSN":["1364-0380","1465-3060"],"container-title-short":"Geom. Topol.","note":"arxivid:1308.3188"},{"id":"manual:-A.-Krasnopeev-and-Yu.-L.-Sag","type":"article-journal","author":[{"family":"Krasnopeev","given":"A."},{"family":"Sagalovich","given":"Yu L."}],"title":"The Kerdock codes and separating systems","container-title":"Eight International Workshop on Algebraic and Combinatorial Coding Theory","volume":"7","issue":"7.2","issued":"2002"},{"id":"doi:10.1109/TIT.2004.838106","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2004.838106","page":"3312-3315","source":"Crossref","title":"On the&amp;lt;tex&amp;gt;$","volume":"50","author":[{"given":"T.H.","family":"Helleseth","sequence":"first","affiliation":[]},{"given":"H.G.","family":"Schaathun","sequence":"additional","affiliation":[]},{"given":"T.H.","family":"Helleseth","sequence":"additional","affiliation":[]},{"given":"H.G.","family":"Schaathun","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2004,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.838106","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1972.1054904","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1972.1054904","page":"825-826","source":"Crossref","title":"A new description of the Nadler code (Corresp.)","volume":"18","author":[{"given":"J.","family":"van Lint","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1972,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1972.1054904","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1977.1055665","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1977.1055665","page":"132-135","source":"Crossref","title":"The extended Nadler code is unique (Corresp.)","volume":"23","author":[{"given":"J.-M.","family":"Goethals","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1977,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1977.1055665","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1962.1057670","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1962.1057670","page":"58-58","source":"Crossref","title":"A 32-point n=12, d=5 code (Corresp.)","volume":"8","author":[{"given":"M.","family":"Nadler","sequence":"first","affiliation":[]}],"container-title":"IRE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1962,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1962.1057670","ISSN":["0096-1000","2168-2712"],"container-title-short":"IRE Trans. Inf. Theory"},{"id":"doi:10.1016/S0019-9958(71)90295-6","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0019-9958(71)90295-6","page":"40-64","source":"Crossref","title":"Coding theory and the Mathieu groups","volume":"18","author":[{"given":"E.R.","family":"Berlekamp","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1971,2]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(71)90295-6","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995871902956"},{"id":"manual:-S.-L.-Snover-(1973).-THE-UNI","type":"book","author":[{"family":"Snover","given":"S.L."}],"title":"THE UNIQUENESS OF THE NORDSTROM-ROBINSON AND THE GOLAY BINARY-CODES","publisher":"Michigan State University","issued":"1973"},{"id":"doi:10.1016/S0019-9958(67)90835-2","type":"journal-article","publisher":"Elsevier BV","issue":"5-6","DOI":"10.1016/s0019-9958(67)90835-2","page":"613-616","source":"Crossref","title":"An optimum nonlinear code","volume":"11","author":[{"given":"Alan W.","family":"Nordstrom","sequence":"first","affiliation":[]},{"given":"John P.","family":"Robinson","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1967,11]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(67)90835-2","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995867908352"},{"id":"manual:-N.-V.-Semakov-and-V.-A.-Zino","type":"article-journal","author":[{"family":"Semakov","given":"N.V."},{"family":"Zinovev","given":"V.A."}],"title":"Complete and Quasi-complete Balanced Codes","volume":"5","container-title":"Problemy Peredachi Informatsii","issue":"2","issued":"1969","page":"14–18"},{"id":"manual:-G.-D.-Forney-Jr.-N.-J.-A.-Sl","type":"chapter","author":[{"family":"Forney","given":"G.D.","suffix":"Jr."},{"family":"Sloane","given":"N.J.A."},{"family":"Trott","given":"M.D."}],"title":"The Nordstrom-Robinson code is the binary image of the octacode","container-title":"Coding and Quantization: DIMACS/IEEE workshop 1992 Oct 19","publisher":"American Mathematical Society","page":"19–26"},{"id":"doi:10.1109/TIT.2021.3114636","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2021.3114636","page":"7769-7781","source":"Crossref","title":"The Geometry of Two-Weight Codes Over ℤ<sub><i>p</i></sub><sup><i>m</i></sup>","volume":"67","author":[{"ORCID":"https://orcid.org/0000-0002-4990-6271","authenticated-orcid":false,"given":"Minjia","family":"Shi","sequence":"first","affiliation":[]},{"given":"Thomas","family":"Honold","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Sole","sequence":"additional","affiliation":[]},{"given":"Yunzhen","family":"Qiu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2361-5126","authenticated-orcid":false,"given":"Rongsheng","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Zahra","family":"Sepasdar","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2021,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3114636","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-V.-I.-Levenshtein-Applicatio","type":"article-journal","author":[{"family":"Levenshtein","given":"V.I."}],"title":"Application of Hadamard matrices to a problem in coding theory","volume":"5","publisher":"GIFML","container-title":"Problems of Cybernetics","issued":"1961","page":"125–136","publisher-place":"Moscow"},{"id":"doi:10.1109/18.333896","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.333896","page":"1693-1698","source":"Crossref","title":"The uniqueness of the Best code","volume":"40","author":[{"given":"S.","family":"Litsyn","sequence":"first","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1994]]},"URL":"http://dx.doi.org/10.1109/18.333896","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Best78","type":"report","author":[{"family":"Best","given":"M.R."}],"title":"Binary codes with minimum distance four","genre":"Report ZW 112/78,","publisher":"Math Centrum","issued":"1978","publisher-place":"Amsterdam"},{"id":"doi:10.1109/TIT.1980.1056269","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1980.1056269","page":"738-742","source":"Crossref","title":"Binary codes with a minimum distance of four (Corresp.)","volume":"26","author":[{"given":"M.","family":"Best","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1980,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1980.1056269","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-J.-A.-Barrau-On-the-combinat","type":"article-journal","author":[{"family":"Barrau","given":"J.A."}],"title":"On the combinatory problem of Steiner, Proc. Section of Sciences","volume":"11","container-title":"Koninklijke Akademie van Wetenschappen te Amsterdam","issued":"1908","page":"352–360"},{"id":"doi:10.4153/CJM-1964-065-1","type":"journal-article","publisher":"Canadian Mathematical Society","abstract":"<jats:p>This paper is concerned with the packing of equal spheres in Euclidean spaces [<jats:italic>n</jats:italic>] of <jats:italic>n</jats:italic> &gt; 8 dimensions. To be precise, a <jats:italic>packing</jats:italic> is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a <jats:italic>lattice packing</jats:italic>. The densest lattice packings are known for spaces of up to eight dimensions <jats:bold>(1, 2)</jats:bold>, but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.</jats:p>","DOI":"10.4153/cjm-1964-065-1","page":"657-682","source":"Crossref","title":"Some Sphere Packings in Higher Space","volume":"16","author":[{"given":"John","family":"Leech","sequence":"first","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1964]]},"URL":"http://dx.doi.org/10.4153/CJM-1964-065-1","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j. math.","note":"alternative-id:S0008414X00035744"},{"id":"doi:10.1109/TIT.1954.1057463","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1954.1057463","page":"23-28","source":"Crossref","title":"Binary coding","volume":"4","author":[{"given":"M.","family":"Golay","sequence":"first","affiliation":[]}],"container-title":"Transactions of the IRE Professional Group on Information Theory","original-title":[],"issued":{"date-parts":[[1954,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1954.1057463","ISSN":["2168-2690","2168-2704"],"container-title-short":"Trans. IRE Prof. Group Inf. Theory"},{"id":"doi:10.1109/TIT.1965.1053779","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1965.1053779","page":"459-459","source":"Crossref","title":"Two improved block codes (Corresp.)","volume":"11","author":[{"given":"D.","family":"Julin","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1965,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1965.1053779","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1964.1053689","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1964.1053689","page":"363-377","source":"Crossref","title":"Nonrandom binary superimposed codes","volume":"10","author":[{"given":"W.","family":"Kautz","sequence":"first","affiliation":[]},{"given":"R.","family":"Singleton","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1964,10]]},"URL":"http://dx.doi.org/10.1109/TIT.1964.1053689","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2303.17524","type":"article-journal","author":[{"given":"Ruizhong","family":"Wei"}],"title":"On Cover-Free Families","issued":{"date-parts":[[2023,3,30]]},"note":"arxivid:2303.17524\narxiv_version_number:1"},{"id":"manual:-C.-N.-Mooers-Application-of-","type":"thesis","author":[{"family":"Mooers","given":"C.N."}],"title":"Application of random codes to the gathering of statistical information","genre":"PhD thesis,","publisher":"Massachusetts Institute of Technology","issued":"1948"},{"id":"doi:10.1108/eb026235","type":"journal-article","publisher":"Emerald","issue":"4","abstract":"<jats:p>We deal here with some problems of superimposed random coding that arise when marking codes into the codefield. Our investigations are concerned with a general model. Systems used in practice are special cases of this, and may be deduced from it by making appropriate simplifying assumptions. We give exact formulas for the distribution of the marked sites in a codefield and for the distribution of multiple marking of a site. These results are obtained by applying a general theorem in probability due to K. Jordan. Some acquaintance with superimposed random coding and knowledge of basic combinatorial and probability theory are assumed.</jats:p>","DOI":"10.1108/eb026235","page":"231-234","source":"Crossref","title":"SOME PROBABILITY PROBLEMS CONCERNING THE MARKING OF CODES INTO THE SUPERIMPOSITION FIELD","volume":"12","author":[{"given":"G.","family":"OROSZ","sequence":"first","affiliation":[]},{"given":"L.","family":"TAKÁCS","sequence":"additional","affiliation":[]}],"container-title":"Journal of Documentation","original-title":[],"language":"en","issued":{"date-parts":[[1956,4,1]]},"URL":"http://dx.doi.org/10.1108/eb026235","ISSN":["0022-0418"],"note":"alternative-id:10.1108/eb026235"},{"id":"doi:10.1002/asi.5090110209","type":"journal-article","publisher":"Wiley","issue":"2","abstract":"<jats:title>Abstract</jats:title><jats:p>The chain‐spelling and random‐number superimposed coding methods and their variants are described. It is shown that for a given field and a given number of words to be punched therein, the above methods are equivalent as far as the probability of a match being false is concerned. For the above methods the optimum number of punches per word is 0.693 x (field size) ‐ (number of words punched) and does not depend on the vocabulary size. The staggered superimposed coding method is also described and analyzed. Charts giving the probabilities of a match being false for various vocabulary sizes, field sizes, and numbers of words punched are provided. Tables extending these figures to multi‐word inquiries are included.</jats:p>","DOI":"10.1002/asi.5090110209","page":"155-169","source":"Crossref","title":"Mathematical analysis of various superimposed coding methods","volume":"11","author":[{"given":"Simon","family":"Stiassny","sequence":"first","affiliation":[]}],"container-title":"American Documentation","original-title":[],"language":"en","issued":{"date-parts":[[1960,4]]},"URL":"http://dx.doi.org/10.1002/asi.5090110209","ISSN":["0096-946X","1936-6108"],"container-title-short":"Amer. Doc.","note":"alternative-id:10.1002/asi.5090110209"},{"id":"doi:10.1145/4078.4080","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"1","abstract":"<jats:p>This paper compares text retrieval methods intended for office systems. The operational requirements of the office environment are discussed, and retrieval methods from database systems and from information retrieval systems are examined. We classify these methods and examine the most interesting representatives of each class. Attempts to speed up retrieval with special purpose hardware are also presented, and issues such as approximate string matching and compression are discussed. A qualitative comparison of the examined methods is presented. The signature file method is discussed in more detail.</jats:p>","DOI":"10.1145/4078.4080","page":"49-74","source":"Crossref","title":"Access methods for text","volume":"17","author":[{"given":"Chris","family":"Faloutsos","sequence":"first","affiliation":[{"name":"Univ. of Toronto, Toronto, Ont., Canada"}]}],"container-title":"ACM Computing Surveys","original-title":[],"language":"en","issued":{"date-parts":[[1985,3]]},"URL":"http://dx.doi.org/10.1145/4078.4080","ISSN":["0360-0300","1557-7341"],"container-title-short":"ACM Comput. Surv.","note":"alternative-id:10.1145/4078.4080"},{"id":"doi:10.1109/SFCS.1997.646101","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/sfcs.1997.646101","page":"127-136","source":"Crossref","title":"Deterministic superimposed coding with applications to pattern matching","author":[{"given":"P.","family":"Indyk","sequence":"first","affiliation":[]}],"event":"38th Annual Symposium on Foundations of Computer Science","container-title":"Proceedings 38th Annual Symposium on Foundations of Computer Science","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.1997.646101"},{"id":"doi:10.18434/T4/1422485","type":"dataset","language":"en","author":[{"literal":"Vadim Okun"}],"contributor":[{"literal":"Vadim Okun"}],"issued":{"date-parts":[[2024]]},"abstract":"The Dictionary of Algorithms and Data Structures (DADS) is an online, publicly\n            accessible dictionary of generally useful algorithms, data structures, algorithmic\n            techniques, archetypal problems, and related definitions. In addition to brief\n            definitions, some entries have links to related entries, links to implementations, and\n            additional information. DADS is meant to be a resource for the practicing programmer,\n            although students and researchers may find it a useful starting point. DADS has\n            fundamental entries in areas such as theory, cryptography and compression, graphs,\n            trees, and searching, for instance, Ackermann's function, quick sort, traveling\n            salesman, big O notation, merge sort, AVL tree, hash table, and Byzantine generals. DADS\n            also has index pages that list entries by area and by type. Currently DADS does not\n            include algorithms particular to business data processing, communications, operating\n            systems or distributed algorithms, programming languages, AI, graphics, or numerical\n            analysis.","DOI":"10.18434/T4/1422485","publisher":"National Institute of Standards and Technology","title":"Dictionary of Algorithms and Data Structures (DADS)","URL":"https://data.nist.gov/od/id/FDB59097469D5200E043065706813E54165","version":"1.0.5"},{"id":"doi:10.1007/s11122-005-0015-5","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s11122-005-0015-5","page":"105-112","source":"Crossref","title":"On the Structure of Symmetry Groups of Vasil’ev Codes","volume":"41","author":[{"given":"S. V.","family":"Avgustinovich","sequence":"first","affiliation":[]},{"given":"F. I.","family":"Solov’eva","sequence":"additional","affiliation":[]},{"given":"O.","family":"Heden","sequence":"additional","affiliation":[]}],"container-title":"Problems of Information Transmission","original-title":[],"language":"en","issued":{"date-parts":[[2005,4]]},"URL":"http://dx.doi.org/10.1007/s11122-005-0015-5","ISSN":["0032-9460","1608-3253"],"container-title-short":"Probl Inf Transm","note":"alternative-id:15"},{"id":"manual:-G.-M.-Tenengolts-Class-of-co","type":"article-journal","author":[{"family":"Tenengolts","given":"G.M."}],"title":"Class of codes correcting bit loss and errors in the preceding bit","translator":[{"family":"English","particle":"to"}],"volume":"37","container-title":"Automation and Remote Control","issue":"5","issued":"1976","page":"797–802"},{"id":"arxiv:math/0207197","type":"article-journal","author":[{"given":"N. J. A.","family":"Sloane"}],"title":"On Single-Deletion-Correcting Codes","issued":{"date-parts":[[2002,7,22]]},"note":"arxivid:math/0207197\narxiv_version_number:1"},{"id":"arxiv:1708.04071","type":"article-journal","author":[{"given":"Mahed","family":"Abroshan"},{"given":"Ramji","family":"Venkataramanan"},{"given":"Albert Guillen i","family":"Fabregas"}],"title":"Efficient Systematic Encoding of Non-binary VT Codes","issued":{"date-parts":[[2018,4,27]]},"note":"arxivid:1708.04071\narxiv_version_number:3"},{"id":"doi:10.1109/TIT.1984.1056962","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1984.1056962","page":"766-769","source":"Crossref","title":"Nonbinary codes, correcting single deletion or insertion (Corresp.)","volume":"30","author":[{"given":"G.","family":"Tenengolts","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1984,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1984.1056962","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1906.07887","type":"article-journal","author":[{"given":"Kedar","family":"Tatwawadi"},{"given":"Shubham","family":"Chandak"}],"title":"Tutorial on algebraic deletion correction codes","issued":{"date-parts":[[2019,6,19]]},"note":"arxivid:1906.07887\narxiv_version_number:1"},{"id":"manual:-R.-R.-Varshamov-and-G.-M.-Te","type":"article-journal","author":[{"family":"Varshamov","given":"R.R."},{"family":"Tenengolts","given":"G.M."}],"title":"Codes which correct single asymmetric errors","translator":[{"family":"English","particle":"to"}],"volume":"26","container-title":"Automation and Remote Control","issue":"2","issued":"1965","page":"286–290"},{"id":"manual:-V.-I.-Levenshtein-Binary-cod","type":"article-journal","author":[{"family":"Levenshtein","given":"V.I."}],"title":"Binary codes capable of correcting deletions, insertions and reversals","translator":[{"family":"English","particle":"to"}],"volume":"10","container-title":"Soviet Physics Doklady","issue":"8","issued":"1966","page":"707–710"},{"id":"manual:-V.-I.-Levenshtein-Binary-cod","type":"article-journal","author":[{"family":"Levenshtein","given":"V.I."}],"title":"Binary codes capable of correcting spurious insertions and deletions of one","translator":[{"family":"English","particle":"to"}],"volume":"1","container-title":"Problems of Information Transmission","issue":"1","issued":"1965","page":"8–17"},{"id":"arxiv:1302.6248","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevb.88.125122","source":"Crossref","title":"Exotic topological order in fractal spin liquids","volume":"88","author":[{"given":"Beni","family":"Yoshida","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2013,9,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.88.125122","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"125122","note":"arxivid:1302.6248"},{"id":"arxiv:2002.11738","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2021.3068359","page":"4504-4516","source":"Crossref","title":"Correcting Spanning Errors With a Fractal Code","volume":"67","author":[{"ORCID":"https://orcid.org/0000-0002-7816-6292","authenticated-orcid":false,"given":"Georgia M.","family":"Nixon","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":false,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2021,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3068359","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2002.11738"},{"id":"arxiv:2405.19293","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>\n                    We show in this paper that a strong and easy connection exists between quantum error correction and Lattice Gauge Theories (LGT) by using the Gauge symmetry to construct an efficient error-correcting code for Abelian\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                        <mml:msub>\n                          <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n                          <mml:mn mathvariant=\"double-struck\">2</mml:mn>\n                        </mml:msub>\n                      </mml:mrow>\n                    </mml:math>\n                    LGTs. We identify the logical operations on this gauge covariant code and show that the corresponding Hamiltonian can be expressed in terms of these logical operations while preserving the locality of the interactions. Furthermore, we demonstrate that these substitutions actually give a new way of writing the LGT as an equivalent hardcore boson model. Finally we demonstrate a method to perform fault-tolerant time evolution of the Hamiltonian within the gauge covariant code using both product formulas and qubitization approaches. This opens up the possibility of inexpensive end to end dynamical simulations that save physical qubits by blurring the lines between simulation algorithms and quantum error correcting codes.\n                  </jats:p>","DOI":"10.22331/q-2026-01-16-1968","page":"1968","source":"Crossref","title":"Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes","volume":"10","author":[{"given":"L.","family":"Spagnoli","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, University of Trento, via Sommarive 14, I–38123, Povo, Trento, Italy"},{"name":"INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento, Italy"}]},{"given":"A.","family":"Roggero","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, University of Trento, via Sommarive 14, I–38123, Povo, Trento, Italy"},{"name":"INFN-TIFPA Trento Institute of Fundamental Physics and Applications, Trento, Italy"}]},{"given":"N.","family":"Wiebe","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Toronto, Toronto, ON M5S 2E4, Canada"},{"name":"Pacific Northwest National Laboratory, Richland, WA 99354, USA"},{"name":"Department of Physics, University of Washington, Seattle, WA 98195, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,16]]},"URL":"http://dx.doi.org/10.22331/q-2026-01-16-1968","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2405.19293"},{"id":"arxiv:2112.05186","type":"article-journal","author":[{"given":"Abhishek","family":"Rajput"},{"given":"Alessandro","family":"Roggero"},{"given":"Nathan","family":"Wiebe"}],"title":"Quantum Error Correction with Gauge Symmetries","issued":{"date-parts":[[2022,11,18]]},"note":"arxivid:2112.05186\narxiv_version_number:2"},{"id":"arxiv:2412.15317","type":"article-journal","author":[{"given":"Sylvain","family":"Carrozza"},{"given":"Aidan","family":"Chatwin-Davies"},{"given":"Philipp A.","family":"Hoehn"},{"given":"Fabio M.","family":"Mele"}],"title":"A correspondence between quantum error correcting codes and quantum reference frames","issued":{"date-parts":[[2025,10,2]]},"note":"arxivid:2412.15317\narxiv_version_number:2"},{"id":"arxiv:2511.13721","type":"article-journal","author":[{"given":"Xiaojun","family":"Yao"}],"title":"Quantum Error Correction Codes for Truncated SU(2) Lattice Gauge Theories","issued":{"date-parts":[[2025,11,17]]},"note":"arxivid:2511.13721\narxiv_version_number:1"},{"id":"doi:10.1109/12.286310","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/12.286310","page":"759-764","source":"Crossref","title":"Design of CAECC - cellular automata based error correcting code","volume":"43","author":[{"given":"D.R.","family":"Chowdhury","sequence":"first","affiliation":[]},{"given":"S.","family":"Basu","sequence":"additional","affiliation":[]},{"given":"I.S.","family":"Gupta","sequence":"additional","affiliation":[]},{"given":"P.P.","family":"Chaudhuri","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Computers","original-title":[],"issued":{"date-parts":[[1994,6]]},"URL":"http://dx.doi.org/10.1109/12.286310","ISSN":["0018-9340"],"container-title-short":"IEEE Trans. Comput."},{"id":"arxiv:cond-mat/9707273","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreve.60.5068","page":"5068-5072","source":"Crossref","title":"Glassy dynamics and aging in an exactly solvable spin model","volume":"60","author":[{"given":"M. E. J.","family":"Newman","sequence":"first","affiliation":[]},{"given":"Cristopher","family":"Moore","sequence":"additional","affiliation":[]}],"container-title":"Physical Review E","original-title":[],"language":"en","issued":{"date-parts":[[1999,11,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevE.60.5068","ISSN":["1063-651X","1095-3787"],"container-title-short":"Phys. Rev. E","note":"arxivid:cond-mat/9707273"},{"id":"arxiv:2402.16831","type":"article-journal","author":[{"given":"Tibor","family":"Rakovszky"},{"given":"Vedika","family":"Khemani"}],"title":"The Physics of (good) LDPC Codes II. Product constructions","issued":{"date-parts":[[2024,2,26]]},"note":"arxivid:2402.16831\narxiv_version_number:1"},{"id":"doi:10.1007/s002220050221","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s002220050221","page":"179-188","source":"Crossref","title":"Quaquaversal tilings and rotations","volume":"132","author":[{"given":"John H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"Charles","family":"Radin","sequence":"additional","affiliation":[]}],"container-title":"Inventiones Mathematicae","original-title":[],"issued":{"date-parts":[[1998,4,14]]},"URL":"http://dx.doi.org/10.1007/s002220050221","ISSN":["0020-9910","1432-1297"],"container-title-short":"Inventiones Mathematicae","note":"alternative-id:GUB2P0HLNMM4VQ9L"},{"id":"manual:-J.-M.-Ziman.-Models-of-disor","type":"document","author":[{"family":"Ziman","given":"J.M."}],"title":"Models of disorder: the theoretical physics of homogeneously disordered systems","note":"Cambridge university press, 1979"},{"id":"arxiv:cond-mat/0309717","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevb.69.104511","source":"Crossref","title":"Geometric effects on<i>T</i>-breaking in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:math>and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>d</mml:mi></mml:math>superconducting arrays","volume":"69","author":[{"given":"J. E.","family":"Moore","sequence":"first","affiliation":[]},{"given":"D.-H.","family":"Lee","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2004,3,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.69.104511","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"104511","note":"arxivid:cond-mat/0309717"},{"id":"arxiv:cond-mat/0312587","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.93.047003","source":"Crossref","title":"Strong-Weak Coupling Self-Duality in the Two-Dimensional Quantum Phase Transition of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:math>Superconducting Arrays","volume":"93","author":[{"given":"Cenke","family":"Xu","sequence":"first","affiliation":[]},{"given":"J. E.","family":"Moore","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2004,7,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.93.047003","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"047003","note":"arxivid:cond-mat/0312587"},{"id":"arxiv:2410.16250","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that\n                    <jats:italic>cohomology invariants</jats:italic>\n                    naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining\n                    <jats:italic>cup products</jats:italic>\n                    on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\Lambda $$</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                          <mml:mi>Λ</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    -fold cup product that can produce a logical operator in the\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\Lambda $$</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                          <mml:mi>Λ</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    -th level of the Clifford hierarchy on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\Lambda $$</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                          <mml:mi>Λ</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    copies of the same quantum code, which we call the\n                    <jats:italic>copy-cup gate</jats:italic>\n                    . For any desired\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\Lambda $$</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                          <mml:mi>Λ</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    , we can construct several families of quantum codes that support gates in the\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\Lambda $$</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                          <mml:mi>Λ</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    -th level with various asymptotic code parameters.\n                  </jats:p>","DOI":"10.1007/s00220-026-05570-z","source":"Crossref","title":"Cups and Gates I: Cohomology Invariants and Logical Quantum Operations","volume":"407","author":[{"ORCID":"https://orcid.org/0000-0002-7211-5515","authenticated-orcid":false,"given":"Nikolas P.","family":"Breuckmann","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8482-5832","authenticated-orcid":false,"given":"Margarita","family":"Davydova","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0577-9159","authenticated-orcid":false,"given":"Jens N.","family":"Eberhardt","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":false,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,4]]},"URL":"http://dx.doi.org/10.1007/s00220-026-05570-z","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","page":"86","note":"alternative-id:5570\narxivid:2410.16250"},{"id":"arxiv:1612.00060","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1140/epjst/e2016-60329-4","page":"749-764","source":"Crossref","title":"Plaquette Ising models, degeneracy and scaling","volume":"226","author":[{"ORCID":"https://orcid.org/0000-0003-0556-3200","authenticated-orcid":false,"given":"Desmond A.","family":"Johnston","sequence":"first","affiliation":[]},{"given":"Marco","family":"Mueller","sequence":"additional","affiliation":[]},{"given":"Wolfhard","family":"Janke","sequence":"additional","affiliation":[]}],"container-title":"The European Physical Journal Special Topics","original-title":[],"language":"en","issued":{"date-parts":[[2017,4]]},"URL":"http://dx.doi.org/10.1140/epjst/e2016-60329-4","ISSN":["1951-6355","1951-6401"],"container-title-short":"Eur. Phys. J. Spec. Top.","note":"alternative-id:2683\narxivid:1612.00060"},{"id":"arxiv:2112.12735","type":"journal-article","publisher":"Stichting SciPost","issue":"1","abstract":"<jats:p>Subsystem symmetry has emerged as a powerful organizing principle for unconventional quantum phases of matter, most prominently fracton topological orders. Here, we focus on a special subclass of such symmetries, known as higher-form subsystem symmetries, which allow us to adapt tools from the study of conventional topological phases to the fracton setting. We demonstrate that certain transitions out of familiar fracton phases, including the X-cube model, can be understood in terms of the spontaneous breaking of higher-form subsystem symmetries. We find simple pictures for these seemingly complicated fracton topological phase transitions by relating them in an exact manner, via gauging, to spontaneous higher-form subsystem symmetry breaking phase transitions of decoupled stacks of lower-dimensional models. We harness this perspective to construct a sequence of unconventional subdimensional critical points in two and three spatial dimensions based on the stacking and gauging of canonical models with higher-form symmetry. Through numerous examples, we illustrate the ubiquity of coupled layer constructions in theories with higher-form subsystem symmetries.</jats:p>","DOI":"10.21468/scipostphys.15.1.017","source":"Crossref","title":"Higher-form subsystem symmetry breaking: Subdimensional criticality and fracton phase transitions","volume":"15","author":[{"given":"Brandon","family":"Rayhaun","sequence":"first","affiliation":[{"name":"Stanford University"}]},{"given":"Dominic","family":"Williamson","sequence":"additional","affiliation":[{"name":"University of Sydney"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2023,7,19]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.15.1.017","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"017","note":"arxivid:2112.12735"},{"id":"arxiv:2505.13604","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/qq9n-16hk","source":"Crossref","title":"String membrane nets from higher-form gauging: An alternate route to \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>p</mml:mi></mml:math>\n-string condensation","volume":"112","author":[{"ORCID":"https://orcid.org/0000-0002-5926-1059","authenticated-orcid":true,"given":"Pranay","family":"Gorantla","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"},{"id":[{"id":"https://ror.org/00hx57361","id-type":"ROR","asserted-by":"publisher"}],"name":"Princeton University"}]},{"ORCID":"https://orcid.org/0000-0003-4438-7107","authenticated-orcid":true,"given":"Abhinav","family":"Prem","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f809463","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Advanced Study"},{"id":[{"id":"https://ror.org/04yrgt058","id-type":"ROR","asserted-by":"publisher"}],"name":"Bard College"}]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":true,"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Sydney"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,11]]},"URL":"http://dx.doi.org/10.1103/qq9n-16hk","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"125124","note":"arxivid:2505.13604"},{"id":"arxiv:1202.3120","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevb.86.115109","source":"Crossref","title":"Braiding statistics approach to symmetry-protected topological phases","volume":"86","author":[{"given":"Michael","family":"Levin","sequence":"first","affiliation":[]},{"given":"Zheng-Cheng","family":"Gu","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2012,9,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.86.115109","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"115109","note":"arxivid:1202.3120"},{"id":"arxiv:1407.1025","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevx.5.011024","source":"Crossref","title":"Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems","volume":"5","author":[{"given":"Jutho","family":"Haegeman","sequence":"first","affiliation":[]},{"given":"Karel","family":"Van Acoleyen","sequence":"additional","affiliation":[]},{"given":"Norbert","family":"Schuch","sequence":"additional","affiliation":[]},{"given":"J. Ignacio","family":"Cirac","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Verstraete","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2015,2,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.5.011024","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011024","note":"arxivid:1407.1025"},{"id":"arxiv:1603.04442","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevb.94.235157","source":"Crossref","title":"Fracton topological order, generalized lattice gauge theory, and duality","volume":"94","author":[{"given":"Sagar","family":"Vijay","sequence":"first","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Fu","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2016,12,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.94.235157","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"235157","note":"arxivid:1603.04442"},{"id":"arxiv:1603.05182","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.94.155128","source":"Crossref","title":"Fractal symmetries: Ungauging the cubic code","volume":"94","author":[{"given":"Dominic J.","family":"Williamson","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2016,10,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.94.155128","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155128","note":"arxivid:1603.05182"},{"id":"arxiv:1605.01640","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/jhep04(2017)096","source":"Crossref","title":"State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter","volume":"2017","author":[{"given":"Lakshya","family":"Bhardwaj","sequence":"first","affiliation":[]},{"given":"Davide","family":"Gaiotto","sequence":"additional","affiliation":[]},{"given":"Anton","family":"Kapustin","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,4]]},"URL":"http://dx.doi.org/10.1007/JHEP04(2017)096","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"96","note":"alternative-id:5882\narxivid:1605.01640"},{"id":"arxiv:1805.01836","type":"article-journal","author":[{"given":"Aleksander","family":"Kubica"},{"given":"Beni","family":"Yoshida"}],"title":"Ungauging quantum error-correcting codes","issued":{"date-parts":[[2018,5,4]]},"note":"arxivid:1805.01836\narxiv_version_number:1"},{"id":"arxiv:1806.08679","type":"journal-article","publisher":"Stichting SciPost","issue":"4","abstract":"<jats:p>Based on several previous examples, we summarize explicitly the\ngeneral procedure to gauge models with subsystem symmetries, which are\nsymmetries with generators that have support within a sub-manifold of\nthe system. The gauging process can be applied to any local quantum\nmodel on a lattice that is invariant under the subsystem symmetry. We\nfocus primarily on simple 3D paramagnetic states with planar symmetries.\nFor these systems, the gauged theory may exhibit foliated fracton order\nand we find that the species of symmetry charges in the paramagnet\ndirectly determine the resulting foliated fracton order. Moreover, we\nfind that gauging linear subsystem symmetries in 2D or 3D models results\nin a self-duality similar to gauging global symmetries in 1D.</jats:p>","DOI":"10.21468/scipostphys.6.4.041","source":"Crossref","title":"Foliated fracton order from gauging subsystem symmetries","volume":"6","author":[{"given":"Wilbur","family":"Shirley","sequence":"first","affiliation":[{"name":"California Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-8036-3447","authenticated-orcid":false,"given":"Kevin","family":"Slagle","sequence":"additional","affiliation":[{"name":"University of Toronto"},{"name":"California Institute of Technology"}]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[{"name":"California Institute of Technology"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2019,4,2]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.6.4.041","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"041","note":"arxivid:1806.08679"},{"id":"arxiv:2108.11402","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>Gauging is a general procedure for mapping a quantum many-body system with a global symmetry to one with a local gauge symmetry. We consider a generalized gauging map that does not enforce gauge symmetry at all lattice sites, and show that it is an isometry on the full input space including all charged sectors. We apply this generalized gauging map to convert global-symmetric bulk systems of holographic codes to gauge-symmetric bulk systems, and vice versa, while preserving duality with a global-symmetric boundary. We separately construct holographic codes with gauge-symmetric bulk systems by directly imposing gauge-invariance constraints onto existing holographic codes, and show that the resulting bulk gauge symmetries are dual to boundary global symmetries. Combining these ideas produces a toy model that captures several interesting features of holography — it exhibits a rudimentary sort of dynamical duality, can be modified to demonstrate the relationship between metric fluctuations and approximate error-correction, and serves as an illustration for certain no-go theorems concerning symmetries in holography. Finally, we apply the generalized gauging map to construct codes with arbitrary transversal gate sets — for any compact Lie group, we use a symmetry-preserving truncation scheme to construct covariant finite-dimensional approximate holographic codes.</jats:p>","DOI":"10.1007/jhep05(2022)158","source":"Crossref","title":"Gauging the bulk: generalized gauging maps and holographic codes","volume":"2022","author":[{"ORCID":"https://orcid.org/0000-0003-0143-1622","authenticated-orcid":false,"given":"Kfir","family":"Dolev","sequence":"first","affiliation":[]},{"given":"Vladimir","family":"Calvera","sequence":"additional","affiliation":[]},{"given":"Samuel S.","family":"Cree","sequence":"additional","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,24]]},"URL":"http://dx.doi.org/10.1007/JHEP05(2022)158","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"158","note":"alternative-id:18446\narxivid:2108.11402"},{"id":"arxiv:2310.16032","type":"article-journal","author":[{"given":"Tibor","family":"Rakovszky"},{"given":"Vedika","family":"Khemani"}],"title":"The Physics of (good) LDPC Codes I. Gauging and dualities","issued":{"date-parts":[[2023,10,24]]},"note":"arxivid:2310.16032\narxiv_version_number:1"},{"id":"arxiv:2410.02213","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1038/s41567-026-03220-8","page":"598-603","source":"Crossref","title":"Low-overhead fault-tolerant quantum computation by gauging logical operators","volume":"22","author":[{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":false,"given":"Dominic J.","family":"Williamson","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9614-2836","authenticated-orcid":false,"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,4]]},"URL":"http://dx.doi.org/10.1038/s41567-026-03220-8","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:3220\narxivid:2410.02213"},{"id":"arxiv:1511.07867","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevb.93.205112","source":"Crossref","title":"Robust topological degeneracy of classical theories","volume":"93","author":[{"given":"Mohammad-Sadegh","family":"Vaezi","sequence":"first","affiliation":[]},{"given":"Gerardo","family":"Ortiz","sequence":"additional","affiliation":[]},{"given":"Zohar","family":"Nussinov","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2016,5,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.93.205112","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"205112","note":"arxivid:1511.07867"},{"id":"doi:10.1109/ACCESS.2017.2784417","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/access.2017.2784417","page":"13729-13757","source":"Crossref","title":"Quantum Topological Error Correction Codes: The Classical-to-Quantum Isomorphism Perspective","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-2406-7229","authenticated-orcid":false,"given":"Daryus","family":"Chandra","sequence":"first","affiliation":[{"name":"School of Electronics and Computer Science, University of Southampton, Southampton, U.K."}]},{"given":"Zunaira","family":"Babar","sequence":"additional","affiliation":[{"name":"School of Electronics and Computer Science, University of Southampton, Southampton, U.K."}]},{"ORCID":"https://orcid.org/0000-0001-6349-1044","authenticated-orcid":false,"given":"Hung Viet","family":"Nguyen","sequence":"additional","affiliation":[{"name":"School of Electronics and Computer Science, University of Southampton, Southampton, U.K."}]},{"ORCID":"https://orcid.org/0000-0002-6654-1702","authenticated-orcid":false,"given":"Dimitrios","family":"Alanis","sequence":"additional","affiliation":[{"name":"School of Electronics and Computer Science, University of Southampton, Southampton, U.K."}]},{"given":"Panagiotis","family":"Botsinis","sequence":"additional","affiliation":[{"name":"School of Electronics and Computer Science, University of Southampton, Southampton, U.K."}]},{"given":"Soon Xin","family":"Ng","sequence":"additional","affiliation":[{"name":"School of Electronics and Computer Science, University of Southampton, Southampton, U.K."}]},{"ORCID":"https://orcid.org/0000-0002-2636-5214","authenticated-orcid":false,"given":"Lajos","family":"Hanzo","sequence":"additional","affiliation":[{"name":"School of Electronics and Computer Science, University of Southampton, Southampton, U.K."}]}],"container-title":"IEEE Access","original-title":[],"issued":{"date-parts":[[2018]]},"URL":"http://dx.doi.org/10.1109/ACCESS.2017.2784417","ISSN":["2169-3536"],"container-title-short":"IEEE Access"},{"id":"arxiv:2202.09981","type":"article-journal","author":[{"given":"Lakshmi Prasad","family":"Natarajan"},{"given":"Prasad","family":"Krishnan"}],"title":"Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity","issued":{"date-parts":[[2023,7,25]]},"note":"arxivid:2202.09981\narxiv_version_number:3"},{"id":"doi:10.1007/BF01119999","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf01119999","page":"17-23","source":"Crossref","title":"Semisimple cyclic and Abelian codes. II","volume":"3","author":[{"given":"S. D.","family":"Berman","sequence":"first","affiliation":[]}],"container-title":"Cybernetics","original-title":[],"language":"en","issued":{"date-parts":[[1970]]},"URL":"http://dx.doi.org/10.1007/BF01119999","ISSN":["0011-4235","1573-8337"],"container-title-short":"Cybern Syst Anal","note":"alternative-id:BF01119999"},{"id":"doi:10.1109/18.904535","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.904535","page":"355-361","source":"Crossref","title":"On a family of Abelian codes and their state complexities","volume":"47","author":[{"given":"T.","family":"Blackmore","sequence":"first","affiliation":[]},{"given":"G.H.","family":"Norton","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.904535","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2303.16729","type":"article-journal","author":[{"given":"Minjia","family":"Shi"},{"given":"Shitao","family":"Li"},{"given":"Tor","family":"Helleseth"},{"given":"Jon-Lark","family":"Kim"}],"title":"Binary self-orthogonal codes which meet the Griesmer bound or have optimal minimum distances","issued":{"date-parts":[[2023,3,29]]},"note":"arxivid:2303.16729\narxiv_version_number:1"},{"id":"preset:Posner68","type":"chapter","author":[{"family":"Posner","given":"E.C."}],"title":"Combinatorial Structures in Planetary Reconnaissance","container-title":"Error Correcting Codes","editor":[{"family":"H. B. Mann","particle":"ed."}],"issued":"1968","publisher-place":"Wiley, NY"},{"id":"doi:10.1109/ITWKSPS.2010.5503223","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itwksps.2010.5503223","page":"1-5","source":"Crossref","title":"A survey of reed-muller codes from polar coding perspective","author":[{"given":"Erdal","family":"Arikan","sequence":"first","affiliation":[]}],"event":"2010 IEEE Information Theory Workshop on Information Theory (ITW)","container-title":"IEEE Information Theory Workshop 2010 (ITW 2010)","original-title":[],"issued":{"date-parts":[[2010,1]]},"URL":"http://dx.doi.org/10.1109/ITWKSPS.2010.5503223"},{"id":"doi:10.1109/TIT.1972.1054732","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1972.1054732","page":"203-207","source":"Crossref","title":"Weight distributions of the cosets of the (32,6) Reed-Muller code","volume":"18","author":[{"given":"E.","family":"Berlekamp","sequence":"first","affiliation":[]},{"given":"L.","family":"Welch","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1972,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1972.1054732","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.720542","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.720542","page":"2384-2415","source":"Crossref","title":"Modulation and coding for linear Gaussian channels","volume":"44","author":[{"given":"G.D.","family":"Forney","sequence":"first","affiliation":[]},{"given":"G.","family":"Ungerboeck","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.720542","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Forney03","type":"document","author":[{"family":"Forney","given":"G.D."}],"title":"6.451 Principles of Digital Communication II","issued":"2003"},{"id":"doi:10.1007/1-4020-2307-3_7","type":"book-chapter","publisher":"Kluwer Academic Publishers","DOI":"10.1007/1-4020-2307-3_7","page":"155-196","source":"Crossref","title":"Pons, Reed-Muller Codes, and Group Algebras","author":[{"given":"Myoung","family":"An","sequence":"first","affiliation":[]},{"given":"Jim","family":"Byrnes","sequence":"additional","affiliation":[]},{"given":"William","family":"Moran","sequence":"additional","affiliation":[]},{"given":"B.","family":"Saffari","sequence":"additional","affiliation":[]},{"given":"Harold S.","family":"Shapiro","sequence":"additional","affiliation":[]},{"given":"Richard","family":"Tolimieri","sequence":"additional","affiliation":[]}],"container-title":"NATO Science Series II: Mathematics, Physics and Chemistry","original-title":[],"language":"en","issued":{"date-parts":[[null]]},"ISBN":["1402019823"],"URL":"http://dx.doi.org/10.1007/1-4020-2307-3_7","note":"publisher-location:Dordrecht"},{"id":"arxiv:quant-ph/9604038","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.54.1862","page":"1862-1868","source":"Crossref","title":"Class of quantum error-correcting codes saturating the quantum Hamming bound","volume":"54","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1996,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.54.1862","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9604038"},{"id":"arxiv:quant-ph/9605021","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.54.4741","page":"4741-4751","source":"Crossref","title":"Simple quantum error-correcting codes","volume":"54","author":[{"given":"A. M.","family":"Steane","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1996,12,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.54.4741","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9605021"},{"id":"arxiv:2204.13176","type":"article-journal","author":[{"given":"Jingzhen","family":"Hu"},{"given":"Qingzhong","family":"Liang"},{"given":"Robert","family":"Calderbank"}],"title":"Divisible Codes for Quantum Computation","issued":{"date-parts":[[2022,4,27]]},"note":"arxivid:2204.13176\narxiv_version_number:1"},{"id":"arxiv:2410.23263","type":"article-journal","author":[{"given":"Anqi","family":"Gong"},{"given":"Joseph M.","family":"Renes"}],"title":"Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays","issued":{"date-parts":[[2024,10,30]]},"note":"arxivid:2410.23263\narxiv_version_number:1"},{"id":"arxiv:2410.07595","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2025.3592631","page":"415-436","source":"Crossref","title":"Geometric Structure and Transversal Logic of Quantum Reed–Muller Codes","volume":"72","author":[{"ORCID":"https://orcid.org/0000-0002-8972-4413","authenticated-orcid":false,"given":"Alexander","family":"Barg","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, Institute for Systems Research, University of Maryland, College Park, MD, USA"}]},{"ORCID":"https://orcid.org/0009-0002-1020-9986","authenticated-orcid":false,"given":"Nolan J.","family":"Coble","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science and NIST, College Park, MD, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4766-7967","authenticated-orcid":false,"given":"Dominik","family":"Hangleiter","sequence":"additional","affiliation":[{"name":"Simons Institute for the Theory of Computing, University of California at Berkeley, Berkeley, CA, USA"}]},{"given":"Christopher","family":"Kang","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Chicago, Chicago, IL, USA"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2026,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2025.3592631","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2410.07595"},{"id":"arxiv:2502.14746","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s10623-025-01749-y","source":"Crossref","title":"Coxeter codes: extending the Reed–Muller family","volume":"94","author":[{"given":"Nolan J.","family":"Coble","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2026,2]]},"URL":"http://dx.doi.org/10.1007/s10623-025-01749-y","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","page":"47","note":"alternative-id:1749\narxivid:2502.14746"},{"id":"doi:10.1214/aos/1176344370","type":"journal-article","publisher":"Institute of Mathematical Statistics","issue":"6","DOI":"10.1214/aos/1176344370","source":"Crossref","title":"Hadamard Matrices and Their Applications","volume":"6","author":[{"given":"A.","family":"Hedayat","sequence":"first","affiliation":[]},{"given":"W. D.","family":"Wallis","sequence":"additional","affiliation":[]}],"container-title":"The Annals of Statistics","original-title":[],"issued":{"date-parts":[[1978,11,1]]},"URL":"http://dx.doi.org/10.1214/aos/1176344370","ISSN":["0090-5364"],"container-title-short":"Ann. Statist."},{"id":"doi:10.1561/0400000030","type":"journal-article","publisher":"Now Publishers","issue":"3","DOI":"10.1561/0400000030","page":"139-255","source":"Crossref","title":"Locally Decodable Codes","volume":"6","author":[{"given":"Sergey","family":"Yekhanin","sequence":"first","affiliation":[]}],"container-title":"Foundations and Trends® in Theoretical Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2012]]},"URL":"http://dx.doi.org/10.1561/0400000030","ISSN":["1551-305X","1551-3068"],"container-title-short":"FNT in Theoretical Computer Science"},{"id":"preset:Gopi18","type":"thesis","author":[{"family":"Gopi","given":"S."}],"title":"Locality in coding theory","genre":"PhD thesis,","publisher":"Princeton University","issued":"2018"},{"id":"arxiv:2311.08653","type":"article-journal","author":[{"given":"Louis","family":"Golowich"},{"given":"Venkatesan","family":"Guruswami"}],"title":"Quantum Locally Recoverable Codes","issued":{"date-parts":[[2023,11,15]]},"note":"arxivid:2311.08653\narxiv_version_number:1"},{"id":"arxiv:0807.3917","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2009.2021379","page":"3051-3073","source":"Crossref","title":"Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels","volume":"55","author":[{"given":"Erdal","family":"Arikan","sequence":"first","affiliation":[{"name":"Dept. of Electr.-Electron. Eng., Bilkent Univ., Ankara"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2009,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2009.2021379","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0807.3917"},{"id":"arxiv:1401.3127","type":"article-journal","author":[{"given":"Marco","family":"Mondelli"},{"given":"S. Hamed","family":"Hassani"},{"given":"Rüdiger","family":"Urbanke"}],"title":"From Polar to Reed-Muller Codes: a Technique to Improve the Finite-Length Performance","issued":{"date-parts":[[2014,9,3]]},"note":"arxivid:1401.3127\narxiv_version_number:3"},{"id":"doi:10.1142/9283","type":"monograph","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/9283","source":"Crossref","title":"Codes from Difference Sets","author":[{"given":"Cunsheng","family":"Ding","sequence":"first","affiliation":[{"name":"The Hong Kong University of Science and TechnologyHong Kong"}]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2014,7,14]]},"ISBN":["9789814619356","9789814619363"],"URL":"http://dx.doi.org/10.1142/9283","note":"alternative-id:10.1142/9283"},{"id":"doi:10.1145/307400.307429","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/307400.307429","page":"145-155","source":"Crossref","title":"Multiclass learning, boosting, and error-correcting codes","author":[{"given":"Venkatesan","family":"Guruswami","sequence":"first","affiliation":[{"name":"MIT Laboratory for Computer Science, Cambridge, MA"}]},{"given":"Amit","family":"Sahai","sequence":"additional","affiliation":[{"name":"MIT Laboratory for Computer Science, Cambridge, MA"}]}],"event":"COLT99: The 12th Annual Conference on Computation Learning Theory","container-title":"Proceedings of the twelfth annual conference on Computational learning theory","original-title":[],"issued":{"date-parts":[[1999,7,6]]},"URL":"http://dx.doi.org/10.1145/307400.307429","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/307400.307429\n10.1145/307400"},{"id":"doi:10.1007/978-3-540-45080-1_51","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-540-45080-1_51","page":"397-404","source":"Crossref","title":"On Hadamard-Type Output Coding in Multiclass Learning","author":[{"given":"Aijun","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Zhi-Li","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Chun-Hung","family":"Li","sequence":"additional","affiliation":[]},{"given":"Kai-Tai","family":"Fang","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2003]]},"ISBN":["9783540405504","9783540450801"],"URL":"http://dx.doi.org/10.1007/978-3-540-45080-1_51","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"preset:Toom74","type":"article-journal","author":[{"family":"Toom","given":"A.L."}],"title":"Nonergodic Multidimensional System of Automata","volume":"10","container-title":"Problemy Peredachi Informatsii","issue":"3","issued":"1974","page":"70–79"},{"id":"preset:Toom80","type":"article-journal","author":[{"family":"Toom","given":"A.L."}],"title":"Stable and attractive trajectories in multicomponent systems","volume":"6","container-title":"Multicomponent random systems","issued":"1980","page":"549–575"},{"id":"doi:10.1007/978-1-4612-2168-5_18","type":"book-chapter","publisher":"Birkhäuser Boston","DOI":"10.1007/978-1-4612-2168-5_18","page":"331-353","source":"Crossref","title":"Toom’s Stability Theorem in Continuous Time","author":[{"given":"Lawrence F.","family":"Gray","sequence":"first","affiliation":[]}],"container-title":"Perplexing Problems in Probability","original-title":[],"language":"en","issued":{"date-parts":[[1999]]},"ISBN":["9781461274421","9781461221685"],"URL":"http://dx.doi.org/10.1007/978-1-4612-2168-5_18","note":"publisher-location:Boston, MA"},{"id":"doi:10.1147/rd.481.0005","type":"journal-article","publisher":"IBM","issue":"1","DOI":"10.1147/rd.481.0005","page":"5-12","source":"Crossref","title":"Can complex structures be generically stable in a noisy world?","volume":"48","author":[{"given":"G.","family":"Grinstein","sequence":"first","affiliation":[{"name":"IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598, USA"}]}],"container-title":"IBM Journal of Research and Development","original-title":[],"issued":{"date-parts":[[2004,1]]},"URL":"http://dx.doi.org/10.1147/rd.481.0005","ISSN":["0018-8646","0018-8646"],"container-title-short":"IBM J. Res. &amp; Dev."},{"id":"doi:10.1145/800061.808730","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/800061.808730","page":"32-41","source":"Crossref","title":"Reliable computation with cellular automata","author":[{"given":"Péter","family":"Gács","sequence":"first","affiliation":[]}],"event":"the fifteenth annual ACM symposium","container-title":"Proceedings of the fifteenth annual ACM symposium on Theory of computing  - STOC '83","original-title":[],"issued":{"date-parts":[[1983]]},"URL":"http://dx.doi.org/10.1145/800061.808730","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"doi:10.1016/0022-0000(86)90002-4","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0022-0000(86)90002-4","page":"15-78","source":"Crossref","title":"Reliable computation with cellular automata","volume":"32","author":[{"given":"Peter","family":"Gács","sequence":"first","affiliation":[]}],"container-title":"Journal of Computer and System Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1986,2]]},"URL":"http://dx.doi.org/10.1016/0022-0000(86)90002-4","ISSN":["0022-0000"],"container-title-short":"Journal of Computer and System Sciences","note":"alternative-id:0022000086900024"},{"id":"preset:Gacs89","type":"article-journal","author":[{"family":"Gács","given":"P."}],"title":"Self-Correcting Two-Dimensional Arrays","volume":"5","container-title":"Adv. Comput. Res","issued":"1989","page":"223–326"},{"id":"arxiv:math/0003117","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1-2","DOI":"10.1023/a:1004823720305","page":"45-267","source":"Crossref","title":"Reliable Cellular Automata with Self-Organization","volume":"103","author":[{"given":"Peter","family":"Gács","sequence":"first","affiliation":[]}],"container-title":"Journal of Statistical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2001,4]]},"URL":"http://dx.doi.org/10.1023/A:1004823720305","ISSN":["0022-4715","1572-9613"],"container-title-short":"Journal of Statistical Physics","note":"alternative-id:294289\narxivid:math/0003117"},{"id":"doi:10.1007/BFb0070081","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/bfb0070081","page":"15-30","source":"Crossref","title":"Reliable storage of information in a system of unreliable components with local interactions","author":[{"given":"B. S.","family":"Cirel’son","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Mathematics","original-title":[],"issued":{"date-parts":[[1978]]},"ISBN":["9783540084501","9783540370444"],"URL":"http://dx.doi.org/10.1007/BFb0070081","ISSN":["0075-8434","1617-9692"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:2408.09524","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.111.012419","source":"Crossref","title":"Enhancing quantum memory lifetime with measurement-free local error correction and reinforcement learning","volume":"111","author":[{"ORCID":"https://orcid.org/0000-0003-4415-9948","authenticated-orcid":true,"given":"Mincheol","family":"Park","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0001-5775-9542","authenticated-orcid":true,"given":"Nishad","family":"Maskara","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0003-0605-8791","authenticated-orcid":true,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.111.012419","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012419","note":"arxivid:2408.09524"},{"id":"doi:10.1147/rd.62.0200","type":"journal-article","publisher":"IBM","issue":"2","DOI":"10.1147/rd.62.0200","page":"200-209","source":"Crossref","title":"The Use of Triple-Modular Redundancy to Improve Computer Reliability","volume":"6","author":[{"given":"R. E.","family":"Lyons","sequence":"first","affiliation":[]},{"given":"W.","family":"Vanderkulk","sequence":"additional","affiliation":[]}],"container-title":"IBM Journal of Research and Development","original-title":[],"issued":{"date-parts":[[1962,4]]},"URL":"http://dx.doi.org/10.1147/rd.62.0200","ISSN":["0018-8646","0018-8646"],"container-title-short":"IBM J. Res. &amp; Dev."},{"id":"arxiv:2111.08894","type":"journal-article","publisher":"Stichting SciPost","abstract":"<jats:p>These lecture notes from the 2019 Les Houches Summer School on “Quantum Information Machines” are intended to provide an introduction to classical and quantum error correction with bits and qubits, and with continuous variable systems (harmonic oscillators). The focus on the latter will be on practical examples that can be realized today or in the near future with a modular architecture based on superconducting electrical circuits and microwave photons. The goal and vision is “hardware-efficient” quantum error correction that does not require exponentially large hardware overhead in order to achieve practical and useful levels of fault tolerance and circuit depth.</jats:p>","DOI":"10.21468/scipostphyslectnotes.70","source":"Crossref","title":"Introduction to quantum error correction and fault tolerance","author":[{"given":"Steven M.","family":"Girvin","sequence":"first","affiliation":[{"name":"Yale University"}]}],"container-title":"SciPost Physics Lecture Notes","original-title":[],"issued":{"date-parts":[[2023,6,12]]},"URL":"http://dx.doi.org/10.21468/SciPostPhysLectNotes.70","ISSN":["2590-1990"],"container-title-short":"SciPost Phys. Lect. Notes","page":"70","note":"arxivid:2111.08894"},{"id":"doi:10.1515/9781400882618-003","type":"book-chapter","publisher":"Princeton University Press","DOI":"10.1515/9781400882618-003","page":"43-98","source":"Crossref","title":"Probabilistic Logics and the Synthesis of Reliable Organisms From Unreliable Components","author":[{"given":"J. von","family":"Neumann","sequence":"first","affiliation":[]}],"container-title":"Automata Studies. (AM-34)","original-title":[],"issued":{"date-parts":[[1956,12,31]]},"ISBN":["9781400882618"],"URL":"http://dx.doi.org/10.1515/9781400882618-003","note":"alternative-id:10.1515/9781400882618-003\n10.1515/9781400882618"},{"id":"manual:-J.-von-Neumann-and-A.-W.-Bur","type":"document","author":[{"family":"Neumann","given":"J.","particle":"von"},{"family":"Burks","given":"A.W."}],"title":"Theory of self-reproducing automata","issued":"1966"},{"id":"doi:10.1109/SFCS.1985.41","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/sfcs.1985.41","page":"30-38","source":"Crossref","title":"On networks of noisy gates","author":[{"given":"Nicholas","family":"Pippenger","sequence":"first","affiliation":[]}],"event":"26th Annual Symposium on Foundations of Computer Science (sfcs 1985)","container-title":"26th Annual Symposium on Foundations of Computer Science (sfcs 1985)","original-title":[],"issued":{"date-parts":[[1985]]},"URL":"http://dx.doi.org/10.1109/SFCS.1985.41"},{"id":"doi:10.1109/18.2628","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.2628","page":"194-197","source":"Crossref","title":"Reliable computation by formulas in the presence of noise","volume":"34","author":[{"given":"N.","family":"Pippenger","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1988,3]]},"URL":"http://dx.doi.org/10.1109/18.2628","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1214/aop/1022855749","type":"journal-article","publisher":"Institute of Mathematical Statistics","issue":"3","DOI":"10.1214/aop/1022855749","source":"Crossref","title":"Unpredictable paths and percolation","volume":"26","author":[{"given":"Itai","family":"Benjamini","sequence":"first","affiliation":[]},{"given":"Robin","family":"Pemantle","sequence":"additional","affiliation":[]},{"given":"Yuval","family":"Peres","sequence":"additional","affiliation":[]}],"container-title":"The Annals of Probability","original-title":[],"issued":{"date-parts":[[1998,7,1]]},"URL":"http://dx.doi.org/10.1214/aop/1022855749","ISSN":["0091-1798"],"container-title-short":"Ann. Probab."},{"id":"doi:10.1214/aoms/1177699266","type":"journal-article","publisher":"Institute of Mathematical Statistics","issue":"5","DOI":"10.1214/aoms/1177699266","page":"1211-1223","source":"Crossref","title":"A Limit Theorem for Multidimensional Galton-Watson Processes","volume":"37","author":[{"given":"H.","family":"Kesten","sequence":"first","affiliation":[]},{"given":"B. P.","family":"Stigum","sequence":"additional","affiliation":[]}],"container-title":"The Annals of Mathematical Statistics","original-title":[],"language":"en","issued":{"date-parts":[[1966,10]]},"URL":"http://dx.doi.org/10.1214/aoms/1177699266","ISSN":["0003-4851"],"container-title-short":"Ann. Math. Statist."},{"id":"doi:10.1109/TCOM.1984.1096094","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tcom.1984.1096094","page":"511-517","source":"Crossref","title":"The Detection of Errors After Error-Correction Decoding","volume":"32","author":[{"given":"T.","family":"Klove","sequence":"first","affiliation":[]},{"given":"M.","family":"Miller","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[1984]]},"URL":"http://dx.doi.org/10.1109/TCOM.1984.1096094","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1016/C2012-0-07058-5","type":"edited-book","publisher":"Elsevier","DOI":"10.1016/c2012-0-07058-5","source":"Crossref","title":"High-Performance Embedded Computing","container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2014]]},"ISBN":["9780124105119"],"URL":"http://dx.doi.org/10.1016/C2012-0-07058-5"},{"id":"doi:10.1007/BF02124328","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf02124328","page":"1-11","source":"Crossref","title":"Bound on the mass gap for finite volume stochastic ising models at low temperature","volume":"126","author":[{"given":"Lawrence E.","family":"Thomas","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1989,11]]},"URL":"http://dx.doi.org/10.1007/BF02124328","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF02124328"},{"id":"arxiv:1411.6643","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/revmodphys.88.045005","source":"Crossref","title":"Quantum memories at finite temperature","volume":"88","author":[{"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]},{"given":"Jiannis K.","family":"Pachos","sequence":"additional","affiliation":[]},{"given":"Chris N.","family":"Self","sequence":"additional","affiliation":[]},{"given":"James R.","family":"Wootton","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2016,11,15]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.88.045005","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","page":"045005","note":"arxivid:1411.6643"},{"id":"preset:Maurice21","type":"thesis","author":[{"family":"Maurice","given":"D."}],"title":"Codes correcteurs quantiques pouvant se décoder itérativement","genre":"PhD thesis,","publisher":"Université Pierre et Marie Curie-Paris VI","issued":"2014"},{"id":"arxiv:2311.01328","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Quantum error correction is crucial for scalable quantum information-processing applications. Traditional discrete-variable quantum codes that use multiple two-level systems to encode logical information can be hardware intensive. An alternative approach is provided by bosonic codes, which use the infinite-dimensional Hilbert space of harmonic oscillators to encode quantum information. Two promising features of bosonic codes are that syndrome measurements are natively analog and that they can be concatenated with discrete-variable codes. In this work, we propose novel decoding methods that explicitly exploit the analog syndrome information obtained from the bosonic qubit readout in a concatenated architecture. Our methods are versatile and can be generally applied to any bosonic code concatenated with a quantum low-density parity-check (QLDPC) code. Furthermore, we introduce the concept of quasi-single shot protocols as a novel approach that significantly reduces the number of repeated syndrome measurements required when decoding under phenomenological noise. To realize the protocol, we present the first implementation of time-domain decoding with the overlapping window method for general QLDPC codes and a novel analog single-shot decoding method. Our results lay the foundation for general decoding algorithms using analog information and demonstrate promising results in the direction of fault-tolerant quantum computation with concatenated bosonic-QLDPC codes.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020349","source":"Crossref","title":"Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-2973-1689","authenticated-orcid":true,"given":"Lucas","family":"Berent","sequence":"first","affiliation":[{"name":"Technical University of Munich"}]},{"ORCID":"https://orcid.org/0000-0002-1476-0647","authenticated-orcid":true,"given":"Timo","family":"Hillmann","sequence":"additional","affiliation":[{"name":"Chalmers University of Technology"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Freie Universität Berlin"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie"}]},{"given":"Robert","family":"Wille","sequence":"additional","affiliation":[{"name":"Technical University of Munich"},{"name":"Software Competence Center Hagenberg"}]},{"ORCID":"https://orcid.org/0000-0001-9202-1156","authenticated-orcid":true,"given":"Joschka","family":"Roffe","sequence":"additional","affiliation":[{"name":"Freie Universität Berlin"},{"name":"University of Edinburgh"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,30]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020349","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020349","note":"arxivid:2311.01328"},{"id":"arxiv:1810.01519","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.122.230501","source":"Crossref","title":"Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates","volume":"122","author":[{"given":"Weilei","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,6,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.122.230501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230501","note":"arxivid:1810.01519"},{"id":"doi:10.1147/rd.41.0043","type":"journal-article","publisher":"IBM","issue":"1","DOI":"10.1147/rd.41.0043","page":"43-57","source":"Crossref","title":"Design Methods for Maximum Minimum-Distance Error-Correcting Codes","volume":"4","author":[{"given":"J. E.","family":"MacDonald","sequence":"first","affiliation":[]}],"container-title":"IBM Journal of Research and Development","original-title":[],"issued":{"date-parts":[[1960,1]]},"URL":"http://dx.doi.org/10.1147/rd.41.0043","ISSN":["0018-8646","0018-8646"],"container-title-short":"IBM J. Res. &amp; Dev."},{"id":"doi:10.1109/TIT.1975.1055315","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1975.1055315","page":"106-110","source":"Crossref","title":"Maximalq-nary linear codes with large minimum distance (Corresp.)","volume":"21","author":[{"given":"A.","family":"Patel","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1975,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1975.1055315","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1111/j.1469-1809.1941.tb02298","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1111/j.1469-1809.1941.tb02298.x","page":"341-353","source":"Crossref","title":"THE THEORY OF CONFOUNDING IN FACTORIAL EXPERIMENTS IN RELATION TO THE THEORY OF GROUPS","volume":"11","author":[{"given":"R. A.","family":"FISHER","sequence":"first","affiliation":[]}],"container-title":"Annals of Eugenics","original-title":[],"language":"en","issued":{"date-parts":[[1941,1]]},"URL":"http://dx.doi.org/10.1111/j.1469-1809.1941.tb02298.x","ISSN":["2050-1420","2050-1439"],"container-title-short":"Annals of Eugenics","note":"alternative-id:10.1111/j.1469-1809.1941.tb02298.x"},{"id":"doi:10.1002/j.1538-7305.1948.tb01338","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1002/j.1538-7305.1948.tb01338.x","page":"379-423","source":"Crossref","title":"A Mathematical Theory of Communication","volume":"27","author":[{"given":"C. E.","family":"Shannon","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1948,7]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x","ISSN":["0005-8580"]},{"id":"manual:-R.-R.-Green-A-serial-orthogo","type":"article-journal","author":[{"family":"Green","given":"R.R."}],"title":"A serial orthogonal decoder","volume":"37–39-IV","container-title":"JPL Space Programs Summary","issued":"1966","page":"247–253,"},{"id":"doi:10.1109/ITW.2003.1216684","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.2003.1216684","page":"18-21","source":"Crossref","title":"Simple MAP decoding of first order Reed-Muller and Hamming codes","author":[{"given":"A.","family":"Ashikhmin","sequence":"first","affiliation":[]},{"given":"S.","family":"Litsyn","sequence":"additional","affiliation":[]}],"event":"2003 IEEE Information Theory Workshop","container-title":"Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ITW.2003.1216684"},{"id":"manual:-A.-Barg-and-S.-Zhou-A-quantu","type":"paper-conference","author":[{"family":"Barg","given":"A."},{"family":"Zhou","given":"S."}],"title":"A quantum decoding algorithm for the simplex code","container-title":"Proceedings of the 36th Annual Allerton Conference on Communication, Control and Computing","issued":"1998-09-23","page":"359–365","publisher-place":"Monticello, IL"},{"id":"arxiv:2502.07150","type":"article-journal","author":[{"given":"Alexander J.","family":"Malcolm"},{"given":"Andrew N.","family":"Glaudell"},{"given":"Patricio","family":"Fuentes"},{"given":"Daryus","family":"Chandra"},{"given":"Alexis","family":"Schotte"},{"given":"Colby","family":"DeLisle"},{"given":"Rafael","family":"Haenel"},{"given":"Amir","family":"Ebrahimi"},{"given":"Joschka","family":"Roffe"},{"given":"Armanda O.","family":"Quintavalle"},{"given":"Stefanie J.","family":"Beale"},{"given":"Nicholas R.","family":"Lee-Hone"},{"given":"Stephanie","family":"Simmons"}],"title":"Computing Efficiently in QLDPC Codes","issued":{"date-parts":[[2025,12,19]]},"note":"arxivid:2502.07150\narxiv_version_number:3"},{"id":"doi:10.1002/j.1538-7305.1950.tb00463","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1002/j.1538-7305.1950.tb00463.x","page":"147-160","source":"Crossref","title":"Error Detecting and Error Correcting Codes","volume":"29","author":[{"given":"R. W.","family":"Hamming","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1950,4]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1950.tb00463.x","ISSN":["0005-8580"]},{"id":"doi:10.1109/SBCCI.2002.1137643","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/sbcci.2002.1137643","page":"95-100","source":"Crossref","title":"Analyzing area and performance penalty of protecting different digital modules with Hamming code and triple modular redundancy","author":[{"given":"R.","family":"Hentschke","sequence":"first","affiliation":[]},{"given":"F.","family":"Marques","sequence":"additional","affiliation":[]},{"given":"F.","family":"Lima","sequence":"additional","affiliation":[]},{"given":"L.","family":"Carro","sequence":"additional","affiliation":[]},{"given":"A.","family":"Susin","sequence":"additional","affiliation":[]},{"given":"R.","family":"Reis","sequence":"additional","affiliation":[]}],"event":"15th Symposium on Integrated Circuits and Systems Design","container-title":"Proceedings. 15th Symposium on Integrated Circuits and Systems Design","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SBCCI.2002.1137643"},{"id":"manual:-R.-Crandall-Some-notes-on-st","type":"chapter","author":[{"family":"Crandall","given":"R."}],"title":"Some notes on steganography","container-title":"Posted on steganography mailing list 1998","issued":"1998","page":"1–6"},{"id":"doi:10.1007/3-540-45496-9_21","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-45496-9_21","page":"289-302","source":"Crossref","title":"F5—A Steganographic Algorithm","author":[{"given":"Andreas","family":"Westfeld","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2001]]},"ISBN":["9783540427339","9783540454960"],"URL":"http://dx.doi.org/10.1007/3-540-45496-9_21","ISSN":["0302-9743"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:1710.07386","type":"article-journal","author":[{"given":"Travis","family":"Baumbaugh"},{"given":"Yariana","family":"Diaz"},{"given":"Sophia","family":"Friesenhahn"},{"given":"Felice","family":"Manganiello"},{"given":"Alexander","family":"Vetter"}],"title":"Batch Codes from Hamming and Reed-Müller Codes","issued":{"date-parts":[[2017,10,20]]},"note":"arxivid:1710.07386\narxiv_version_number:1"},{"id":"arxiv:0710.1052","type":"article-journal","author":[{"given":"Andrew S.","family":"Fletcher"},{"given":"Peter W.","family":"Shor"},{"given":"Moe Z.","family":"Win"}],"title":"Channel-Adapted Quantum Error Correction for the Amplitude Damping Channel","issued":{"date-parts":[[2007,10,4]]},"note":"arxivid:0710.1052\narxiv_version_number:1"},{"id":"arxiv:2411.14389","type":"article-journal","author":[{"given":"Priya J.","family":"Nadkarni"},{"given":"Serge","family":"Adonsou"},{"given":"Guillaume","family":"Dauphinais"},{"given":"David W.","family":"Kribs"},{"given":"Michael","family":"Vasmer"}],"title":"Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction","issued":{"date-parts":[[2024,11,21]]},"note":"arxivid:2411.14389\narxiv_version_number:1"},{"id":"arxiv:2411.14448","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    While stabilizer tableaus have proven useful as a descriptive tool for additive quantum codes, they otherwise offer little guidance for concrete constructions or algorithm analysis. We introduce a representation of stabilizer codes as graphs with certain structures, and prove via the ZX calculus that this representation is related to stabilizer tableaus by an efficiently computable bijection. This gives a new universal recipe for code construction by way of finding graphs with nice properties. The graph representation gives insight into both code construction and algorithms. We construct as examples families of\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mtext>⟦</a:mtext>\n                      <a:mi>n</a:mi>\n                      <a:mo>,</a:mo>\n                      <a:mspace width=\"0.25em\"/>\n                      <a:mi mathvariant=\"normal\">Θ</a:mi>\n                      <a:mo stretchy=\"false\">(</a:mo>\n                      <a:mfrac>\n                        <a:mi>n</a:mi>\n                        <a:mrow>\n                          <a:mi>log</a:mi>\n                          <a:mo></a:mo>\n                          <a:mi>n</a:mi>\n                        </a:mrow>\n                      </a:mfrac>\n                      <a:mo stretchy=\"false\">)</a:mo>\n                      <a:mo>,</a:mo>\n                      <a:mspace width=\"0.25em\"/>\n                      <a:mi mathvariant=\"normal\">Θ</a:mi>\n                      <a:mo stretchy=\"false\">(</a:mo>\n                      <a:mi>log</a:mi>\n                      <a:mo></a:mo>\n                      <a:mi>n</a:mi>\n                      <a:mo stretchy=\"false\">)</a:mo>\n                      <a:mtext>⟧</a:mtext>\n                    </a:math>\n                    and\n                    <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <k:mtext>⟦</k:mtext>\n                      <k:mi>n</k:mi>\n                      <k:mo>,</k:mo>\n                      <k:mspace width=\"0.25em\"/>\n                      <k:mi mathvariant=\"normal\">Ω</k:mi>\n                      <k:mo stretchy=\"false\">(</k:mo>\n                      <k:msup>\n                        <k:mi>n</k:mi>\n                        <k:mrow>\n                          <k:mn>4</k:mn>\n                          <k:mo>/</k:mo>\n                          <k:mn>5</k:mn>\n                        </k:mrow>\n                      </k:msup>\n                      <k:mo stretchy=\"false\">)</k:mo>\n                      <k:mo>,</k:mo>\n                      <k:mspace width=\"0.25em\"/>\n                      <k:mi mathvariant=\"normal\">Θ</k:mi>\n                      <k:mo stretchy=\"false\">(</k:mo>\n                      <k:msup>\n                        <k:mi>n</k:mi>\n                        <k:mrow>\n                          <k:mn>1</k:mn>\n                          <k:mo>/</k:mo>\n                          <k:mn>5</k:mn>\n                        </k:mrow>\n                      </k:msup>\n                      <k:mo stretchy=\"false\">)</k:mo>\n                      <k:mtext>⟧</k:mtext>\n                    </k:math>\n                    codes. We use graphs in a probabilistic analysis to extend the quantum Gilbert-Varshamov bound into a three-way distance-rate-weight trade-off. Moreover, code properties such as distance and encoding circuit depth are bounded by simple functions of the graph degree. We prove that key coding algorithms—distance approximation, minimum-weight generator selection, and decoding—are unified as instances of one optimization game on a graph. By studying this game, we construct an efficient greedy decoder and prove that it corrects all recoverable errors for all graphs with cycle lengths no shorter than 13 (reducible to 5 with mild extra constraints); these include the above two families. Our results suggest that graphs are generically useful for the study of stabilizer codes.\n                  </jats:p>","DOI":"10.1103/1gjs-2rhx","source":"Crossref","title":"Universal Graph Representation of Stabilizer Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-8620-4092","authenticated-orcid":true,"given":"Andrey Boris","family":"Khesin","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-2806-8248","authenticated-orcid":true,"given":"Jonathan Z.","family":"Lu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,5]]},"URL":"http://dx.doi.org/10.1103/1gjs-2rhx","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040325","note":"arxivid:2411.14448"},{"id":"arxiv:1611.08012","type":"journal-article","publisher":"IOP Publishing","issue":"4","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>We introduce a high-level graphical framework for designing and analysing quantum error correcting codes, centred on what we term the coherent parity check (CPC). The graphical formulation is based on the diagrammatic tools of the <jats:italic>ZX</jats:italic>-calculus of quantum observables. The resulting framework leads to a construction for stabilizer codes that allows us to design and verify a broad range of quantum codes based on classical ones, and that gives a means of discovering large classes of codes using both analytical and numerical methods. We focus in particular on the smaller codes that will be the first used by near-term devices. We show how CSS codes form a subset of CPC codes and, more generally, how to compute stabilizers for a CPC code. As an explicit example of this framework, we give a method for turning almost any pair of classical <jats:inline-formula>\n                     <jats:tex-math/>\n                     <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                        <mml:mo stretchy=\"false\">[</mml:mo>\n                        <mml:mi>n</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mn>3</mml:mn>\n                        <mml:mo stretchy=\"false\">]</mml:mo>\n                     </mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"qstacf157ieqn1.gif\" xlink:type=\"simple\"/>\n                  </jats:inline-formula> codes into a <jats:inline-formula>\n                     <jats:tex-math/>\n                     <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                        <mml:mo stretchy=\"false\">[</mml:mo>\n                        <mml:mo stretchy=\"false\">[</mml:mo>\n                        <mml:mn>2</mml:mn>\n                        <mml:mi>n</mml:mi>\n                        <mml:mo>−</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>+</mml:mo>\n                        <mml:mn>2</mml:mn>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mn>3</mml:mn>\n                        <mml:mo stretchy=\"false\">]</mml:mo>\n                        <mml:mo stretchy=\"false\">]</mml:mo>\n                     </mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"qstacf157ieqn2.gif\" xlink:type=\"simple\"/>\n                  </jats:inline-formula> CPC code. Further, we give a simple technique for machine search which yields thousands of potential codes, and demonstrate its operation for distance 3 and 5 codes. Finally, we use the graphical tools to demonstrate how Clifford computation can be performed within CPC codes. As our framework gives a new tool for constructing small- to medium-sized codes with relatively high code rates, it provides a new source for codes that could be suitable for emerging devices, while its <jats:italic>ZX</jats:italic>-calculus foundations enable natural integration of error correction with graphical compiler toolchains. It also provides a powerful framework for reasoning about all stabilizer quantum error correction codes of any size.</jats:p>","DOI":"10.1088/2058-9565/acf157","page":"045028","source":"Crossref","title":"Graphical structures for design and verification of quantum error correction","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-1293-0761","authenticated-orcid":true,"given":"Nicholas","family":"Chancellor","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6090-9684","authenticated-orcid":true,"given":"Aleks","family":"Kissinger","sequence":"additional","affiliation":[]},{"given":"Stefan","family":"Zohren","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9202-1156","authenticated-orcid":true,"given":"Joschka","family":"Roffe","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4965-0584","authenticated-orcid":true,"given":"Dominic","family":"Horsman","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2023,9,22]]},"URL":"http://dx.doi.org/10.1088/2058-9565/acf157","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1611.08012"},{"id":"preset:HKSgraphs","type":"chapter","author":[{"family":"Kelley","given":"C.A."}],"title":"Codes over Graphs","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1109/TIT.2010.2080990","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2010.2080990","page":"6253-6264","source":"Crossref","title":"Polar Codes: Characterization of Exponent, Bounds, and Constructions","volume":"56","author":[{"given":"Satish Babu","family":"Korada","sequence":"first","affiliation":[]},{"given":"Eren","family":"Sasoglu","sequence":"additional","affiliation":[]},{"given":"Rüdiger","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2080990","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1601.06215","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2016.7541295","page":"230-234","source":"Crossref","title":"Algebraic properties of polar codes from a new polynomial formalism","author":[{"given":"Magali","family":"Bardet","sequence":"first","affiliation":[]},{"given":"Vlad","family":"Dragoi","sequence":"additional","affiliation":[]},{"given":"Ayoub","family":"Otmani","sequence":"additional","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]}],"event":"2016 IEEE International Symposium on Information Theory (ISIT)","container-title":"2016 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2016,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2016.7541295","note":"arxivid:1601.06215"},{"id":"arxiv:2112.12495","type":"article-journal","author":[{"given":"Kirill","family":"Ivanov"},{"given":"Rüdiger","family":"Urbanke"}],"title":"Polar Codes Do Not Have Many Affine Automorphisms","issued":{"date-parts":[[2022,4,25]]},"note":"arxivid:2112.12495\narxiv_version_number:3"},{"id":"manual:-N.-Stolte.-Rekursive-Codes-m","type":"thesis","author":[{"family":"Stolte","given":"N."}],"title":"Rekursive Codes mit der Plotkin-Konstruktion und ihre Decodierung","genre":"PhD thesis,","publisher":"Darmstadt University of Technology","issued":"2002","publisher-place":"Germany"},{"id":"doi:10.1109/ISIT.2014.6875073","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2014.6875073","page":"1451-1455","source":"Crossref","title":"Universal polar codes","author":[{"given":"S. Hamed","family":"Hassani","sequence":"first","affiliation":[]},{"given":"Rudiger","family":"Urbanke","sequence":"additional","affiliation":[]}],"event":"2014 IEEE International Symposium on Information Theory (ISIT)","container-title":"2014 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2014.6875073"},{"id":"doi:10.1109/ISIT.2009.5205860","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2009.5205860","page":"1488-1492","source":"Crossref","title":"Performance of polar codes for channel and source coding","author":[{"given":"Nadine","family":"Hussami","sequence":"first","affiliation":[]},{"given":"Satish Babu","family":"Korada","sequence":"additional","affiliation":[]},{"given":"Rudiger","family":"Urbanke","sequence":"additional","affiliation":[]}],"event":"2009 IEEE International Symposium on Information Theory - ISIT","container-title":"2009 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2009,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2009.5205860"},{"id":"doi:10.1109/TIT.2016.2616117","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2016.2616117","page":"6698-6712","source":"Crossref","title":"Unified Scaling of Polar Codes: Error Exponent, Scaling Exponent, Moderate Deviations, and Error Floors","volume":"62","author":[{"ORCID":"https://orcid.org/0000-0002-3242-7020","authenticated-orcid":false,"given":"Marco","family":"Mondelli","sequence":"first","affiliation":[]},{"given":"S. Hamed","family":"Hassani","sequence":"additional","affiliation":[]},{"given":"Rudiger L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2016,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2016.2616117","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ISIT.2010.5513579","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2010.5513579","page":"884-888","source":"Crossref","title":"An empirical scaling law for polar codes","author":[{"given":"Satish Babu","family":"Korada","sequence":"first","affiliation":[]},{"given":"Andrea","family":"Montanari","sequence":"additional","affiliation":[]},{"given":"Emre","family":"Telatar","sequence":"additional","affiliation":[]},{"given":"Rudiger","family":"Urbanke","sequence":"additional","affiliation":[]}],"event":"2010 IEEE International Symposium on Information Theory - ISIT","container-title":"2010 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2010,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2010.5513579"},{"id":"arxiv:2205.08857","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/tsp.2022.3216921","page":"5592-5607","source":"Crossref","title":"A Sequence Repetition Node-Based Successive Cancellation List Decoder for 5G Polar Codes: Algorithm and Implementation","volume":"70","author":[{"ORCID":"https://orcid.org/0000-0003-2750-2011","authenticated-orcid":false,"given":"Yuqing","family":"Ren","sequence":"first","affiliation":[{"name":"Telecommunications Circuits Laboratory (TCL), &#x00C9;cole Polytechnique F&#x00E9;d&#x00E9;rale de Lausanne (EPFL), Lausanne, Switzerland"}]},{"ORCID":"https://orcid.org/0000-0002-6421-9234","authenticated-orcid":false,"given":"Andreas Toftegaard","family":"Kristensen","sequence":"additional","affiliation":[{"name":"Telecommunications Circuits Laboratory (TCL), &#x00C9;cole Polytechnique F&#x00E9;d&#x00E9;rale de Lausanne (EPFL), Lausanne, Switzerland"}]},{"given":"Yifei","family":"Shen","sequence":"additional","affiliation":[{"name":"Telecommunications Circuits Laboratory (TCL), &#x00C9;cole Polytechnique F&#x00E9;d&#x00E9;rale de Lausanne (EPFL), Lausanne, Switzerland"}]},{"ORCID":"https://orcid.org/0000-0002-6721-4666","authenticated-orcid":false,"given":"Alexios","family":"Balatsoukas-Stimming","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0002-7736-6487","authenticated-orcid":false,"given":"Chuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"LEADS of Southeast University, the National Mobile Communications Research Laboratory, and the Purple Mountain Laboratories, Nanjing, China"}]},{"ORCID":"https://orcid.org/0000-0002-7270-5558","authenticated-orcid":false,"given":"Andreas","family":"Burg","sequence":"additional","affiliation":[{"name":"Telecommunications Circuits Laboratory (TCL), &#x00C9;cole Polytechnique F&#x00E9;d&#x00E9;rale de Lausanne (EPFL), Lausanne, Switzerland"}]}],"container-title":"IEEE Transactions on Signal Processing","original-title":[],"issued":{"date-parts":[[2022]]},"URL":"http://dx.doi.org/10.1109/TSP.2022.3216921","ISSN":["1053-587X","1941-0476"],"container-title-short":"IEEE Trans. Signal Process.","note":"arxivid:2205.08857"},{"id":"doi:10.1109/JSAC.2014.140515","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/jsac.2014.140515","page":"958-966","source":"Crossref","title":"Low-Complexity Soft-Output Decoding of Polar Codes","volume":"32","author":[{"given":"Ubaid U.","family":"Fayyaz","sequence":"first","affiliation":[]},{"given":"John R.","family":"Barry","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[2014,5]]},"URL":"http://dx.doi.org/10.1109/JSAC.2014.140515","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1109/ICC.2013.6655247","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.2013.6655247","page":"4337-4341","source":"Crossref","title":"Polar codes for partial response channels","author":[{"given":"Ubaid U.","family":"Fayyaz","sequence":"first","affiliation":[]},{"given":"John R.","family":"Barry","sequence":"additional","affiliation":[]}],"event":"ICC 2013 - 2013 IEEE International Conference on Communications","container-title":"2013 IEEE International Conference on Communications (ICC)","original-title":[],"issued":{"date-parts":[[2013,6]]},"URL":"http://dx.doi.org/10.1109/ICC.2013.6655247"},{"id":"doi:10.1109/LCOMM.2008.080017","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/lcomm.2008.080017","page":"447-449","source":"Crossref","title":"A performance comparison of polar codes and Reed-Muller codes","volume":"12","author":[{"given":"E.","family":"Arkan","sequence":"first","affiliation":[]}],"container-title":"IEEE Communications Letters","original-title":[],"issued":{"date-parts":[[2008,6]]},"URL":"http://dx.doi.org/10.1109/LCOMM.2008.080017","ISSN":["1089-7798"],"container-title-short":"IEEE Commun. Lett."},{"id":"arxiv:2210.10854","type":"article-journal","author":[{"given":"Srikar","family":"Kasi"},{"given":"John","family":"Kaewell"},{"given":"Shahab","family":"Hamidi-Rad"},{"given":"Kyle","family":"Jamieson"}],"title":"Decoding Polar Codes via Noisy Quantum Gates: Quantum Circuits and Insights","issued":{"date-parts":[[2022,10,19]]},"note":"arxivid:2210.10854\narxiv_version_number:1"},{"id":"manual:-3rd-Generation-Partnership-P","type":"document","citation-number":"3rd","title":"Generation Partnership Project","volume":"38","issue":"212","issued":"3GPP"},{"id":"preset:HKSpolar","type":"chapter","author":[{"family":"Presman","given":"N."},{"family":"Litsyn","given":"S."}],"title":"Polar codes","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"preset:PGCdatabase","type":"document","author":[{"family":"Liva","given":"G."},{"family":"Steiner","given":"F."}],"title":"pretty-good-codes.org: Online library of good channel codes","URL":"https://web.archive.org/web/20220524005723/http://pretty-good-codes.org/"},{"id":"doi:10.1016/j.softx.2019.100345","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.softx.2019.100345","page":"100345","source":"Crossref","title":"AFF3CT: A Fast Forward Error Correction Toolbox!","volume":"10","author":[{"given":"Adrien","family":"Cassagne","sequence":"first","affiliation":[]},{"given":"Olivier","family":"Hartmann","sequence":"additional","affiliation":[]},{"given":"Mathieu","family":"Léonardon","sequence":"additional","affiliation":[]},{"given":"Kun","family":"He","sequence":"additional","affiliation":[]},{"given":"Camille","family":"Leroux","sequence":"additional","affiliation":[]},{"given":"Romain","family":"Tajan","sequence":"additional","affiliation":[]},{"given":"Olivier","family":"Aumage","sequence":"additional","affiliation":[]},{"given":"Denis","family":"Barthou","sequence":"additional","affiliation":[]},{"given":"Thibaud","family":"Tonnellier","sequence":"additional","affiliation":[]},{"given":"Vincent","family":"Pignoly","sequence":"additional","affiliation":[]},{"given":"Bertrand","family":"Le Gal","sequence":"additional","affiliation":[]},{"given":"Christophe","family":"Jégo","sequence":"additional","affiliation":[]}],"container-title":"SoftwareX","original-title":[],"language":"en","issued":{"date-parts":[[2019,7]]},"URL":"http://dx.doi.org/10.1016/j.softx.2019.100345","ISSN":["2352-7110"],"container-title-short":"SoftwareX","note":"special_numbering:C\nalternative-id:S2352711019300457"},{"id":"doi:10.1109/TIT.2010.2040961","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2010.2040961","page":"1751-1768","source":"Crossref","title":"Polar Codes are Optimal for Lossy Source Coding","volume":"56","author":[{"given":"Satish Babu","family":"Korada","sequence":"first","affiliation":[]},{"given":"Rüdiger L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2040961","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1312.4575","type":"article-journal","author":[{"given":"Andrew J.","family":"Ferris"},{"given":"David","family":"Poulin"}],"title":"Branching MERA codes: a natural extension of polar codes","issued":{"date-parts":[[2013,12,16]]},"note":"arxivid:1312.4575\narxiv_version_number:1"},{"id":"doi:10.1109/ISIT.2014.6874999","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2014.6874999","page":"1081-1085","source":"Crossref","title":"Branching MERA codes: A natural extension of classical and quantum polar codes","author":[{"given":"Andrew James","family":"Ferris","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"event":"2014 IEEE International Symposium on Information Theory (ISIT)","container-title":"2014 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2014.6874999"},{"id":"arxiv:2002.03317","type":"article-journal","author":[{"given":"Emmanuel","family":"Abbe"},{"given":"Amir","family":"Shpilka"},{"given":"Min","family":"Ye"}],"title":"Reed-Muller Codes: Theory and Algorithms","issued":{"date-parts":[[2020,6,10]]},"note":"arxivid:2002.03317\narxiv_version_number:2"},{"id":"doi:10.1109/irepgelc.1954.6499441","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/irepgelc.1954.6499441","page":"6-12","source":"Crossref","title":"Application of Boolean algebra to switching circuit design and to error detection","volume":"EC-3","author":[{"given":"D. E.","family":"Muller","sequence":"first","affiliation":[{"name":"Digital Computer Lab., University of Illinois, Urbana, Illinois"}]}],"container-title":"Transactions of the I.R.E. Professional Group on Electronic Computers","original-title":[],"issued":{"date-parts":[[1954,9]]},"URL":"http://dx.doi.org/10.1109/irepgelc.1954.6499441","ISSN":["2168-1740","2168-1759"],"container-title-short":"Trans. I.R.E. Prof. Group Electron. Comput."},{"id":"doi:10.1109/tit.1954.1057465","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1954.1057465","page":"38-49","source":"Crossref","title":"A class of multiple-error-correcting codes and the decoding scheme","volume":"4","author":[{"given":"I.","family":"Reed","sequence":"first","affiliation":[]}],"container-title":"Transactions of the IRE Professional Group on Information Theory","original-title":[],"issued":{"date-parts":[[1954,9]]},"URL":"http://dx.doi.org/10.1109/tit.1954.1057465","ISSN":["2168-2690","2168-2704"],"container-title-short":"Trans. IRE Prof. Group Inf. Theory"},{"id":"manual:-N.-Mitani-On-the-transmissio","type":"article-journal","author":[{"family":"Mitani","given":"N."}],"title":"On the transmission of numbers in a sequential computer, delivered at the National Convention of the Inst","container-title":"Elect. Engineers of Japan","issued":"1951-11"},{"id":"arxiv:2506.08651","type":"article-journal","author":[{"given":"Dina","family":"Abdelhadi"},{"given":"Colin","family":"Sandon"},{"given":"Emmanuel","family":"Abbe"},{"given":"Ruediger","family":"Urbanke"}],"title":"Reed-Muller Codes for Quantum Pauli and Multiple Access Channels","issued":{"date-parts":[[2025,6,10]]},"note":"arxivid:2506.08651\narxiv_version_number:1"},{"id":"arxiv:1505.05123","type":"article-journal","author":[{"given":"Santhosh","family":"Kumar"},{"given":"Henry D.","family":"Pfister"}],"title":"Reed-Muller Codes Achieve Capacity on Erasure Channels","issued":{"date-parts":[[2015,6,15]]},"note":"arxivid:1505.05123\narxiv_version_number:2"},{"id":"arxiv:1601.04689","type":"article-journal","author":[{"given":"Shrinivas","family":"Kudekar"},{"given":"Santhosh","family":"Kumar"},{"given":"Marco","family":"Mondelli"},{"given":"Henry D.","family":"Pfister"},{"given":"Eren","family":"Şaşoğlu"},{"given":"Rüdiger","family":"Urbanke"}],"title":"Reed-Muller Codes Achieve Capacity on Erasure Channels","issued":{"date-parts":[[2016,1,18]]},"note":"arxivid:1601.04689\narxiv_version_number:1"},{"id":"arxiv:2110.14631","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2023.3286452","page":"920-949","source":"Crossref","title":"Reed–Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for all Rates Below Capacity","volume":"70","author":[{"ORCID":"https://orcid.org/0000-0003-4230-0688","authenticated-orcid":false,"given":"Galen","family":"Reeves","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering and the Department of Statistical Science, Duke University, Durham, NC, USA"}]},{"ORCID":"https://orcid.org/0000-0001-5521-4397","authenticated-orcid":false,"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering and the Department of Mathematics, Duke University, Durham, NC, USA"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2024,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2023.3286452","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2110.14631"},{"id":"arxiv:1411.4590","type":"article-journal","author":[{"given":"Emmanuel","family":"Abbe"},{"given":"Amir","family":"Shpilka"},{"given":"Avi","family":"Wigderson"}],"title":"Reed-Muller codes for random erasures and errors","issued":{"date-parts":[[2014,11,17]]},"note":"arxivid:1411.4590\narxiv_version_number:1"},{"id":"arxiv:2304.02509","type":"article-journal","author":[{"given":"Emmanuel","family":"Abbe"},{"given":"Colin","family":"Sandon"}],"title":"A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels","issued":{"date-parts":[[2023,4,5]]},"note":"arxivid:2304.02509\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.1973.1055023","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1973.1055023","page":"549-555","source":"Crossref","title":"Decoding by sequential code reduction","volume":"19","author":[{"given":"L.","family":"Rudolph","sequence":"first","affiliation":[]},{"given":"C.","family":"Hartmann","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1973,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1973.1055023","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1137/0209059","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","DOI":"10.1137/0209059","page":"758-767","source":"Crossref","title":"Factorization of Symmetric Matrices and Trace-Orthogonal Bases in Finite Fields","volume":"9","author":[{"given":"Gadiel","family":"Seroussi","sequence":"first","affiliation":[]},{"given":"Abraham","family":"Lempel","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[1980,11]]},"URL":"http://dx.doi.org/10.1137/0209059","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/0209059"},{"id":"arxiv:1606.01904","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.95.022316","source":"Crossref","title":"Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost","volume":"95","author":[{"given":"Earl T.","family":"Campbell","sequence":"first","affiliation":[]},{"given":"Mark","family":"Howard","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,2,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.95.022316","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022316","note":"arxivid:1606.01904"},{"id":"doi:10.1007/bfb0036046","type":"book-chapter","publisher":"Springer-Verlag","DOI":"10.1007/bfb0036046","page":"1-17","source":"Crossref","title":"Deep-space communications and coding: A marriage made in heaven","author":[{"given":"James L.","family":"Massey","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Control and Information Sciences","original-title":[],"language":"en","issued":{"date-parts":[[null]]},"ISBN":["3540558519"],"URL":"http://dx.doi.org/10.1007/bfb0036046","note":"publisher-location:London"},{"id":"doi:10.1017/CBO9781316529836","type":"monograph","publisher":"Cambridge University Press","DOI":"10.1017/cbo9781316529836","source":"Crossref","title":"Designs and their Codes","author":[{"given":"E. F.","family":"Assmus","sequence":"first","affiliation":[]},{"given":"J. D.","family":"Key","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1992,8,28]]},"ISBN":["9780521413619","9780521458399","9781316529836"],"URL":"http://dx.doi.org/10.1017/CBO9781316529836","note":"edition-number:1"},{"id":"doi:10.1007/3-540-45624-4_17","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-45624-4_17","page":"159-171","source":"Crossref","title":"On Hyperbolic Codes","author":[{"given":"Olav","family":"Geil","sequence":"first","affiliation":[]},{"given":"Tom","family":"Høholdt","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2001]]},"ISBN":["9783540429111","9783540456247"],"URL":"http://dx.doi.org/10.1007/3-540-45624-4_17","ISSN":["0302-9743"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1016/0097-3165(71)90066-5","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(71)90066-5","page":"80-91","source":"Crossref","title":"On periodic sequences from GF(q)","volume":"10","author":[{"given":"Robert J","family":"McEliece","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1971,1]]},"URL":"http://dx.doi.org/10.1016/0097-3165(71)90066-5","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316571900665"},{"id":"doi:10.1016/0012-365X(72)90032-5","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(72)90032-5","page":"177-192","source":"Crossref","title":"Weight congruences for p-ary cyclic codes","volume":"3","author":[{"given":"Robert J.","family":"McEliece","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1972]]},"URL":"http://dx.doi.org/10.1016/0012-365X(72)90032-5","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X72900325"},{"id":"doi:10.1007/BF01072842","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01072842","page":"25-31","source":"Crossref","title":"On the theory of group codes","volume":"3","author":[{"given":"S. D.","family":"Berman","sequence":"first","affiliation":[]}],"container-title":"Cybernetics","original-title":[],"language":"en","issued":{"date-parts":[[1969]]},"URL":"http://dx.doi.org/10.1007/BF01072842","ISSN":["0011-4235","1573-8337"],"container-title-short":"Cybern Syst Anal","note":"alternative-id:BF01072842"},{"id":"preset:Charpin82","type":"thesis","author":[{"family":"Charpin","given":"P."}],"title":"Codes idéaux de certaines algèbres modulaires","genre":"PhD thesis,","issued":"1982"},{"id":"arxiv:1601.07363","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2019.2906374","page":"4771-4784","source":"Crossref","title":"T-Count Optimization and Reed–Muller Codes","volume":"65","author":[{"ORCID":"https://orcid.org/0000-0003-3514-420X","authenticated-orcid":false,"given":"Matthew","family":"Amy","sequence":"first","affiliation":[]},{"given":"Michele","family":"Mosca","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2019,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2019.2906374","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1601.07363"},{"id":"arxiv:2011.05500","type":"article-journal","author":[{"given":"Fernando Granha","family":"Jeronimo"},{"given":"Dylan","family":"Quintana"},{"given":"Shashank","family":"Srivastava"},{"given":"Madhur","family":"Tulsiani"}],"title":"Unique Decoding of Explicit $ε$-balanced Codes Near the Gilbert-Varshamov Bound","issued":{"date-parts":[[2020,11,11]]},"note":"arxivid:2011.05500\narxiv_version_number:1"},{"id":"doi:10.1145/3055399.3055408","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3055399.3055408","page":"238-251","source":"Crossref","title":"Explicit, almost optimal, epsilon-balanced codes","author":[{"given":"Amnon","family":"Ta-Shma","sequence":"first","affiliation":[{"name":"Tel Aviv University, Israel"}]}],"event":"STOC '17: Symposium on Theory of Computing","container-title":"Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[2017,6,19]]},"URL":"http://dx.doi.org/10.1145/3055399.3055408","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3055399.3055408\n10.1145/3055399"},{"id":"doi:10.1007/BF02579283","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf02579283","page":"71-78","source":"Crossref","title":"Explicit constructions of graphs without short cycles and low density codes","volume":"2","author":[{"given":"G. A.","family":"Margulis","sequence":"first","affiliation":[]}],"container-title":"Combinatorica","original-title":[],"language":"en","issued":{"date-parts":[[1982,3]]},"URL":"http://dx.doi.org/10.1007/BF02579283","ISSN":["0209-9683","1439-6912"],"container-title-short":"Combinatorica","note":"alternative-id:BF02579283"},{"id":"doi:10.1109/TIT.2007.892774","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2007.892774","page":"1460-1478","source":"Crossref","title":"Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights","volume":"53","author":[{"given":"Christine A.","family":"Kelley","sequence":"first","affiliation":[]},{"given":"Deepak","family":"Sridhara","sequence":"additional","affiliation":[]},{"given":"Joachim","family":"Rosenthal","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2007,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2007.892774","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2004.828066","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2004.828066","page":"1156-1176","source":"Crossref","title":"Combinatorial Constructions of Low-Density Parity-Check Codes for Iterative Decoding","volume":"50","author":[{"given":"B.","family":"Vasic","sequence":"first","affiliation":[]},{"given":"O.","family":"Milenkovic","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2004,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.828066","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ISIT.2002.1023582","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2002.1023582","page":"310","source":"Crossref","title":"Codes for iterative decoding from partial geometries","author":[{"given":"S.J.","family":"Johnson","sequence":"first","affiliation":[]},{"given":"S.R.","family":"Weller","sequence":"additional","affiliation":[]}],"event":"IEEE International Symposium on Information Theory","container-title":"Proceedings IEEE International Symposium on Information Theory,","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2002.1023582"},{"id":"doi:10.1109/ISIT.2001.936086","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2001.936086","page":"223","source":"Crossref","title":"Construction of codes based on finite generalized quadrangles for iterative decoding","author":[{"given":"P.O.","family":"Vontobel","sequence":"first","affiliation":[]},{"given":"R.M.","family":"Tanner","sequence":"additional","affiliation":[]}],"event":"2001 IEEE International Symposium on Information Theory","container-title":"Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2001.936086"},{"id":"doi:10.1109/TIT.2005.856936","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2005.856936","page":"3890-3898","source":"Crossref","title":"LDPC Codes From Generalized Polygons","volume":"51","author":[{"given":"Z.","family":"Liu","sequence":"first","affiliation":[]},{"given":"D.A.","family":"Pados","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2005,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.856936","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ICC.2004.1312607","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.2004.1312607","page":"777-781 Vol.2","source":"Crossref","title":"Analysis of the cycle-structure of LDPC codes based on Latin squares","author":[{"given":"O.","family":"Milenkovic","sequence":"first","affiliation":[]},{"given":"S.","family":"Laendner","sequence":"additional","affiliation":[]}],"event":"2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577)","container-title":"2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577)","original-title":[],"issued":{"date-parts":[[2004]]},"URL":"http://dx.doi.org/10.1109/ICC.2004.1312607"},{"id":"doi:10.1109/TCOMM.2006.888633","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tcomm.2006.888633","page":"303-312","source":"Crossref","title":"LDPC Codes Based on Latin Squares: Cycle Structure, Stopping Set, and Trapping Set Analysis","volume":"55","author":[{"given":"S.","family":"Laendner","sequence":"first","affiliation":[]},{"given":"O.","family":"Milenkovic","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2007,2]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2006.888633","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/TCOMM.2010.091710.090721","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tcomm.2010.091710.090721","page":"3126-3139","source":"Crossref","title":"Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares","volume":"58","author":[{"given":"Li","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Qin","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Shu","family":"Lin","sequence":"additional","affiliation":[]},{"given":"Khaled","family":"Abdel-Ghaffar","sequence":"additional","affiliation":[]},{"given":"Ian F.","family":"Blake","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2010,11]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2010.091710.090721","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"preset:Tanner01","type":"paper-conference","author":[{"family":"Tanner","given":"R.M."},{"family":"Sridhara","given":"D."},{"family":"Fuja","given":"T."}],"title":"A class of group-structured LDPC codes","container-title":"Proc. ISTA","issued":"2001"},{"id":"doi:10.1109/ALLERTON.2008.4797704","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/allerton.2008.4797704","page":"1254-1261","source":"Crossref","title":"On codes designed via algebraic lifts of graphs","author":[{"given":"Christine A.","family":"Kelley","sequence":"first","affiliation":[]}],"event":"2008 46th Annual Allerton Conference on Communication, Control, and Computing","container-title":"2008 46th Annual Allerton Conference on Communication, Control, and Computing","original-title":[],"issued":{"date-parts":[[2008,9]]},"URL":"http://dx.doi.org/10.1109/ALLERTON.2008.4797704"},{"id":"doi:10.1109/ISIT.2008.4595095","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2008.4595095","page":"792-796","source":"Crossref","title":"LDPC codes from voltage graphs","author":[{"given":"Christine A.","family":"Kelley","sequence":"first","affiliation":[]},{"given":"Judy L.","family":"Walker","sequence":"additional","affiliation":[]}],"event":"2008 IEEE International Symposium on Information Theory - ISIT","container-title":"2008 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2008.4595095"},{"id":"doi:10.1017/CBO9781139087223","type":"book","publisher":"Cambridge University Press","abstract":"<jats:p>The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature. Other books cover portions of the material here, but there are no other books with such a wide scope. This book contains fifteen expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory and the topology of surfaces. Each chapter concludes with an extensive list of references.</jats:p>","DOI":"10.1017/cbo9781139087223","source":"Crossref","title":"Topics in Topological Graph Theory","container-title":[],"original-title":[],"editor":[{"given":"Lowell W.","family":"Beineke","sequence":"first","affiliation":[]},{"given":"Robin J.","family":"Wilson","sequence":"additional","affiliation":[]},{"given":"Jonathan L.","family":"Gross","sequence":"additional","affiliation":[]},{"given":"Thomas W.","family":"Tucker","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2009,7,9]]},"ISBN":["9780521802307","9781139087223"],"URL":"http://dx.doi.org/10.1017/CBO9781139087223","note":"edition-number:1"},{"id":"arxiv:0812.5104","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"}],"title":"On Quantum and Classical Error Control Codes: Constructions and Applications","issued":{"date-parts":[[2008,12,30]]},"note":"arxivid:0812.5104\narxiv_version_number:1"},{"id":"doi:10.1109/18.910576","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910576","page":"585-598","source":"Crossref","title":"Improved low-density parity-check codes using irregular graphs","volume":"47","author":[{"given":"M.G.","family":"Luby","sequence":"first","affiliation":[]},{"given":"M.","family":"Mitzenmacher","sequence":"additional","affiliation":[]},{"given":"M.A.","family":"Shokrollahi","sequence":"additional","affiliation":[]},{"given":"D.A.","family":"Spielman","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910576","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.910578","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910578","page":"619-637","source":"Crossref","title":"Design of capacity-approaching irregular low-density parity-check codes","volume":"47","author":[{"given":"T.J.","family":"Richardson","sequence":"first","affiliation":[]},{"given":"M.A.","family":"Shokrollahi","sequence":"additional","affiliation":[]},{"given":"R.L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910578","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/4234.905935","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/4234.905935","page":"58-60","source":"Crossref","title":"On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit","volume":"5","author":[{"family":"Sae-Young Chung","sequence":"first","affiliation":[]},{"given":"G.D.","family":"Forney","sequence":"additional","affiliation":[]},{"given":"T.J.","family":"Richardson","sequence":"additional","affiliation":[]},{"given":"R.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Letters","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/4234.905935","ISSN":["1089-7798"],"container-title-short":"IEEE Commun. Lett."},{"id":"doi:10.1109/TIT.2006.883541","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2006.883541","page":"4839-4855","source":"Crossref","title":"Weight Distribution of Low-Density Parity-Check Codes","volume":"52","author":[{"given":"C.","family":"Di","sequence":"first","affiliation":[]},{"given":"T.J.","family":"Richardson","sequence":"additional","affiliation":[]},{"given":"R.L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2006,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2006.883541","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TCOMM.2004.826370","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tcomm.2004.826370","page":"670-678","source":"Crossref","title":"Design of Low-Density Parity-Check Codes for Modulation and Detection","volume":"52","author":[{"given":"S.","family":"ten Brink","sequence":"first","affiliation":[]},{"given":"G.","family":"Kramer","sequence":"additional","affiliation":[]},{"given":"A.","family":"Ashikhmin","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[2004,4]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2004.826370","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/18.910580","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910580","page":"657-670","source":"Crossref","title":"Analysis of sum-product decoding of low-density parity-check codes using a Gaussian approximation","volume":"47","author":[{"family":"Sae-Young Chung","sequence":"first","affiliation":[]},{"given":"T.J.","family":"Richardson","sequence":"additional","affiliation":[]},{"given":"R.L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910580","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-R.-Urbanke-LdpcOpt---a-fast-","type":"document","author":[{"family":"Urbanke","given":"R."}],"title":"LdpcOpt - a fast and accurate degree distribution optimizer for LDPC code ensembles","issued":"2001"},{"id":"preset:Richardson02","type":"chapter","author":[{"family":"Richardson","given":"T."},{"family":"Urbanke","given":"R."}],"title":"Multi-edge type LDPC codes","container-title":"Workshop honoring Prof. Bob McEliece on his 60th birthday","publisher":"California Institute of Technology","issued":"2002","publisher-place":"Pasadena, California"},{"id":"doi:10.1109/18.748992","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.748992","page":"399-431","source":"Crossref","title":"Good error-correcting codes based on very sparse matrices","volume":"45","author":[{"given":"D.J.C.","family":"MacKay","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999,3]]},"URL":"http://dx.doi.org/10.1109/18.748992","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.910577","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910577","page":"599-618","source":"Crossref","title":"The capacity of low-density parity-check codes under message-passing decoding","volume":"47","author":[{"given":"T.J.","family":"Richardson","sequence":"first","affiliation":[]},{"given":"R.L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910577","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1702.07740","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The speed at which two remote parties can exchange secret keys in continuous-variable quantum key distribution (CV-QKD) is currently limited by the computational complexity of key reconciliation. Multi-dimensional reconciliation using multi-edge low-density parity-check (LDPC) codes with low code rates and long block lengths has been shown to improve error-correction performance and extend the maximum reconciliation distance. We introduce a quasi-cyclic code construction for multi-edge codes that is highly suitable for hardware-accelerated decoding on a graphics processing unit (GPU). When combined with an 8-dimensional reconciliation scheme, our LDPC decoder achieves an information throughput of 7.16 Kbit/s on a single NVIDIA GeForce GTX 1080 GPU, at a maximum distance of 142 km with a secret key rate of 6.64 × 10<jats:sup>−8</jats:sup> bits/pulse for a rate 0.02 code with block length of 10<jats:sup>6</jats:sup> bits. The LDPC codes presented in this work can be used to extend the previous maximum CV-QKD distance of 100 km to 142 km, while delivering up to 3.50× higher information throughput over the tight upper bound on secret key rate for a lossy channel.</jats:p>","DOI":"10.1038/s41534-018-0070-6","source":"Crossref","title":"Quasi-cyclic multi-edge LDPC codes for long-distance quantum cryptography","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-6365-6195","authenticated-orcid":false,"given":"Mario","family":"Milicevic","sequence":"first","affiliation":[]},{"given":"Chen","family":"Feng","sequence":"additional","affiliation":[]},{"given":"Lei M.","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"P. Glenn","family":"Gulak","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2018,4,27]]},"URL":"http://dx.doi.org/10.1038/s41534-018-0070-6","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"21","note":"alternative-id:70\narxivid:1702.07740"},{"id":"doi:10.1103/PhysRevA.103.062419","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.103.062419","source":"Crossref","title":"Multiedge-type low-density parity-check codes for continuous-variable quantum key distribution","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0002-0408-9458","authenticated-orcid":true,"given":"Hossein","family":"Mani","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4311-2593","authenticated-orcid":true,"given":"Tobias","family":"Gehring","sequence":"additional","affiliation":[]},{"given":"Philipp","family":"Grabenweger","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3150-5998","authenticated-orcid":true,"given":"Bernhard","family":"Ömer","sequence":"additional","affiliation":[]},{"given":"Christoph","family":"Pacher","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1990-7687","authenticated-orcid":true,"given":"Ulrik Lund","family":"Andersen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,6,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.062419","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062419"},{"id":"arxiv:2402.16044","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>To establish a scalable and secure quantum network, a critical milestone is advancing from basic point-to-point quantum key distribution (QKD) systems to the development of inherently multi-user protocols designed to maximize network capacity. Here, we propose a quantum passive optical network (QPON) protocol based on continuous-variable (CV) systems, particularly the quadrature of the coherent state, which enables deterministic, simultaneous, and high-rate secret key generation among all network users. We implement two protocols with different trust levels assigned to the network users and experimentally demonstrate key generation in a quantum access network with 8 users, each with an 11 km span of access link. Depending on the trust assumptions about the users, we reach 1.5 and 2.1 Mbits/s of total network key generation (or 0.4 and 1.0 Mbits/s with finite-size channels estimation). Demonstrating the potential to expand the network’s capacity to accommodate tens of users at a high rate, our CV-QPON protocols open up new possibilities in establishing low-cost, high-rate, and scalable secure quantum access networks serving as a stepping stone towards a quantum internet.</jats:p>","DOI":"10.1038/s41377-024-01633-9","source":"Crossref","title":"Continuous-variable quantum passive optical network","volume":"13","author":[{"given":"Adnan A. E.","family":"Hajomer","sequence":"first","affiliation":[]},{"given":"Ivan","family":"Derkach","sequence":"additional","affiliation":[]},{"given":"Radim","family":"Filip","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1990-7687","authenticated-orcid":false,"given":"Ulrik L.","family":"Andersen","sequence":"additional","affiliation":[]},{"given":"Vladyslav","family":"C. Usenko","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4311-2593","authenticated-orcid":false,"given":"Tobias","family":"Gehring","sequence":"additional","affiliation":[]}],"container-title":"Light: Science &amp; Applications","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,16]]},"URL":"http://dx.doi.org/10.1038/s41377-024-01633-9","ISSN":["2047-7538"],"container-title-short":"Light Sci Appl","page":"291","note":"alternative-id:1633\narxivid:2402.16044"},{"id":"doi:10.1109/ITW.2011.6089477","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.2011.6089477","page":"350-354","source":"Crossref","title":"Quasi-cyclic LDPC codes based on pre-lifted protographs","author":[{"given":"David G. M.","family":"Mitchell","sequence":"first","affiliation":[]},{"given":"Roxana","family":"Smarandache","sequence":"additional","affiliation":[]},{"given":"Daniel J.","family":"Costello","sequence":"additional","affiliation":[]}],"event":"2011 IEEE Information Theory Workshop (ITW)","container-title":"2011 IEEE Information Theory Workshop","original-title":[],"issued":{"date-parts":[[2011,10]]},"URL":"http://dx.doi.org/10.1109/ITW.2011.6089477"},{"id":"arxiv:1102.4612","type":"journal-article","publisher":"Institute of Electronics, Information and Communications Engineers (IEICE)","issue":"11","DOI":"10.1587/transfun.e94.a.2161","page":"2161-2168","source":"Crossref","title":"Spatially-Coupled MacKay-Neal Codes and Hsu-Anastasopoulos Codes","volume":"E94-A","author":[{"given":"Kenta","family":"KASAI","sequence":"first","affiliation":[{"name":"Dept. of Communications and Integrated Systems, Tokyo Institute of Technology"}]},{"given":"Kohichi","family":"SAKANIWA","sequence":"additional","affiliation":[{"name":"Dept. of Communications and Integrated Systems, Tokyo Institute of Technology"}]}],"container-title":"IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2011]]},"URL":"http://dx.doi.org/10.1587/transfun.E94.A.2161","ISSN":["0916-8508","1745-1337"],"container-title-short":"IEICE Trans. Fundamentals","note":"arxivid:1102.4612"},{"id":"manual:-J.-Thorpe-Low-density-parity","type":"report","author":[{"family":"Thorpe","given":"J."}],"title":"Low-density parity-check (LDPC) codes constructed from protographs","genre":"IPN progress report 42.154","issued":"2003","page":"42–154"},{"id":"doi:10.1109/GLOCOM.2005.1577834","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/glocom.2005.1577834","page":"5 pp.","source":"Crossref","title":"Protograph based LDPC codes with minimum distance linearly growing with block size","author":[{"given":"D.","family":"Divsalar","sequence":"first","affiliation":[]},{"given":"C.","family":"Jones","sequence":"additional","affiliation":[]},{"given":"S.","family":"Dolinar","sequence":"additional","affiliation":[]},{"given":"J.","family":"Thorpe","sequence":"additional","affiliation":[]}],"event":"GLOBECOM '05. IEEE Global Telecommunications Conference, 2005.","container-title":"GLOBECOM '05. IEEE Global Telecommunications Conference, 2005.","original-title":[],"issued":{"date-parts":[[2005]]},"URL":"http://dx.doi.org/10.1109/GLOCOM.2005.1577834"},{"id":"doi:10.1109/MILCOM.2006.302313","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/milcom.2006.302313","page":"1-7","source":"Crossref","title":"Protograph LDPC Codes over Burst Erasure Channels","author":[{"given":"Dariush","family":"Divsalar","sequence":"first","affiliation":[]},{"given":"Sam","family":"Dolinar","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Jones","sequence":"additional","affiliation":[]}],"event":"MILCOM 2006","container-title":"MILCOM 2006","original-title":[],"issued":{"date-parts":[[2006,10]]},"URL":"http://dx.doi.org/10.1109/MILCOM.2006.302313"},{"id":"doi:10.1109/JSAC.2009.090806","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/jsac.2009.090806","page":"876-888","source":"Crossref","title":"Capacity-approaching protograph codes","volume":"27","author":[{"given":"Dariush","family":"Divsalar","sequence":"first","affiliation":[]},{"given":"Sam","family":"Dolinar","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Jones","sequence":"additional","affiliation":[]},{"given":"Kenneth","family":"Andrews","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[2009,8]]},"URL":"http://dx.doi.org/10.1109/JSAC.2009.090806","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1109/COMST.2015.2436705","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/comst.2015.2436705","page":"1989-2016","source":"Crossref","title":"A Survey on Protograph LDPC Codes and Their Applications","volume":"17","author":[{"given":"Yi","family":"Fang","sequence":"first","affiliation":[]},{"given":"Guoan","family":"Bi","sequence":"additional","affiliation":[]},{"given":"Yong Liang","family":"Guan","sequence":"additional","affiliation":[]},{"given":"Francis C. M.","family":"Lau","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Surveys &amp; Tutorials","original-title":[],"issued":{"date-parts":[[2015]]},"URL":"http://dx.doi.org/10.1109/COMST.2015.2436705","ISSN":["1553-877X"],"container-title-short":"IEEE Commun. Surv. Tutorials"},{"id":"doi:10.1109/ICC.2005.1494433","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.2005.1494433","page":"658-662","source":"Crossref","title":"Constructing LDPC codes from simple loop-free encoding modules","volume":"1","author":[{"given":"D.","family":"Divsalar","sequence":"first","affiliation":[]},{"given":"S.","family":"Dolinar","sequence":"additional","affiliation":[]},{"given":"J.","family":"Thorpe","sequence":"additional","affiliation":[]},{"given":"C.","family":"Jones","sequence":"additional","affiliation":[]}],"event":"IEEE International Conference on Communications, 2005. ICC 2005. 2005","container-title":"IEEE International Conference on Communications, 2005. ICC 2005. 2005","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ICC.2005.1494433"},{"id":"arxiv:1710.03619","type":"article-journal","author":[{"given":"Allison","family":"Beemer"},{"given":"Salman","family":"Habib"},{"given":"Christine A.","family":"Kelley"},{"given":"Joerg","family":"Kliewer"}],"title":"A Generalized Algebraic Approach to Optimizing SC-LDPC Codes","issued":{"date-parts":[[2017,10,10]]},"note":"arxivid:1710.03619\narxiv_version_number:1"},{"id":"doi:10.1109/TCOMM.2007.894109","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tcomm.2007.894109","page":"692-702","source":"Crossref","title":"Accumulate-Repeat-Accumulate Codes","volume":"55","author":[{"given":"Aliazam","family":"Abbasfar","sequence":"first","affiliation":[]},{"given":"Dariush","family":"Divsalar","sequence":"additional","affiliation":[]},{"given":"Kung","family":"Yao","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2007,4]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2007.894109","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"manual:-M.-Yang-Y.-Li-and-W.-E.-Ryan","type":"paper-conference","author":[{"family":"Yang","given":"M."},{"family":"Li","given":"Y."},{"family":"Ryan","given":"W.E."}],"title":"Design of efficiently-encodable moderate-length high-rate irregular LDPC codes","container-title":"Proc. 40th Annual Allerton Conference on Communication, Control, and Computing","issued":"2002-10","page":"1415–1424,","publisher-place":"Champaign, IL"},{"id":"doi:10.1109/ISIT.2003.1228251","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2003.1228251","page":"237-237","source":"Crossref","title":"Lowering the error-rate floors of moderate-length high-rate irregular ldpc codes","author":[{"given":"M.","family":"Yang","sequence":"first","affiliation":[]},{"given":"W.E.","family":"Ryan","sequence":"additional","affiliation":[]}],"event":"IEEE International Symposium on Information Theory, 2003. Proceedings.","container-title":"IEEE International Symposium on Information Theory, 2003. Proceedings.","original-title":[],"issued":{"date-parts":[[2003]]},"URL":"http://dx.doi.org/10.1109/ISIT.2003.1228251"},{"id":"doi:10.1109/TCOMM.2004.826367","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tcomm.2004.826367","page":"564-571","source":"Crossref","title":"Design of Efficiently Encodable Moderate-Length High-Rate Irregular LDPC Codes","volume":"52","author":[{"given":"M.","family":"Yang","sequence":"first","affiliation":[]},{"given":"W.E.","family":"Ryan","sequence":"additional","affiliation":[]},{"given":"Y.","family":"Li","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[2004,4]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2004.826367","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/TIT.2004.831778","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2004.831778","page":"1711-1727","source":"Crossref","title":"Design Methods for Irregular Repeat–Accumulate Codes","volume":"50","author":[{"given":"A.","family":"Roumy","sequence":"first","affiliation":[]},{"given":"S.","family":"Guemghar","sequence":"additional","affiliation":[]},{"given":"G.","family":"Caire","sequence":"additional","affiliation":[]},{"given":"S.","family":"Verdu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2004,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.831778","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-H.-Jin-A.-Khandekar-and-R.-J","type":"paper-conference","author":[{"family":"Jin","given":"H."},{"family":"Khandekar","given":"A."},{"family":"McEliece","given":"R.J."}],"title":"Irregular repeat-accumulate codes","container-title":"Proc. 2nd Int. Symp. Turbo Codes and Related Topics","issued":"2000"},{"id":"doi:10.7907/Q06G-MW38","type":"thesis","categories":["typical set bound","erasure channel","gaussian approximation","Electrical Engineering","irregular repeat-accumulate codes","IRA codes","irregular LDPC codes","consistency condition","iterative decoding","stability condition","binary adder channel"],"language":"en","author":[{"family":"Khandekar","given":"Aamod Dinkar"}],"issued":{"date-parts":[[2003]]},"abstract":"The field of error correcting codes was revolutionized by the introduction of turbo codes [7] in 1993. These codes demonstrated dramatic performance improvements over any previously known codes, with significantly lower complexity. Since then, much progress has been made towards understanding the performance of these codes, as well as in using this understanding to design even better codes.\n\nThis thesis takes a few more steps in both these directions. We develop a new technique, called the typical set bound, for analyzing the asymptotic performance of code ensembles based on their weight enumerators. This technique yields very tight bounds on the maximum-likelihood decoding threshold of code ensembles, and is powerful enough to reproduce Shannon's noisy coding theorem for the class of binary-input symmetric channels.\n\nWe also introduce a new class of codes called irregular repeat-accumulate~(IRA) codes, which are adapted from the previously known class of repeat-accumulate~(RA) codes. These codes are competitive in terms of decoding performance with the class of irregular low-density parity-check~(LDPC) codes, which are arguably the best class of codes known today, at least for long block lengths. In addition, IRA codes have a significant advantage over irregular LDPC codes in terms of encoding complexity.\n\nWe also derive an analytical bound regarding iterative decoding thresholds of code ensembles on general binary-input symmetric channels, an area in which theoretical results are currently lacking","DOI":"10.7907/Q06G-MW38","publisher":"California Institute of Technology","title":"Graph-based codes and iterative decoding","URL":"https://resolver.caltech.edu/CaltechETD:etd-06202002-170522","version":"Final","note":"copyright:No commercial reproduction, distribution, display or performance rights in this work are provided."},{"id":"manual:-H.-Jin-A.-Khandekar-and-R.-J","type":"article-journal","author":[{"family":"Jin","given":"H."},{"family":"Khandekar","given":"A."},{"family":"McEliece","given":"R.J."}],"title":"SERIAL CONCATENATION OF INTERLEAVED CONVOLUTIONAL CODES FORMING TURBO-LIKE CODES","volume":"Number 7116710B1","container-title":"United States Patent","issued":"2023"},{"id":"doi:10.1109/TIT.2005.850079","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2005.850079","page":"2352-2379","source":"Crossref","title":"Capacity-Achieving Ensembles for the Binary Erasure Channel With Bounded Complexity","volume":"51","author":[{"given":"H.D.","family":"Pfister","sequence":"first","affiliation":[]},{"given":"I.","family":"Sason","sequence":"additional","affiliation":[]},{"given":"R.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2005,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.850079","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ICARES.2014.7024407","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icares.2014.7024407","page":"83-88","source":"Crossref","title":"Design and implementation of LDPC(Low Density Parity Check) coding technique on FPGA (Field Programmable Gate Array) for DVB-S2 (Digital Video Broadcasting-Satellite)","author":[{"given":"Rita","family":"Purnamasari","sequence":"first","affiliation":[]},{"given":"Heroe","family":"Wijanto","sequence":"additional","affiliation":[]},{"given":"Iswahyudi","family":"Hidayat","sequence":"additional","affiliation":[]}],"event":"2014 IEEE International Conference on Aerospace Electronics and Remote Sensing Technology (ICARES)","container-title":"2014 IEEE International Conference on Aerospace Electronics and Remote Sensing Technology","original-title":[],"issued":{"date-parts":[[2014,11]]},"URL":"http://dx.doi.org/10.1109/ICARES.2014.7024407"},{"id":"manual:-ETSI-ETSI.-Digital-video-bro","type":"article-journal","author":[{"family":"ETSI","given":"E.T.S.I."}],"title":"Digital video broadcasting (dvb); second generation framing structure, channel coding and modulation systems for broadcasting, interactive services, news gathering and other broadband satellite applications","volume":"II","container-title":"Part","issued":"2005","page":"2– – 2"},{"id":"doi:10.1109/18.669119","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.669119","page":"909-926","source":"Crossref","title":"Serial concatenation of interleaved codes: performance analysis, design, and iterative decoding","volume":"44","author":[{"given":"S.","family":"Benedetto","sequence":"first","affiliation":[]},{"given":"D.","family":"Divsalar","sequence":"additional","affiliation":[]},{"given":"G.","family":"Montorsi","sequence":"additional","affiliation":[]},{"given":"F.","family":"Pollara","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998,5]]},"URL":"http://dx.doi.org/10.1109/18.669119","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-H.-D.-Pfister-private-commun","type":"document","author":[{"family":"Pfister","given":"H.D."}],"genre":"private communication,","issued":"2022"},{"id":"preset:Johnson06","type":"article-journal","author":[{"family":"Johnson","given":"S.J."}],"title":"Introducing low-density parity-check codes","volume":"1","container-title":"University of Newcastle, Australia","issued":"2006","page":"2006"},{"id":"manual:-D.-Divsalar-H.-Jin-and-R.-J.","type":"paper-conference","author":[{"family":"Divsalar","given":"D."},{"family":"Jin","given":"H."},{"family":"McEliece","given":"R.J."}],"title":"Coding theorems for\" turbo-like\" codes","container-title":"Proceedings of the annual Allerton Conference on Communication control and Computing","volume":"36","publisher":"University Of Illinois","issued":"1998"},{"id":"doi:10.1109/TIT.2007.892805","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2007.892805","page":"1479-1483","source":"Crossref","title":"Construction of Irregular LDPC Codes by Quasi-Cyclic Extension","volume":"53","author":[{"given":"Jinghu","family":"Chen","sequence":"first","affiliation":[]},{"given":"R. Michael","family":"Tanner","sequence":"additional","affiliation":[]},{"given":"Juntan","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Marc P. C.","family":"Fossorier","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2007,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2007.892805","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-R.-M.-Tanner-On-quasi-cyclic","type":"paper-conference","author":[{"family":"Tanner","given":"R.M."}],"title":"On quasi-cyclic repeat-accumulate codes","container-title":"Proceedings of the 37th Annual Allerton Conference on Communication, Control, and Computing","volume":"37","issued":"1999-09","publisher-place":"Monticello, IL"},{"id":"doi:10.1109/TIT.2008.2008114","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2008.2008114","page":"6-15","source":"Crossref","title":"The Minimum Distance of Turbo-Like Codes","volume":"55","author":[{"given":"Louay","family":"Bazzi","sequence":"first","affiliation":[]},{"given":"Mohammad","family":"Mahdian","sequence":"additional","affiliation":[]},{"given":"Daniel A.","family":"Spielman","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2009,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.2008114","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:MacKay","type":"book","author":[{"family":"MacKay","given":"D.J.C."}],"title":"\\emph{Information Theory, Inference and Learning Algorithms}","publisher":"Cambridge University Press","issued":"2003"},{"id":"doi:10.1109/ISIT.2008.4595333","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2008.4595333","page":"1968-1972","source":"Crossref","title":"Explicit interleavers for a Repeat Accumulate Accumulate (RAA) code construction","author":[{"given":"Venkatesan","family":"Guruswami","sequence":"first","affiliation":[]},{"given":"Widad","family":"Machmouchi","sequence":"additional","affiliation":[]}],"event":"2008 IEEE International Symposium on Information Theory - ISIT","container-title":"2008 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2008.4595333"},{"id":"doi:10.1109/LCOMM.2003.813816","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/lcomm.2003.813816","page":"266-268","source":"Crossref","title":"Approaching Shannon performance by iterative decoding of linear codes with low-density generator matrix","volume":"7","author":[{"given":"J.","family":"Garcia-Frias","sequence":"first","affiliation":[]},{"family":"Wei Zhong","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Letters","original-title":[],"issued":{"date-parts":[[2003,6]]},"URL":"http://dx.doi.org/10.1109/LCOMM.2003.813816","ISSN":["1089-7798"],"container-title-short":"IEEE Commun. Lett."},{"id":"doi:10.1109/SPAWC.2005.1506298","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/spawc.2005.1506298","page":"1043-1047","source":"Crossref","title":"Quantum error-correction using codes with low density generator matrix","author":[{"family":"Hatuying Lou","sequence":"first","affiliation":[]},{"given":"J.","family":"Garcia-Frias","sequence":"additional","affiliation":[]}],"event":"IEEE 6th Workshop on Signal Processing Advances in Wireless Communications, 2005.","container-title":"IEEE 6th Workshop on Signal Processing Advances in Wireless Communications, 2005.","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SPAWC.2005.1506298"},{"id":"manual:-H.-Lou-and-J.-Garcia-Frias-O","type":"paper-conference","author":[{"family":"Lou","given":"H."},{"family":"Garcia-Frias","given":"J."}],"title":"On the Application of Error-Correcting Codes with Low-Density Generator Matrix over Different Quantum Channels","container-title":"4th International Symposium on Turbo Codes & Related Topics; 6th International ITG-Conference on Source and Channel Coding","issued":"2006","page":"1–6","publisher-place":"Munich, Germany"},{"id":"doi:10.1109/CISS.2008.4558588","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/ciss.2008.4558588","page":"562-567","source":"Crossref","title":"Design of near-optimum quantum error-correcting codes based on generator and parity-check matrices of LDGM codes","author":[{"given":"Javier","family":"Garcia-Frias","sequence":"first","affiliation":[]},{"family":"Kejing Liu","sequence":"additional","affiliation":[]}],"event":"2008 42nd Annual Conference on Information Sciences and Systems (CISS)","container-title":"2008 42nd Annual Conference on Information Sciences and Systems","original-title":[],"issued":{"date-parts":[[2008,3]]},"URL":"http://dx.doi.org/10.1109/CISS.2008.4558588"},{"id":"doi:10.1103/PhysRevA.103.022617","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.103.022617","source":"Crossref","title":"Design of low-density-generator-matrix–based quantum codes for asymmetric quantum channels","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0003-0576-0239","authenticated-orcid":true,"given":"Patricio","family":"Fuentes","sequence":"first","affiliation":[]},{"given":"Josu Etxezarreta","family":"Martinez","sequence":"additional","affiliation":[]},{"given":"Pedro M.","family":"Crespo","sequence":"additional","affiliation":[]},{"given":"Javier","family":"Garcia-Frias","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.022617","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022617"},{"id":"doi:10.1103/PhysRevA.102.012423","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.102.012423","source":"Crossref","title":"Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0003-0576-0239","authenticated-orcid":true,"given":"Patricio","family":"Fuentes","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7058-8426","authenticated-orcid":true,"given":"Josu","family":"Etxezarreta Martinez","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8259-3353","authenticated-orcid":true,"given":"Pedro M.","family":"Crespo","sequence":"additional","affiliation":[]},{"given":"Javier","family":"Garcia-Frias","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.012423","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012423"},{"id":"doi:10.1109/QCE49297.2020.00022","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce49297.2020.00022","page":"93-101","source":"Crossref","title":"Performance of non-CSS LDGM-based quantum codes over the misidentified depolarizing channel","author":[{"given":"Patricio","family":"Fuentes","sequence":"first","affiliation":[]},{"given":"Josu Etxezarreta","family":"Martinez","sequence":"additional","affiliation":[]},{"given":"Pedro M.","family":"Crespo","sequence":"additional","affiliation":[]},{"given":"Javier","family":"Garcia-Frias","sequence":"additional","affiliation":[]}],"event":"2020 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2020 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2020,10]]},"URL":"http://dx.doi.org/10.1109/QCE49297.2020.00022"},{"id":"arxiv:0903.0566","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2013.2292061","page":"1193-1202","source":"Crossref","title":"Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength","volume":"60","author":[{"given":"Jean-Pierre","family":"Tillich","sequence":"first","affiliation":[]},{"given":"Gilles","family":"Zemor","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2292061","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0903.0566"},{"id":"doi:10.1109/TIT.1962.1057683","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1962.1057683","page":"21-28","source":"Crossref","title":"Low-density parity-check codes","volume":"8","author":[{"given":"R.","family":"Gallager","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1962,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1962.1057683","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Gallager63","type":"thesis","author":[{"family":"Gallager","given":"R."}],"title":"Low-density parity check codes","genre":"\\href{https://dspace.mit.edu/bitstream/handle/1721.1/11804/32786367-MIT.pdf}{PhD thesis},","publisher":"MIT","issued":"1963","publisher-place":"Cambridge, MA"},{"id":"manual:-L.-A.-Bassalygo-and-M.-S.-Pi","type":"article-journal","author":[{"family":"Bassalygo","given":"L.A."},{"family":"Pinsker","given":"M.S."}],"title":"The complexity of an optimal non-blocking commutation scheme without reorganization","volume":"9","container-title":"Problemy Peredaci Informacii","issue":"1","issued":"1973","page":"84–87"},{"id":"arxiv:cs/0610022","type":"article-journal","author":[{"given":"Venkatesan","family":"Guruswami"}],"title":"Iterative Decoding of Low-Density Parity Check Codes (A Survey)","issued":{"date-parts":[[2006,10,5]]},"note":"arxivid:cs/0610022\narxiv_version_number:1"},{"id":"arxiv:1201.2999","type":"article-journal","author":[{"given":"Shrinivas","family":"Kudekar"},{"given":"Tom","family":"Richardson"},{"given":"Ruediger","family":"Urbanke"}],"title":"Spatially Coupled Ensembles Universally Achieve Capacity under Belief Propagation","issued":{"date-parts":[[2012,1,14]]},"note":"arxivid:1201.2999\narxiv_version_number:1"},{"id":"arxiv:1909.06430","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","DOI":"10.1137/20m1365934","page":"FOCS20-38-FOCS20-73","source":"Crossref","title":"Low-Density Parity-Check Codes Achieve List-Decoding Capacity","volume":"53","author":[{"ORCID":"https://orcid.org/0000-0002-7947-1205","authenticated-orcid":true,"given":"Jonathan","family":"Mosheiff","sequence":"first","affiliation":[]},{"given":"Nicolas","family":"Resch","sequence":"additional","affiliation":[]},{"given":"Noga","family":"Ron-Zewi","sequence":"additional","affiliation":[]},{"given":"Shashwat","family":"Silas","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2345-2531","authenticated-orcid":true,"given":"Mary","family":"Wootters","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[2021,11,23]]},"URL":"http://dx.doi.org/10.1137/20M1365934","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/20M1365934\narxivid:1909.06430"},{"id":"doi:10.1109/18.910579","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910579","page":"638-656","source":"Crossref","title":"Efficient encoding of low-density parity-check codes","volume":"47","author":[{"given":"T.J.","family":"Richardson","sequence":"first","affiliation":[]},{"given":"R.L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910579","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-J.-Pearl-(2022).-Reverend-Ba","type":"chapter","author":[{"family":"Pearl","given":"J."}],"title":"Reverend Bayes on inference engines: A distributed hierarchical approach","container-title":"Probabilistic and Causal Inference: The Works of Judea Pearl","issued":"2022","page":"129–138"},{"id":"doi:10.1016/C2009-0-27609-4","type":"edited-book","publisher":"Elsevier","DOI":"10.1016/c2009-0-27609-4","source":"Crossref","title":"Probabilistic Reasoning in Intelligent Systems","container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[1988]]},"ISBN":["9780080514895"],"URL":"http://dx.doi.org/10.1016/C2009-0-27609-4"},{"id":"doi:10.1109/18.910572","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910572","page":"498-519","source":"Crossref","title":"Factor graphs and the sum-product algorithm","volume":"47","author":[{"given":"F.R.","family":"Kschischang","sequence":"first","affiliation":[]},{"given":"B.J.","family":"Frey","sequence":"additional","affiliation":[]},{"given":"H.-A.","family":"Loeliger","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910572","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ISCAS.2006.1693779","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/iscas.2006.1693779","page":"4","source":"Crossref","title":"A parallel LSI architecture for LDPC decoder improving message-passing schedule","author":[{"given":"K.","family":"Shimizu","sequence":"first","affiliation":[]},{"given":"T.","family":"Ishikawa","sequence":"additional","affiliation":[]},{"given":"N.","family":"Togawa","sequence":"additional","affiliation":[]},{"given":"T.","family":"Ikenaga","sequence":"additional","affiliation":[]},{"given":"S.","family":"Goto","sequence":"additional","affiliation":[]}],"event":"2006 IEEE International Symposium on Circuits and Systems","container-title":"2006 IEEE International Symposium on Circuits and Systems","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISCAS.2006.1693779"},{"id":"doi:10.1109/TCOMM.2005.852852","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tcomm.2005.852852","page":"1288-1299","source":"Crossref","title":"Reduced-Complexity Decoding of LDPC Codes","volume":"53","author":[{"given":"J.","family":"Chen","sequence":"first","affiliation":[]},{"given":"A.","family":"Dholakia","sequence":"additional","affiliation":[]},{"given":"E.","family":"Eleftheriou","sequence":"additional","affiliation":[]},{"given":"M.P.C.","family":"Fossorier","sequence":"additional","affiliation":[]},{"given":"X.-Y.","family":"Hu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[2005,8]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2005.852852","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"manual:-J.-Feldman.-Decoding-Error-C","type":"thesis","author":[{"family":"Feldman","given":"J."}],"title":"Decoding Error-Correcting Codes via Linear Programming","genre":"PhD thesis,","publisher":"Massachusetts Institute of Technology","issued":"2003"},{"id":"doi:10.1109/TIT.2004.842696","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2004.842696","page":"954-972","source":"Crossref","title":"Using Linear Programming to Decode Binary Linear Codes","volume":"51","author":[{"given":"J.","family":"Feldman","sequence":"first","affiliation":[]},{"given":"M.J.","family":"Wainwright","sequence":"additional","affiliation":[]},{"given":"D.R.","family":"Karger","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2005,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.842696","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/978-1-4939-2864-4_216","type":"book-chapter","publisher":"Springer New York","DOI":"10.1007/978-1-4939-2864-4_216","page":"1177-1182","source":"Crossref","title":"LP Decoding","author":[{"given":"Jonathan","family":"Feldman","sequence":"first","affiliation":[]}],"container-title":"Encyclopedia of Algorithms","original-title":[],"language":"en","issued":{"date-parts":[[2016]]},"ISBN":["9781493928637","9781493928644"],"URL":"http://dx.doi.org/10.1007/978-1-4939-2864-4_216","note":"publisher-location:New York, NY"},{"id":"doi:10.1049/el:20030217","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"3","DOI":"10.1049/el:20030217","page":"299-301","source":"Crossref","title":"Iterative decoding using stochastic computation","volume":"39","author":[{"given":"V.C.","family":"Gaudet","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Alberta, 2nd Floor ECERF Building (Main Office), Edmonton, Alberta T6G 2V4, Canada"}]},{"given":"A.C.","family":"Rapley","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Alberta, 2nd Floor ECERF Building (Main Office), Edmonton, Alberta T6G 2V4, Canada"}]}],"container-title":"Electronics Letters","original-title":[],"language":"en","issued":{"date-parts":[[2003,2,6]]},"URL":"http://dx.doi.org/10.1049/el:20030217","ISSN":["0013-5194","1350-911X"],"container-title-short":"Electron. Lett.","note":"alternative-id:10.1049/el:20030217"},{"id":"doi:10.1109/TIT.2002.1003839","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2002.1003839","page":"1570-1579","source":"Crossref","title":"Finite-length analysis of low-density parity-check codes on the binary erasure channel","volume":"48","author":[{"family":"Changyan Di","sequence":"first","affiliation":[]},{"given":"D.","family":"Proietti","sequence":"additional","affiliation":[]},{"given":"I.E.","family":"Telatar","sequence":"additional","affiliation":[]},{"given":"T.J.","family":"Richardson","sequence":"additional","affiliation":[]},{"given":"R.L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2002,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.1003839","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2005.864441","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2005.864441","page":"922-932","source":"Crossref","title":"On the stopping distance and the stopping redundancy of codes","volume":"52","author":[{"given":"M.","family":"Schwartz","sequence":"first","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2006,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.864441","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2112.13934","type":"article-journal","author":[{"given":"Salman","family":"Habib"},{"given":"Allison","family":"Beemer"},{"given":"Joerg","family":"Kliewer"}],"title":"RELDEC: Reinforcement Learning-Based Decoding of Moderate Length LDPC Codes","issued":{"date-parts":[[2023,7,27]]},"note":"arxivid:2112.13934\narxiv_version_number:3"},{"id":"arxiv:2412.08596","type":"article-journal","author":[{"given":"Sheila M.","family":"Perez-Garcia"},{"given":"Ashley","family":"Montanaro"}],"title":"Quantum-enhanced belief propagation for LDPC decoding","issued":{"date-parts":[[2024,12,11]]},"note":"arxivid:2412.08596\narxiv_version_number:1"},{"id":"doi:10.1049/el:19970362","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"6","DOI":"10.1049/el:19970362","page":"457-458","source":"Crossref","title":"Near Shannon limit performance of low density paritycheck codes","volume":"33","author":[{"given":"D.J.C.","family":"MacKay","sequence":"first","affiliation":[{"name":"Cavendish Laboratory, Cambridge, CB3 0HE, United Kingdom"}]},{"given":"R.M.","family":"Neal","sequence":"additional","affiliation":[{"name":"Departments of Statistics and Computer Science, University of Toronto, MS 1A4, Canada"}]}],"container-title":"Electronics Letters","original-title":[],"language":"en","issued":{"date-parts":[[1997,3,13]]},"URL":"http://dx.doi.org/10.1049/el:19970362","ISSN":["0013-5194","1350-911X"],"container-title-short":"Electron. Lett.","note":"alternative-id:10.1049/el:19970362"},{"id":"doi:10.1093/acprof:oso/9780198570837","type":"edited-book","publisher":"Oxford University PressOxford","abstract":"<jats:title>Abstract</jats:title><jats:p>This book presents a unified approach to a rich and rapidly evolving research domain at the interface between statistical physics, theoretical computer science/discrete mathematics, and coding/information theory. The topics which have been selected, including spin glasses, error correcting codes, satisfiability, are central to each field. The approach focuses on the limit of large random instances, adopting a common formulation in terms of graphical models. It presents message passing algorithms like belief propagation and survey propagation, and their use in decoding and constraint satisfaction solving. It also explains analysis techniques like density evolution and the cavity method, and uses them to derive phase diagrams and study phase transitions.</jats:p>","DOI":"10.1093/acprof:oso/9780198570837.001.0001","source":"Crossref","title":"Information, Physics, and Computation","author":[{"given":"Marc","family":"Mézard","sequence":"first","affiliation":[{"name":"Laboratoire de Physique Théorique et Modeles Statistiques, Université de Paris Sud, Orsay, France"}]},{"given":"Andrea","family":"Montanari","sequence":"additional","affiliation":[{"name":"Electrical Engineering and Statistics Department, Stanford University, USA"}]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2009,1,22]]},"ISBN":["019857083X","9780198570837","9780191718755"],"URL":"http://dx.doi.org/10.1093/acprof:oso/9780198570837.001.0001","note":"edition-number:1"},{"id":"doi:10.1093/acprof:oso/9780198509417","type":"edited-book","publisher":"Oxford University PressOxford","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Spin glasses are magnetic materials with strong disorder. Statistical mechanics has been a powerful tool to theoretically analyse various unique properties of spin glasses. A number of new analytical techniques have been developed to establish a theory of spin glasses. Surprisingly, these techniques have offered new tools and viewpoints for the understanding of information processing problems, including neural networks, error-correcting codes, image restoration, and optimization problems. A vast, interdisciplinary field has consequently been developing between physics and information, or more specifically, between the statistical physics of spin glasses and several important aspects of information processing tasks. This book provides a broad overview of this new field. It also contains detailed descriptions of the theory of spin glasses.</jats:p>","DOI":"10.1093/acprof:oso/9780198509417.001.0001","source":"Crossref","title":"Statistical Physics of Spin Glasses and Information Processing","author":[{"given":"Hidetoshi","family":"Nishimori","sequence":"first","affiliation":[{"name":"Department of Physics, Tokyo Institute of Technology"}]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2001,7,19]]},"ISBN":["0198509413","9780198509417","9780191709081"],"URL":"http://dx.doi.org/10.1093/acprof:oso/9780198509417.001.0001","note":"edition-number:1"},{"id":"doi:10.1088/0305-4470/37/6/R01","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/0305-4470/37/6/r01","page":"R1-R43","source":"Crossref","title":"Statistical mechanics of low-density parity-check codes","volume":"37","author":[{"given":"Yoshiyuki","family":"Kabashima","sequence":"first","affiliation":[]},{"given":"David","family":"Saad","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[2004,1,28]]},"URL":"http://dx.doi.org/10.1088/0305-4470/37/6/R01","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen.","note":"alternative-id:S0305-4470(04)34343-X"},{"id":"doi:10.1016/S1076-5670(02)80018-0","type":"book-chapter","publisher":"Elsevier","DOI":"10.1016/s1076-5670(02)80018-0","page":"231-353","source":"Crossref","title":"Low-density parity-check codes—A statistical physics perspective","author":[{"given":"Renato","family":"Vicente","sequence":"first","affiliation":[]},{"given":"David","family":"Saad","sequence":"additional","affiliation":[]},{"given":"Yoshiyuki","family":"Kabashima","sequence":"additional","affiliation":[]}],"container-title":"Advances in Imaging and Electron Physics","original-title":[],"issued":{"date-parts":[[2003]]},"URL":"http://dx.doi.org/10.1016/S1076-5670(02)80018-0","ISSN":["1076-5670"]},{"id":"doi:10.1016/S0924-8099(07)80009-6","type":"book-chapter","publisher":"Elsevier","DOI":"10.1016/s0924-8099(07)80009-6","page":"67-130","source":"Crossref","title":"Course 2 Modern coding theory: The statistical mechanics and computer science point of view","author":[{"given":"Andrea","family":"Montanari","sequence":"first","affiliation":[]},{"given":"Rüdiger","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"Les Houches","original-title":[],"issued":{"date-parts":[[2007]]},"URL":"http://dx.doi.org/10.1016/S0924-8099(07)80009-6","ISSN":["0924-8099"]},{"id":"doi:10.36471/JCCM_August_2025_01","type":"journal-article","publisher":"Journal Club for Condensed Matter Physics","DOI":"10.36471/jccm_august_2025_01","source":"Crossref","title":"Classical and quantum glasses based on expanders","author":[{"name":"MPI-PKS Dresden","sequence":"first","affiliation":[]},{"given":"Roderich","family":"Moessner","sequence":"first","affiliation":[]}],"container-title":"Journal Club for Condensed Matter Physics","original-title":[],"issued":{"date-parts":[[2025,8,31]]},"URL":"http://dx.doi.org/10.36471/JCCM_August_2025_01","container-title-short":"JCCMP"},{"id":"doi:10.1109/MCOM.2003.1222728","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/mcom.2003.1222728","page":"126-131","source":"Crossref","title":"The renaissance of gallager's low-density parity-check codes","volume":"41","author":[{"given":"T.","family":"Richardson","sequence":"first","affiliation":[]},{"given":"R.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Magazine","original-title":[],"language":"en","issued":{"date-parts":[[2003,8]]},"URL":"http://dx.doi.org/10.1109/MCOM.2003.1222728","ISSN":["0163-6804"],"container-title-short":"IEEE Commun. Mag."},{"id":"doi:10.1201/9780203490310","type":"book","publisher":"CRC Press","DOI":"10.1201/9780203490310","source":"Crossref","title":"Coding and Signal Processing for Magnetic Recording Systems","container-title":[],"original-title":[],"language":"en","editor":[{"given":"Bane","family":"Vasic","sequence":"first","affiliation":[]},{"given":"Erozan M.","family":"Kurtas","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2004,11,9]]},"ISBN":["9780203490310"],"URL":"http://dx.doi.org/10.1201/9780203490310","note":"edition-number:0"},{"id":"doi:10.1007/978-3-0348-7865-4_5","type":"book-chapter","publisher":"Birkhäuser Basel","DOI":"10.1007/978-3-0348-7865-4_5","page":"85-110","source":"Crossref","title":"LDPC Codes: An Introduction","author":[{"given":"Amin","family":"Shokrollahi","sequence":"first","affiliation":[]}],"container-title":"Coding, Cryptography and Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2004]]},"ISBN":["9783034896023","9783034878654"],"URL":"http://dx.doi.org/10.1007/978-3-0348-7865-4_5","note":"publisher-location:Basel"},{"id":"manual:-D.-J.-C.-MacKay.-Encyclopedi","type":"document","author":[{"family":"MacKay","given":"D.J.C."}],"title":"Encyclopedia of sparse graph codes","issued":"2005"},{"id":"arxiv:2212.01121","type":"journal-article","publisher":"MDPI AG","issue":"1","abstract":"<jats:p>We develop a new approach for asymmetric LDPC-based information reconciliation in order to adapt to the current channel state and achieve better performance and scalability in practical resource-constrained QKD systems. The new scheme combines the advantages of LDPC codes, a priori error rate estimation, rate-adaptive and blind information reconciliation techniques. We compare the performance of several asymmetric and symmetric error correction schemes using a real industrial QKD setup. The proposed asymmetric algorithm achieves significantly higher throughput, providing a secret key rate that is close to the symmetric one in a wide range of error rates. Thus, our approach is found to be particularly efficient for applications with high key rates, limited classical channel capacity and asymmetric computational resource allocation.</jats:p>","DOI":"10.3390/e25010031","page":"31","source":"Crossref","title":"Asymmetric Adaptive LDPC-Based Information Reconciliation for Industrial Quantum Key Distribution","volume":"25","author":[{"given":"Nikolay","family":"Borisov","sequence":"first","affiliation":[{"name":"Laboratory of Quantum Information Technologies, National University of Science and Technology “MISIS”, Moscow 119049, Russia"}]},{"given":"Ivan","family":"Petrov","sequence":"additional","affiliation":[{"name":"Laboratory of Quantum Information Technologies, National University of Science and Technology “MISIS”, Moscow 119049, Russia"}]},{"given":"Andrey","family":"Tayduganov","sequence":"additional","affiliation":[{"name":"Laboratory of Quantum Information Technologies, National University of Science and Technology “MISIS”, Moscow 119049, Russia"}]}],"container-title":"Entropy","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,23]]},"URL":"http://dx.doi.org/10.3390/e25010031","ISSN":["1099-4300"],"container-title-short":"Entropy","note":"alternative-id:e25010031\narxivid:2212.01121"},{"id":"arxiv:2005.07016","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevresearch.2.043423","source":"Crossref","title":"Decoding across the quantum low-density parity-check code landscape","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0001-9202-1156","authenticated-orcid":true,"given":"Joschka","family":"Roffe","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6317-4348","authenticated-orcid":true,"given":"David R.","family":"White","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8932-3492","authenticated-orcid":true,"given":"Simon","family":"Burton","sequence":"additional","affiliation":[]},{"given":"Earl","family":"Campbell","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.043423","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"043423","note":"arxivid:2005.07016"},{"id":"preset:Roffe22","type":"document","author":[{"family":"Roffe","given":"J."}],"title":"LDPC: Python tools for low density parity check codes","issued":"2022"},{"id":"doi:10.1007/11830924_29","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/11830924_29","page":"304-315","source":"Crossref","title":"Robust Local Testability of Tensor Products of LDPC Codes","author":[{"given":"Irit","family":"Dinur","sequence":"first","affiliation":[]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[]},{"given":"Avi","family":"Wigderson","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2006]]},"ISBN":["9783540380443","9783540380450"],"URL":"http://dx.doi.org/10.1007/11830924_29","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1007/978-3-540-85363-3_24","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-540-85363-3_24","page":"290-302","source":"Crossref","title":"Tensor Products of Weakly Smooth Codes Are Robust","author":[{"given":"Eli","family":"Ben-Sasson","sequence":"first","affiliation":[]},{"given":"Michael","family":"Viderman","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2008]]},"ISBN":["9783540853626","9783540853633"],"URL":"http://dx.doi.org/10.1007/978-3-540-85363-3_24","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:cond-mat/0205051","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreve.66.046120","source":"Crossref","title":"Dynamic phase transition for decoding algorithms","volume":"66","author":[{"family":"Franz","sequence":"first","affiliation":[]},{"given":"Michele","family":"Leone","sequence":"additional","affiliation":[]},{"given":"Andrea","family":"Montanari","sequence":"additional","affiliation":[]},{"given":"Federico","family":"Ricci-Tersenghi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review E","original-title":[],"language":"en","issued":{"date-parts":[[2002,10,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevE.66.046120","ISSN":["1063-651X","1095-3787"],"container-title-short":"Phys. Rev. E","page":"046120","note":"arxivid:cond-mat/0205051"},{"id":"manual:-P.-Gaborit-G.-Murat-O.-Ruatt","type":"paper-conference","author":[{"family":"Gaborit","given":"P."},{"family":"Murat","given":"G."},{"family":"Ruatta","given":"O."},{"family":"Zemor","given":"G."}],"title":"Low rank parity check codes and their application to cryptography","container-title":"Proceedings of the Workshop on Coding and Cryptography WCC","volume":"2013","issued":"2013-04"},{"id":"doi:10.1109/ACCESS.2020.3021700","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/access.2020.3021700","page":"162892-162902","source":"Crossref","title":"Error Rate-Based Log-Likelihood Ratio Processing for Low-Density Parity-Check Codes in DNA Storage","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0003-1056-2421","authenticated-orcid":false,"given":"Xiaozhou","family":"Lu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1055-618X","authenticated-orcid":false,"given":"Jaeho","family":"Jeong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1608-5849","authenticated-orcid":false,"given":"Jae-Won","family":"Kim","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3946-0958","authenticated-orcid":false,"given":"Jong-Seon","family":"No","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7854-7792","authenticated-orcid":false,"given":"Hosung","family":"Park","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6346-4182","authenticated-orcid":false,"given":"Albert","family":"No","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1762-5915","authenticated-orcid":false,"given":"Sunghwan","family":"Kim","sequence":"additional","affiliation":[]}],"container-title":"IEEE Access","original-title":[],"issued":{"date-parts":[[2020]]},"URL":"http://dx.doi.org/10.1109/ACCESS.2020.3021700","ISSN":["2169-3536"],"container-title-short":"IEEE Access"},{"id":"arxiv:2401.09541","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n          <jats:p>The main obstacle to large scale quantum computing are the errors present in every physical qubit realization. Correcting these errors requires a large number of additional qubits. Two main avenues to reduce this overhead are (i) low-density parity check (LDPC) codes requiring very few additional qubits to correct errors (ii) cat qubits where bit-flip errors are exponentially suppressed by design. In this work, we combine both approaches to obtain an extremely low overhead architecture. Assuming a physical phase-flip error probability <jats:italic>ϵ</jats:italic> ≈ 0.1% per qubit and operation, one hundred logical qubits can be implemented on a 758 cat qubit chip, with a total logical error probability per cycle and per logical qubit <jats:italic>ϵ</jats:italic>\n            <jats:sub>\n              <jats:italic>L</jats:italic>\n            </jats:sub> ≤ 10<jats:sup>−8</jats:sup>. Our architecture also features two major advantages. First, the hardware implementation of the code can be realised with short-range qubit interactions in 2D and low-weight stabilizers, under constraints similar to those of the popular surface code architecture. Second, we demonstrate how to implement a fault-tolerant universal set of logical gates with an additional layer of routing cat qubits stacked on top of the LDPC layer, while maintaining the local connectivity. Furthermore, our architecture benefits from a high capacity of parallelization for these logical gates.</jats:p>","DOI":"10.1038/s41467-025-56298-8","source":"Crossref","title":"LDPC-cat codes for low-overhead quantum computing in 2D","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0003-2003-7030","authenticated-orcid":false,"given":"Diego","family":"Ruiz","sequence":"first","affiliation":[]},{"given":"Jérémie","family":"Guillaud","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9471-6031","authenticated-orcid":false,"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3445-0179","authenticated-orcid":false,"given":"Christophe","family":"Vuillot","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,26]]},"URL":"http://dx.doi.org/10.1038/s41467-025-56298-8","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"1040","note":"alternative-id:56298\narxivid:2401.09541"},{"id":"arxiv:2403.10599","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41467-024-55570-7","source":"Crossref","title":"Quantum memory at nonzero temperature in a thermodynamically trivial system","volume":"16","author":[{"given":"Yifan","family":"Hong","sequence":"first","affiliation":[]},{"given":"Jinkang","family":"Guo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3098-5683","authenticated-orcid":false,"given":"Andrew","family":"Lucas","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,2]]},"URL":"http://dx.doi.org/10.1038/s41467-024-55570-7","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"316","note":"alternative-id:55570\narxivid:2403.10599"},{"id":"arxiv:2403.10268","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>In fault-tolerant quantum computing, quantum algorithms are implemented through quantum circuits capable of error correction. These circuits are typically constructed based on specific quantum error correction codes, with consideration given to the characteristics of the underlying physical platforms. Optimizing these circuits within the constraints of today's quantum computing technologies, particularly in terms of error rates, qubit counts, and network topologies, holds substantial implications for the feasibility of quantum applications in the near future. This paper presents a toolkit for designing and analyzing fault-tolerant quantum circuits. We introduce a framework for representing stabilizer circuits using classical low-density parity-check (LDPC) codes. Each codeword in the representation corresponds to a quantum-mechanical equation regarding the circuit, formalizing the correlations utilized in parity checks and delineating logical operations within the circuit. Consequently, the LDPC code provides a means of quantifying fault tolerance and verifying logical operations. We outline the procedure for generating LDPC codes from circuits using the Tanner graph notation, alongside proposing graph-theory tools for constructing fault-tolerant quantum circuits from classical LDPC codes. These findings offer a systematic approach to applying classical error correction techniques in optimizing existing fault-tolerant protocols and developing new ones. As an example, we develop a resource-efficient scheme for universal fault-tolerant quantum computing on hypergraph product codes based on the LDPC representation.</jats:p>","DOI":"10.1103/physrevresearch.7.013115","source":"Crossref","title":"Low-density parity-check representation of fault-tolerant quantum circuits","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-1705-2494","authenticated-orcid":true,"given":"Ying","family":"Li","sequence":"first","affiliation":[{"name":"China Academy of Engineering Physics"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.7.013115","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013115","note":"arxivid:2403.10268"},{"id":"arxiv:1108.0679","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2013.2247461","page":"3347-3353","source":"Crossref","title":"A Characterization of Entanglement-Assisted Quantum Low-Density Parity-Check Codes","volume":"59","author":[{"given":"Yuichiro","family":"Fujiwara","sequence":"first","affiliation":[]},{"given":"Vladimir D.","family":"Tonchev","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2247461","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1108.0679"},{"id":"arxiv:1108.0840","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2011.2176717","page":"2265-2279","source":"Crossref","title":"Searching for Voltage Graph-Based LDPC Tailbiting Codes With Large Girth","volume":"58","author":[{"given":"Irina E.","family":"Bocharova","sequence":"first","affiliation":[]},{"given":"Florian","family":"Hug","sequence":"additional","affiliation":[]},{"given":"Rolf","family":"Johannesson","sequence":"additional","affiliation":[]},{"given":"Boris D.","family":"Kudryashov","sequence":"additional","affiliation":[]},{"given":"Roman V.","family":"Satyukov","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2012,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2011.2176717","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1108.0840"},{"id":"doi:10.1109/ISIT.2006.261870","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2006.261870","page":"674-678","source":"Crossref","title":"A Combining Method of Structured LDPC Codes from Affine Permutation Matrices","author":[{"given":"Seho","family":"Myung","sequence":"first","affiliation":[{"name":"Dept. of Electronics and Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Kyungbuk 790-784, Korea. Email: bighead@postech.ac.kr"}]},{"given":"Kyeongcheol","family":"Yang","sequence":"additional","affiliation":[{"name":"Dept. of Electronics and Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Kyungbuk 790-784, Korea. Email: kcyang@postech.ac.kr"}]},{"given":"Dong Seek","family":"Park","sequence":"additional","affiliation":[{"name":"Advanced Research Lab., Global Standards and Research Team, Samsung Electronics Co., LTD, Korea. Email: dspark@samsung.com"}]}],"event":"2006 IEEE International Symposium on Information Theory","container-title":"2006 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2006,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2006.261870"},{"id":"arxiv:2504.17790","type":"article-journal","author":[{"given":"Kenta","family":"Kasai"}],"title":"Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction","issued":{"date-parts":[[2025,4,28]]},"note":"arxivid:2504.17790\narxiv_version_number:2"},{"id":"doi:10.1109/ISIT.2002.1023554","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2002.1023554","page":"282","source":"Crossref","title":"Efficient encoding and minimum distance bounds of Reed-Solomon-type array codes","author":[{"given":"T.","family":"Mittelholzer","sequence":"first","affiliation":[]}],"event":"IEEE International Symposium on Information Theory","container-title":"Proceedings IEEE International Symposium on Information Theory,","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2002.1023554"},{"id":"manual:-J.-L.-Fan-Array-codes-as-low","type":"paper-conference","author":[{"family":"Fan","given":"J.L."}],"title":"Array codes as low-density parity-check codes","container-title":"Proc. 2nd Int. Symp","publisher":"on Turbo Codes & Related Topics","issued":"2000-09","publisher-place":"Brest, France"},{"id":"doi:10.1007/978-1-4615-1525-8_7","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-1-4615-1525-8_7","page":"195-203","source":"Crossref","title":"Array Codes as LDPC Codes","author":[{"given":"John L.","family":"Fan","sequence":"first","affiliation":[]}],"container-title":"Constrained Coding and Soft Iterative Decoding","original-title":[],"issued":{"date-parts":[[2001]]},"ISBN":["9781461355977","9781461515258"],"URL":"http://dx.doi.org/10.1007/978-1-4615-1525-8_7","note":"publisher-location:Boston, MA"},{"id":"doi:10.1109/ICC.2002.997149","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.2002.997149","page":"1752-1757","source":"Crossref","title":"Low-density parity-check codes for digital subscriber lines","volume":"3","author":[{"given":"E.","family":"Eleftheriou","sequence":"first","affiliation":[]},{"given":"S.","family":"Olcer","sequence":"additional","affiliation":[]}],"event":"IEEE International Conference on Communications","container-title":"2002 IEEE International Conference on Communications. Conference Proceedings. ICC 2002 (Cat. No.02CH37333)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ICC.2002.997149"},{"id":"doi:10.1109/TIT.2005.851753","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2005.851753","page":"2894-2901","source":"Crossref","title":"Quasi-cyclic LDPC codes for fast encoding","volume":"51","author":[{"family":"Seho Myung","sequence":"first","affiliation":[{"name":"Dept. of Electron. & Electr. Eng., Pohang Univ. of Sci. & Technol., Kyungbuk, South Korea"}]},{"family":"Kyeongcheol Yang","sequence":"additional","affiliation":[{"name":"Dept. of Electron. & Electr. Eng., Pohang Univ. of Sci. & Technol., Kyungbuk, South Korea"}]},{"family":"Jaeyoel Kim","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2005,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.851753","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1002/j.1538-7305.1966.tb01686","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1002/j.1538-7305.1966.tb01686.x","page":"1045-1055","source":"Crossref","title":"Difference-Set Cyclic Codes","volume":"45","author":[{"suffix":"Jr.","given":"E. J.","family":"Weldon","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1966,9]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1966.tb01686.x","ISSN":["0005-8580"]},{"id":"doi:10.1023/A:1008264606494","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1023/a:1008264606494","page":"89-98","source":"Crossref","title":"Difference Sets and Hyperovals","volume":"14","author":[{"given":"Antonio","family":"Maschietti","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1998,4]]},"URL":"http://dx.doi.org/10.1023/A:1008264606494","ISSN":["0925-1022","1573-7586"],"container-title-short":"Designs, Codes and Cryptography","note":"alternative-id:153105"},{"id":"manual:-H.-Jin-T.-Richardson-and-V.-","type":"document","author":[{"family":"Jin","given":"H."},{"family":"Richardson","given":"T."},{"family":"Novichkov","given":"V."},{"family":"Patent","given":"E.R.R.O.R.C.O.R.R.E.C.T.I.O.N.O.F.A.L.G.E.B.R.A.I.C.B.L.O.C.K.C.O.D.E.S.U.S."}],"volume":"Number US8751902B2","issued":"2002"},{"id":"doi:10.1109/ISIEA.2009.5356472","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isiea.2009.5356472","page":"239-242","source":"Crossref","title":"A new Quasi-Cyclic low density parity check codes","author":[{"given":"Abid","family":"Yahya","sequence":"first","affiliation":[]},{"given":"Othman","family":"Sidek","sequence":"additional","affiliation":[]},{"given":"M. F.","family":"M. Salleh","sequence":"additional","affiliation":[]},{"given":"Farid","family":"Ghani","sequence":"additional","affiliation":[]}],"event":"Applications (ISIEA 2009)","container-title":"2009 IEEE Symposium on Industrial Electronics &amp; Applications","original-title":[],"issued":{"date-parts":[[2009,10]]},"URL":"http://dx.doi.org/10.1109/ISIEA.2009.5356472"},{"id":"doi:10.1109/ISIT.2002.1023584","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2002.1023584","page":"312","source":"Crossref","title":"Combinatorial constructions of low-density parity check codes for iterative decoding","author":[{"given":"B.","family":"Vasic","sequence":"first","affiliation":[]}],"event":"IEEE International Symposium on Information Theory","container-title":"Proceedings IEEE International Symposium on Information Theory,","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2002.1023584"},{"id":"doi:10.1109/LCOMM.2003.814716","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/lcomm.2003.814716","page":"317-319","source":"Crossref","title":"A class of low-density parity-check codes constructed based on Reed-Solomon codes with two information symbols","volume":"7","author":[{"given":"I.","family":"Djurdjevic","sequence":"first","affiliation":[]},{"family":"Jun Xu","sequence":"additional","affiliation":[]},{"given":"K.","family":"Abdel-Ghaffar","sequence":"additional","affiliation":[]},{"family":"Shu Lin","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Letters","original-title":[],"issued":{"date-parts":[[2003,7]]},"URL":"http://dx.doi.org/10.1109/LCOMM.2003.814716","ISSN":["1089-7798"],"container-title-short":"IEEE Commun. Lett."},{"id":"doi:10.1109/ISIT.2003.1228165","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2003.1228165","page":"151","source":"Crossref","title":"Designing LDPC codes using cyclic shifts","author":[{"given":"T.","family":"Okamura","sequence":"first","affiliation":[]}],"event":"IEEE International Symposium on Information Theory, 2003. Proceedings.","container-title":"IEEE International Symposium on Information Theory, 2003. Proceedings.","original-title":[],"issued":{"date-parts":[[2003]]},"URL":"http://dx.doi.org/10.1109/ISIT.2003.1228165"},{"id":"doi:10.1109/TIT.2004.831841","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2004.831841","page":"1788-1793","source":"Crossref","title":"Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices","volume":"50","author":[{"given":"M.P.C.","family":"Fossorier","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2004,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.831841","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2011.2173244","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2011.2173244","page":"585-607","source":"Crossref","title":"Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on Minimum Hamming Distance Upper Bounds","volume":"58","author":[{"given":"Roxana","family":"Smarandache","sequence":"first","affiliation":[]},{"given":"Pascal O.","family":"Vontobel","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2012,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2011.2173244","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/MCOM.2018.1700839","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/mcom.2018.1700839","page":"28-34","source":"Crossref","title":"Design of Low-Density Parity Check Codes for 5G New Radio","volume":"56","author":[{"given":"Tom","family":"Richardson","sequence":"first","affiliation":[]},{"given":"Shrinivas","family":"Kudekar","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Magazine","original-title":[],"issued":{"date-parts":[[2018,3]]},"URL":"http://dx.doi.org/10.1109/MCOM.2018.1700839","ISSN":["0163-6804","1558-1896"],"container-title-short":"IEEE Commun. Mag."},{"id":"doi:10.1109/CSCITA47329.2020.9137797","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/cscita47329.2020.9137797","page":"208-213","source":"Crossref","title":"Coding Techniques for 5G Networks: A Review","author":[{"given":"Mrinmayi V","family":"Patil","sequence":"first","affiliation":[]},{"given":"Sanjay","family":"Pawar","sequence":"additional","affiliation":[]},{"given":"Zia","family":"Saquib","sequence":"additional","affiliation":[]}],"event":"2020 3rd International Conference on Communication System, Computing and IT Applications (CSCITA)","container-title":"2020 3rd International Conference on Communication System, Computing and IT Applications (CSCITA)","original-title":[],"issued":{"date-parts":[[2020,4]]},"URL":"http://dx.doi.org/10.1109/CSCITA47329.2020.9137797"},{"id":"manual:-LDPC-coding-for-OFDMA-PHY.-8","type":"document","title":"LDPC coding for OFDMA PHY. 802.16REVe Sponsor Ballot Recirculation comment","note":"IEEE C802.16e04/141r2","issued":"2004-07"},{"id":"doi:10.1109/PIMRC.2006.254126","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/pimrc.2006.254126","page":"1-5","source":"Crossref","title":"A Synthesizable IP Core for WIMAX 802.16E LDPC Code Decoding","author":[{"given":"Torben","family":"Brack","sequence":"first","affiliation":[]},{"given":"Matthias","family":"Alles","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Kienle","sequence":"additional","affiliation":[]},{"given":"Norbert","family":"Wehn","sequence":"additional","affiliation":[]}],"event":"2006 IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications","container-title":"2006 IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications","original-title":[],"issued":{"date-parts":[[2006,9]]},"URL":"http://dx.doi.org/10.1109/PIMRC.2006.254126"},{"id":"doi:10.1049/el:20081927","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"24","DOI":"10.1049/el:20081927","page":"1415-1417","source":"Crossref","title":"High coded data rate and multicodeword WiMAX LDPC decoding on Cell/BE","volume":"44","author":[{"given":"G.","family":"Falcão","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Coimbra/Instituto de Telecomunicações, Polo II Universidade de Coimbra, Coimbra, 3030-290, Portugal"}]},{"given":"V.","family":"Silva","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Coimbra/Instituto de Telecomunicações, Polo II Universidade de Coimbra, Coimbra, 3030-290, Portugal"}]},{"given":"L.","family":"Sousa","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Technical University of Lisbon, Instituto Superior Técnico/INESC-ID, R. Alves Redol, 9, Lisboa, 1000-029, Portugal"}]},{"given":"J.","family":"Marinho","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Coimbra/Instituto de Telecomunicações, Polo II Universidade de Coimbra, Coimbra, 3030-290, Portugal"}]}],"container-title":"Electronics Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,11,20]]},"URL":"http://dx.doi.org/10.1049/el:20081927","ISSN":["0013-5194","1350-911X"],"container-title-short":"Electron. Lett.","note":"alternative-id:10.1049/el:20081927"},{"id":"doi:10.1109/ICCT.2013.6820375","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icct.2013.6820375","page":"218-222","source":"Crossref","title":"Implementation of IEEE 802.11n LDPC codes based on general purpose processors","author":[{"family":"Xiao Han","sequence":"first","affiliation":[]},{"family":"Kai Niu","sequence":"additional","affiliation":[]},{"family":"Zhiqiang He","sequence":"additional","affiliation":[]}],"event":"2013 15th IEEE International Conference on Communication Technology (ICCT)","container-title":"2013 15th IEEE International Conference on Communication Technology","original-title":[],"issued":{"date-parts":[[2013,11]]},"URL":"http://dx.doi.org/10.1109/ICCT.2013.6820375"},{"id":"doi:10.1109/ASICON.2015.7517004","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/asicon.2015.7517004","page":"1-4","source":"Crossref","title":"A full layer parallel QC-LDPC decoder for WiMAX and Wi-Fi","author":[{"given":"Wenchao","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Song","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Xuefei","family":"Bai","sequence":"additional","affiliation":[]},{"given":"Dajiang","family":"Zhou","sequence":"additional","affiliation":[]}],"event":"2015 IEEE 11th International Conference on ASIC (ASICON )","container-title":"2015 IEEE 11th International Conference on ASIC (ASICON)","original-title":[],"issued":{"date-parts":[[2015,11]]},"URL":"http://dx.doi.org/10.1109/ASICON.2015.7517004"},{"id":"doi:10.1109/TIT.2004.838370","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2004.838370","page":"2966-2984","source":"Crossref","title":"LDPC Block and Convolutional Codes Based on Circulant Matrices","volume":"50","author":[{"given":"R.M.","family":"Tanner","sequence":"first","affiliation":[]},{"given":"D.","family":"Sridhara","sequence":"additional","affiliation":[]},{"given":"A.","family":"Sridharan","sequence":"additional","affiliation":[]},{"given":"T.E.","family":"Fuja","sequence":"additional","affiliation":[]},{"given":"D.J.","family":"Costello","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2004,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.838370","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.959255","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/18.959255","page":"2711-2736","source":"Crossref","title":"Low-density parity-check codes based on finite geometries: a rediscovery and new results","volume":"47","author":[{"given":"Y.","family":"Kou","sequence":"first","affiliation":[]},{"given":"S.","family":"Lin","sequence":"additional","affiliation":[]},{"given":"M.P.C.","family":"Fossorier","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.959255","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2004.840867","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2004.840867","page":"572-596","source":"Crossref","title":"Codes on finite geometries","volume":"51","author":[{"family":"Heng Tang","sequence":"first","affiliation":[{"name":"PMC-Sierra Inc., Portland, OR, USA"}]},{"family":"Jun Xu","sequence":"additional","affiliation":[]},{"given":"S.","family":"Lin","sequence":"additional","affiliation":[]},{"given":"K.A.S.","family":"Abdel-Ghaffar","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2005,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.840867","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/C2014-0-03412-0","type":"edited-book","publisher":"Elsevier","DOI":"10.1016/c2014-0-03412-0","source":"Crossref","title":"Latin Squares and their Applications","container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2015]]},"ISBN":["9780444635556"],"URL":"http://dx.doi.org/10.1016/C2014-0-03412-0"},{"id":"doi:10.1109/TIT.2009.2034794","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2009.2034794","page":"476-491","source":"Crossref","title":"Efficient Quantum Stabilizer Codes: LDPC and LDPC-Convolutional Constructions","volume":"56","author":[{"given":"Peiyu","family":"Tan","sequence":"first","affiliation":[]},{"given":"Jing","family":"Li","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2009.2034794","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2501.13444","type":"article-journal","author":[{"given":"Daiki","family":"Komoto"},{"given":"Kenta","family":"Kasai"}],"title":"Explicit Construction of Quantum Quasi-Cyclic Low-Density Parity-Check Codes with Column Weight 2 and Girth 12","issued":{"date-parts":[[2025,5,3]]},"note":"arxivid:2501.13444\narxiv_version_number:2"},{"id":"arxiv:2407.18490","type":"article-journal","author":[{"given":"Qian","family":"Xu"},{"given":"Hengyun","family":"Zhou"},{"given":"Guo","family":"Zheng"},{"given":"Dolev","family":"Bluvstein"},{"given":"J. Pablo Bonilla","family":"Ataides"},{"given":"Mikhail D.","family":"Lukin"},{"given":"Liang","family":"Jiang"}],"title":"Fast and Parallelizable Logical Computation with Homological Product Codes","issued":{"date-parts":[[2024,7,26]]},"note":"arxivid:2407.18490\narxiv_version_number:1"},{"id":"arxiv:2111.07029","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. In particular, several researchers have considered the individual code qubits being encoded with the continuous variable GottesmanKitaev-Preskill (GKP) code, and then imposed an outer discrete-variable code such as the surface code on these GKP qubits. Under such a concatenation scheme, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and demands a lot of resources with growing distance. In this work, we concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information in iterative decoding algorithms. We first show the noise thresholds for two lifted product QLDPC code families, and then show the improvements of noise thresholds when the iterative decoder – a hardware-friendly min-sum algorithm (MSA) – utilizes the GKP analog information. We also show that, when the GKP analog information is combined with a sequential update schedule for MSA, the scheme surpasses the well-known CSS Hamming bound for these code families. Furthermore, we observe that the GKP analog information helps the iterative decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves. Finally, we discuss new fundamental and practical questions that arise from this work on channel capacity under GKP analog information, and on improving decoder design and analysis.</jats:p>","DOI":"10.22331/q-2022-07-20-767","page":"767","source":"Crossref","title":"Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound","volume":"6","author":[{"given":"Nithin","family":"Raveendran","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"given":"Filip","family":"Rozpędek","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637, USA"}]},{"given":"Ankur","family":"Raina","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637, USA"}]},{"given":"Bane","family":"Vasić","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,20]]},"URL":"http://dx.doi.org/10.22331/q-2022-07-20-767","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2111.07029"},{"id":"arxiv:2401.06874","type":"article-journal","author":[{"given":"Sisi","family":"Miao"},{"given":"Jonathan","family":"Mandelbaum"},{"given":"Holger","family":"Jäkel"},{"given":"Laurent","family":"Schmalen"}],"title":"A Joint Code and Belief Propagation Decoder Design for Quantum LDPC Codes","issued":{"date-parts":[[2024,5,5]]},"note":"arxivid:2401.06874\narxiv_version_number:3"},{"id":"arxiv:2406.14445","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>\n                    We present a family of quantum low-density parity-check codes, which we call radial codes, obtained from the lifted product of a specific subset of classical quasicyclic codes. The codes are defined using a pair of integers\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mo stretchy=\"false\">(</a:mo>\n                      <a:mi>r</a:mi>\n                      <a:mo>,</a:mo>\n                      <a:mi>s</a:mi>\n                      <a:mo stretchy=\"false\">)</a:mo>\n                    </a:math>\n                    and have parameters\n                    <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <e:mtext>⟦</e:mtext>\n                      <e:mn>2</e:mn>\n                      <e:msup>\n                        <e:mi>r</e:mi>\n                        <e:mn>2</e:mn>\n                      </e:msup>\n                      <e:mi>s</e:mi>\n                      <e:mo>,</e:mo>\n                      <e:mn>2</e:mn>\n                      <e:mo stretchy=\"false\">(</e:mo>\n                      <e:mi>r</e:mi>\n                      <e:mo>−</e:mo>\n                      <e:mn>1</e:mn>\n                      <e:msup>\n                        <e:mo stretchy=\"false\">)</e:mo>\n                        <e:mn>2</e:mn>\n                      </e:msup>\n                      <e:mo>,</e:mo>\n                      <e:mo>≤</e:mo>\n                      <e:mn>2</e:mn>\n                      <e:mi>s</e:mi>\n                      <e:mtext>⟧</e:mtext>\n                    </e:math>\n                    , with numerical studies suggesting average-case distance linear in\n                    <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <i:mi>s</i:mi>\n                    </i:math>\n                    . In simulations of circuit-level noise, we observe comparable error suppression to surface codes of similar distance while using approximately five times fewer physical qubits. This is true even when radial codes are decoded using a single-shot approach, which can allow for faster logical clock speeds and reduced decoding complexity. We describe an intuitive visual representation, canonical basis of logical operators and optimal-length stabilizer measurement circuits for these codes, and argue that their error correction capabilities, tunable parameters and small size make them promising candidates for implementation on near-term quantum devices.\n                  </jats:p>","DOI":"10.1103/67xf-zdjb","source":"Crossref","title":"High-Threshold, Low-Overhead and Single-Shot Decodable Fault-Tolerant Quantum Memory","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-8750-4820","authenticated-orcid":true,"given":"Thomas R.","family":"Scruby","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02qg15b79","id-type":"ROR","asserted-by":"publisher"}],"name":"Okinawa Institute of Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0002-1476-0647","authenticated-orcid":true,"given":"Timo","family":"Hillmann","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/040wg7k59","id-type":"ROR","asserted-by":"publisher"}],"name":"Chalmers University of Technology"},{"name":"School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]},{"ORCID":"https://orcid.org/0000-0001-9202-1156","authenticated-orcid":true,"given":"Joschka","family":"Roffe","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01nrxwf90","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Edinburgh"},{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,17]]},"URL":"http://dx.doi.org/10.1103/67xf-zdjb","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020310","note":"arxivid:2406.14445"},{"id":"arxiv:2112.01647","type":"article-journal","author":[{"given":"Fernando Granha","family":"Jeronimo"},{"given":"Tushant","family":"Mittal"},{"given":"Ryan","family":"O'Donnell"},{"given":"Pedro","family":"Paredes"},{"given":"Madhur","family":"Tulsiani"}],"title":"Explicit Abelian Lifts and Quantum LDPC Codes","issued":{"date-parts":[[2021,12,2]]},"note":"arxivid:2112.01647\narxiv_version_number:1"},{"id":"doi:10.1109/ICC.1999.767979","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.1999.767979","source":"Crossref","title":"Generalized low density (Tanner) codes","author":[{"given":"J.","family":"Boutros","sequence":"first","affiliation":[]},{"given":"O.","family":"Pothier","sequence":"additional","affiliation":[]},{"given":"G.","family":"Zemor","sequence":"additional","affiliation":[]}],"event":"1999 IEEE International Conference on Communications","container-title":"1999 IEEE International Conference on Communications (Cat. No. 99CH36311)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ICC.1999.767979"},{"id":"doi:10.1109/TIT.1981.1056404","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1981.1056404","page":"533-547","source":"Crossref","title":"A recursive approach to low complexity codes","volume":"27","author":[{"given":"R.","family":"Tanner","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1981,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1981.1056404","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.910591","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910591","page":"808-821","source":"Crossref","title":"Minimum-distance bounds by graph analysis","volume":"47","author":[{"given":"R.M.","family":"Tanner","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910591","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1504/IJICoT.2010.032546","type":"journal-article","publisher":"Inderscience Publishers","issue":"3","DOI":"10.1504/ijicot.2010.032546","page":"313","source":"Crossref","title":"Minimum distance and pseudodistance lower bounds for generalised LDPC codes","volume":"1","author":[{"given":"Christine A.","family":"Kelley","sequence":"first","affiliation":[]}],"container-title":"International Journal of Information and Coding Theory","original-title":[],"language":"en","issued":{"date-parts":[[2010]]},"URL":"http://dx.doi.org/10.1504/IJICoT.2010.032546","ISSN":["1753-7703","1753-7711"],"container-title-short":"IJICOT"},{"id":"arxiv:0708.2462","type":"article-journal","author":[{"given":"Christine A.","family":"Kelley"},{"given":"Deepak","family":"Sridhara"}],"title":"Eigenvalue bounds on the pseudocodeword weight of expander codes","issued":{"date-parts":[[2007,8,18]]},"note":"arxivid:0708.2462\narxiv_version_number:1"},{"id":"doi:10.1145/258533.258575","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/258533.258575","page":"160-167","source":"Crossref","title":"Spectral techniques for expander codes","author":[{"given":"John D.","family":"Lafferty","sequence":"first","affiliation":[]},{"given":"Daniel N.","family":"Rockmore","sequence":"additional","affiliation":[]}],"event":"the twenty-ninth annual ACM symposium","container-title":"Proceedings of the twenty-ninth annual ACM symposium on Theory of computing  - STOC '97","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1145/258533.258575","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"doi:10.1109/18.556668","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.556668","page":"1723-1731","source":"Crossref","title":"Linear-time encodable and decodable error-correcting codes","volume":"42","author":[{"given":"D.A.","family":"Spielman","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1996]]},"URL":"http://dx.doi.org/10.1109/18.556668","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.556667","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.556667","page":"1710-1722","source":"Crossref","title":"Expander codes","volume":"42","author":[{"given":"M.","family":"Sipser","sequence":"first","affiliation":[]},{"given":"D.A.","family":"Spielman","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1996]]},"URL":"http://dx.doi.org/10.1109/18.556667","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.910593","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.910593","page":"835-837","source":"Crossref","title":"On expander codes","volume":"47","author":[{"given":"G.","family":"Zemor","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.910593","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2005.846392","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.2005.846392","page":"1625-1634","source":"Crossref","title":"Concatenated Codes: Serial and Parallel","volume":"51","author":[{"given":"A.","family":"Barg","sequence":"first","affiliation":[]},{"given":"G.","family":"Zemor","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2005,5]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.846392","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/0166-218X(91)90111-9","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0166-218x(91)90111-9","page":"109-128","source":"Crossref","title":"A semi-systolic decoder for the PDSC-73 error-correcting code","volume":"33","author":[{"given":"Kevin","family":"Karplus","sequence":"first","affiliation":[]},{"given":"Habib","family":"Krit","sequence":"additional","affiliation":[]}],"container-title":"Discrete Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1991,11]]},"URL":"http://dx.doi.org/10.1016/0166-218X(91)90111-9","ISSN":["0166-218X"],"container-title-short":"Discrete Applied Mathematics","note":"alternative-id:0166218X91901119"},{"id":"doi:10.1109/ISIT.1994.394711","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.1994.394711","page":"307","source":"Crossref","title":"On the error-correcting capabilities of cycle codes of graphs","author":[{"given":"L.","family":"Decreusefond","sequence":"first","affiliation":[]},{"given":"G.","family":"Zemor","sequence":"additional","affiliation":[]}],"event":"1994 IEEE International Symposium on Information Theory","container-title":"Proceedings of 1994 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.1994.394711"},{"id":"doi:10.1109/LCOMM.2006.1633326","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/lcomm.2006.1633326","page":"369-371","source":"Crossref","title":"Linear time encoding of cycle GF(2/sup P)/ codes through graph analysis","volume":"10","author":[{"family":"Jie Huang","sequence":"first","affiliation":[]},{"family":"Jinkang Zhu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,5]]},"URL":"http://dx.doi.org/10.1109/LCOMM.2006.1633326","ISSN":["1089-7798"],"container-title-short":"IEEE Commun. Lett."},{"id":"doi:10.1109/TWC.2008.070057","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/twc.2008.070057","page":"2020-2026","source":"Crossref","title":"Application of Nonbinary LDPC Cycle Codes to MIMO Channels","volume":"7","author":[{"family":"Ronghui Peng","sequence":"first","affiliation":[]},{"family":"Rong-Rong Chen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Wireless Communications","original-title":[],"issued":{"date-parts":[[2008,6]]},"URL":"http://dx.doi.org/10.1109/TWC.2008.070057","ISSN":["1536-1276"],"container-title-short":"IEEE Trans. Wireless Commun."},{"id":"doi:10.1007/BF02126799","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf02126799","page":"261-277","source":"Crossref","title":"Ramanujan graphs","volume":"8","author":[{"given":"A.","family":"Lubotzky","sequence":"first","affiliation":[]},{"given":"R.","family":"Phillips","sequence":"additional","affiliation":[]},{"given":"P.","family":"Sarnak","sequence":"additional","affiliation":[]}],"container-title":"Combinatorica","original-title":[],"language":"en","issued":{"date-parts":[[1988,9]]},"URL":"http://dx.doi.org/10.1007/BF02126799","ISSN":["0209-9683","1439-6912"],"container-title-short":"Combinatorica","note":"alternative-id:BF02126799"},{"id":"doi:10.1017/CBO9780511615825","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>This text is a self contained treatment of expander graphs and in particular their explicit construction. Expander graphs are both highly connected but sparse, and besides their interest within combinatorics and graph theory, they also find various applications in computer science and engineering. The reader needs only a background in elementary algebra, analysis and combinatorics; the authors supply the necessary background material from graph theory, number theory, group theory and representation theory. The text can therefore be used as a brief introduction to these subjects as well as an illustration of how such topics are synthesised in modern mathematics.</jats:p>","DOI":"10.1017/cbo9780511615825","source":"Crossref","title":"Elementary Number Theory, Group Theory and Ramanujan Graphs","author":[{"given":"Giuliana","family":"Davidoff","sequence":"first","affiliation":[]},{"given":"Peter","family":"Sarnak","sequence":"additional","affiliation":[]},{"given":"Alain","family":"Valette","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2001,1,1]]},"ISBN":["9780521531436","9780521824262","9780511615825"],"URL":"http://dx.doi.org/10.1017/CBO9780511615825","note":"edition-number:1"},{"id":"doi:10.1145/509907.510003","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/509907.510003","page":"659-668","source":"Crossref","title":"Randomness conductors and constant-degree lossless expanders","author":[{"given":"Michael","family":"Capalbo","sequence":"first","affiliation":[{"name":"Institute for Advanced Study, Princeton, NJ"}]},{"given":"Omer","family":"Reingold","sequence":"additional","affiliation":[{"name":"AT&amp;T Labs - Research, Florham Park, NJ"}]},{"given":"Salil","family":"Vadhan","sequence":"additional","affiliation":[{"name":"Harvard University, Cambridge, MA"}]},{"given":"Avi","family":"Wigderson","sequence":"additional","affiliation":[{"name":"Hebrew University, Jerusalem and Institute for Advanced Study, Princeton, NJ"}]}],"event":"STOC02: Symposium on the Theory of Computing","container-title":"Proceedings of the thiry-fourth annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2002,5,19]]},"URL":"http://dx.doi.org/10.1145/509907.510003","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/509907.510003\n10.1145/509907"},{"id":"arxiv:cs/0406048","type":"article-journal","author":[{"given":"H. L. Janwa A. K.","family":"Lal"}],"title":"On Expanders Graphs: Parameters and Applications","issued":{"date-parts":[[2004,6,25]]},"note":"arxivid:cs/0406048\narxiv_version_number:1"},{"id":"arxiv:0805.2427","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2010.2040962","page":"1600-1611","source":"Crossref","title":"On Trapping Sets and Guaranteed Error Correction Capability of LDPC Codes and GLDPC Codes","volume":"56","author":[{"given":"S.K.","family":"Chilappagari","sequence":"first","affiliation":[]},{"family":"Dung Viet Nguyen","sequence":"additional","affiliation":[]},{"given":"B.","family":"Vasic","sequence":"additional","affiliation":[]},{"given":"M.W.","family":"Marcellin","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2040962","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0805.2427"},{"id":"doi:10.1109/TIT.2017.2726064","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2017.2726064","page":"972-978","source":"Crossref","title":"Fast Decoding of Expander Codes","volume":"64","author":[{"ORCID":"https://orcid.org/0000-0001-7941-6073","authenticated-orcid":false,"given":"Michael","family":"Dowling","sequence":"first","affiliation":[]},{"given":"Shuhong","family":"Gao","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2017.2726064","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2312.16087","type":"article-journal","author":[{"given":"Kuan","family":"Cheng"},{"given":"Minghui","family":"Ouyang"},{"given":"Chong","family":"Shangguan"},{"given":"Yuanting","family":"Shen"}],"title":"When can an expander code correct $Ω(n)$ errors in $O(n)$ time?","issued":{"date-parts":[[2024,7,17]]},"note":"arxivid:2312.16087\narxiv_version_number:2"},{"id":"doi:10.1109/TIT.2006.887523","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2006.887523","page":"82-89","source":"Crossref","title":"LP Decoding Corrects a Constant Fraction of Errors","volume":"53","author":[{"given":"J.","family":"Feldman","sequence":"first","affiliation":[]},{"given":"T.","family":"Malkin","sequence":"additional","affiliation":[]},{"given":"R.A.","family":"Servedio","sequence":"additional","affiliation":[]},{"given":"C.","family":"Stein","sequence":"additional","affiliation":[]},{"given":"M.J.","family":"Wainwright","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2007,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2006.887523","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1145/2493252.2493255","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"3","abstract":"<jats:p>\n            Sipser and Spielman (IEEE IT, [1996]) showed that any\n            <jats:italic>c</jats:italic>\n            ,\n            <jats:italic>d</jats:italic>\n            )-regular expander code with expansion parameter &gt;¾ is decodable in linear time from a constant fraction of errors. Feldman et al. (IEEE IT, [2007]) proved that expansion parameter &gt;⅔ + 1/3c is sufficient to correct a constant fraction of errors in polynomial time using LP decoding.\n          </jats:p>\n          <jats:p>In this work, we give a simple combinatorial algorithm that achieves even better parameters. In particular, our algorithm runs in linear time and works for any expansion parameter &gt;⅔ − 1/6c. We also prove that our decoding algorithm can be executed in logarithmic time on a linear number of parallel processors.</jats:p>","DOI":"10.1145/2493252.2493255","page":"1-25","source":"Crossref","title":"Linear-time decoding of regular expander codes","volume":"5","author":[{"given":"Michael","family":"Viderman","sequence":"first","affiliation":[{"name":"Technion — Israel Institute of Technology Andmicrosoft Research, New England"}]}],"container-title":"ACM Transactions on Computation Theory","original-title":[],"language":"en","issued":{"date-parts":[[2013,8]]},"URL":"http://dx.doi.org/10.1145/2493252.2493255","ISSN":["1942-3454","1942-3462"],"container-title-short":"ACM Trans. Comput. Theory","note":"alternative-id:10.1145/2493252.2493255"},{"id":"arxiv:1304.8129","type":"article-journal","author":[{"given":"Brett","family":"Hemenway"},{"given":"Rafail","family":"Ostrovsky"},{"given":"Mary","family":"Wootters"}],"title":"Local Correctability of Expander Codes","issued":{"date-parts":[[2015,1,7]]},"note":"arxivid:1304.8129\narxiv_version_number:2"},{"id":"manual:-F.-G.-Jeronimo-Fast-decoding","type":"chapter","author":[{"family":"Jeronimo","given":"F.G."}],"title":"Fast decoding of explicit almost optimal ε-balanced q-ary codes and fast approximation of expanding k-CSPs","container-title":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Schloss Dagstuhl–Leibniz-Zentrum für Informatik","issued":"2023"},{"id":"manual:-A.-Rao-Lecture-6:-Expander-C","type":"article-journal","author":[{"family":"Rao","given":"A."}],"title":"Lecture 6: Expander Codes: Tanner Codes","publisher":"University of Washington","container-title":"Coding Theory lecture notes","issued":"2019-10-14"},{"id":"arxiv:2410.14662","type":"article-journal","author":[{"given":"Louis","family":"Golowich"},{"given":"Ting-Chun","family":"Lin"}],"title":"Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes","issued":{"date-parts":[[2024,10,18]]},"note":"arxivid:2410.14662\narxiv_version_number:1"},{"id":"arxiv:2111.03654","type":"article-journal","author":[{"given":"Pavel","family":"Panteleev"},{"given":"Gleb","family":"Kalachev"}],"title":"Asymptotically Good Quantum and Locally Testable Classical LDPC Codes","issued":{"date-parts":[[2022,1,21]]},"note":"arxivid:2111.03654\narxiv_version_number:2"},{"id":"manual:-V.-V.-Zyablov-and-M.-S.-Pins","type":"article-journal","author":[{"family":"Zyablov","given":"V.V."},{"family":"Pinsker","given":"M.S."}],"title":"Estimation of the error-correction complexity for Gallager low-density codes","volume":"11","container-title":"Problemy Peredachi Informatsii","issue":"1","issued":"1975","page":"23–36"},{"id":"arxiv:cs/0509062","type":"article-journal","author":[{"given":"Chun-Hao","family":"Hsu"},{"given":"Achilleas","family":"Anastasopoulos"}],"title":"Capacity-Achieving Codes with Bounded Graphical Complexity on Noisy Channels","issued":{"date-parts":[[2005,9,24]]},"note":"arxivid:cs/0509062\narxiv_version_number:3"},{"id":"doi:10.1016/0166-218X(94)00058-L","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0166-218x(94)00058-l","page":"275-284","source":"Crossref","title":"Explicit construction of graphs with an arbitrary large girth and of large size","volume":"60","author":[{"given":"Felix","family":"Lazebnik","sequence":"first","affiliation":[]},{"given":"Vasiliy A.","family":"Ustimenko","sequence":"additional","affiliation":[]}],"container-title":"Discrete Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1995,6]]},"URL":"http://dx.doi.org/10.1016/0166-218X(94)00058-L","ISSN":["0166-218X"],"container-title-short":"Discrete Applied Mathematics","note":"alternative-id:0166218X9400058L"},{"id":"doi:10.1109/TIT.2004.834760","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2004.834760","page":"2378-2388","source":"Crossref","title":"Explicit Construction of Families of LDPC Codes With No&amp;lt;tex&amp;gt;$4$&amp;lt;/tex&amp;gt;-Cycles","volume":"50","author":[{"given":"J.-L.","family":"Kim","sequence":"first","affiliation":[]},{"given":"U.N.","family":"Peled","sequence":"additional","affiliation":[]},{"given":"I.","family":"Perepelitsa","sequence":"additional","affiliation":[]},{"given":"V.","family":"Pless","sequence":"additional","affiliation":[]},{"given":"S.","family":"Friedland","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2004,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.834760","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/0022-0000(81)90040-4","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0022-0000(81)90040-4","page":"407-420","source":"Crossref","title":"Explicit constructions of linear-sized superconcentrators","volume":"22","author":[{"given":"Ofer","family":"Gabber","sequence":"first","affiliation":[]},{"given":"Zvi","family":"Galil","sequence":"additional","affiliation":[]}],"container-title":"Journal of Computer and System Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1981,6]]},"URL":"http://dx.doi.org/10.1016/0022-0000(81)90040-4","ISSN":["0022-0000"],"container-title-short":"Journal of Computer and System Sciences","note":"alternative-id:0022000081900404"},{"id":"doi:10.1109/ISIT.2001.935867","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2001.935867","page":"4","source":"Crossref","title":"Constructions of regular and irregular LDPC codes using Ramanujan graphs and ideas from Margulis","author":[{"given":"J.","family":"Rosenthal","sequence":"first","affiliation":[]},{"given":"P.O.","family":"Vontobel","sequence":"additional","affiliation":[]}],"event":"2001 IEEE International Symposium on Information Theory","container-title":"Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2001.935867"},{"id":"doi:10.1007/3-540-60693-9_13","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-60693-9_13","page":"100-111","source":"Crossref","title":"Good codes based on very sparse matrices","author":[{"given":"David J. C.","family":"MacKay","sequence":"first","affiliation":[]},{"given":"Radford M.","family":"Neal","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[1995]]},"ISBN":["9783540606932","9783540492801"],"URL":"http://dx.doi.org/10.1007/3-540-60693-9_13","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1017/CBO9780511800467","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>The need to transmit and store massive amounts of data reliably and without error is a vital part of modern communications systems. Error-correcting codes play a fundamental role in minimising data corruption caused by defects such as noise, interference, crosstalk and packet loss. This book provides an accessible introduction to the basic elements of algebraic codes, and discusses their use in a variety of applications. The author describes a range of important coding techniques, including Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. Throughout the book, mathematical theory is illustrated by reference to many practical examples. The book was first published in 2003 and is aimed at graduate students of electrical and computer engineering, and at practising engineers whose work involves communications or signal processing.</jats:p>","DOI":"10.1017/cbo9780511800467","source":"Crossref","title":"Algebraic Codes for Data Transmission","author":[{"given":"Richard E.","family":"Blahut","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2003,2,6]]},"ISBN":["9780521556590","9780521553742","9780511800467"],"URL":"http://dx.doi.org/10.1017/CBO9780511800467","note":"edition-number:1"},{"id":"doi:10.1098/rspa.2008.0439","type":"journal-article","publisher":"The Royal Society","issue":"2105","abstract":"<jats:p>Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit- and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank–Shor–Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.</jats:p>","DOI":"10.1098/rspa.2008.0439","page":"1645-1672","source":"Crossref","title":"Asymmetric quantum codes: constructions, bounds and performance","volume":"465","author":[{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"first","affiliation":[{"name":"Texas A&amp;M UniversityCollege Station, TX 77843, USA"}]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[{"name":"Texas A&amp;M UniversityCollege Station, TX 77843, USA"}]},{"given":"Martin","family":"Rötteler","sequence":"additional","affiliation":[{"name":"NEC Laboratories America, Inc.4 Independence Way, Suite 200, Princeton, NJ 08540, USA"}]}],"container-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,4]]},"URL":"http://dx.doi.org/10.1098/rspa.2008.0439","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. A.","note":"alternative-id:10.1098/rspa.2008.0439"},{"id":"arxiv:0804.4316","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2008.4594997","page":"305-309","source":"Crossref","title":"Asymmetric quantum LDPC codes","author":[{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"2008 IEEE International Symposium on Information Theory - ISIT","container-title":"2008 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2008.4594997","note":"arxivid:0804.4316"},{"id":"arxiv:0712.4115","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/glocom.2008.ecp.217","page":"1-5","source":"Crossref","title":"A Class of Quantum LDPC Codes Constructed From Finite Geometries","author":[{"given":"Salah A.","family":"Aly","sequence":"first","affiliation":[]}],"event":"IEEE GLOBECOM 2008 - 2008 IEEE Global Telecommunications Conference","container-title":"IEEE GLOBECOM 2008 - 2008 IEEE Global Telecommunications Conference","original-title":[],"issued":{"date-parts":[[2008]]},"URL":"http://dx.doi.org/10.1109/GLOCOM.2008.ECP.217","note":"arxivid:0712.4115"},{"id":"arxiv:1207.0732","type":"article-journal","author":[{"given":"Jacob","family":"Farinholt"}],"title":"Quantum LDPC Codes Constructed from Point-Line Subsets of the Finite Projective Plane","issued":{"date-parts":[[2012,7,3]]},"note":"arxivid:1207.0732\narxiv_version_number:1"},{"id":"arxiv:1005.1062","type":"article-journal","author":[{"given":"Michael","family":"Lentmaier"},{"given":"David G. M.","family":"Mitchell"},{"given":"Gerhard P.","family":"Fettweis"},{"given":"Daniel J.","family":"Costello"}],"title":"Asymptotically Regular LDPC Codes with Linear Distance Growth and Thresholds Close to Capacity","issued":{"date-parts":[[2010,5,6]]},"note":"arxivid:1005.1062\narxiv_version_number:1"},{"id":"doi:10.1109/TCOMM.2018.2867493","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tcomm.2018.2867493","page":"3-16","source":"Crossref","title":"Finite-Length Construction of High Performance Spatially-Coupled Codes via Optimized Partitioning and Lifting","volume":"67","author":[{"ORCID":"https://orcid.org/0000-0003-1217-692X","authenticated-orcid":false,"given":"Homa","family":"Esfahanizadeh","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8523-6754","authenticated-orcid":false,"given":"Ahmed","family":"Hareedy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3736-4345","authenticated-orcid":false,"given":"Lara","family":"Dolecek","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2019,1]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2018.2867493","ISSN":["0090-6778","1558-0857"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/18.782171","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.782171","page":"2181-2191","source":"Crossref","title":"Time-varying periodic convolutional codes with low-density parity-check matrix","volume":"45","author":[{"given":"A.","family":"Jimenez Felstrom","sequence":"first","affiliation":[]},{"given":"K.S.","family":"Zigangirov","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999]]},"URL":"http://dx.doi.org/10.1109/18.782171","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Engdahl99","type":"article-journal","author":[{"family":"Engdahl","given":"K."},{"family":"Zigangirov","given":"K.Sh"}],"title":"To the Theory of Low-Density Convolutional Codes. I","volume":"35","container-title":"Problemy Peredachi Informatsii","issue":"4","issued":"1999","page":"12–28"},{"id":"doi:10.1023/A:1013815115684","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1023/a:1013815115684","page":"288-306","source":"Crossref","title":[],"volume":"37","author":[{"given":"M.","family":"Lentmaier","sequence":"first","affiliation":[]},{"given":"D. V.","family":"Truhachev","sequence":"additional","affiliation":[]},{"given":"K. Sh.","family":"Zigangirov","sequence":"additional","affiliation":[]}],"container-title":"Problems of Information Transmission","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1023/A:1013815115684","ISSN":["0032-9460"],"note":"alternative-id:370157"},{"id":"doi:10.1109/TIT.2010.2095072","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2010.2095072","page":"803-834","source":"Crossref","title":"Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform So Well over the BEC","volume":"57","author":[{"given":"Shrinivas","family":"Kudekar","sequence":"first","affiliation":[]},{"given":"Thomas J.","family":"Richardson","sequence":"additional","affiliation":[]},{"given":"Rüdiger L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2011,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2095072","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2013.2280915","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2013.2280915","page":"7761-7813","source":"Crossref","title":"Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation","volume":"59","author":[{"given":"Shrinivas","family":"Kudekar","sequence":"first","affiliation":[]},{"given":"Tom","family":"Richardson","sequence":"additional","affiliation":[]},{"given":"Rudiger L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2280915","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ISTC.2012.6325232","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/istc.2012.6325232","page":"225-229","source":"Crossref","title":"Comparison of LDPC block and LDPC convolutional codes based on their decoding latency","author":[{"given":"Najeeb","family":"ul Hassan","sequence":"first","affiliation":[]},{"given":"Michael","family":"Lentmaier","sequence":"additional","affiliation":[]},{"given":"Gerhard P.","family":"Fettweis","sequence":"additional","affiliation":[]}],"event":"2012 7th International Symposium on Turbo Codes & Iterative Information Processing (ISTC)","container-title":"2012 7th International Symposium on Turbo Codes and Iterative Information Processing (ISTC)","original-title":[],"issued":{"date-parts":[[2012,8]]},"URL":"http://dx.doi.org/10.1109/ISTC.2012.6325232"},{"id":"doi:10.1109/ITA.2007.4357569","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/ita.2007.4357569","page":"111-119","source":"Crossref","title":"A Comparison of ARA- and Protograph-Based LDPC Block and Convolutional Codes","author":[{"given":"Daniel J.","family":"Costello","sequence":"first","affiliation":[]},{"given":"Ali Emre","family":"Pusane","sequence":"additional","affiliation":[]},{"given":"Christopher R.","family":"Jones","sequence":"additional","affiliation":[]},{"given":"Dariush","family":"Divsalar","sequence":"additional","affiliation":[]}],"event":"2007 Information Theory and Applications Workshop","container-title":"2007 Information Theory and Applications Workshop","original-title":[],"issued":{"date-parts":[[2007,1]]},"URL":"http://dx.doi.org/10.1109/ITA.2007.4357569"},{"id":"arxiv:cs/0508030","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2005.1523567","page":"1372-1376","source":"Crossref","title":"Terminated LDPC convolutional codes with thresholds close to capacity","author":[{"given":"M.","family":"Lentmaier","sequence":"first","affiliation":[]},{"given":"A.","family":"Sridharan","sequence":"additional","affiliation":[]},{"given":"K.Sh.","family":"Zigangirov","sequence":"additional","affiliation":[]},{"given":"D.J.","family":"Costello","sequence":"additional","affiliation":[]}],"event":"Proceedings. International Symposium on Information Theory, 2005. ISIT 2005.","container-title":"Proceedings. International Symposium on Information Theory, 2005. ISIT 2005.","original-title":[],"issued":{"date-parts":[[2005]]},"URL":"http://dx.doi.org/10.1109/ISIT.2005.1523567","note":"arxivid:cs/0508030"},{"id":"arxiv:1309.7543","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2014.2360692","page":"7389-7415","source":"Crossref","title":"Threshold Saturation for Spatially Coupled LDPC and LDGM Codes on BMS Channels","volume":"60","author":[{"given":"Santhosh","family":"Kumar","sequence":"first","affiliation":[]},{"given":"Andrew J.","family":"Young","sequence":"additional","affiliation":[]},{"given":"Nicolas","family":"Macris","sequence":"additional","affiliation":[]},{"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2360692","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1309.7543"},{"id":"doi:10.1080/00029890.1970.11992464","type":"journal-article","publisher":"Informa UK Limited","issue":"3","DOI":"10.1080/00029890.1970.11992464","page":"249-258","source":"Crossref","title":"Coding Theory: A Counterexample to G. H. Hardy's Conception of Applied Mathematics","volume":"77","author":[{"given":"Norman","family":"Levinson","sequence":"first","affiliation":[{"name":"Massachusetts Institute of Technology"}]}],"container-title":"The American Mathematical Monthly","original-title":[],"language":"en","issued":{"date-parts":[[1970,3]]},"URL":"http://dx.doi.org/10.1080/00029890.1970.11992464","ISSN":["0002-9890","1930-0972"],"container-title-short":"The American Mathematical Monthly","note":"aliases:10.2307/2317708\nalternative-id:10.1080/00029890.1970.11992464"},{"id":"arxiv:2009.07161","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2022.3169438","page":"282-317","source":"Crossref","title":"Fault-Tolerant Coding for Quantum Communication","volume":"70","author":[{"ORCID":"https://orcid.org/0000-0003-2281-3355","authenticated-orcid":false,"given":"Matthias","family":"Christandl","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark"}]},{"ORCID":"https://orcid.org/0000-0002-9252-2695","authenticated-orcid":false,"given":"Alexander","family":"Müller-Hermes","sequence":"additional","affiliation":[{"name":"Institut Camille Jordan, Universit&#x00E9; Claude Bernard Lyon 1, Villeurbanne, France"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2024,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2022.3169438","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2009.07161"},{"id":"manual:-V.-Strassen-Asymptotische-Ab","type":"paper-conference","author":[{"family":"Strassen","given":"V."}],"title":"Asymptotische Absch ̈atzungen in Shannons Informationstheorie","container-title":"Trans. Third Prague Conference on Information Theory, Prague","issued":"1962","page":"689–723,"},{"id":"arxiv:0801.2242","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2009.2030478","page":"4947-4966","source":"Crossref","title":"Information Spectrum Approach to Second-Order Coding Rate in Channel Coding","volume":"55","author":[{"given":"Masahito","family":"Hayashi","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2009,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2009.2030478","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0801.2242"},{"id":"doi:10.1109/TIT.2010.2043769","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.2010.2043769","page":"2307-2359","source":"Crossref","title":"Channel Coding Rate in the Finite Blocklength Regime","volume":"56","author":[{"given":"Yury","family":"Polyanskiy","sequence":"first","affiliation":[]},{"given":"H. Vincent","family":"Poor","sequence":"additional","affiliation":[]},{"given":"Sergio","family":"Verdu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,5]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2043769","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1208.1924","type":"article-journal","author":[{"given":"Yucel","family":"Altug"},{"given":"Aaron B.","family":"Wagner"}],"title":"Moderate Deviations in Channel Coding","issued":{"date-parts":[[2012,8,9]]},"note":"arxivid:1208.1924\narxiv_version_number:1"},{"id":"doi:10.1109/ALLERTON.2010.5707068","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/allerton.2010.5707068","page":"1334-1339","source":"Crossref","title":"Channel dispersion and moderate deviations limits for memoryless channels","author":[{"given":"Yury","family":"Polyanskiy","sequence":"first","affiliation":[]},{"given":"Sergio","family":"Verdu","sequence":"additional","affiliation":[]}],"event":"2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)","container-title":"2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)","original-title":[],"issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1109/ALLERTON.2010.5707068"},{"id":"arxiv:1701.03114","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-017-2971-1","page":"1283-1315","source":"Crossref","title":"Moderate Deviation Analysis for Classical Communication over Quantum Channels","volume":"355","author":[{"given":"Christopher T.","family":"Chubb","sequence":"first","affiliation":[]},{"given":"Vincent Y. F.","family":"Tan","sequence":"additional","affiliation":[]},{"given":"Marco","family":"Tomamichel","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,8,2]]},"URL":"http://dx.doi.org/10.1007/s00220-017-2971-1","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2971\narxivid:1701.03114"},{"id":"doi:10.1017/CBO9780511921889","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Csiszár and Körner's book is widely regarded as a classic in the field of information theory, providing deep insights and expert treatment of the key theoretical issues. It includes in-depth coverage of the mathematics of reliable information transmission, both in two-terminal and multi-terminal network scenarios. Updated and considerably expanded, this new edition presents unique discussions of information theoretic secrecy and of zero-error information theory, including the deep connections of the latter with extremal combinatorics. The presentations of all core subjects are self contained, even the advanced topics, which helps readers to understand the important connections between seemingly different problems. Finally, 320 end-of-chapter problems, together with helpful solving hints, allow readers to develop a full command of the mathematical techniques. It is an ideal resource for graduate students and researchers in electrical and electronic engineering, computer science and applied mathematics.</jats:p>","DOI":"10.1017/cbo9780511921889","source":"Crossref","title":"Information Theory","author":[{"given":"Imre","family":"Csiszár","sequence":"first","affiliation":[]},{"given":"János","family":"Körner","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2011,6,30]]},"ISBN":["9780511921889","9780521196819","9781107565043"],"URL":"http://dx.doi.org/10.1017/CBO9780511921889","note":"edition-number:2"},{"id":"doi:10.1109/TIT.1973.1055007","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1973.1055007","page":"357-359","source":"Crossref","title":"On the converse to the coding theorem for discrete memoryless channels (Corresp.)","volume":"19","author":[{"given":"S.","family":"Arimoto","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1973,5]]},"URL":"http://dx.doi.org/10.1109/TIT.1973.1055007","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1979.1056003","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1979.1056003","page":"82-85","source":"Crossref","title":"Reliability function of a discrete memoryless channel at rates above capacity (Corresp.)","volume":"25","author":[{"given":"G.","family":"Dueck","sequence":"first","affiliation":[]},{"given":"J.","family":"Korner","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1979,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1979.1056003","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2014.2341919","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2014.2341919","page":"5875-5898","source":"Crossref","title":"Finite-Length Scaling for Polar Codes","volume":"60","author":[{"given":"Seyed Hamed","family":"Hassani","sequence":"first","affiliation":[]},{"given":"Kasra","family":"Alishahi","sequence":"additional","affiliation":[]},{"given":"Rudiger L.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2341919","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Doeblin37","type":"article-journal","author":[{"family":"Doeblin","given":"W."}],"title":"Sur les propriétés asymptotiques de mouvement régis par certains types de chaines simples","volume":"39","container-title":"Bulletin mathématique de la Société roumaine des sciences","issue":"1","issued":"1937","page":"57–115"},{"id":"arxiv:2309.08475","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2024.3367856","page":"4667-4692","source":"Crossref","title":"Doeblin Coefficients and Related Measures","volume":"70","author":[{"ORCID":"https://orcid.org/0000-0002-2978-8116","authenticated-orcid":false,"given":"Anuran","family":"Makur","sequence":"first","affiliation":[{"name":"Department of Computer Science and the Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4953-1465","authenticated-orcid":false,"given":"Japneet","family":"Singh","sequence":"additional","affiliation":[{"name":"Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2024,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3367856","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2309.08475"},{"id":"doi:10.1090/conm/634/12692","type":"other","publisher":"American Mathematical Society","DOI":"10.1090/conm/634/12692","page":"79-99","source":"Crossref","title":"Open Problems in Coding Theory","author":[{"given":"Steven","family":"Dougherty","sequence":"first","affiliation":[]},{"given":"Jon-Lark","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Solé","sequence":"additional","affiliation":[]}],"container-title":"Contemporary Mathematics","original-title":[],"issued":{"date-parts":[[2015]]},"URL":"http://dx.doi.org/10.1090/conm/634/12692","ISSN":["0271-4132","1098-3627"],"note":"publisher-location:Providence, Rhode Island"},{"id":"doi:10.1145/1897816.1897844","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"2","abstract":"<jats:p>Errors in dynamic random access memory (DRAM) are a common form of hardware failure in modern compute clusters. Failures are costly both in terms of hardware replacement costs and service disruption. While a large body of work exists on DRAM in laboratory conditions, little has been reported on real DRAM failures in large production clusters. In this paper, we analyze measurements of memory errors in a large fleet of commodity servers over a period of 2.5 years. The collected data covers multiple vendors, DRAM capacities and technologies, and comprises many millions of dual in-line memory module (DIMM) days.</jats:p>\n          <jats:p>The goal of this paper is to answer questions such as the following: How common are memory errors in practice? What are their statistical properties? How are they affected by external factors, such as temperature and utilization, and by chip-specific factors, such as chip density, memory technology, and DIMM age?</jats:p>\n          <jats:p>We find that DRAM error behavior in the field differs in many key aspects from commonly held assumptions. For example, we observe DRAM error rates that are orders of magnitude higher than previously reported, with 25,000--70,000 errors per billion device hours per Mb and more than 8% of DIMMs affected by errors per year. We provide strong evidence that memory errors are dominated by hard errors, rather than soft errors, which previous work suspects to be the dominant error mode. We find that temperature, known to strongly impact DIMM error rates in lab conditions, has a surprisingly small effect on error behavior in the field, when taking all other factors into account. Finally, unlike commonly feared, we do not observe any indication that newer generations of DIMMs have worse error behavior.</jats:p>","DOI":"10.1145/1897816.1897844","page":"100-107","source":"Crossref","title":"DRAM errors in the wild","volume":"54","author":[{"given":"Bianca","family":"Schroeder","sequence":"first","affiliation":[{"name":"University of Toronto, Toronto, Canada"}]},{"given":"Eduardo","family":"Pinheiro","sequence":"additional","affiliation":[{"name":"Google Inc., Mountain View, CA"}]},{"given":"Wolf-Dietrich","family":"Weber","sequence":"additional","affiliation":[{"name":"Google Inc., Mountain View, CA"}]}],"container-title":"Communications of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[2011,2]]},"URL":"http://dx.doi.org/10.1145/1897816.1897844","ISSN":["0001-0782","1557-7317"],"container-title-short":"Commun. ACM","note":"alternative-id:10.1145/1897816.1897844"},{"id":"doi:10.1109/SC.2018.00046","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/sc.2018.00046","page":"554-565","source":"Crossref","title":"Lessons Learned from Memory Errors Observed Over the Lifetime of Cielo","author":[{"given":"Scott","family":"Levy","sequence":"first","affiliation":[]},{"given":"Kurt B.","family":"Ferreira","sequence":"additional","affiliation":[]},{"given":"Nathan","family":"DeBardeleben","sequence":"additional","affiliation":[]},{"given":"Taniya","family":"Siddiqua","sequence":"additional","affiliation":[]},{"given":"Vilas","family":"Sridharan","sequence":"additional","affiliation":[]},{"given":"Elisabeth","family":"Baseman","sequence":"additional","affiliation":[]}],"event":"SC18: International Conference for High Performance Computing, Networking, Storage and Analysis","container-title":"SC18: International Conference for High Performance Computing, Networking, Storage and Analysis","original-title":[],"issued":{"date-parts":[[2018,11]]},"URL":"http://dx.doi.org/10.1109/SC.2018.00046"},{"id":"doi:10.1145/3502181.3531465","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3502181.3531465","page":"84-96","source":"Crossref","title":"Understanding Memory Failures on a Petascale Arm System","author":[{"given":"Kurt B.","family":"Ferreira","sequence":"first","affiliation":[{"name":"Sandia National Laboratories, Albuquerque, NM, USA"}]},{"given":"Scott","family":"Levy","sequence":"additional","affiliation":[{"name":"Sandia National Laboratories, Albuquerque, NM, USA"}]},{"given":"Joshua","family":"Hemmert","sequence":"additional","affiliation":[{"name":"Sandia National Laboratories, Albuquerque, NM, USA"}]},{"given":"Kevin","family":"Pedretti","sequence":"additional","affiliation":[{"name":"Sandia National Laboratories, Albuquerque, NM, USA"}]}],"event":"HPDC '22: The 31st International Symposium on High-Performance Parallel and Distributed Computing","container-title":"Proceedings of the 31st International Symposium on High-Performance Parallel and Distributed Computing","original-title":[],"issued":{"date-parts":[[2022,6,27]]},"URL":"http://dx.doi.org/10.1145/3502181.3531465","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3502181.3531465\n10.1145/3502181"},{"id":"doi:10.1109/DSN.2014.62","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/dsn.2014.62","page":"610-621","source":"Crossref","title":"Lessons Learned from the Analysis of System Failures at Petascale: The Case of Blue Waters","author":[{"given":"Catello","family":"Di Martino","sequence":"first","affiliation":[]},{"given":"Zbigniew","family":"Kalbarczyk","sequence":"additional","affiliation":[]},{"given":"Ravishankar K.","family":"Iyer","sequence":"additional","affiliation":[]},{"given":"Fabio","family":"Baccanico","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Fullop","sequence":"additional","affiliation":[]},{"given":"William","family":"Kramer","sequence":"additional","affiliation":[]}],"event":"2014 44th Annual IEEE/IFIP International Conference on Dependable Systems and Networks (DSN)","container-title":"2014 44th Annual IEEE/IFIP International Conference on Dependable Systems and Networks","original-title":[],"issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1109/DSN.2014.62"},{"id":"doi:10.1109/DSN.2015.57","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/dsn.2015.57","source":"Crossref","title":"Revisiting Memory Errors in Large-Scale Production Data Centers: Analysis and Modeling of New Trends from the Field","author":[{"given":"Justin","family":"Meza","sequence":"first","affiliation":[]},{"given":"Qiang","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Sanjeev","family":"Kumar","sequence":"additional","affiliation":[]},{"given":"Onur","family":"Mutlu","sequence":"additional","affiliation":[]}],"event":"2015 45th Annual IEEE/IFIP International Conference on Dependable Systems and Networks (DSN)","container-title":"2015 45th Annual IEEE/IFIP International Conference on Dependable Systems and Networks","original-title":[],"issued":{"date-parts":[[2015,6]]},"URL":"http://dx.doi.org/10.1109/DSN.2015.57"},{"id":"arxiv:0910.1536","type":"article-journal","author":[{"given":"Manas K.","family":"Patra"},{"given":"Samuel L.","family":"Braunstein"}],"title":"An algebraic framework for information theory: Classical Information","issued":{"date-parts":[[2009,10,12]]},"note":"arxivid:0910.1536\narxiv_version_number:2"},{"id":"arxiv:1005.0353","type":"article-journal","author":[{"given":"Greg","family":"Kuperberg"},{"given":"Nik","family":"Weaver"}],"title":"A von Neumann algebra approach to quantum metrics","issued":{"date-parts":[[2010,9,29]]},"note":"arxivid:1005.0353\narxiv_version_number:2"},{"id":"doi:10.1017/CBO9780511610882","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Association schemes are of interest to both mathematicians and statisticians and this book was written with both audiences in mind. For statisticians, it shows how to construct designs for experiments in blocks, how to compare such designs, and how to analyse data from them. The reader is only assumed to know very basic abstract algebra. For pure mathematicians, it tells why association schemes are important and develops the theory to the level of advanced research. Originally published in 2004, this book arose from a course successfully taught by the author and as such the material is thoroughly class-tested. There are a great number of examples and exercises that will increase the book's appeal to both graduate students and their instructors. It is ideal for those coming either from pure mathematics or statistics backgrounds who wish to develop their understanding of association schemes.</jats:p>","DOI":"10.1017/cbo9780511610882","source":"Crossref","title":"Association Schemes","author":[{"given":"R. A.","family":"Bailey","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2004,2,26]]},"ISBN":["9780521824460","9780511610882","9780521188012"],"URL":"http://dx.doi.org/10.1017/CBO9780511610882","note":"edition-number:1"},{"id":"doi:10.1038/299802a0","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5886","DOI":"10.1038/299802a0","page":"802-803","source":"Crossref","title":"A single quantum cannot be cloned","volume":"299","author":[{"given":"W. K.","family":"Wootters","sequence":"first","affiliation":[]},{"given":"W. H.","family":"Zurek","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[1982,10]]},"URL":"http://dx.doi.org/10.1038/299802a0","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BF299802a0"},{"id":"arxiv:2109.08691","type":"article-journal","author":[{"given":"Beni","family":"Yoshida"}],"title":"Decoding the Entanglement Structure of Monitored Quantum Circuits","issued":{"date-parts":[[2021,9,17]]},"note":"arxivid:2109.08691\narxiv_version_number:1"},{"id":"manual:-J.-Verhoeff-Error-detecting-","type":"document","author":[{"family":"Verhoeff","given":"J."}],"title":"Error detecting decimal codes","issued":"1969"},{"id":"doi:10.1109/TIT.1985.1056991","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1985.1056991","page":"102-105","source":"Crossref","title":"A new class of check-digit methods for arbitrary number systems (Corresp.)","volume":"31","author":[{"given":"H.","family":"Gumm","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1985,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1985.1056991","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/BF03025254","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf03025254","page":"20-26","source":"Crossref","title":"At the Dawn of the Theory of Codes","volume":"15","author":[{"given":"Alexander","family":"Barg","sequence":"first","affiliation":[]}],"container-title":"The Mathematical Intelligencer","original-title":[],"language":"en","issued":{"date-parts":[[1993,12]]},"URL":"http://dx.doi.org/10.1007/BF03025254","ISSN":["0343-6993"],"container-title-short":"The Mathematical Intelligencer","note":"alternative-id:BF03025254"},{"id":"doi:10.1080/07468342.1991.11973381","type":"journal-article","publisher":"Informa UK Limited","issue":"3","DOI":"10.1080/07468342.1991.11973381","page":"194-202","source":"Crossref","title":"The Mathematics of Identification Numbers","volume":"22","author":[{"given":"Joseph A.","family":"Gallian","sequence":"first","affiliation":[]}],"container-title":"The College Mathematics Journal","original-title":[],"language":"en","issued":{"date-parts":[[1991,5]]},"URL":"http://dx.doi.org/10.1080/07468342.1991.11973381","ISSN":["0746-8342","1931-1346"],"container-title-short":"The College Mathematics Journal","note":"aliases:10.2307/2686641\nalternative-id:10.1080/07468342.1991.11973381"},{"id":"manual:-K.-Conrad-Dihedral-groups.-L","type":"article-journal","author":[{"family":"Conrad","given":"K."}],"title":"Dihedral groups","container-title":"Lecture notes","issued":"2017"},{"id":"doi:10.1109/ISIT.2012.6284269","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2012.6284269","page":"631-635","source":"Crossref","title":"Codes over non-Abelian groups: Point-to-point communications and computation over MAC","author":[{"given":"Aria G.","family":"Sahebi","sequence":"first","affiliation":[]},{"given":"S. Sandeep","family":"Pradhan","sequence":"additional","affiliation":[]}],"event":"2012 IEEE International Symposium on Information Theory - ISIT","container-title":"2012 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2012,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2012.6284269"},{"id":"preset:Diaconis88","type":"book","author":[{"family":"Diaconis","given":"P."}],"title":"\\emph{Group Representations in Probability and Statistics}","collection-title":"Lecture Notes-Monograph Series","volume":"11","publisher":"Institute of Mathematical Statistics","issued":"1988"},{"id":"arxiv:2205.04965","type":"article-journal","author":[{"given":"Laith","family":"Rastanawi"},{"given":"Günter","family":"Rote"}],"title":"Towards a Geometric Understanding of the 4-Dimensional Point Groups","issued":{"date-parts":[[2022,5,10]]},"note":"arxivid:2205.04965\narxiv_version_number:1"},{"id":"doi:10.1109/49.29613","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/49.29613","page":"900-913","source":"Crossref","title":"Block coset codes for M-ary phase shift keying","volume":"7","author":[{"given":"F.R.","family":"Kschischang","sequence":"first","affiliation":[]},{"given":"P.G.","family":"de Buda","sequence":"additional","affiliation":[]},{"given":"S.","family":"Pasupathy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[1989]]},"URL":"http://dx.doi.org/10.1109/49.29613","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1109/18.133243","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.133243","page":"1241-1260","source":"Crossref","title":"Geometrically uniform codes","volume":"37","author":[{"given":"G.D.","family":"Forney","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1991]]},"URL":"http://dx.doi.org/10.1109/18.133243","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.104333","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.104333","page":"1675-1682","source":"Crossref","title":"Signal sets matched to groups","volume":"37","author":[{"given":"H.-A.","family":"Loeliger","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1991]]},"URL":"http://dx.doi.org/10.1109/18.104333","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.165454","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.165454","page":"1797-1801","source":"Crossref","title":"On the Hamming distance properties of group codes","volume":"38","author":[{"given":"G.D.","family":"Forney","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1992]]},"URL":"http://dx.doi.org/10.1109/18.165454","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/BF02997785","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"9-10","DOI":"10.1007/bf02997785","page":"817-823","source":"Crossref","title":"On the construction of group block codes","volume":"50","author":[{"given":"Ezio","family":"Biglieri","sequence":"first","affiliation":[]},{"given":"Michele","family":"Elia","sequence":"additional","affiliation":[]}],"container-title":"Annales Des Télécommunications","original-title":[],"language":"en","issued":{"date-parts":[[1995,9]]},"URL":"http://dx.doi.org/10.1007/BF02997785","ISSN":["0003-4347","1958-9395"],"container-title-short":"Ann. Télécommun.","note":"alternative-id:BF02997785"},{"id":"doi:10.1109/18.508859","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/18.508859","page":"1277-1280","source":"Crossref","title":"Group block codes over nonabelian groups are asymptotically bad","volume":"42","author":[{"given":"J.C.","family":"Interlando","sequence":"first","affiliation":[{"name":"Dept. of Commun., Univ. Estadual de Campinas, Brazil"}]},{"given":"R.","family":"Palazzo","sequence":"additional","affiliation":[{"name":"Dept. of Commun., Univ. Estadual de Campinas, Brazil"}]},{"given":"M.","family":"Elia","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1996,7]]},"URL":"http://dx.doi.org/10.1109/18.508859","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.259635","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.259635","page":"1491-1513","source":"Crossref","title":"The dynamics of group codes: state spaces, trellis diagrams, and canonical encoders","volume":"39","author":[{"given":"G.D.","family":"Forney","sequence":"first","affiliation":[]},{"given":"M.D.","family":"Trott","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1993]]},"URL":"http://dx.doi.org/10.1109/18.259635","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/0097-3165(91)90024-B","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(91)90024-b","page":"84-95","source":"Crossref","title":"Upper bounds for football pool problems and mixed covering codes","volume":"56","author":[{"given":"Heikki","family":"Hämäläinen","sequence":"first","affiliation":[]},{"given":"Seppo","family":"Rankinen","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1991,1]]},"URL":"http://dx.doi.org/10.1016/0097-3165(91)90024-B","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:009731659190024B"},{"id":"doi:10.1109/18.651001","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.651001","page":"140-161","source":"Crossref","title":"Bounds on mixed binary/ternary codes","volume":"44","author":[{"given":"A.E.","family":"Brouwer","sequence":"first","affiliation":[]},{"given":"H.O.","family":"Hamalainen","sequence":"additional","affiliation":[]},{"given":"P.R.J.","family":"Ostergard","sequence":"additional","affiliation":[]},{"given":"N.J.A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.651001","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1606.06930","type":"journal-article","publisher":"Elsevier BV","issue":"6","DOI":"10.1016/j.disc.2018.03.013","page":"1740-1748","source":"Crossref","title":"Semidefinite bounds for mixed binary/ternary codes","volume":"341","author":[{"given":"Bart","family":"Litjens","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2018,6]]},"URL":"http://dx.doi.org/10.1016/j.disc.2018.03.013","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X18300840\narxivid:1606.06930"},{"id":"doi:10.1016/0097-3165(91)90011-5","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(91)90011-5","page":"130-143","source":"Crossref","title":"Generalized bounds on binary/ternary mixed packing and covering codes","volume":"57","author":[{"suffix":"Jr.","given":"J.H.","family":"van Lint","sequence":"first","affiliation":[]},{"given":"G.J.M.","family":"van Wee","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1991,5]]},"URL":"http://dx.doi.org/10.1016/0097-3165(91)90011-5","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316591900115"},{"id":"manual:-P.-J.-Cameron-Some-bridges-b","type":"manuscript","author":[{"family":"Cameron","given":"P.J."}],"title":"Some bridges between codes and designs","genre":"Unpublished manuscript,","publisher":"Queen Mary and Westfield College","issued":"1998","publisher-place":"London"},{"id":"manual:-S.-Addelman-and-O.-Kempthorn","type":"report","author":[{"family":"Addelman","given":"S."},{"family":"Kempthorne","given":"O."}],"title":"Orthogonal Main-Effect Plans","genre":"Technical Report ARL 79,","publisher":"Aeronautical Research Lab., Wright-Patterson Air Force Base","issued":"1961","publisher-place":"Ohio"},{"id":"doi:10.1016/B978-0-7204-2262-7.50034","type":"book-chapter","publisher":"Elsevier","DOI":"10.1016/b978-0-7204-2262-7.50034-x","page":"349-359","source":"Crossref","title":"Some Combinatorial Problems of Arrays and Applications to Design of Experiments††Paper read at the International Symposium on Combinatorial Mathematics and its Applications, Fort Collins, Colorado, September 1971.","author":[{"given":"C. RADHAKRISHNA","family":"RAO","sequence":"first","affiliation":[]}],"container-title":"A Survey of Combinatorial Theory","original-title":[],"language":"en","issued":{"date-parts":[[1973]]},"ISBN":["9780720422627"],"URL":"http://dx.doi.org/10.1016/B978-0-7204-2262-7.50034-X"},{"id":"doi:10.1007/978-1-4612-1478-6","type":"book","publisher":"Springer New York","DOI":"10.1007/978-1-4612-1478-6","source":"Crossref","title":"Orthogonal Arrays","author":[{"given":"A. S.","family":"Hedayat","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]},{"given":"John","family":"Stufken","sequence":"additional","affiliation":[]}],"container-title":"Springer Series in Statistics","original-title":[],"issued":{"date-parts":[[1999]]},"ISBN":["9781461271581","9781461214786"],"URL":"http://dx.doi.org/10.1007/978-1-4612-1478-6","ISSN":["0172-7397"],"note":"publisher-location:New York, NY"},{"id":"doi:10.1016/S0378-3758(96)00025-0","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0378-3758(96)00025-0","page":"295-305","source":"Crossref","title":"A linear programming bound for orthogonal arrays with mixed levels","volume":"56","author":[{"given":"N.J.A.","family":"Sloane","sequence":"first","affiliation":[]},{"given":"J.","family":"Stufken","sequence":"additional","affiliation":[]}],"container-title":"Journal of Statistical Planning and Inference","original-title":[],"language":"en","issued":{"date-parts":[[1996,12]]},"URL":"http://dx.doi.org/10.1016/S0378-3758(96)00025-0","ISSN":["0378-3758"],"container-title-short":"Journal of Statistical Planning and Inference","note":"alternative-id:S0378375896000250"},{"id":"arxiv:0908.4094","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2010.2048455","page":"3158-3165","source":"Crossref","title":"Codes in Permutations and Error Correction for Rank Modulation","volume":"56","author":[{"given":"Alexander","family":"Barg","sequence":"first","affiliation":[]},{"given":"Arya","family":"Mazumdar","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2048455","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0908.4094"},{"id":"doi:10.1109/PROC.1965.3680","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/proc.1965.3680","page":"228-236","source":"Crossref","title":"Permutation modulation","volume":"53","author":[{"given":"D.","family":"Slepian","sequence":"first","affiliation":[]}],"container-title":"Proceedings of the IEEE","original-title":[],"issued":{"date-parts":[[1965]]},"URL":"http://dx.doi.org/10.1109/PROC.1965.3680","ISSN":["0018-9219"],"container-title-short":"Proc. IEEE"},{"id":"doi:10.1109/TIT.1969.1054291","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1969.1054291","page":"306-315","source":"Crossref","title":"Rank permutation group codes based on Kendall's correlation statistic","volume":"15","author":[{"given":"H.","family":"Chadwick","sequence":"first","affiliation":[]},{"given":"L.","family":"Kurz","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1969,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1969.1054291","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/0378-3758(78)90008-3","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0378-3758(78)90008-3","page":"197-209","source":"Crossref","title":"Bounds for permutation arrays","volume":"2","author":[{"given":"M.","family":"Deza","sequence":"first","affiliation":[]},{"given":"S.A.","family":"Vanstone","sequence":"additional","affiliation":[]}],"container-title":"Journal of Statistical Planning and Inference","original-title":[],"language":"en","issued":{"date-parts":[[1978,1]]},"URL":"http://dx.doi.org/10.1016/0378-3758(78)90008-3","ISSN":["0378-3758"],"container-title-short":"Journal of Statistical Planning and Inference","note":"alternative-id:0378375878900083"},{"id":"doi:10.1016/0097-3165(77)90009-7","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0097-3165(77)90009-7","page":"352-360","source":"Crossref","title":"On the maximum number of permutations with given maximal or minimal distance","volume":"22","author":[{"given":"Peter","family":"Frankl","sequence":"first","affiliation":[]},{"given":"Mikhail","family":"Deza","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1977,5]]},"URL":"http://dx.doi.org/10.1016/0097-3165(77)90009-7","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316577900097"},{"id":"doi:10.1016/S0019-9958(79)90076-7","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0019-9958(79)90076-7","page":"1-19","source":"Crossref","title":"Coding with permutations","volume":"43","author":[{"given":"Ian F.","family":"Blake","sequence":"first","affiliation":[]},{"given":"Gérard","family":"Cohen","sequence":"additional","affiliation":[]},{"given":"Mikhail","family":"Deza","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1979,10]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(79)90076-7","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995879900767"},{"id":"doi:10.1016/0097-3165(79)90012-8","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(79)90012-8","page":"119","source":"Crossref","title":"An inequality for finite permutation groups","volume":"27","author":[{"given":"Masao","family":"Kiyota","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1979,7]]},"URL":"http://dx.doi.org/10.1016/0097-3165(79)90012-8","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316579900128"},{"id":"doi:10.1006/eujc.1998.0272","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1006/eujc.1998.0272","page":"101-114","source":"Crossref","title":"Upper Bounds on Permutation Codes via Linear Programming","volume":"20","author":[{"given":"H.","family":"Tarnanen","sequence":"first","affiliation":[]}],"container-title":"European Journal of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[1999,1]]},"URL":"http://dx.doi.org/10.1006/eujc.1998.0272","ISSN":["0195-6698"],"container-title-short":"European Journal of Combinatorics","note":"alternative-id:S0195669898902724"},{"id":"arxiv:1912.04500","type":"article-journal","author":[{"given":"Peter J.","family":"Dukes"},{"given":"Ferdinand","family":"Ihringer"},{"given":"Nathan","family":"Lindzey"}],"title":"On the Algebraic Combinatorics of Injections and its Applications to Injection Codes","issued":{"date-parts":[[2019,12,10]]},"note":"arxivid:1912.04500\narxiv_version_number:1"},{"id":"doi:10.1515/dma.1992.2.3.241","type":"journal-article","publisher":"Walter de Gruyter GmbH","issue":"3","DOI":"10.1515/dma.1992.2.3.241","source":"Crossref","title":"On perfect codes in deletion and insertion metric","volume":"2","author":[{"given":"V. I.","family":"LEVENSHTEIN","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics and Applications","original-title":[],"issued":{"date-parts":[[1992]]},"URL":"http://dx.doi.org/10.1515/dma.1992.2.3.241","ISSN":["0924-9265","1569-3929"]},{"id":"arxiv:2401.17235","type":"article-journal","author":[{"given":"Elazar","family":"Goldenberg"},{"given":"Mursalin","family":"Habib"},{"given":"Karthik C.","family":"S"}],"title":"Explicit Good Codes Approaching Distance 1 in Ulam Metric","issued":{"date-parts":[[2024,5,11]]},"note":"arxivid:2401.17235\narxiv_version_number:2"},{"id":"doi:10.1016/j.ejc.2009.03.044","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/j.ejc.2009.03.044","page":"482-490","source":"Crossref","title":"Permutation codes","volume":"31","author":[{"given":"Peter J.","family":"Cameron","sequence":"first","affiliation":[]}],"container-title":"European Journal of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2010,2]]},"URL":"http://dx.doi.org/10.1016/j.ejc.2009.03.044","ISSN":["0195-6698"],"container-title-short":"European Journal of Combinatorics","note":"alternative-id:S0195669809000894"},{"id":"doi:10.1109/TCOMM.2005.858683","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tcomm.2005.858683","page":"1782-1789","source":"Crossref","title":"Permutation Trellis Codes","volume":"53","author":[{"given":"H.C.","family":"Ferreira","sequence":"first","affiliation":[]},{"given":"A.J.H.","family":"Vinck","sequence":"additional","affiliation":[]},{"given":"T.G.","family":"Swart","sequence":"additional","affiliation":[]},{"given":"I.","family":"deBeer","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[2005,11]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2005.858683","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"manual:-G.-Kuperberg-private-communi","type":"document","author":[{"family":"Kuperberg","given":"G."}],"genre":"private communication,","issued":"2024"},{"id":"arxiv:1110.2557","type":"article-journal","author":[{"given":"Arya","family":"Mazumdar"},{"given":"Alexander","family":"Barg"},{"given":"Gilles","family":"Zémor"}],"title":"Constructions of Rank Modulation Codes","issued":{"date-parts":[[2011,10,12]]},"note":"arxivid:1110.2557\narxiv_version_number:1"},{"id":"doi:10.1109/ISIT.2008.4595285","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2008.4595285","page":"1736-1740","source":"Crossref","title":"Error-correcting codes for rank modulation","author":[{"family":"Anxiao Jiang","sequence":"first","affiliation":[]},{"given":"Moshe","family":"Schwartz","sequence":"additional","affiliation":[]},{"given":"Jehoshua","family":"Bruck","sequence":"additional","affiliation":[]}],"event":"2008 IEEE International Symposium on Information Theory - ISIT","container-title":"2008 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2008.4595285"},{"id":"doi:10.1109/ISIT.2010.5513604","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2010.5513604","page":"854-858","source":"Crossref","title":"Codes in permutations and error correction for rank modulation","author":[{"given":"Alexander","family":"Barg","sequence":"first","affiliation":[]},{"given":"Arya","family":"Mazumdar","sequence":"additional","affiliation":[]}],"event":"2010 IEEE International Symposium on Information Theory - ISIT","container-title":"2010 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2010,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2010.5513604"},{"id":"doi:10.2307/1969252","type":"journal-article","publisher":"JSTOR","issue":"3","DOI":"10.2307/1969252","page":"647","source":"Crossref","title":"Hilbert Distances and Positive Definite Functions","volume":"42","author":[{"given":"S.","family":"Bochner","sequence":"first","affiliation":[]}],"container-title":"The Annals of Mathematics","original-title":[],"issued":{"date-parts":[[1941,7]]},"URL":"http://dx.doi.org/10.2307/1969252","ISSN":["0003-486X"],"container-title-short":"The Annals of Mathematics"},{"id":"arxiv:math/0701533","type":"journal-article","publisher":"Wiley","issue":"2","DOI":"10.1112/plms/pdp027","page":"348-376","source":"Crossref","title":"Harmonic analysis on a finite homogeneous space","volume":"100","author":[{"given":"Fabio","family":"Scarabotti","sequence":"first","affiliation":[{"name":"Dipartimento MeMoMat; Università degli Studi di Roma “La Sapienza”; via A. Scarpa 8 Roma 00161 Italy"}]},{"given":"Filippo","family":"Tolli","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica; Università Roma TRE; L. San Leonardo Murialdo 1 Roma 00146 Italy"}]}],"container-title":"Proceedings of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,17]]},"URL":"http://dx.doi.org/10.1112/plms/pdp027","ISSN":["0024-6115"],"container-title-short":"Proceedings of the London Mathematical Society","note":"arxivid:math/0701533"},{"id":"doi:10.1007/978-1-4939-3082-1","type":"book","publisher":"Springer New York","DOI":"10.1007/978-1-4939-3082-1","source":"Crossref","title":"The Grassmannian Variety","author":[{"given":"V.","family":"Lakshmibai","sequence":"first","affiliation":[]},{"given":"Justin","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Developments in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2015]]},"ISBN":["9781493930814","9781493930821"],"URL":"http://dx.doi.org/10.1007/978-1-4939-3082-1","ISSN":["1389-2177","2197-795X"],"note":"publisher-location:New York, NY"},{"id":"manual:-A.-Bezdek-Solid-packing-of-c","type":"document","author":[{"family":"Bezdek","given":"A."}],"title":"Solid packing of circles in the hyperbolic plane","issued":"1979"},{"id":"doi:10.1016/j.jfranklin.2005.09.001","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.jfranklin.2005.09.001","page":"69-82","source":"Crossref","title":"Signal constellations in the hyperbolic plane: A proposal for new communication systems","volume":"343","author":[{"given":"Eduardo Brandani","family":"da Silva","sequence":"first","affiliation":[]},{"given":"Marcelo","family":"Firer","sequence":"additional","affiliation":[]},{"given":"Sueli R.","family":"Costa","sequence":"additional","affiliation":[]},{"suffix":"Jr.","given":"Reginaldo","family":"Palazzo","sequence":"additional","affiliation":[]}],"container-title":"Journal of the Franklin Institute","original-title":[],"language":"en","issued":{"date-parts":[[2006,1]]},"URL":"http://dx.doi.org/10.1016/j.jfranklin.2005.09.001","ISSN":["0016-0032"],"container-title-short":"Journal of the Franklin Institute","note":"alternative-id:S0016003205000815"},{"id":"doi:10.1137/1104038","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","DOI":"10.1137/1104038","page":"391-398","source":"Crossref","title":"Waveguides with Random Inhomogeneities and Brownian Motion In the Lobachevsky Plane","volume":"4","author":[{"given":"M. E.","family":"Gertsenshtein","sequence":"first","affiliation":[]},{"given":"V. B.","family":"Vasil’ev","sequence":"additional","affiliation":[]}],"container-title":"Theory of Probability &amp; Its Applications","original-title":[],"language":"en","issued":{"date-parts":[[1959,1]]},"URL":"http://dx.doi.org/10.1137/1104038","ISSN":["0040-585X","1095-7219"],"container-title-short":"Theory Probab. Appl.","note":"alternative-id:10.1137/1104038"},{"id":"doi:10.1023/B:GEOM.0000006580.47816.","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1023/b:geom.0000006580.47816.e9","page":"213-236","source":"Crossref","title":"Periodicity and Circle Packings of the Hyperbolic Plane","volume":"102","author":[{"given":"Lewis","family":"Bowen","sequence":"first","affiliation":[]}],"container-title":"Geometriae Dedicata","original-title":[],"language":"en","issued":{"date-parts":[[2003,12]]},"URL":"http://dx.doi.org/10.1023/B:GEOM.0000006580.47816.e9","ISSN":["0046-5755","1572-9168"],"container-title-short":"Geometriae Dedicata","note":"alternative-id:5150849"},{"id":"arxiv:1212.5966","type":"journal-article","publisher":"Duke University Press","issue":"10","DOI":"10.1215/00127094-2738857","source":"Crossref","title":"Sphere packing bounds via spherical codes","volume":"163","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[]},{"given":"Yufei","family":"Zhao","sequence":"additional","affiliation":[]}],"container-title":"Duke Mathematical Journal","original-title":[],"issued":{"date-parts":[[2014,7,15]]},"URL":"http://dx.doi.org/10.1215/00127094-2738857","ISSN":["0012-7094"],"container-title-short":"Duke Math. J.","note":"arxivid:1212.5966"},{"id":"arxiv:2302.02540","type":"article-journal","author":[{"given":"Maxime Fortier","family":"Bourque"},{"given":"Bram","family":"Petri"}],"title":"Linear programming bounds for hyperbolic surfaces","issued":{"date-parts":[[2026,2,9]]},"note":"arxivid:2302.02540\narxiv_version_number:3"},{"id":"arxiv:2411.07139","type":"journal-article","publisher":"MathDoc/Centre Mersenne","issue":"G1","abstract":"<jats:p>We prove sphere packing density bounds in hyperbolic space (and more generally irreducible symmetric spaces of noncompact type), which were conjectured by Cohn and Zhao and generalize Euclidean bounds by Cohn and Elkies. We work within the Bowen–Radin framework of packing density and replace the use of the Poisson summation formula in the proof of the Euclidean bound by Cohn and Elkies with an analogous formula arising from methods used in the theory of mathematical quasicrystals.</jats:p>","DOI":"10.5802/crmath.821","page":"237-242","source":"Crossref","title":"Bounds on hyperbolic sphere packings: on a conjecture by Cohn and Zhao","volume":"364","author":[{"given":"Maximilian","family":"Wackenhuth","sequence":"first","affiliation":[{"name":"KIT, Germany"}]}],"container-title":"Comptes Rendus. Mathématique","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,17]]},"URL":"http://dx.doi.org/10.5802/crmath.821","ISSN":["1631-073X","1778-3569"],"note":"alternative-id:10.5802/crmath.821\narxivid:2411.07139"},{"id":"arxiv:2505.23572","type":"article-journal","author":[{"given":"Maximilian","family":"Wackenhuth"}],"title":"Linear programming bounds in homogeneous spaces, I: Optimal packing density","issued":{"date-parts":[[2025,5,29]]},"note":"arxivid:2505.23572\narxiv_version_number:1"},{"id":"manual:-E.-B.-Silva-and-R.-Palazzo-J","type":"chapter","author":[{"family":"Silva","given":"E.B."},{"family":"Palazzo","given":"R.","suffix":"Jr"}],"title":"M-PSK signal constellations in hyperbolic space achieving better performance than the M-PSK signal constellations in Euclidean space","container-title":"1999 IEEE Information Theory Workshop","issued":"1999","publisher-place":"Metsovo, Greece"},{"id":"doi:10.1109/GLOCOM.2015.7417496","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/glocom.2015.7417496","page":"1-7","source":"Crossref","title":"Space-Time Block Codes over the Stiefel Manifold","author":[{"given":"Mohammad T.","family":"Hussien","sequence":"first","affiliation":[]},{"given":"Karim G.","family":"Seddik","sequence":"additional","affiliation":[]},{"given":"Ramy H.","family":"Gohary","sequence":"additional","affiliation":[]},{"given":"Mohammad","family":"Shaqfeh","sequence":"additional","affiliation":[]},{"given":"Hussein","family":"Alnuweiri","sequence":"additional","affiliation":[]},{"given":"Halim","family":"Yanikomeroglu","sequence":"additional","affiliation":[]}],"event":"GLOBECOM 2015 - 2015 IEEE Global Communications Conference","container-title":"2015 IEEE Global Communications Conference (GLOBECOM)","original-title":[],"issued":{"date-parts":[[2015,12]]},"URL":"http://dx.doi.org/10.1109/GLOCOM.2015.7417496"},{"id":"arxiv:math/0308110","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2005.855594","page":"3445-3456","source":"Crossref","title":"Sphere-packing bounds in the Grassmann and Stiefel manifolds","volume":"51","author":[{"given":"O.","family":"Henkel","sequence":"first","affiliation":[{"name":"Fraunhofer German-Sino Lab for Mobile Commun. MCI, Berlin, Germany"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2005,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.855594","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:math/0308110"},{"id":"arxiv:math/0610813","type":"article-journal","author":[{"given":"Christine","family":"Bachoc"},{"given":"Yael","family":"Ben-Haim"},{"given":"Simon","family":"Litsyn"}],"title":"Bounds for codes in products of spaces, Grassmann and Stiefel manifolds","issued":{"date-parts":[[2006,10,26]]},"note":"arxivid:math/0610813\narxiv_version_number:1"},{"id":"arxiv:2407.08423","type":"article-journal","author":[{"given":"Miguel","family":"Casanova"},{"given":"Kentaro","family":"Ohki"},{"given":"Francesco","family":"Ticozzi"}],"title":"Finding Quantum Codes via Riemannian Optimization","issued":{"date-parts":[[2024,10,28]]},"note":"arxivid:2407.08423\narxiv_version_number:2"},{"id":"doi:10.2307/1969427","type":"journal-article","publisher":"JSTOR","issue":"1","DOI":"10.2307/1969427","page":"177","source":"Crossref","title":"Two-Point Homogeneous Spaces","volume":"55","author":[{"given":"Hsien-Chung","family":"Wang","sequence":"first","affiliation":[]}],"container-title":"The Annals of Mathematics","original-title":[],"issued":{"date-parts":[[1952,1]]},"URL":"http://dx.doi.org/10.2307/1969427","ISSN":["0003-486X"],"container-title-short":"The Annals of Mathematics"},{"id":"doi:10.2307/1970789","type":"journal-article","publisher":"JSTOR","issue":"2","DOI":"10.2307/1970789","page":"277","source":"Crossref","title":"Compact Homogeneous Riemannian Manifolds with Strictly Positive Curvature","volume":"96","author":[{"given":"Nolan R.","family":"Wallach","sequence":"first","affiliation":[]}],"container-title":"The Annals of Mathematics","original-title":[],"issued":{"date-parts":[[1972,9]]},"URL":"http://dx.doi.org/10.2307/1970789","ISSN":["0003-486X"],"container-title-short":"The Annals of Mathematics"},{"id":"doi:10.1016/S0926-2245(00)00038-3","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0926-2245(00)00038-3","page":"57-78","source":"Crossref","title":"Isometry groups of homogeneous spaces with positive sectional curvature","volume":"14","author":[{"given":"Krishnan","family":"Shankar","sequence":"first","affiliation":[]}],"container-title":"Differential Geometry and its Applications","original-title":[],"language":"en","issued":{"date-parts":[[2001,1]]},"URL":"http://dx.doi.org/10.1016/S0926-2245(00)00038-3","ISSN":["0926-2245"],"container-title-short":"Differential Geometry and its Applications","note":"alternative-id:S0926224500000383"},{"id":"doi:10.1007/978-94-010-9787-1_2","type":"book-chapter","publisher":"Springer Netherlands","DOI":"10.1007/978-94-010-9787-1_2","page":"87-128","source":"Crossref","title":"Orthogonal Polynomials and Chevalley Groups","author":[{"given":"Dennis","family":"Stanton","sequence":"first","affiliation":[]}],"container-title":"Special Functions: Group Theoretical Aspects and Applications","original-title":[],"language":"en","issued":{"date-parts":[[1984]]},"ISBN":["9781402003196","9789401097871"],"URL":"http://dx.doi.org/10.1007/978-94-010-9787-1_2","note":"publisher-location:Dordrecht"},{"id":"doi:10.1017/CBO9780511619823","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Line up a deck of 52 cards on a table. Randomly choose two cards and switch them. How many switches are needed in order to mix up the deck? Starting from a few concrete problems such as random walks on the discrete circle and the finite ultrametric space this book develops the necessary tools for the asymptotic analysis of these processes. This detailed study culminates with the case-by-case analysis of the cut-off phenomenon discovered by Persi Diaconis. This self-contained text is ideal for graduate students and researchers working in the areas of representation theory, group theory, harmonic analysis and Markov chains. Its topics range from the basic theory needed for students new to this area, to advanced topics such as the theory of Green's algebras, the complete analysis of the random matchings, and the representation theory of the symmetric group.</jats:p>","DOI":"10.1017/cbo9780511619823","source":"Crossref","title":"Harmonic Analysis on Finite Groups","author":[{"given":"Tullio","family":"Ceccherini-Silberstein","sequence":"first","affiliation":[]},{"given":"Fabio","family":"Scarabotti","sequence":"additional","affiliation":[]},{"given":"Filippo","family":"Tolli","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2008,3,6]]},"ISBN":["9780521883368","9780511619823"],"URL":"http://dx.doi.org/10.1017/CBO9780511619823","note":"edition-number:1"},{"id":"doi:10.1016/0097-3165(78)90015-8","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0097-3165(78)90015-8","page":"226-241","source":"Crossref","title":"Bilinear forms over a finite field, with applications to coding theory","volume":"25","author":[{"given":"Ph","family":"Delsarte","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1978,11]]},"URL":"http://dx.doi.org/10.1016/0097-3165(78)90015-8","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316578900158"},{"id":"doi:10.1016/j.jcta.2010.05.006","type":"journal-article","publisher":"Elsevier BV","issue":"8","DOI":"10.1016/j.jcta.2010.05.006","page":"1011-1026","source":"Crossref","title":"Symmetric bilinear forms over finite fields of even characteristic","volume":"117","author":[{"given":"Kai-Uwe","family":"Schmidt","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[2010,11]]},"URL":"http://dx.doi.org/10.1016/j.jcta.2010.05.006","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S0097316510000981"},{"id":"arxiv:1907.07848","type":"article-journal","author":[{"given":"John","family":"Jasper"},{"given":"Emily J.","family":"King"},{"given":"Dustin G.","family":"Mixon"}],"title":"Game of Sloanes: Best known packings in complex projective space","issued":{"date-parts":[[2019,7,18]]},"note":"arxivid:1907.07848\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0310075","type":"journal-article","publisher":"AIP Publishing","issue":"6","abstract":"<jats:p>We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d2 rank-one operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC–POVM) and is equivalent to a set of d2 equiangular lines in Cd. SIC–POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.</jats:p>","DOI":"10.1063/1.1737053","page":"2171-2180","source":"Crossref","title":"Symmetric informationally complete quantum measurements","volume":"45","author":[{"given":"Joseph M.","family":"Renes","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131-1156"}]},{"given":"Robin","family":"Blume-Kohout","sequence":"additional","affiliation":[{"name":"Theoretical Division, LANL, MS-B213, Los Alamos, New Mexico 87545"}]},{"given":"A. J.","family":"Scott","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131-1156"}]},{"given":"Carlton M.","family":"Caves","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131-1156"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2004,6,1]]},"URL":"http://dx.doi.org/10.1063/1.1737053","ISSN":["0022-2488","1089-7658"],"note":"arxivid:quant-ph/0310075"},{"id":"arxiv:quant-ph/0701126","type":"article-journal","author":[{"given":"Andris","family":"Ambainis"},{"given":"Joseph","family":"Emerson"}],"title":"Quantum t-designs: t-wise independence in the quantum world","issued":{"date-parts":[[2007,2,6]]},"note":"arxivid:quant-ph/0701126\narxiv_version_number:2"},{"id":"doi:10.1017/9781139207010","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Quantum information theory is a branch of science at the frontier of physics, mathematics, and information science, and offers a variety of solutions that are impossible using classical theory. This book provides a detailed introduction to the key concepts used in processing quantum information and reveals that quantum mechanics is a generalisation of classical probability theory. The second edition contains new sections and entirely new chapters: the hot topic of multipartite entanglement; in-depth discussion of the discrete structures in finite dimensional Hilbert space, including unitary operator bases, mutually unbiased bases, symmetric informationally complete generalized measurements, discrete Wigner function, and unitary designs; the Gleason and Kochen–Specker theorems; the proof of the Lieb conjecture; the measure concentration phenomenon; and the Hastings' non-additivity theorem. This richly-illustrated book will be useful to a broad audience of graduates and researchers interested in quantum information theory. Exercises follow each chapter, with hints and answers supplied.</jats:p>","DOI":"10.1017/9781139207010","source":"Crossref","title":"Geometry of Quantum States","author":[{"given":"Ingemar","family":"Bengtsson","sequence":"first","affiliation":[]},{"given":"Karol","family":"Życzkowski","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2017,8,30]]},"ISBN":["9781107026254","9781139207010","9781107656147"],"URL":"http://dx.doi.org/10.1017/9781139207010","note":"edition-number:2"},{"id":"manual:-G.-Zauner-(1999).-Grundz-ge-","type":"thesis","author":[{"family":"Zauner","given":"G."}],"title":"Grundzüge einer nichtkommutativen Designtheorie","genre":"PhD thesis","issued":"1999"},{"id":"arxiv:quant-ph/0309120","type":"article-journal","author":[{"given":"Andreas","family":"Klappenecker"},{"given":"Martin","family":"Roetteler"}],"title":"Constructions of Mutually Unbiased Bases","issued":{"date-parts":[[2003,9,15]]},"note":"arxivid:quant-ph/0309120\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0502031","type":"article-journal","author":[{"given":"Andreas","family":"Klappenecker"},{"given":"Martin","family":"Roetteler"}],"title":"Mutually Unbiased Bases are Complex Projective 2-Designs","issued":{"date-parts":[[2005,2,11]]},"note":"arxivid:quant-ph/0502031\narxiv_version_number:2"},{"id":"arxiv:0711.1017","type":"journal-article","publisher":"IOP Publishing","issue":"5","DOI":"10.1088/1751-8113/41/5/055308","page":"055308","source":"Crossref","title":"Optimizing quantum process tomography with unitary<b>2</b>-designs","volume":"41","author":[{"given":"A J","family":"Scott","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2008,1,23]]},"URL":"http://dx.doi.org/10.1088/1751-8113/41/5/055308","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(08)64633-8\narxivid:0711.1017"},{"id":"arxiv:1004.3348","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"04","abstract":"<jats:p>Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p<jats:sup>M</jats:sup>, with p prime and M a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12,…).</jats:p><jats:p>In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, …, N - 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two–q-bit Bell states) that are related to the mutually unbiased bases.</jats:p><jats:p>There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime.</jats:p><jats:p>The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist — one can find no more than three pairwise unbiased bases — although there is currently no clear-cut demonstration of the case.</jats:p>","DOI":"10.1142/s0219749910006502","page":"535-640","source":"Crossref","title":"ON MUTUALLY UNBIASED BASES","volume":"08","author":[{"given":"THOMAS","family":"DURT","sequence":"first","affiliation":[{"name":"TONA Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium"}]},{"given":"BERTHOLD-GEORG","family":"ENGLERT","sequence":"additional","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore"},{"name":"Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore"}]},{"given":"INGEMAR","family":"BENGTSSON","sequence":"additional","affiliation":[{"name":"Stockholms Universitet, Fysikum, Alba Nova, 106 91 Stockholm, Sweden"}]},{"given":"KAROL","family":"ŻYCZKOWSKI","sequence":"additional","affiliation":[{"name":"Instytut Fizyki Uniwersytetu Jagiellońskiego, ul. Reymonta 4, 30-059 Kraków, Poland"},{"name":"Centrum Fizyki Teoretycznej PAN, Al. Lotników 32/44, 02-668 Warszawa, Poland"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2010,6]]},"URL":"http://dx.doi.org/10.1142/S0219749910006502","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749910006502\narxivid:1004.3348"},{"id":"arxiv:1505.01123","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.91.060301","source":"Crossref","title":"Mutually unbiased bases as minimal Clifford covariant 2-designs","volume":"91","author":[{"given":"Huangjun","family":"Zhu","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,6,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.060301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"060301","note":"arxivid:1505.01123"},{"id":"arxiv:2410.23997","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.</jats:p>","DOI":"10.22331/q-2026-04-01-2051","page":"2051","source":"Crossref","title":"Mutually Unbiased Bases in Composite Dimensions – A Review","volume":"10","author":[{"given":"Daniel","family":"McNulty","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Università di Bari, Bari, Italy"}]},{"given":"Stefan","family":"Weigert","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of York, York, UK"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,1]]},"URL":"http://dx.doi.org/10.22331/q-2026-04-01-2051","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2410.23997"},{"id":"arxiv:1510.02767","type":"article-journal","author":[{"given":"Richard","family":"Kueng"},{"given":"David","family":"Gross"}],"title":"Qubit stabilizer states are complex projective 3-designs","issued":{"date-parts":[[2015,10,9]]},"note":"arxivid:1510.02767\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0103098","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.985948","page":"580-598","source":"Crossref","title":"Quantum data hiding","volume":"48","author":[{"given":"D.P.","family":"DiVincenzo","sequence":"first","affiliation":[]},{"given":"D.W.","family":"Leung","sequence":"additional","affiliation":[]},{"given":"B.M.","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2002,3]]},"URL":"http://dx.doi.org/10.1109/18.985948","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0103098"},{"id":"arxiv:math/0502080","type":"article-journal","author":[{"given":"Robert M.","family":"Guralnick"},{"given":"Pham Huu","family":"Tiep"}],"title":"Decompositions of Small Tensor Powers and Larsen's Conjecture","issued":{"date-parts":[[2005,2,3]]},"note":"arxivid:math/0502080\narxiv_version_number:1"},{"id":"arxiv:1510.02619","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.96.062336","source":"Crossref","title":"Multiqubit Clifford groups are unitary 3-designs","volume":"96","author":[{"given":"Huangjun","family":"Zhu","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,12,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.96.062336","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062336","note":"arxivid:1510.02619"},{"id":"arxiv:1510.02769","type":"article-journal","author":[{"given":"Zak","family":"Webb"}],"title":"The Clifford group forms a unitary 3-design","issued":{"date-parts":[[2016,11,8]]},"note":"arxivid:1510.02769\narxiv_version_number:3"},{"id":"arxiv:1609.08172","type":"article-journal","author":[{"given":"Huangjun","family":"Zhu"},{"given":"Richard","family":"Kueng"},{"given":"Markus","family":"Grassl"},{"given":"David","family":"Gross"}],"title":"The Clifford group fails gracefully to be a unitary 4-design","issued":{"date-parts":[[2016,9,26]]},"note":"arxivid:1609.08172\narxiv_version_number:1"},{"id":"doi:10.1112/plms/s2-35.1.23","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1112/plms/s2-35.1.23","page":"23-74","source":"Crossref","title":"On the Directrices of a Set of Points in a Plane","volume":"s2-35","author":[{"given":"Patrick","family":"du Val","sequence":"first","affiliation":[]}],"container-title":"Proceedings of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1933]]},"URL":"http://dx.doi.org/10.1112/plms/s2-35.1.23","ISSN":["0024-6115"],"container-title-short":"Proceedings of the London Mathematical Society"},{"id":"preset:Arnold99","type":"chapter","author":[{"family":"Arnold","given":"V.I."}],"title":"Symplectization, complexification and mathematical trinities","container-title":"The Arnoldfest","issued":"1999","page":"23–37"},{"id":"arxiv:1505.06742","type":"article-journal","author":[{"given":"Yang-Hui","family":"He"},{"given":"John","family":"McKay"}],"title":"Sporadic and Exceptional","issued":{"date-parts":[[2015,5,25]]},"note":"arxivid:1505.06742\narxiv_version_number:1"},{"id":"arxiv:2108.04200","type":"article-journal","author":[{"given":"Matthew A.","family":"Graydon"},{"given":"Joshua","family":"Skanes-Norman"},{"given":"Joel J.","family":"Wallman"}],"title":"Clifford groups are not always 2-designs","issued":{"date-parts":[[2021,8,9]]},"note":"arxivid:2108.04200\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0405016","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2005.844076","page":"1451-1468","source":"Crossref","title":"Unconditionally Secure Key Distribution in Higher Dimensions by Depolarization","volume":"51","author":[{"given":"H.F.","family":"Chau","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2005,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.844076","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0405016"},{"id":"arxiv:math/0301135","type":"article-journal","author":[{"given":"Thomas","family":"Strohmer"},{"given":"Robert","family":"Heath"}],"title":"Grassmannian Frames with Applications to Coding and Communication","issued":{"date-parts":[[2003,1,13]]},"note":"arxivid:math/0301135\narxiv_version_number:1"},{"id":"arxiv:1504.00253","type":"article-journal","author":[{"given":"Matthew","family":"Fickus"},{"given":"Dustin G.","family":"Mixon"}],"title":"Tables of the existence of equiangular tight frames","issued":{"date-parts":[[2016,6,16]]},"note":"arxivid:1504.00253\narxiv_version_number:2"},{"id":"doi:10.1109/TIT.1974.1055219","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1974.1055219","page":"397-399","source":"Crossref","title":"Lower bounds on the maximum cross correlation of signals (Corresp.)","volume":"20","author":[{"given":"L.","family":"Welch","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1974,5]]},"URL":"http://dx.doi.org/10.1109/TIT.1974.1055219","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2004.839492","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2004.839492","page":"188-209","source":"Crossref","title":"Designing structured tight frames via an alternating projection method","volume":"51","author":[{"given":"J.A.","family":"Tropp","sequence":"first","affiliation":[]},{"given":"I.S.","family":"Dhillon","sequence":"additional","affiliation":[]},{"given":"R.W.","family":"Heath","sequence":"additional","affiliation":[]},{"given":"T.","family":"Strohmer","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2005,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.839492","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-J.-A.-Tropp.-Topics-in-spars","type":"book","author":[{"family":"Tropp","given":"J.A."}],"title":"Topics in sparse approximation","publisher":"The University of Texas at Austin","issued":"2004"},{"id":"arxiv:math/0208002","type":"article-journal","author":[{"given":"A. R.","family":"Calderbank"},{"given":"R. H.","family":"Hardin"},{"given":"E. M.","family":"Rains"},{"given":"P. W.","family":"Shor"},{"given":"N. J. A.","family":"Sloane"}],"title":"A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces","issued":{"date-parts":[[2002,8,1]]},"note":"arxivid:math/0208002\narxiv_version_number:1"},{"id":"arxiv:math/0208003","type":"article-journal","author":[{"given":"P. W.","family":"Shor"},{"given":"N. J. A.","family":"Sloane"}],"title":"A Family of Optimal Packings in Grassmannian Manifolds","issued":{"date-parts":[[2002,8,1]]},"note":"arxivid:math/0208003\narxiv_version_number:1"},{"id":"arxiv:math/0208004","type":"article-journal","author":[{"given":"J. H.","family":"Conway"},{"given":"R. H.","family":"Hardin"},{"given":"N. J. A.","family":"Sloane"}],"title":"Packing Lines, Planes, etc.: Packings in Grassmannian Space","issued":{"date-parts":[[2002,8,1]]},"note":"arxivid:math/0208004\narxiv_version_number:1"},{"id":"arxiv:0806.2317","type":"article-journal","author":[{"given":"Aidan","family":"Roy"}],"title":"Bounds for codes and designs in complex subspaces","issued":{"date-parts":[[2008,6,13]]},"note":"arxivid:0806.2317\narxiv_version_number:1"},{"id":"doi:10.1016/j.acha.2006.07.003","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/j.acha.2006.07.003","page":"274-285","source":"Crossref","title":"Optimal linear transmission by loss-insensitive packet encoding","volume":"22","author":[{"given":"Bernhard G.","family":"Bodmann","sequence":"first","affiliation":[]}],"container-title":"Applied and Computational Harmonic Analysis","original-title":[],"language":"en","issued":{"date-parts":[[2007,5]]},"URL":"http://dx.doi.org/10.1016/j.acha.2006.07.003","ISSN":["1063-5203"],"container-title-short":"Applied and Computational Harmonic Analysis","note":"alternative-id:S1063520306000972"},{"id":"doi:10.1109/TIT.2002.801469","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2002.801469","page":"2450-2454","source":"Crossref","title":"Bounds on packings of spheres in the Grassmann manifold","volume":"48","author":[{"given":"A.","family":"Barg","sequence":"first","affiliation":[]},{"given":"D.Yu.","family":"Nogin","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2002,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.801469","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:math/0610812","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.2006.872973","page":"2111-2125","source":"Crossref","title":"Linear programming bounds for codes in grassmannian spaces","volume":"52","author":[{"given":"C.","family":"Bachoc","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2006,5]]},"URL":"http://dx.doi.org/10.1109/TIT.2006.872973","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:math/0610812"},{"id":"arxiv:1111.3376","type":"article-journal","author":[{"given":"Dustin G.","family":"Mixon"},{"given":"Christopher J.","family":"Quinn"},{"given":"Negar","family":"Kiyavash"},{"given":"Matthew","family":"Fickus"}],"title":"Fingerprinting with Equiangular Tight Frames","issued":{"date-parts":[[2011,11,14]]},"note":"arxivid:1111.3376\narxiv_version_number:1"},{"id":"manual:-N.-J.-A.-Sloane-How-to-pack-","type":"article-journal","author":[{"family":"Sloane","given":"N.J.A."}],"title":"How to pack lines, planes, 3-spaces, etc","volume":"l","container-title":"Online","issue":"ttp://www2.research.att.com/~njas/grass/index.html","issued":"2006"},{"id":"doi:10.1023/A:1020826329555","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1023/a:1020826329555","page":"5-19","source":"Crossref","title":"Designs in Grassmannian Spaces and Lattices","volume":"16","author":[{"given":"Christine","family":"Bachoc","sequence":"first","affiliation":[]},{"given":"Renaud","family":"Coulangeon","sequence":"additional","affiliation":[]},{"given":"Gabriele","family":"Nebe","sequence":"additional","affiliation":[]}],"container-title":"Journal of Algebraic Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2002,7]]},"URL":"http://dx.doi.org/10.1023/A:1020826329555","ISSN":["0925-9899","1572-9192"],"container-title-short":"Journal of Algebraic Combinatorics","note":"alternative-id:5100492"},{"id":"doi:10.1016/S0012-365X(03)00151-1","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/s0012-365x(03)00151-1","page":"15-28","source":"Crossref","title":"Codes and designs in Grassmannian spaces","volume":"277","author":[{"given":"Christine","family":"Bachoc","sequence":"first","affiliation":[]},{"given":"Eiichi","family":"Bannai","sequence":"additional","affiliation":[]},{"given":"Renaud","family":"Coulangeon","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2004,2]]},"URL":"http://dx.doi.org/10.1016/S0012-365X(03)00151-1","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X03001511"},{"id":"arxiv:1612.06729","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00605-018-1174-y","page":"235-248","source":"Crossref","title":"Asymptotically optimal designs on compact algebraic manifolds","volume":"186","author":[{"ORCID":"https://orcid.org/0000-0002-7310-2978","authenticated-orcid":false,"given":"Ujué","family":"Etayo","sequence":"first","affiliation":[]},{"given":"Jordi","family":"Marzo","sequence":"additional","affiliation":[]},{"given":"Joaquim","family":"Ortega-Cerdà","sequence":"additional","affiliation":[]}],"container-title":"Monatshefte für Mathematik","original-title":[],"language":"en","issued":{"date-parts":[[2018,3,2]]},"URL":"http://dx.doi.org/10.1007/s00605-018-1174-y","ISSN":["0026-9255","1436-5081"],"container-title-short":"Monatsh Math","note":"alternative-id:1174\narxivid:1612.06729"},{"id":"arxiv:1705.02978","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-69802-1_8","page":"235-259","source":"Crossref","title":"Cubatures on Grassmannians: Moments, Dimension Reduction, and Related Topics","author":[{"given":"Anna","family":"Breger","sequence":"first","affiliation":[]},{"given":"Martin","family":"Ehler","sequence":"additional","affiliation":[]},{"given":"Manuel","family":"Gräf","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Peter","sequence":"additional","affiliation":[]}],"container-title":"Applied and Numerical Harmonic Analysis","original-title":[],"issued":{"date-parts":[[2017]]},"ISBN":["9783319698014","9783319698021"],"URL":"http://dx.doi.org/10.1007/978-3-319-69802-1_8","ISSN":["2296-5009","2296-5017"],"note":"publisher-location:Cham\narxivid:1705.02978"},{"id":"doi:10.1109/18.825818","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.825818","page":"543-564","source":"Crossref","title":"Unitary space-time modulation for multiple-antenna communications in Rayleigh flat fading","volume":"46","author":[{"given":"B.M.","family":"Hochwald","sequence":"first","affiliation":[]},{"given":"T.L.","family":"Marzetta","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000,3]]},"URL":"http://dx.doi.org/10.1109/18.825818","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.746779","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.746779","page":"139-157","source":"Crossref","title":"Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading","volume":"45","author":[{"given":"T.L.","family":"Marzetta","sequence":"first","affiliation":[]},{"given":"B.M.","family":"Hochwald","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999]]},"URL":"http://dx.doi.org/10.1109/18.746779","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.978730","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.978730","page":"359-383","source":"Crossref","title":"Communication on the Grassmann manifold: a geometric approach to the noncoherent multiple-antenna channel","volume":"48","author":[{"family":"Lizhong Zheng","sequence":"first","affiliation":[]},{"given":"D.N.C.","family":"Tse","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2002]]},"URL":"http://dx.doi.org/10.1109/18.978730","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.24033/asens.781","type":"journal-article","publisher":"Societe Mathematique de France","DOI":"10.24033/asens.781","page":"345-467","source":"Crossref","title":"Sur certaines formes Riemanniennes remarquables des géométries à groupe fondamental simple","volume":"44","author":[{"given":"Elie","family":"Cartan","sequence":"first","affiliation":[]}],"container-title":"Annales scientifiques de l'École normale supérieure","original-title":[],"issued":{"date-parts":[[1927]]},"URL":"http://dx.doi.org/10.24033/asens.781","ISSN":["0012-9593","1873-2151"],"container-title-short":"Ann. Sci. École Norm. Sup."},{"id":"arxiv:1904.11159","type":"article-journal","author":[{"given":"Alexey","family":"Glazyrin"}],"title":"Moments of isotropic measures and optimal projective codes","issued":{"date-parts":[[2020,12,30]]},"note":"arxivid:1904.11159\narxiv_version_number:2"},{"id":"doi:10.1016/S0195-6698(85)80039-1","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s0195-6698(85)80039-1","page":"279-285","source":"Crossref","title":"On the Nonbinary Johnson Scheme","volume":"6","author":[{"given":"Hannu","family":"Tarnanen","sequence":"first","affiliation":[]},{"given":"Matti J.","family":"Aaltonen","sequence":"additional","affiliation":[]},{"given":"Jean-Marie","family":"Goethals","sequence":"additional","affiliation":[]}],"container-title":"European Journal of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[1985,9]]},"URL":"http://dx.doi.org/10.1016/S0195-6698(85)80039-1","ISSN":["0195-6698"],"container-title-short":"European Journal of Combinatorics","note":"alternative-id:S0195669885800391"},{"id":"doi:10.4153/CJM-1999-017-5","type":"journal-article","publisher":"Canadian Mathematical Society","issue":"2","abstract":"<jats:title>Abstract</jats:title><jats:p>In an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and (<jats:italic>T, M, S</jats:italic>)-nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the LP bound is always at least as strong as the generalized Rao bound.</jats:p>","DOI":"10.4153/cjm-1999-017-5","page":"326-346","source":"Crossref","title":"Association Schemes for Ordered Orthogonal Arrays and (<i>T, M, S</i>)-Nets","volume":"51","author":[{"given":"W. J.","family":"Martin","sequence":"first","affiliation":[]},{"given":"D. R.","family":"Stinson","sequence":"additional","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1999,4,1]]},"URL":"http://dx.doi.org/10.4153/CJM-1999-017-5","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j. math.","note":"alternative-id:S0008414X0000465X"},{"id":"arxiv:cs/0702033","type":"article-journal","author":[{"given":"Alexander","family":"Barg"},{"given":"Punarbasu","family":"Purkayastha"}],"title":"Bounds on ordered codes and orthogonal arrays","issued":{"date-parts":[[2009,7,10]]},"note":"arxivid:cs/0702033\narxiv_version_number:3"},{"id":"doi:10.1080/01969728208927711","type":"journal-article","publisher":"Informa UK Limited","issue":"4","DOI":"10.1080/01969728208927711","page":"331-343","source":"Crossref","title":"THE LEE-SCHEME AND BOUNDS FOR LEE-CODES","volume":"13","author":[{"given":"JAAKKO","family":"ASTOLA","sequence":"first","affiliation":[]}],"container-title":"Cybernetics and Systems","original-title":[],"language":"en","issued":{"date-parts":[[1982,10]]},"URL":"http://dx.doi.org/10.1080/01969728208927711","ISSN":["0196-9722","1087-6553"],"container-title-short":"Cybernetics and Systems","note":"alternative-id:10.1080/01969728208927711"},{"id":"doi:10.1007/3-540-19368-5_5","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-19368-5_5","page":"45-55","source":"Crossref","title":"The les association scheme","author":[{"given":"Patrick","family":"Sole","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[1988]]},"ISBN":["9783540193685","9783540392439"],"URL":"http://dx.doi.org/10.1007/3-540-19368-5_5","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"preset:HPArray","type":"chapter","author":[{"family":"Blaum","given":"M."},{"family":"Farrell","given":"P.G."},{"family":"Tilborg","given":"H.C.A.","particle":"van"}],"title":"Array codes","container-title":"\\emph{Handbook of Coding Theory}","volume":"II, Part 3","editor":[{"family":"V. S. Pless","particle":"eds."},{"family":"Huffman","given":"W.C."}],"publisher":"Elsevier","issued":"1998","page":"1855–1909"},{"id":"doi:10.1002/0471792748","type":"monograph","publisher":"Wiley","DOI":"10.1002/0471792748","source":"Crossref","title":"Code Design for Dependable Systems","author":[{"given":"Eiji","family":"Fujiwara","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2005,10,7]]},"ISBN":["9780471756187","9780471792741"],"URL":"http://dx.doi.org/10.1002/0471792748","note":"edition-number:1\nalternative-id:10.1002/0471792748"},{"id":"preset:Anvin07","type":"document","author":[{"family":"Anvin","given":"H.P."}],"title":"The mathematics of RAID-6","issued":"2007"},{"id":"preset:DDFspec09","type":"document","author":[{"family":"Position","given":"S.T."}],"title":"Common raid disk data format specification","genre":"[Online].","note":"Available:","issued":"2009","URL":"https://www.snia.org/tech_activities/standards/curr_standards/ddf"},{"id":"doi:10.1109/18.485722","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.485722","page":"529-542","source":"Crossref","title":"MDS array codes with independent parity symbols","volume":"42","author":[{"given":"M.","family":"Blaum","sequence":"first","affiliation":[]},{"given":"J.","family":"Bruck","sequence":"additional","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1996,3]]},"URL":"http://dx.doi.org/10.1109/18.485722","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/12.364531","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/12.364531","page":"192-202","source":"Crossref","title":"EVENODD: an efficient scheme for tolerating double disk failures in RAID architectures","volume":"44","author":[{"given":"M.","family":"Blaum","sequence":"first","affiliation":[]},{"given":"J.","family":"Brady","sequence":"additional","affiliation":[]},{"given":"J.","family":"Bruck","sequence":"additional","affiliation":[]},{"family":"Jai Menon","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Computers","original-title":[],"issued":{"date-parts":[[1995]]},"URL":"http://dx.doi.org/10.1109/12.364531","ISSN":["0018-9340"],"container-title-short":"IEEE Trans. Comput."},{"id":"arxiv:1112.0371","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2012.2227110","page":"1597-1616","source":"Crossref","title":"Zigzag Codes: MDS Array Codes With Optimal Rebuilding","volume":"59","author":[{"given":"Itzhak","family":"Tamo","sequence":"first","affiliation":[]},{"given":"Zhiying","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Jehoshua","family":"Bruck","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2012.2227110","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1112.0371"},{"id":"doi:10.1109/18.179343","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.179343","page":"66-77","source":"Crossref","title":"New array codes for multiple phased burst correction","volume":"39","author":[{"given":"M.","family":"Blaum","sequence":"first","affiliation":[]},{"given":"R.M.","family":"Roth","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1993]]},"URL":"http://dx.doi.org/10.1109/18.179343","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-P.-Corbett-B.-English-A.-Goe","type":"paper-conference","author":[{"family":"Corbett","given":"P."},{"family":"English","given":"B."},{"family":"Goel","given":"A."},{"family":"Grcanac","given":"T."},{"family":"Kleiman","given":"S."},{"family":"Leong","given":"J."},{"family":"Sankar","given":"S."}],"title":"Row-diagonal parity for double disk failure correction","container-title":"Proceedings of the 3rd USENIX Conference on File and Storage Technologies","issued":"2004-03","page":"1–14"},{"id":"doi:10.1109/TC.2007.70830","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tc.2007.70830","page":"889-901","source":"Crossref","title":"STAR : An Efficient Coding Scheme for Correcting Triple Storage Node Failures","volume":"57","author":[{"given":"Cheng","family":"Huang","sequence":"first","affiliation":[]},{"given":"Lihao","family":"Xu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Computers","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1109/TC.2007.70830","ISSN":["0018-9340","1557-9956","2326-3814"],"container-title-short":"IEEE Trans. Comput."},{"id":"doi:10.1109/18.746809","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.746809","page":"272-276","source":"Crossref","title":"X-code: MDS array codes with optimal encoding","volume":"45","author":[{"family":"Lihao Xu","sequence":"first","affiliation":[]},{"given":"J.","family":"Bruck","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999]]},"URL":"http://dx.doi.org/10.1109/18.746809","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1604.00454","type":"article-journal","author":[{"given":"Min","family":"Ye"},{"given":"Alexander","family":"Barg"}],"title":"Explicit constructions of high-rate MDS array codes with optimal repair bandwidth","issued":{"date-parts":[[2016,4,8]]},"note":"arxivid:1604.00454\narxiv_version_number:2"},{"id":"arxiv:1605.08630","type":"article-journal","author":[{"given":"Min","family":"Ye"},{"given":"Alexander","family":"Barg"}],"title":"Explicit constructions of optimal-access MDS codes with nearly optimal sub-packetization","issued":{"date-parts":[[2017,7,28]]},"note":"arxivid:1605.08630\narxiv_version_number:4"},{"id":"arxiv:2408.16584","type":"article-journal","author":[{"given":"Vinayak","family":"Ramkumar"},{"given":"Netanel","family":"Raviv"},{"given":"Itzhak","family":"Tamo"}],"title":"$\\varepsilon$-MSR Codes for Any Set of Helper Nodes","issued":{"date-parts":[[2024,8,29]]},"note":"arxivid:2408.16584\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.2016.2616137","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2016.2616137","page":"6683-6697","source":"Crossref","title":"Determinant Coding: A Novel Framework for Exact-Repair Regenerating Codes","volume":"62","author":[{"ORCID":"https://orcid.org/0000-0002-8743-9969","authenticated-orcid":false,"given":"Mehran","family":"Elyasi","sequence":"first","affiliation":[]},{"given":"Soheil","family":"Mohajer","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2016,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2016.2616137","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ISIT.2016.7541489","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2016.7541489","page":"1202-1206","source":"Crossref","title":"Explicit constructions of MDS array codes and RS codes with optimal repair bandwidth","author":[{"given":"Min","family":"Ye","sequence":"first","affiliation":[{"name":"Dept. of ECE and ISR, University of Maryland, College Park, 20742, United States"}]},{"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[{"name":"Dept. of ECE and ISR, University of Maryland, College Park, 20742, United States"}]}],"event":"2016 IEEE International Symposium on Information Theory (ISIT)","container-title":"2016 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2016,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2016.7541489"},{"id":"doi:10.1109/TIT.2010.2054295","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2010.2054295","page":"4539-4551","source":"Crossref","title":"Network Coding for Distributed Storage Systems","volume":"56","author":[{"given":"Alexandros G.","family":"Dimakis","sequence":"first","affiliation":[]},{"given":"P. Brighten","family":"Godfrey","sequence":"additional","affiliation":[]},{"given":"Yunnan","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Martin J.","family":"Wainwright","sequence":"additional","affiliation":[]},{"given":"Kannan","family":"Ramchandran","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2054295","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1005.4178","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2011.2159049","page":"5227-5239","source":"Crossref","title":"Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product-Matrix Construction","volume":"57","author":[{"given":"K. V.","family":"Rashmi","sequence":"first","affiliation":[]},{"given":"Nihar B.","family":"Shah","sequence":"additional","affiliation":[]},{"given":"P. Vijay","family":"Kumar","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2011,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2011.2159049","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1005.4178"},{"id":"arxiv:1512.02673","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2017.2736066","page":"1514-1529","source":"Crossref","title":"Speeding Up Distributed Machine Learning Using Codes","volume":"64","author":[{"ORCID":"https://orcid.org/0000-0002-3360-9678","authenticated-orcid":false,"given":"Kangwook","family":"Lee","sequence":"first","affiliation":[{"name":"School of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, South Korea"}]},{"given":"Maximilian","family":"Lam","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA, USA"}]},{"ORCID":"https://orcid.org/0000-0002-1126-0292","authenticated-orcid":false,"given":"Ramtin","family":"Pedarsani","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of California at Santa Barbara, Santa Barbara, CA, USA"}]},{"given":"Dimitris","family":"Papailiopoulos","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Wisconsin&#x2013;Madison, Madison, WI, USA"}]},{"given":"Kannan","family":"Ramchandran","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA, USA"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2017.2736066","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1512.02673"},{"id":"doi:10.1109/4234.846498","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/4234.846498","page":"161-163","source":"Crossref","title":"Lattice code decoder for space-time codes","volume":"4","author":[{"given":"O.","family":"Damen","sequence":"first","affiliation":[]},{"given":"A.","family":"Chkeif","sequence":"additional","affiliation":[]},{"given":"J.-C.","family":"Belfiore","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Letters","original-title":[],"issued":{"date-parts":[[2000,5]]},"URL":"http://dx.doi.org/10.1109/4234.846498","ISSN":["1089-7798"],"container-title-short":"IEEE Commun. Lett."},{"id":"doi:10.1109/TSP.2005.850352","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tsp.2005.850352","page":"2806-2818","source":"Crossref","title":"On the sphere-decoding algorithm I. Expected complexity","volume":"53","author":[{"given":"B.","family":"Hassibi","sequence":"first","affiliation":[]},{"given":"H.","family":"Vikalo","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Signal Processing","original-title":[],"issued":{"date-parts":[[2005,8]]},"URL":"http://dx.doi.org/10.1109/TSP.2005.850352","ISSN":["1053-587X"],"container-title-short":"IEEE Trans. Signal Process."},{"id":"doi:10.1109/18.771234","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.771234","page":"1639-1642","source":"Crossref","title":"A universal lattice code decoder for fading channels","volume":"45","author":[{"given":"E.","family":"Viterbo","sequence":"first","affiliation":[]},{"given":"J.","family":"Bouros","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999,7]]},"URL":"http://dx.doi.org/10.1109/18.771234","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/49.730453","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/49.730453","page":"1451-1458","source":"Crossref","title":"A simple transmit diversity technique for wireless communications","volume":"16","author":[{"given":"S.M.","family":"Alamouti","sequence":"first","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/49.730453","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1109/18.771146","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.771146","page":"1456-1467","source":"Crossref","title":"Space-time block codes from orthogonal designs","volume":"45","author":[{"given":"V.","family":"Tarokh","sequence":"first","affiliation":[]},{"given":"H.","family":"Jafarkhani","sequence":"additional","affiliation":[]},{"given":"A.R.","family":"Calderbank","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999,7]]},"URL":"http://dx.doi.org/10.1109/18.771146","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2003.817426","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2003.817426","page":"2468-2503","source":"Crossref","title":"Orthogonal designs with maximal rates","volume":"49","author":[{"family":"Xue-Bin Liang","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2003,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2003.817426","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:HKSstc","type":"chapter","author":[{"family":"Oggier","given":"F."}],"title":"Space-Time Coding","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1109/49.753730","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/49.753730","page":"451-460","source":"Crossref","title":"Space-time block coding for wireless communications: performance results","volume":"17","author":[{"given":"V.","family":"Tarokh","sequence":"first","affiliation":[]},{"given":"H.","family":"Jafarkhani","sequence":"additional","affiliation":[]},{"given":"A.R.","family":"Calderbank","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[1999,3]]},"URL":"http://dx.doi.org/10.1109/49.753730","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1109/18.661517","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.661517","page":"744-765","source":"Crossref","title":"Space-time codes for high data rate wireless communication: performance criterion and code construction","volume":"44","author":[{"given":"V.","family":"Tarokh","sequence":"first","affiliation":[]},{"given":"N.","family":"Seshadri","sequence":"additional","affiliation":[]},{"given":"A.R.","family":"Calderbank","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998,3]]},"URL":"http://dx.doi.org/10.1109/18.661517","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1002/bltj.2015","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1002/bltj.2015","page":"41-59","source":"Crossref","title":"Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas","volume":"1","author":[{"given":"Gerard J.","family":"Foschini","sequence":"first","affiliation":[]}],"container-title":"Bell Labs Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[2002,8,14]]},"URL":"http://dx.doi.org/10.1002/bltj.2015","ISSN":["1089-7089"],"container-title-short":"Bell Labs Tech. J."},{"id":"doi:10.1002/ett.4460100604","type":"journal-article","publisher":"Wiley","issue":"6","abstract":"<jats:title>Abstract</jats:title><jats:p>We investigate the use of multiple transmitting and/or receiving antennas for single user communications over the additive Gaussian channel with and without fading. We derive formulas for the capacities and error exponents of such channels, and describe computational procedures to evaluate such formulas. We show that the potential gains of such multi‐antenna systems over single‐antenna systems is rather large under independenceassumptions for the fades and noises at different receiving antennas.</jats:p>","DOI":"10.1002/ett.4460100604","page":"585-595","source":"Crossref","title":"Capacity of Multi‐antenna Gaussian Channels","volume":"10","author":[{"given":"Emre","family":"Telatar","sequence":"first","affiliation":[]}],"container-title":"European Transactions on Telecommunications","original-title":[],"language":"en","issued":{"date-parts":[[1999,11]]},"URL":"http://dx.doi.org/10.1002/ett.4460100604","ISSN":["1124-318X","1541-8251"],"container-title-short":"Trans Emerging Tel Tech","note":"alternative-id:10.1002/ett.4460100604"},{"id":"doi:10.1023/A:1008889222784","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1023/a:1008889222784","page":"311-335","source":"Crossref","title":"On Limits of Wireless Communications in a Fading Environment when Using Multiple Antennas","volume":"6","author":[{"given":"G.J.","family":"Foschini","sequence":"first","affiliation":[]},{"given":"M.J.","family":"Gans","sequence":"additional","affiliation":[]}],"container-title":"Wireless Personal Communications","original-title":[],"language":"en","issued":{"date-parts":[[1998,3]]},"URL":"http://dx.doi.org/10.1023/A:1008889222784","ISSN":["0929-6212","1572-834X"],"container-title-short":"Wireless Personal Communications","note":"alternative-id:144738"},{"id":"doi:10.1109/JSAC.2003.810294","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/jsac.2003.810294","page":"684-702","source":"Crossref","title":"Capacity limits of MIMO channels","volume":"21","author":[{"given":"A.","family":"Goldsmith","sequence":"first","affiliation":[]},{"given":"S.A.","family":"Jafar","sequence":"additional","affiliation":[]},{"given":"N.","family":"Jindal","sequence":"additional","affiliation":[]},{"given":"S.","family":"Vishwanath","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"language":"en","issued":{"date-parts":[[2003,6]]},"URL":"http://dx.doi.org/10.1109/JSAC.2003.810294","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1016/B978-0-12-385055-3.00002","type":"book-chapter","publisher":"Elsevier","DOI":"10.1016/b978-0-12-385055-3.00002-x","page":"29-58","source":"Crossref","title":"From Multi-Dimensional Propagation to Multi-Link MIMO Channels","author":[{"given":"Bruno","family":"Clerckx","sequence":"first","affiliation":[]},{"given":"Claude","family":"Oestges","sequence":"additional","affiliation":[]}],"container-title":"Mimo Wireless Networks","original-title":[],"language":"en","issued":{"date-parts":[[2013]]},"ISBN":["9780123850553"],"URL":"http://dx.doi.org/10.1016/B978-0-12-385055-3.00002-X"},{"id":"arxiv:1210.0913","type":"journal-article","publisher":"IOP Publishing","issue":"17","DOI":"10.1088/1751-8113/49/17/175304","page":"175304","source":"Crossref","title":"Summoning information in spacetime, or where and when can a qubit be?","volume":"49","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"given":"Alex","family":"May","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2016,3,22]]},"URL":"http://dx.doi.org/10.1088/1751-8113/49/17/175304","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1210.0913"},{"id":"arxiv:1601.02544","type":"journal-article","publisher":"IOP Publishing","issue":"8","DOI":"10.1088/1367-2630/18/8/083043","page":"083043","source":"Crossref","title":"Spacetime replication of continuous variable quantum information","volume":"18","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"given":"Sepehr","family":"Nezami","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3191-0325","authenticated-orcid":false,"given":"Grant","family":"Salton","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8326-8912","authenticated-orcid":false,"given":"Barry C","family":"Sanders","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2016,8,25]]},"URL":"http://dx.doi.org/10.1088/1367-2630/18/8/083043","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1601.02544"},{"id":"doi:10.1109/18.887864","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/18.887864","page":"2567-2578","source":"Crossref","title":"Differential space-time modulation","volume":"46","author":[{"given":"B.L.","family":"Hughes","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000]]},"URL":"http://dx.doi.org/10.1109/18.887864","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.868472","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.868472","page":"1962-1973","source":"Crossref","title":"Systematic design of unitary space-time constellations","volume":"46","author":[{"given":"B.M.","family":"Hochwald","sequence":"first","affiliation":[]},{"given":"T.L.","family":"Marzetta","sequence":"additional","affiliation":[]},{"given":"T.J.","family":"Richardson","sequence":"additional","affiliation":[]},{"given":"W.","family":"Sweldens","sequence":"additional","affiliation":[]},{"given":"R.","family":"Urbanke","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000]]},"URL":"http://dx.doi.org/10.1109/18.868472","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/26.891215","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/26.891215","page":"2041-2052","source":"Crossref","title":"Differential unitary space-time modulation","volume":"48","author":[{"given":"B.M.","family":"Hochwald","sequence":"first","affiliation":[]},{"given":"W.","family":"Sweldens","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2000]]},"URL":"http://dx.doi.org/10.1109/26.891215","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/18.720551","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.720551","page":"2619-2692","source":"Crossref","title":"Fading channels: information-theoretic and communications aspects","volume":"44","author":[{"given":"E.","family":"Biglieri","sequence":"first","affiliation":[]},{"given":"J.","family":"Proakis","sequence":"additional","affiliation":[]},{"given":"S.","family":"Shamai","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.720551","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s11235-013-9846-3","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s11235-013-9846-3","page":"327-334","source":"Crossref","title":"Wimax (IEEE 802.16 m) system based on space time block code and discrete multiwavelet transform and implementation in FPGA","volume":"56","author":[{"given":"R.","family":"Vidhya Lavanya","sequence":"first","affiliation":[]},{"given":"M.","family":"Madheswaran","sequence":"additional","affiliation":[]}],"container-title":"Telecommunication Systems","original-title":[],"language":"en","issued":{"date-parts":[[2013,8,23]]},"URL":"http://dx.doi.org/10.1007/s11235-013-9846-3","ISSN":["1018-4864","1572-9451"],"container-title-short":"Telecommun Syst","note":"alternative-id:9846"},{"id":"doi:10.1109/JSTSP.2008.922508","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/jstsp.2008.922508","page":"181-190","source":"Crossref","title":"Performance Evaluation of MIMO in IEEE802.16e/WiMAX","volume":"2","author":[{"given":"Sam P.","family":"Alex","sequence":"first","affiliation":[]},{"given":"Louay M. A.","family":"Jalloul","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal of Selected Topics in Signal Processing","original-title":[],"issued":{"date-parts":[[2008,4]]},"URL":"http://dx.doi.org/10.1109/JSTSP.2008.922508","ISSN":["1932-4553"],"container-title-short":"IEEE J. Sel. Top. Signal Process."},{"id":"doi:10.1109/MWC.2006.1678163","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/mwc.2006.1678163","page":"31-37","source":"Crossref","title":"Advances in smart antennas - MIMO-OFDM wireless systems: basics, perspectives, and challenges","volume":"13","container-title":"IEEE Wireless Communications","original-title":[],"language":"en","issued":{"date-parts":[[2006,8]]},"URL":"http://dx.doi.org/10.1109/MWC.2006.1678163","ISSN":["1536-1284"],"container-title-short":"IEEE Wireless Commun."},{"id":"doi:10.1109/TIT.2003.809567","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2003.809567","page":"866-872","source":"Crossref","title":"Linear authentication codes: bounds and constructions","volume":"49","author":[{"family":"Huaxiong Wang","sequence":"first","affiliation":[]},{"family":"Chaoping Xing","sequence":"additional","affiliation":[]},{"given":"R.","family":"Safavi-Naini","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2003,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2003.809567","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/978-3-642-10868-6_1","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-10868-6_1","page":"1-21","source":"Crossref","title":"Subspace Codes","author":[{"given":"Azadeh","family":"Khaleghi","sequence":"first","affiliation":[]},{"given":"Danilo","family":"Silva","sequence":"additional","affiliation":[]},{"given":"Frank R.","family":"Kschischang","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2009]]},"ISBN":["9783642108679","9783642108686"],"URL":"http://dx.doi.org/10.1007/978-3-642-10868-6_1","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:cs/0703061","type":"article-journal","author":[{"given":"Ralf","family":"Koetter"},{"given":"Frank","family":"Kschischang"}],"title":"Coding for Errors and Erasures in Random Network Coding","issued":{"date-parts":[[2008,3,25]]},"note":"arxivid:cs/0703061\narxiv_version_number:2"},{"id":"arxiv:0709.1074","type":"article-journal","author":[{"given":"Shu-Tao","family":"Xia"},{"given":"Fang-Wei","family":"Fu"}],"title":"Johnson Type Bounds on Constant Dimension Codes","issued":{"date-parts":[[2007,9,7]]},"note":"arxivid:0709.1074\narxiv_version_number:1"},{"id":"arxiv:0711.0708","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2008.928291","page":"3951-3967","source":"Crossref","title":"A Rank-Metric Approach to Error Control in Random Network Coding","volume":"54","author":[{"given":"Danilo","family":"Silva","sequence":"first","affiliation":[]},{"given":"Frank R.","family":"Kschischang","sequence":"additional","affiliation":[]},{"given":"Ralf","family":"Koetter","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.928291","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0711.0708"},{"id":"preset:HKSnetwork","type":"chapter","author":[{"family":"Kschischang","given":"F.R."}],"title":"Network Codes","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1515/9783110811056.145","type":"book-chapter","publisher":"DE GRUYTER","DOI":"10.1515/9783110811056.145","source":"Crossref","title":"Codes associated to Grassmannians","author":[{"given":"D. Yu.","family":"Nogin","sequence":"first","affiliation":[]}],"container-title":"Arithmetic, Geometry, and Coding Theory","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1515/9783110811056.145","note":"publisher-location:Berlin, New York"},{"id":"arxiv:0805.3824","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2009.2032817","page":"5479-5490","source":"Crossref","title":"On Metrics for Error Correction in Network Coding","volume":"55","author":[{"given":"Danilo","family":"Silva","sequence":"first","affiliation":[]},{"given":"Frank R.","family":"Kschischang","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2009,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2009.2032817","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0805.3824"},{"id":"arxiv:1103.3117","type":"article-journal","author":[{"given":"Michael","family":"Braun"},{"given":"Tuvi","family":"Etzion"},{"given":"Alexander","family":"Vardy"}],"title":"Linearity and Complements in Projective Space","issued":{"date-parts":[[2011,3,16]]},"note":"arxivid:1103.3117\narxiv_version_number:1"},{"id":"doi:10.1145/2488608.2488715","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/2488608.2488715","page":"843-852","source":"Crossref","title":"List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound","author":[{"given":"Venkatesan","family":"Guruswami","sequence":"first","affiliation":[{"name":"Carnegie Mellon University, Pittsburgh, PA, USA"}]},{"given":"Chaoping","family":"Xing","sequence":"additional","affiliation":[{"name":"Nanyang Technological University, Singapore, Singapore"}]}],"event":"STOC'13: Symposium on Theory of Computing","container-title":"Proceedings of the forty-fifth annual ACM symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[2013,6]]},"URL":"http://dx.doi.org/10.1145/2488608.2488715","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/2488608.2488715\n10.1145/2488608"},{"id":"preset:HKSprojective","type":"chapter","author":[{"family":"Storme","given":"L."}],"title":"Coding Theory and Galois Geometries","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"preset:HKSmetrics","type":"chapter","author":[{"family":"Firer","given":"M."}],"title":"Alternative Metrics","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1016/j.jcta.2014.06.001","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.jcta.2014.06.001","page":"149-160","source":"Crossref","title":"Nontrivial t-designs over finite fields exist for all t","volume":"127","author":[{"given":"Arman","family":"Fazeli","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4552-1443","authenticated-orcid":false,"given":"Shachar","family":"Lovett","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Vardy","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[2014,9]]},"URL":"http://dx.doi.org/10.1016/j.jcta.2014.06.001","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"special_numbering:C\nalternative-id:S0097316514000879"},{"id":"arxiv:2212.00870","type":"article-journal","author":[{"given":"Peter","family":"Keevash"},{"given":"Ashwin","family":"Sah"},{"given":"Mehtaab","family":"Sawhney"}],"title":"The existence of subspace designs","issued":{"date-parts":[[2023,2,14]]},"note":"arxivid:2212.00870\narxiv_version_number:2"},{"id":"doi:10.1109/FOCS.2013.71","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/focs.2013.71","page":"608-617","source":"Crossref","title":"Explicit Subspace Designs","author":[{"given":"Venkatesan","family":"Guruswami","sequence":"first","affiliation":[]},{"given":"Swastik","family":"Kopparty","sequence":"additional","affiliation":[]}],"event":"2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS)","container-title":"2013 IEEE 54th Annual Symposium on Foundations of Computer Science","original-title":[],"issued":{"date-parts":[[2013,10]]},"URL":"http://dx.doi.org/10.1109/FOCS.2013.71"},{"id":"doi:10.1007/BFb0066199","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/bfb0066199","page":"278-287","source":"Crossref","title":"Unsolved problems","author":[{"given":"Claude","family":"Berge","sequence":"first","affiliation":[]},{"given":"Dijen","family":"Ray-Chaudhuri","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Mathematics","original-title":[],"issued":{"date-parts":[[1974]]},"ISBN":["9783540068464","9783540378037"],"URL":"http://dx.doi.org/10.1007/BFb0066199","ISSN":["0075-8434","1617-9692"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"preset:Cameron73","type":"article-journal","author":[{"family":"Cameron","given":"P.J."}],"title":"Generalisation of Fisher’s inequality to fields with more than one element","volume":"13","container-title":"Combinatorics, London Math. Soc. Lecture Note Ser","issued":"1973","page":"9–13"},{"id":"doi:10.1007/978-3-319-70293-3_8","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-70293-3_8","page":"171-211","source":"Crossref","title":"q-Analogs of Designs: Subspace Designs","author":[{"given":"Michael","family":"Braun","sequence":"first","affiliation":[]},{"given":"Michael","family":"Kiermaier","sequence":"additional","affiliation":[]},{"given":"Alfred","family":"Wassermann","sequence":"additional","affiliation":[]}],"container-title":"Signals and Communication Technology","original-title":[],"issued":{"date-parts":[[2018]]},"ISBN":["9783319702926","9783319702933"],"URL":"http://dx.doi.org/10.1007/978-3-319-70293-3_8","ISSN":["1860-4862","1860-4870"],"note":"publisher-location:Cham"},{"id":"doi:10.1007/s10623-012-9704-4","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s10623-012-9704-4","page":"405-431","source":"Crossref","title":"Linear codes using skew polynomials with automorphisms and derivations","volume":"70","author":[{"given":"D.","family":"Boucher","sequence":"first","affiliation":[]},{"given":"F.","family":"Ulmer","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2012,6,12]]},"URL":"http://dx.doi.org/10.1007/s10623-012-9704-4","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9704"},{"id":"doi:10.1109/CWIT.2015.7255141","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/cwit.2015.7255141","page":"9-13","source":"Crossref","title":"Construction and decoding of generalized skew-evaluation codes","author":[{"given":"Siyu","family":"Liu","sequence":"first","affiliation":[]},{"given":"Felice","family":"Manganiello","sequence":"additional","affiliation":[]},{"given":"Frank R.","family":"Kschischang","sequence":"additional","affiliation":[]}],"event":"2015 IEEE 14th Canadian Workshop on Information Theory (CWIT)","container-title":"2015 IEEE 14th Canadian Workshop on Information Theory (CWIT)","original-title":[],"issued":{"date-parts":[[2015,7]]},"URL":"http://dx.doi.org/10.1109/CWIT.2015.7255141"},{"id":"arxiv:1805.03789","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2019.2912165","page":"4785-4803","source":"Crossref","title":"Reliable and Secure Multishot Network Coding Using Linearized Reed-Solomon Codes","volume":"65","author":[{"ORCID":"https://orcid.org/0000-0003-2626-8139","authenticated-orcid":false,"given":"Umberto","family":"Martinez-Penas","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4274-1785","authenticated-orcid":false,"given":"Frank R.","family":"Kschischang","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2019,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2019.2912165","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1805.03789"},{"id":"manual:-F.-H-rmann-H.-Bartz-and-A.-L","type":"document","author":[{"family":"Hörmann","given":"F."},{"family":"Bartz","given":"H."},{"family":"Horlemann","given":"A.L."}],"title":"Security considerations for Mceliece-like cryptosystems based on linearized Reed-Solomon codes in the sum-rank metric","issued":"2022"},{"id":"arxiv:2304.00627","type":"book-chapter","publisher":"Springer Nature Switzerland","DOI":"10.1007/978-3-031-29689-5_1","page":"1-20","source":"Crossref","title":"Distinguishing and Recovering Generalized Linearized Reed–Solomon Codes","author":[{"ORCID":"https://orcid.org/0000-0003-2217-9753","authenticated-orcid":false,"given":"Felicitas","family":"Hörmann","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7767-1513","authenticated-orcid":false,"given":"Hannes","family":"Bartz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2685-2343","authenticated-orcid":false,"given":"Anna-Lena","family":"Horlemann","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2023]]},"ISBN":["9783031296888","9783031296895"],"URL":"http://dx.doi.org/10.1007/978-3-031-29689-5_1","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Cham\narxivid:2304.00627"},{"id":"arxiv:1809.11158","type":"article-journal","author":[{"given":"Umberto","family":"Martínez-Peñas"},{"given":"Frank R.","family":"Kschischang"}],"title":"Universal and Dynamic Locally Repairable Codes with Maximal Recoverability via Sum-Rank Codes","issued":{"date-parts":[[2019,4,24]]},"note":"arxivid:1809.11158\narxiv_version_number:2"},{"id":"preset:Gabidulin85","type":"article-journal","author":[{"family":"Gabidulin","given":"E.M."}],"title":"Theory of Codes with Maximum Rank Distance","volume":"21","container-title":"Problemy Peredachi Informatsii","issue":"ue 1","issued":"1985","page":"3–16"},{"id":"doi:10.1109/18.75248","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.75248","page":"328-336","source":"Crossref","title":"Maximum-rank array codes and their application to crisscross error correction","volume":"37","author":[{"given":"R.M.","family":"Roth","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1991,3]]},"URL":"http://dx.doi.org/10.1109/18.75248","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:0901.2483","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2009.5205272","page":"2858-2862","source":"Crossref","title":"Fast encoding and decoding of Gabidulin codes","author":[{"given":"Danilo","family":"Silva","sequence":"first","affiliation":[]},{"given":"Frank R.","family":"Kschischang","sequence":"additional","affiliation":[]}],"event":"2009 IEEE International Symposium on Information Theory - ISIT","container-title":"2009 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2009,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2009.5205272","note":"arxivid:0901.2483"},{"id":"doi:10.1007/s10623-003-6151-2","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-003-6151-2","page":"63-79","source":"Crossref","title":"How to Mask the Structure of Codes for a Cryptographic Use","volume":"35","author":[{"given":"Thierry P.","family":"Berger","sequence":"first","affiliation":[]},{"given":"Pierre","family":"Loidreau","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2005,4]]},"URL":"http://dx.doi.org/10.1007/s10623-003-6151-2","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:6151"},{"id":"doi:10.3390/cryptography2040032","type":"journal-article","publisher":"MDPI AG","issue":"4","abstract":"<jats:p>We propose a rank metric codes based encryption based on the hard problem of rank syndrome decoding problem. We propose a new encryption with a public key matrix by considering the adding of a random distortion matrix over     F  q m      of full column rank n. We show that IND-CPA security is achievable for our encryption under assumption of the Decisional Rank Syndrome Decoding problem. Furthermore, we also prove some bounds for the number of matrices of a fixed rank with entries over a finite field. Our proposal allows the choice of the error terms with rank up to     r 2    , where r is the error-correcting capability of a code. Our encryption based on Gabidulin codes has public key size of     13 . 68     KB, which is 82 times smaller than the public key size of McEliece Cryptosystem based on Goppa codes. For similar post-quantum security level of     2 140     bits, our encryption scheme has a smaller public key size than the key size suggested by LOI17 Encryption.</jats:p>","DOI":"10.3390/cryptography2040032","page":"32","source":"Crossref","title":"A New Technique in Rank Metric Code-Based Encryption","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-1218-9133","authenticated-orcid":false,"given":"Terry","family":"Lau","sequence":"first","affiliation":[{"name":"Temasek Laboratories, National University of Singapore, T-Lab Building, 5A, Engineering Drive 1, #09-02, Singapore 117411, Singapore"}]},{"given":"Chik","family":"Tan","sequence":"additional","affiliation":[{"name":"Temasek Laboratories, National University of Singapore, T-Lab Building, 5A, Engineering Drive 1, #09-02, Singapore 117411, Singapore"}]}],"container-title":"Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2018,10,15]]},"URL":"http://dx.doi.org/10.3390/cryptography2040032","ISSN":["2410-387X"],"container-title-short":"Cryptography","note":"alternative-id:cryptography2040032"},{"id":"doi:10.1109/ICASSP.2018.8462647","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icassp.2018.8462647","page":"2087-2091","source":"Crossref","title":"Watermarking and Rank Metric Codes","author":[{"given":"Pascal","family":"Lefevre","sequence":"first","affiliation":[]},{"given":"Philippe","family":"Carre","sequence":"additional","affiliation":[]},{"given":"Philippe","family":"Gaborit","sequence":"additional","affiliation":[]}],"event":"ICASSP 2018 - 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)","container-title":"2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)","original-title":[],"issued":{"date-parts":[[2018,4]]},"URL":"http://dx.doi.org/10.1109/ICASSP.2018.8462647"},{"id":"arxiv:1410.1333","type":"article-journal","author":[{"given":"Alberto","family":"Ravagnani"}],"title":"Rank-metric codes and their duality theory","issued":{"date-parts":[[2015,4,2]]},"note":"arxivid:1410.1333\narxiv_version_number:3"},{"id":"arxiv:2411.09173","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Gilles","family":"Zémor"}],"title":"Correction of circuit faults in a stacked quantum memory using rank-metric codes","issued":{"date-parts":[[2024,11,14]]},"note":"arxivid:2411.09173\narxiv_version_number:1"},{"id":"doi:10.1007/978-3-319-11659-4_6","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-11659-4_6","page":"88-107","source":"Crossref","title":"RankSign: An Efficient Signature Algorithm Based on the Rank Metric","author":[{"given":"Philippe","family":"Gaborit","sequence":"first","affiliation":[]},{"given":"Olivier","family":"Ruatta","sequence":"additional","affiliation":[]},{"given":"Julien","family":"Schrek","sequence":"additional","affiliation":[]},{"given":"Gilles","family":"Zémor","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2014]]},"ISBN":["9783319116587","9783319116594"],"URL":"http://dx.doi.org/10.1007/978-3-319-11659-4_6","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Cham"},{"id":"doi:10.1007/BFb0054868","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/bfb0054868","page":"267-288","source":"Crossref","title":"NTRU: A ring-based public key cryptosystem","author":[{"given":"Jeffrey","family":"Hoffstein","sequence":"first","affiliation":[]},{"given":"Jill","family":"Pipher","sequence":"additional","affiliation":[]},{"given":"Joseph H.","family":"Silverman","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[1998]]},"ISBN":["9783540646570","9783540691136"],"URL":"http://dx.doi.org/10.1007/BFb0054868","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/ISIT.2013.6620590","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2013.6620590","page":"2069-2073","source":"Crossref","title":"MDPC-McEliece: New McEliece variants from Moderate Density Parity-Check codes","author":[{"given":"Rafael","family":"Misoczki","sequence":"first","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]},{"given":"Nicolas","family":"Sendrier","sequence":"additional","affiliation":[]},{"given":"Paulo S. L. M.","family":"Barreto","sequence":"additional","affiliation":[]}],"event":"2013 IEEE International Symposium on Information Theory (ISIT)","container-title":"2013 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2013,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2013.6620590"},{"id":"arxiv:1404.3482","type":"article-journal","author":[{"given":"Gaborit","family":"Philippe"},{"given":"Zemor","family":"Gilles"}],"title":"On the hardness of the decoding and the minimum distance problems for rank codes","issued":{"date-parts":[[2014,4,14]]},"note":"arxivid:1404.3482\narxiv_version_number:1"},{"id":"doi:10.1007/11779360_4","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/11779360_4","page":"36-45","source":"Crossref","title":"A Welch–Berlekamp Like Algorithm for Decoding Gabidulin Codes","author":[{"given":"Pierre","family":"Loidreau","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2006]]},"ISBN":["9783540354819","9783540354826"],"URL":"http://dx.doi.org/10.1007/11779360_4","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/ISIT.2004.1365435","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2004.1365435","page":"398-398","source":"Crossref","title":"Fast decoding of rank-codes with rank errors and column erasures","author":[{"given":"G.","family":"Richter","sequence":"first","affiliation":[]},{"given":"S.","family":"Plass","sequence":"additional","affiliation":[]}],"event":"International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings.","container-title":"International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings.","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2004.1365435"},{"id":"doi:10.1007/978-3-319-63697-9_7","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-63697-9_7","page":"194-224","source":"Crossref","title":"Identity-Based Encryption from Codes with Rank Metric","author":[{"given":"Philippe","family":"Gaborit","sequence":"first","affiliation":[]},{"given":"Adrien","family":"Hauteville","sequence":"additional","affiliation":[]},{"given":"Duong Hieu","family":"Phan","sequence":"additional","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2017]]},"ISBN":["9783319636962","9783319636979"],"URL":"http://dx.doi.org/10.1007/978-3-319-63697-9_7","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Cham"},{"id":"doi:10.1016/j.image.2018.12.015","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.image.2018.12.015","page":"119-128","source":"Crossref","title":"Application of rank metric codes in digital image watermarking","volume":"74","author":[{"given":"Pascal","family":"Lefèvre","sequence":"first","affiliation":[]},{"given":"Philippe","family":"Carré","sequence":"additional","affiliation":[]},{"given":"Philippe","family":"Gaborit","sequence":"additional","affiliation":[]}],"container-title":"Signal Processing: Image Communication","original-title":[],"language":"en","issued":{"date-parts":[[2019,5]]},"URL":"http://dx.doi.org/10.1016/j.image.2018.12.015","ISSN":["0923-5965"],"container-title-short":"Signal Processing: Image Communication","note":"special_numbering:C\nalternative-id:S0923596518302005"},{"id":"doi:10.1109/CWIT.2007.375706","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/cwit.2007.375706","page":"81-84","source":"Crossref","title":"Rank-Metric Codes for Priority Encoding Transmission with Network Coding","author":[{"given":"Danilo","family":"Silva","sequence":"first","affiliation":[]},{"given":"Frank R.","family":"Kschischang","sequence":"additional","affiliation":[]}],"event":"2007 10th Canadian Workshop on Information Theory","container-title":"2007 10th Canadian Workshop on Information Theory (CWIT)","original-title":[],"issued":{"date-parts":[[2007,6]]},"URL":"http://dx.doi.org/10.1109/CWIT.2007.375706"},{"id":"doi:10.14459/2021md1601193","type":"book","author":[{"family":"Gabidulin","given":"Ernst M."}],"contributor":[{"family":"Sidorenko","given":"Vladimir"}],"issued":{"date-parts":[[2021,3,11]]},"abstract":"This textbook contains the theory and some applications of the rank metric codes developed by the author and called Gabidulin Codes by the scientific community. A matrix code C is a set of matrices (codewords) of fixed size over a finite field. The code is considered in the rank metric that is defined as follows: The distance between two matrices is the rank of their difference. The code distance of a code C is the minimum distance between different code matrices. Given a metric, the main directions of coding theory are to design codes with a maximum number of codewords for a fixed code distance, to obtain the properties of the codes, to construct efficient decoding algorithms that find a code matrix nearest to a given matrix. This book can be recommended to students and researchers working with rank metric codes.","DOI":"10.14459/2021MD1601193","publisher":"Technical University of Munich","title":"Rank Codes","URL":"https://mediatum.ub.tum.de/1601193"},{"id":"doi:10.1561/0100000119","type":"journal-article","publisher":"Emerald","issue":"3","DOI":"10.1561/0100000119","page":"390-546","source":"Crossref","title":"Rank-Metric Codes and Their Applications","volume":"19","author":[{"given":"Hannes","family":"Bartz","sequence":"first","affiliation":[]},{"given":"Lukas","family":"Holzbaur","sequence":"first","affiliation":[]},{"given":"Hedongliang","family":"Liu","sequence":"first","affiliation":[]},{"given":"Sven","family":"Puchinger","sequence":"first","affiliation":[]},{"given":"Julian","family":"Renner","sequence":"first","affiliation":[]},{"given":"Antonia","family":"Wachter-Zeh","sequence":"first","affiliation":[]}],"container-title":"Foundations and Trends® in Communications and Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2022]]},"URL":"http://dx.doi.org/10.1561/0100000119","ISSN":["1567-2190","1567-2328"],"container-title-short":"FNT in Communications and Information Theory"},{"id":"arxiv:1001.2059","type":"article-journal","author":[{"given":"Roberto W.","family":"Nobrega"},{"given":"Bartolomeu F.","family":"Uchoa-Filho"}],"title":"Multishot Codes for Network Coding using Rank-Metric Codes","issued":{"date-parts":[[2010,4,12]]},"note":"arxivid:1001.2059\narxiv_version_number:2"},{"id":"arxiv:2406.15926","type":"article-journal","author":[{"given":"Aida","family":"Abiad"},{"given":"Alexander L.","family":"Gavrilyuk"},{"given":"Antonina P.","family":"Khramova"},{"given":"Ilia","family":"Ponomarenko"}],"title":"A linear programming bound for sum-rank metric codes","issued":{"date-parts":[[2024,6,22]]},"note":"arxivid:2406.15926\narxiv_version_number:1"},{"id":"arxiv:2209.11405","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In this paper, we present two constructions of quantum locally testable codes (QLTC) with constant soundness. In the first approach, we introduce an operation called check product, and show how this operation gives rise to QLTCs of constant soundness, constant rate, and distance scaling with locality. In the second approach, we consider hypergraph product of a quantum code and a classical repetition code, and observe a special case in which the soundness of component codes is preserved. This insight leads us to construct QLTCs of constant soundness, scalable rate and distance, and constant average locality. Our work marks a step towards constructing QLTCs of high soundness and distance, which would give a different construction to the No Low-Energy Trivial States (NLTS) theorem.</jats:p>","DOI":"10.22331/q-2024-10-18-1501","page":"1501","source":"Crossref","title":"Quantum Locally Testable Code with Constant Soundness","volume":"8","author":[{"given":"Andrew","family":"Cross","sequence":"first","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, United States."}]},{"given":"Zhiyang","family":"He","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, United States."},{"name":"Department of Mathematics, Massachusetts Institute of Technology."}]},{"given":"Anand","family":"Natarajan","sequence":"additional","affiliation":[{"name":"Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology."}]},{"given":"Mario","family":"Szegedy","sequence":"additional","affiliation":[{"name":"Rutgers, The State University of New Jersey, New Brunswick, NJ, United States."}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, United States."}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,18]]},"URL":"http://dx.doi.org/10.22331/q-2024-10-18-1501","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2209.11405"},{"id":"doi:10.1109/TIT.1954.1057464","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1954.1057464","page":"29-37","source":"Crossref","title":"Error-free Coding","volume":"4","author":[{"given":"P.","family":"Elias","sequence":"first","affiliation":[]}],"container-title":"Transactions of the IRE Professional Group on Information Theory","original-title":[],"issued":{"date-parts":[[1954,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1954.1057464","ISSN":["2168-2690","2168-2704"],"container-title-short":"Trans. IRE Prof. Group Inf. Theory"},{"id":"doi:10.1109/TIT.1965.1053802","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1965.1053802","page":"433-439","source":"Crossref","title":"Cyclic product codes","volume":"11","author":[{"given":"H.","family":"Burton","sequence":"first","affiliation":[]},{"given":"E.","family":"Weldon","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1965,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1965.1053802","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Forney","type":"book","author":[{"family":"Forney","given":"G.D.","suffix":"Jr"}],"title":"\\emph{Concatenated Codes}","publisher":"MIT Press","issued":"1966","publisher-place":"Cambridge, MA"},{"id":"doi:10.1109/TIT.1970.1054477","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1970.1054477","page":"446-451","source":"Crossref","title":"Further results on product codes","volume":"16","author":[{"given":"W.","family":"Gore","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1970,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1970.1054477","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1971.1054690","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1971.1054690","page":"594-600","source":"Crossref","title":"Single- and multiple-burst-correcting properties of a class of cyclic product codes","volume":"17","author":[{"given":"L.","family":"Bahl","sequence":"first","affiliation":[]},{"given":"R.","family":"Chien","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1971,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1971.1054690","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1973.1055085","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1973.1055085","page":"672-677","source":"Crossref","title":"Dual product codes for correction of multiple low-density burst errors","volume":"19","author":[{"given":"R.","family":"Chien","sequence":"first","affiliation":[]},{"given":"S.","family":"Ng","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1973,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1973.1055085","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/11538462_40","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/11538462_40","page":"472-481","source":"Crossref","title":"The Tensor Product of Two Codes Is Not Necessarily Robustly Testable","author":[{"given":"Paul","family":"Valiant","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2005]]},"ISBN":["9783540282396","9783540318743"],"URL":"http://dx.doi.org/10.1007/11538462_40","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:2206.09973","type":"article-journal","author":[{"given":"Gleb","family":"Kalachev"},{"given":"Pavel","family":"Panteleev"}],"title":"Two-sided Robustly Testable Codes","issued":{"date-parts":[[2023,8,10]]},"note":"arxivid:2206.09973\narxiv_version_number:3"},{"id":"arxiv:2206.07750","type":"article-journal","author":[{"given":"Irit","family":"Dinur"},{"given":"Min-Hsiu","family":"Hsieh"},{"given":"Ting-Chun","family":"Lin"},{"given":"Thomas","family":"Vidick"}],"title":"Good Quantum LDPC Codes with Linear Time Decoders","issued":{"date-parts":[[2022,6,15]]},"note":"arxivid:2206.07750\narxiv_version_number:1"},{"id":"arxiv:2208.05537","type":"article-journal","author":[{"given":"Anthony","family":"Leverrier"},{"given":"Gilles","family":"Zémor"}],"title":"Decoding quantum Tanner codes","issued":{"date-parts":[[2022,12,8]]},"note":"arxivid:2208.05537\narxiv_version_number:3"},{"id":"doi:10.1109/TIT.1966.1053873","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1966.1053873","page":"125-131","source":"Crossref","title":"Generalized minimum distance decoding","volume":"12","author":[{"given":"G.","family":"Forney","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1966,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1966.1053873","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TMAG.2005.861043","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tmag.2005.861043","page":"350-352","source":"Crossref","title":"Tensor-product parity code for magnetic recording","volume":"42","author":[{"given":"P.","family":"Chaichanavong","sequence":"first","affiliation":[]},{"given":"P.H.","family":"Siegel","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Magnetics","original-title":[],"issued":{"date-parts":[[2006,2]]},"URL":"http://dx.doi.org/10.1109/TMAG.2005.861043","ISSN":["0018-9464"],"container-title-short":"IEEE Trans. Magn."},{"id":"manual:-J.-K.-Wolf-An-introduction-t","type":"chapter","author":[{"family":"Wolf","given":"J.K."}],"title":"An introduction to tensor product codes and applications to digital storage systems","container-title":"2006 IEEE Information Theory Workshop-ITW 2006","publisher":"IEEE","issued":"2006","publisher-place":"Chengdu"},{"id":"doi:10.1145/195058.195132","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/195058.195132","page":"194-203","source":"Crossref","title":"Nearly-linear size holographic proofs","author":[{"given":"Alexander","family":"Polishchuk","sequence":"first","affiliation":[]},{"given":"Daniel A.","family":"Spielman","sequence":"additional","affiliation":[]}],"event":"the twenty-sixth annual ACM symposium","container-title":"Proceedings of the twenty-sixth annual ACM symposium on Theory of computing  - STOC '94","original-title":[],"issued":{"date-parts":[[1994]]},"URL":"http://dx.doi.org/10.1145/195058.195132","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"arxiv:quant-ph/0304125","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.68.042318","source":"Crossref","title":"Clifford group, stabilizer states, and linear and quadratic operations over GF(2)","volume":"68","author":[{"given":"Jeroen","family":"Dehaene","sequence":"first","affiliation":[]},{"given":"Bart","family":"De Moor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,10,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.042318","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042318","note":"arxivid:quant-ph/0304125"},{"id":"arxiv:0811.0898","type":"article-journal","author":[{"given":"M. Van den","family":"Nest"}],"title":"Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond","issued":{"date-parts":[[2009,10,14]]},"note":"arxivid:0811.0898\narxiv_version_number:2"},{"id":"arxiv:1310.6813","type":"journal-article","publisher":"Centre pour la Communication Scientifique Directe (CCSD)","abstract":"<jats:p>We define a normal form for Clifford circuits, and we prove that every\nClifford operator has a unique normal form. Moreover, we present a rewrite\nsystem by which any Clifford circuit can be reduced to normal form. This yields\na presentation of Clifford operators in terms of generators and relations.</jats:p>","DOI":"10.2168/lmcs-11(2:10)2015","source":"Crossref","title":"Generators and relations for n-qubit Clifford operators","volume":"Volume 11, Issue 2","author":[{"ORCID":"https://orcid.org/0000-0003-3161-856X","authenticated-orcid":false,"given":"Peter","family":"Selinger","sequence":"first","affiliation":[]}],"container-title":"Logical Methods in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2015,6,19]]},"URL":"http://dx.doi.org/10.2168/LMCS-11(2:10)2015","ISSN":["1860-5974"],"page":"1570","note":"arxivid:1310.6813"},{"id":"arxiv:1810.10259","type":"journal-article","publisher":"IOP Publishing","DOI":"10.1088/1742-6596/1071/1/012022","page":"012022","source":"Crossref","title":"On Clifford groups in quantum computing","volume":"1071","author":[{"given":"J","family":"Tolar","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics: Conference Series","original-title":[],"issued":{"date-parts":[[2018,8]]},"URL":"http://dx.doi.org/10.1088/1742-6596/1071/1/012022","ISSN":["1742-6588","1742-6596"],"container-title-short":"J. Phys.: Conf. Ser.","note":"arxivid:1810.10259"},{"id":"preset:GottesmanBook","type":"document","author":[{"family":"Gottesman","given":"D."}],"title":"\\emph{Surviving as a Quantum Computer in a Classical World}","issued":"2024","URL":"https://www.cs.umd.edu/class/spring2024/cmsc858G"},{"id":"arxiv:2406.09951","type":"article-journal","author":[{"given":"Simon","family":"Burton"},{"given":"Elijah","family":"Durso-Sabina"},{"given":"Natalie C.","family":"Brown"}],"title":"Genons, Double Covers and Fault-tolerant Clifford Gates","issued":{"date-parts":[[2024,6,14]]},"note":"arxivid:2406.09951\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0412001","type":"journal-article","publisher":"AIP Publishing","issue":"5","abstract":"<jats:p>We describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg group as it is often called)]. We also obtain a number of results concerning the structure of the Clifford group proper (i.e., the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete–positive operator valued measures (or SIC–POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al.) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjecture of Zauner’s. We give a complete characterization of the orbits and stability groups in dimensions 2–7. Finally, we show that the problem of constructing fiducial vectors may be expected to simplify in the infinite sequence of dimensions 7,13,19,21,31,… . We illustrate this point by constructing exact expressions for fiducial vectors in dimensions 7 and 19.</jats:p>","DOI":"10.1063/1.1896384","source":"Crossref","title":"Symmetric informationally complete–positive operator valued measures and the extended Clifford group","volume":"46","author":[{"given":"D. M.","family":"Appleby","sequence":"first","affiliation":[{"name":"Queen Mary University of London Department of Physics, , London E1 4NS, United Kingdom"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2005,4,19]]},"URL":"http://dx.doi.org/10.1063/1.1896384","ISSN":["0022-2488","1089-7658"],"page":"052107","note":"arxivid:quant-ph/0412001"},{"id":"arxiv:2409.14624","type":"article-journal","author":[{"given":"Eric","family":"Kubischta"},{"given":"Ian","family":"Teixeira"}],"title":"Classification of the Subgroups of the Two-Qubit Clifford Group","issued":{"date-parts":[[2024,9,22]]},"note":"arxivid:2409.14624\narxiv_version_number:1"},{"id":"arxiv:1609.08188","type":"journal-article","publisher":"AIP Publishing","issue":"7","abstract":"<jats:p>The q-qubit Clifford group, that is, the normalizer of the q-qubit Pauli group in U(2q), is a fundamental structure in quantum information with a wide variety of applications. We characterize all irreducible subrepresentations of the two-copy representation φ⊗2 of the Clifford group on the two-fold tensor product of the space of linear operators M2q⊗2. In the companion paper [Helsen et al., e-print arXiv:1701.04299 (2017)], we apply this result to improve the statistics of randomized benchmarking, a method for characterizing quantum systems.</jats:p>","DOI":"10.1063/1.4997688","source":"Crossref","title":"Representations of the multi-qubit Clifford group","volume":"59","author":[{"ORCID":"https://orcid.org/0000-0001-7218-2585","authenticated-orcid":false,"given":"Jonas","family":"Helsen","sequence":"first","affiliation":[{"name":"QuTech, Delft University of Technology 1 , Lorentzweg 1, 2628 CJ Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0001-6943-5334","authenticated-orcid":false,"given":"Joel J.","family":"Wallman","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo 2 , Waterloo, Ontario N2L 3G1, Canada"},{"name":"Department of Applied Mathematics, University of Waterloo 3 , Waterloo, Ontario N2L 3G1, Canada"}]},{"given":"Stephanie","family":"Wehner","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology 1 , Lorentzweg 1, 2628 CJ Delft, The Netherlands"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2018,7,1]]},"URL":"http://dx.doi.org/10.1063/1.4997688","ISSN":["0022-2488","1089-7658"],"page":"072201","note":"arxivid:1609.08188"},{"id":"arxiv:2504.12263","type":"article-journal","author":[{"given":"Lennart","family":"Bittel"},{"given":"Jens","family":"Eisert"},{"given":"Lorenzo","family":"Leone"},{"given":"Antonio A.","family":"Mele"},{"given":"Salvatore F. E.","family":"Oliviero"}],"title":"A complete theory of the Clifford commutant","issued":{"date-parts":[[2025,4,16]]},"note":"arxivid:2504.12263\narxiv_version_number:1"},{"id":"arxiv:1406.2170","type":"journal-article","publisher":"AIP Publishing","issue":"12","abstract":"<jats:p>We give an algorithm which produces a unique element of the Clifford group on n qubits (Cn) from an integer 0≤i&amp;lt;Cn (the number of elements in the group). The algorithm involves O(n3) operations and provides, in addition to a canonical mapping from the integers to group elements g, a factorization of g into a sequence of at most 4n symplectic transvections. The algorithm can be used to efficiently select random elements of Cn which are often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O(n3).</jats:p>","DOI":"10.1063/1.4903507","source":"Crossref","title":"How to efficiently select an arbitrary Clifford group element","volume":"55","author":[{"given":"Robert","family":"Koenig","sequence":"first","affiliation":[{"name":"University of Waterloo 1 Institute for Quantum Computing and Department of Applied Mathematics, , Waterloo, Ontario N2L 3G1, Canada"}]},{"given":"John A.","family":"Smolin","sequence":"additional","affiliation":[{"name":"2IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,1]]},"URL":"http://dx.doi.org/10.1063/1.4903507","ISSN":["0022-2488","1089-7658"],"page":"122202","note":"arxivid:1406.2170"},{"id":"manual:-M.-Oura.-The-dimension-formu","type":"article-journal","author":[{"family":"Oura","given":"M."}],"title":"The dimension formula for the ring of code polynomials in genus 4","volume":"34","container-title":"Osaka J. Math","issue":"1","issued":"1997","page":"53 – 72,"},{"id":"doi:10.1515/crll.1993.436.57","type":"journal-article","publisher":"Walter de Gruyter GmbH","issue":"436","DOI":"10.1515/crll.1993.436.57","page":"57-86","source":"Crossref","title":"On Siegel modular forms. Part I.","volume":"1993","container-title":"Journal für die reine und angewandte Mathematik (Crelles Journal)","original-title":[],"issued":{"date-parts":[[1993,3,1]]},"URL":"http://dx.doi.org/10.1515/crll.1993.436.57","ISSN":["0075-4102","1435-5345"]},{"id":"doi:10.1017/S0027763000005237","type":"journal-article","publisher":"Cambridge University Press (CUP)","abstract":"<jats:p>In this paper we compute dimension formulas for rings of Siegel modular forms of genus <jats:italic>g</jats:italic> = 3. Let denote <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0027763000005237_inline1\" /> the main congruence subgroup of level two, <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0027763000005237_inline2\" /> the Hecke subgroup of level two and <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0027763000005237_inline3\" /> the full modular group. We give the dimension formulas for genus <jats:italic>g =</jats:italic> 3 for the above mentioned groups <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0027763000005237_inline4\" /> and determine the graded ring <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0027763000005237_inline5\" /> of modular forms with respect to <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0027763000005237_inline6\" />.</jats:p>","DOI":"10.1017/s0027763000005237","page":"179-197","source":"Crossref","title":"On Siegel modular forms part II","volume":"138","author":[{"given":"Bernhard","family":"Runge","sequence":"first","affiliation":[]}],"container-title":"Nagoya Mathematical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1995,6]]},"URL":"http://dx.doi.org/10.1017/S0027763000005237","ISSN":["0027-7630","2152-6842"],"container-title-short":"Nagoya Mathematical Journal","note":"alternative-id:S0027763000005237"},{"id":"doi:10.1155/S1073792893000121","type":"journal-article","publisher":"Oxford University Press (OUP)","issue":"5","DOI":"10.1155/s1073792893000121","page":"125","source":"Crossref","title":[],"volume":"1993","author":[{"given":"W.","family":"Duke","sequence":"first","affiliation":[]}],"container-title":"International Mathematics Research Notices","original-title":[],"issued":{"date-parts":[[1993]]},"URL":"http://dx.doi.org/10.1155/S1073792893000121","ISSN":["1073-7928"],"container-title-short":"Internat. Math. Res. Notices","note":"alternative-id:S1073792893000121"},{"id":"arxiv:quant-ph/9608006","type":"article-journal","author":[{"given":"A. R.","family":"Calderbank"},{"given":"E. M","family":"Rains"},{"given":"P. W.","family":"Shor"},{"given":"N. J. A.","family":"Sloane"}],"title":"Quantum Error Correction via Codes over GF(4)","issued":{"date-parts":[[1997,9,10]]},"note":"arxivid:quant-ph/9608006\narxiv_version_number:5"},{"id":"arxiv:quant-ph/9807006","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"The Heisenberg Representation of Quantum Computers","issued":{"date-parts":[[1998,7,1]]},"note":"arxivid:quant-ph/9807006\narxiv_version_number:1"},{"id":"manual:-E.-Knill-private-communicati","type":"document","author":[{"given":"E."}],"title":"Knill, private communication, ca","issued":"1998"},{"id":"arxiv:2003.09412","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2021.3081415","page":"4546-4563","source":"Crossref","title":"Hadamard-Free Circuits Expose the Structure of the Clifford Group","volume":"67","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7381-4556","authenticated-orcid":false,"given":"Dmitri","family":"Maslov","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2021,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3081415","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2003.09412"},{"id":"arxiv:2408.15202","type":"article-journal","author":[{"given":"Dimiter","family":"Ostrev"}],"title":"Canonical Form and Finite Blocklength Bounds for Stabilizer Codes","issued":{"date-parts":[[2026,3,16]]},"note":"arxivid:2408.15202\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0406196","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.70.052328","source":"Crossref","title":"Improved simulation of stabilizer circuits","volume":"70","author":[{"given":"Scott","family":"Aaronson","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,11,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.70.052328","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052328","note":"arxivid:quant-ph/0406196"},{"id":"arxiv:1712.03554","type":"article-journal","author":[{"given":"Héctor J.","family":"García"},{"given":"Igor L.","family":"Markov"}],"title":"Simulation of Quantum Circuits via Stabilizer Frames","issued":{"date-parts":[[2017,12,10]]},"note":"arxivid:1712.03554\narxiv_version_number:1"},{"id":"arxiv:2301.02356","type":"article-journal","author":[{"given":"Andrey Boris","family":"Khesin"},{"given":"Jonathan Z.","family":"Lu"},{"given":"Peter W.","family":"Shor"}],"title":"Graphical quantum Clifford-encoder compilers from the ZX calculus","issued":{"date-parts":[[2025,2,8]]},"note":"arxivid:2301.02356\narxiv_version_number:3"},{"id":"arxiv:quant-ph/9503016","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.52.3457","page":"3457-3467","source":"Crossref","title":"Elementary gates for quantum computation","volume":"52","author":[{"given":"Adriano","family":"Barenco","sequence":"first","affiliation":[]},{"given":"Charles H.","family":"Bennett","sequence":"additional","affiliation":[]},{"given":"Richard","family":"Cleve","sequence":"additional","affiliation":[]},{"given":"David P.","family":"DiVincenzo","sequence":"additional","affiliation":[]},{"given":"Norman","family":"Margolus","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Shor","sequence":"additional","affiliation":[]},{"given":"Tycho","family":"Sleator","sequence":"additional","affiliation":[]},{"given":"John A.","family":"Smolin","sequence":"additional","affiliation":[]},{"given":"Harald","family":"Weinfurter","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1995,11,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.52.3457","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9503016"},{"id":"doi:10.1070/RM1997v052n06ABEH002155","type":"journal-article","publisher":"Steklov Mathematical Institute","issue":"6","DOI":"10.1070/rm1997v052n06abeh002155","page":"1191-1249","source":"Crossref","title":"Quantum computations: algorithms and error correction","volume":"52","author":[{"given":"A Yu","family":"Kitaev","sequence":"first","affiliation":[]}],"container-title":"Russian Mathematical Surveys","original-title":[],"issued":{"date-parts":[[1997,12,31]]},"URL":"http://dx.doi.org/10.1070/RM1997v052n06ABEH002155","ISSN":["0036-0279","1468-4829"],"container-title-short":"Russ. Math. Surv."},{"id":"arxiv:quant-ph/0505030","type":"article-journal","author":[{"given":"Christopher M.","family":"Dawson"},{"given":"Michael A.","family":"Nielsen"}],"title":"The Solovay-Kitaev algorithm","issued":{"date-parts":[[2005,8,23]]},"note":"arxivid:quant-ph/0505030\narxiv_version_number:2"},{"id":"doi:10.1017/cbo9780511976667.019","type":"other","publisher":"Cambridge University Press","DOI":"10.1017/cbo9780511976667.019","page":"617-624","source":"Crossref","title":"The Solovay–Kitaev theorem","container-title":"Quantum Computation and Quantum Information","original-title":[],"issued":{"date-parts":[[2012,6,5]]},"URL":"http://dx.doi.org/10.1017/cbo9780511976667.019","note":"edition-number:1"},{"id":"arxiv:0809.3813","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-009-9290-2","page":"13-31","source":"Crossref","title":"Unitary designs and codes","volume":"53","author":[{"given":"Aidan","family":"Roy","sequence":"first","affiliation":[]},{"given":"A. J.","family":"Scott","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2009,4,23]]},"URL":"http://dx.doi.org/10.1007/s10623-009-9290-2","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9290\narxivid:0809.3813"},{"id":"arxiv:0803.1227","type":"article-journal","author":[{"given":"Jean","family":"Creignou"},{"given":"Hervé","family":"Diet"}],"title":"Linear programming bounds for unitary space time codes","issued":{"date-parts":[[2008,3,8]]},"note":"arxivid:0803.1227\narxiv_version_number:1"},{"id":"arxiv:2407.07754","type":"article-journal","author":[{"given":"Thomas","family":"Schuster"},{"given":"Jonas","family":"Haferkamp"},{"given":"Hsin-Yuan","family":"Huang"}],"title":"Random unitaries in extremely low depth","issued":{"date-parts":[[2025,1,4]]},"note":"arxivid:2407.07754\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0611002","type":"journal-article","publisher":"AIP Publishing","issue":"5","abstract":"<jats:p>We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.</jats:p>","DOI":"10.1063/1.2716992","source":"Crossref","title":"Evenly distributed unitaries: On the structure of unitary designs","volume":"48","author":[{"given":"D.","family":"Gross","sequence":"first","affiliation":[{"name":"Institute for Mathematical Sciences , Imperial College London, Princes Gate, London SW7 2PE, United Kingdom and QOLS, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom"}]},{"given":"K.","family":"Audenaert","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Sciences , Imperial College London, Princes Gate, London SW7 2PE, United Kingdom and QOLS, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom"}]},{"given":"J.","family":"Eisert","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Sciences , Imperial College London, Princes Gate, London SW7 2PE, United Kingdom and QOLS, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW, United Kingdom"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2007,5,1]]},"URL":"http://dx.doi.org/10.1063/1.2716992","ISSN":["0022-2488","1089-7658"],"page":"052104","note":"arxivid:quant-ph/0611002"},{"id":"arxiv:quant-ph/0512217","type":"article-journal","author":[{"given":"Christoph","family":"Dankert"}],"title":"Efficient Simulation of Random Quantum States and Operators","issued":{"date-parts":[[2005,12,24]]},"note":"arxivid:quant-ph/0512217\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0606161","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.80.012304","source":"Crossref","title":"Exact and approximate unitary 2-designs and their application to fidelity estimation","volume":"80","author":[{"given":"Christoph","family":"Dankert","sequence":"first","affiliation":[]},{"given":"Richard","family":"Cleve","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Emerson","sequence":"additional","affiliation":[]},{"given":"Etera","family":"Livine","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,7,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.012304","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012304","note":"arxivid:quant-ph/0606161"},{"id":"arxiv:2307.08956","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The Haar measure plays a vital role in quantum information, but its study often requires a deep understanding of representation theory, posing a challenge for beginners. This tutorial aims to provide a basic introduction to Haar measure tools in quantum information, utilizing only basic knowledge of linear algebra and thus aiming to make this topic more accessible. The tutorial begins by introducing the Haar measure with a specific emphasis on characterizing the moment operator, an essential element for computing integrals over the Haar measure. It also covers properties of the symmetric subspace and introduces helpful tools like tensor network diagrammatic notation, which aid in visualizing and simplifying calculations. Next, the tutorial explores the concept of unitary designs, providing equivalent definitions, and subsequently explores approximate notions of unitary designs, shedding light on the relationships between these different notions. Practical examples of Haar measure calculations are illustrated, including the derivation of well-known formulas such as the twirling of a quantum channel. Lastly, the tutorial showcases the applications of Haar measure calculations in quantum machine learning and classical shadow tomography.</jats:p>","DOI":"10.22331/q-2024-05-08-1340","page":"1340","source":"Crossref","title":"Introduction to Haar Measure Tools in Quantum Information: A Beginner&amp;apos;s Tutorial","volume":"8","author":[{"given":"Antonio Anna","family":"Mele","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,8]]},"URL":"http://dx.doi.org/10.22331/q-2024-05-08-1340","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2307.08956"},{"id":"arxiv:2510.08434","type":"article-journal","author":[{"given":"Laura","family":"Cui"},{"given":"Thomas","family":"Schuster"},{"given":"Liang","family":"Mao"},{"given":"Hsin-Yuan","family":"Huang"},{"given":"Fernando","family":"Brandao"}],"title":"Random unitaries from Hamiltonian dynamics","issued":{"date-parts":[[2025,10,9]]},"note":"arxivid:2510.08434\narxiv_version_number:1"},{"id":"arxiv:1208.0692","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-016-2706-8","page":"397-434","source":"Crossref","title":"Local Random Quantum Circuits are Approximate Polynomial-Designs","volume":"346","author":[{"given":"Fernando G. S. L.","family":"Brandão","sequence":"first","affiliation":[]},{"given":"Aram W.","family":"Harrow","sequence":"additional","affiliation":[]},{"given":"Michał","family":"Horodecki","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2016,8,3]]},"URL":"http://dx.doi.org/10.1007/s00220-016-2706-8","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2706\narxivid:1208.0692"},{"id":"arxiv:2412.04414","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Unitary <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>k</a:mi></a:math>-designs are distributions of unitary gates that match the Haar distribution up to its <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>k</c:mi><c:mi>th</c:mi></c:math> statistical moment. They are a crucial resource for randomized quantum protocols. However, their implementation on encoded logical qubits is nontrivial due to the need for magic gates, which can require a large resource overhead. In this work, we propose an efficient approach to generate unitary designs for encoded qubits in surface codes by applying local unitary rotations (“coherent errors”) on the physical qubits followed by syndrome measurement and error correction. We prove that, under some conditions on the coherent errors (notably including all single-qubit unitaries) and on the error-correcting code, this process induces a unitary transformation of the logical subspace. We numerically show that the ensemble of logical unitaries (indexed by the random syndrome outcomes) converges to a unitary design in the thermodynamic limit, provided that the density or strength of coherent errors is above a finite threshold. This “unitary design” phase transition coincides with the code’s coherent error threshold under optimal decoding. Furthermore, we propose a classical algorithm to simulate the protocol based on a “staircase” implementation of the surface code encoder and decoder circuits. This enables a mapping to a <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mo stretchy=\"false\">(</e:mo><e:mn>1</e:mn><e:mo>+</e:mo><e:mn>1</e:mn><e:mo stretchy=\"false\">)</e:mo></e:math>-dimensional monitored circuit, where we observe an entanglement phase transition (and thus a classical complexity phase transition of the decoding algorithm) coinciding with the aforementioned unitary design phase transition. Our results provide a practical way to realize unitary designs on encoded qubits, with applications including quantum state tomography and benchmarking in error-correcting codes.</jats:p>","DOI":"10.1103/bnld-2chd","source":"Crossref","title":"Emergent Unitary Designs for Encoded Qubits from Coherent Errors and Syndrome Measurements","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-0267-1815","authenticated-orcid":true,"given":"Zihan","family":"Cheng","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00hj54h04","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Texas at Austin"}]},{"ORCID":"https://orcid.org/0000-0003-3843-9935","authenticated-orcid":true,"given":"Eric","family":"Huang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland"}]},{"given":"Vedika","family":"Khemani","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f54p054","id-type":"ROR","asserted-by":"publisher"}],"name":"Stanford University"}]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland"}]},{"ORCID":"https://orcid.org/0000-0003-4095-0038","authenticated-orcid":true,"given":"Matteo","family":"Ippoliti","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00hj54h04","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Texas at Austin"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,22]]},"URL":"http://dx.doi.org/10.1103/bnld-2chd","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030333","note":"arxivid:2412.04414"},{"id":"arxiv:2402.01638","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.133.030602","source":"Crossref","title":"Quantum Codes from Twisted Unitary \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>t</mml:mi></mml:math>\n-Groups","volume":"133","author":[{"ORCID":"https://orcid.org/0000-0001-5451-0290","authenticated-orcid":true,"given":"Eric","family":"Kubischta","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST/University of Maryland"}]},{"given":"Ian","family":"Teixeira","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST/University of Maryland"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.030602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030602","note":"arxivid:2402.01638"},{"id":"arxiv:1810.02507","type":"article-journal","author":[{"given":"Eiichi","family":"Bannai"},{"given":"Gabriel","family":"Navarro"},{"given":"Noelia","family":"Rizo"},{"given":"Pham Huu","family":"Tiep"}],"title":"Unitary t-groups","issued":{"date-parts":[[2018,10,5]]},"note":"arxivid:1810.02507\narxiv_version_number:1"},{"id":"doi:10.1007/978-3-642-58575-3","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-58575-3","source":"Crossref","title":"Introduction to Coding Theory","author":[{"given":"J. H.","family":"van Lint","sequence":"first","affiliation":[]}],"container-title":"Graduate Texts in Mathematics","original-title":[],"issued":{"date-parts":[[1999]]},"ISBN":["9783642636530","9783642585753"],"URL":"http://dx.doi.org/10.1007/978-3-642-58575-3","ISSN":["0072-5285"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1016/S0019-9958(67)90052-6","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0019-9958(67)90052-6","page":"65-103","source":"Crossref","title":"Lower bounds to error probability for coding on discrete memoryless channels. I","volume":"10","author":[{"given":"C.E.","family":"Shannon","sequence":"first","affiliation":[]},{"given":"R.G.","family":"Gallager","sequence":"additional","affiliation":[]},{"given":"E.R.","family":"Berlekamp","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1967,1]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(67)90052-6","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995867900526"},{"id":"manual:-R.-M.-Fano.-The-transmission","type":"book","author":[{"family":"Fano","given":"R.M."}],"title":"The transmission of information","volume":"65","publisher":"Massachusetts Institute of Technology, Research Laboratory of Electronics","issued":"1949","publisher-place":"Cambridge, MA, USA"},{"id":"doi:10.1038/339693a0","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6227","DOI":"10.1038/339693a0","page":"693-695","source":"Crossref","title":"Spin-glass models as error-correcting codes","volume":"339","author":[{"given":"Nicolas","family":"Sourlas","sequence":"first","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[1989,6]]},"URL":"http://dx.doi.org/10.1038/339693a0","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BF339693a0"},{"id":"manual:-T.-H.-Cormen-C.-E.-Leiserson","type":"book","author":[{"family":"Cormen","given":"T.H."},{"family":"Leiserson","given":"C.E."},{"family":"Rivest","given":"R.L."},{"family":"Stein","given":"C."}],"title":"Introduction to algorithms","publisher":"MIT press","issued":"2022"},{"id":"doi:10.1147/sj.141.0016","type":"journal-article","publisher":"IBM","issue":"1","DOI":"10.1147/sj.141.0016","page":"16-34","source":"Crossref","title":"The characteristics and decodability of the Universal Product Code symbol","volume":"14","author":[{"given":"D.","family":"Savir","sequence":"first","affiliation":[]},{"given":"G. J.","family":"Laurer","sequence":"additional","affiliation":[]}],"container-title":"IBM Systems Journal","original-title":[],"issued":{"date-parts":[[1975]]},"URL":"http://dx.doi.org/10.1147/sj.141.0016","ISSN":["0018-8670"],"container-title-short":"IBM Syst. J."},{"id":"doi:10.1007/3-540-44750-4_36","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-44750-4_36","page":"452-465","source":"Crossref","title":"Collusion-Secure Fingerprinting for Digital Data","author":[{"given":"Dan","family":"Boneh","sequence":"first","affiliation":[]},{"given":"James","family":"Shaw","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[1995]]},"ISBN":["9783540602217","9783540447504"],"URL":"http://dx.doi.org/10.1007/3-540-44750-4_36","ISSN":["0302-9743"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1007/s10623-021-00988-z","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","abstract":"<jats:title>Abstract</jats:title><jats:p>According to their strength, the tracing properties of a code can be categorized as frameproof, separating, IPP and TA. It is known that, if the minimum distance of the code is larger than a certain threshold then the TA property implies the rest. Silverberg et al. ask if there is some kind of tracing capability left when the minimum distance falls below the threshold. Under different assumptions, several papers have given a negative answer to the question. In this paper, further progress is made. We establish values of the minimum distance for which Reed-Solomon codes do not posses the separating property.</jats:p>","DOI":"10.1007/s10623-021-00988-z","page":"427-442","source":"Crossref","title":"A study of the separating property in Reed-Solomon codes by bounding the minimum distance","volume":"90","author":[{"ORCID":"https://orcid.org/0000-0001-7655-135X","authenticated-orcid":false,"given":"Marcel","family":"Fernandez","sequence":"first","affiliation":[]},{"given":"Jorge J.","family":"Urroz","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2022,1,5]]},"URL":"http://dx.doi.org/10.1007/s10623-021-00988-z","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:988"},{"id":"doi:10.1016/S0378-3758(99)00131-7","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0378-3758(99)00131-7","page":"595-617","source":"Crossref","title":"Secure frameproof codes, key distribution patterns, group testing algorithms and related structures","volume":"86","author":[{"given":"D.R.","family":"Stinson","sequence":"first","affiliation":[]},{"given":"Tran","family":"van Trung","sequence":"additional","affiliation":[]},{"given":"R.","family":"Wei","sequence":"additional","affiliation":[]}],"container-title":"Journal of Statistical Planning and Inference","original-title":[],"language":"en","issued":{"date-parts":[[2000,5]]},"URL":"http://dx.doi.org/10.1016/S0378-3758(99)00131-7","ISSN":["0378-3758"],"container-title-short":"Journal of Statistical Planning and Inference","note":"alternative-id:S0378375899001317"},{"id":"doi:10.1137/S0895480101384633","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/s0895480101384633","page":"499-510","source":"Crossref","title":"Frameproof Codes","volume":"16","author":[{"given":"Simon R.","family":"Blackburn","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2003,1]]},"URL":"http://dx.doi.org/10.1137/S0895480101384633","ISSN":["0895-4801","1095-7146"],"container-title-short":"SIAM J. Discrete Math.","note":"alternative-id:10.1137/S0895480101384633"},{"id":"doi:10.1109/18.705568","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.705568","page":"1897-1905","source":"Crossref","title":"Collusion-secure fingerprinting for digital data","volume":"44","author":[{"given":"D.","family":"Boneh","sequence":"first","affiliation":[]},{"given":"J.","family":"Shaw","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.705568","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1411.5782","type":"article-journal","author":[{"given":"Chong","family":"Shangguan"},{"given":"Xin","family":"Wang"},{"given":"Gennian","family":"Ge"},{"given":"Ying","family":"Miao"}],"title":"New Bounds For Frameproof Codes","issued":{"date-parts":[[2014,11,21]]},"note":"arxivid:1411.5782\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.2002.804111","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2002.804111","page":"2991-2995","source":"Crossref","title":"Asymptotic bounds on frameproof codes","volume":"48","author":[{"family":"Chaoping Xing","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2002,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.804111","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1010.5764","type":"article-journal","author":[{"given":"Hugues","family":"Randriambololona"}],"title":"(2,1)-separating systems beyond the probabilistic bound","issued":{"date-parts":[[2012,1,10]]},"note":"arxivid:1010.5764\narxiv_version_number:7"},{"id":"preset:HKSag","type":"chapter","author":[{"family":"Couvreur","given":"A."},{"family":"Randriambololona","given":"H."}],"title":"Algebraic Geometry Codes and Some Applications","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1109/18.915661","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.915661","page":"1042-1049","source":"Crossref","title":"Combinatorial properties of frameproof and traceability codes","volume":"47","author":[{"given":"J.N.","family":"Staddon","sequence":"first","affiliation":[]},{"given":"D.R.","family":"Stinson","sequence":"additional","affiliation":[]},{"family":"Ruizhong Wei","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001,3]]},"URL":"http://dx.doi.org/10.1109/18.915661","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/j.endm.2005.07.012","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.endm.2005.07.012","page":"143-147","source":"Crossref","title":"Codes with the identifiable parent property and the multiple–access channel","volume":"21","author":[{"given":"Rudolf","family":"Ahlswede","sequence":"first","affiliation":[]},{"given":"Ning","family":"Cai","sequence":"additional","affiliation":[]}],"container-title":"Electronic Notes in Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2005,8]]},"URL":"http://dx.doi.org/10.1016/j.endm.2005.07.012","ISSN":["1571-0653"],"container-title-short":"Electronic Notes in Discrete Mathematics","note":"special_numbering:C\nalternative-id:S1571065305051048"},{"id":"doi:10.1006/jcta.1997.2851","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1006/jcta.1997.2851","page":"121-133","source":"Crossref","title":"On Codes with the Identifiable Parent Property","volume":"82","author":[{"given":"Henk D.L","family":"Hollmann","sequence":"first","affiliation":[]},{"given":"Jack H","family":"van Lint","sequence":"additional","affiliation":[]},{"given":"Jean-Paul","family":"Linnartz","sequence":"additional","affiliation":[]},{"given":"Ludo M.G.M","family":"Tolhuizen","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1998,5]]},"URL":"http://dx.doi.org/10.1006/jcta.1997.2851","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S009731659792851X"},{"id":"doi:10.1137/S0895480100376848","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/s0895480100376848","page":"423-431","source":"Crossref","title":"A Hypergraph Approach to the Identifying Parent Property: The Case of Multiple Parents","volume":"14","author":[{"given":"Alexander","family":"Barg","sequence":"first","affiliation":[]},{"given":"Gérard","family":"Cohen","sequence":"additional","affiliation":[]},{"given":"Sylvia","family":"Encheva","sequence":"additional","affiliation":[]},{"given":"Gregory","family":"Kabatiansky","sequence":"additional","affiliation":[]},{"given":"Gilles","family":"Zémor","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2001,1]]},"URL":"http://dx.doi.org/10.1137/S0895480100376848","ISSN":["0895-4801","1095-7146"],"container-title-short":"SIAM J. Discrete Math.","note":"alternative-id:10.1137/S0895480100376848"},{"id":"doi:10.1109/18.841169","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.841169","page":"893-910","source":"Crossref","title":"Tracing traitors","volume":"46","author":[{"given":"B.","family":"Chor","sequence":"first","affiliation":[]},{"given":"A.","family":"Fiat","sequence":"additional","affiliation":[]},{"given":"M.","family":"Naor","sequence":"additional","affiliation":[]},{"given":"B.","family":"Pinkas","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000,5]]},"URL":"http://dx.doi.org/10.1109/18.841169","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:HKSdist","type":"chapter","author":[{"family":"Ramkumar","given":"V."},{"family":"Vajha","given":"M."},{"family":"Balaji","given":"S.B."},{"family":"Krishnan","given":"M.N."},{"family":"Sasidharan","given":"B."},{"family":"Kumar","given":"P.Vijay"}],"title":"Codes for Distributed Storage","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"arxiv:1612.01361","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2018.2827942","page":"6567-6582","source":"Crossref","title":"Repairing Reed-Solomon Codes With Multiple Erasures","volume":"64","author":[{"ORCID":"https://orcid.org/0000-0002-2276-017X","authenticated-orcid":false,"given":"Hoang","family":"Dau","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2436-3944","authenticated-orcid":false,"given":"Iwan M.","family":"Duursma","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5611-0848","authenticated-orcid":false,"given":"Han Mao","family":"Kiah","sequence":"additional","affiliation":[]},{"given":"Olgica","family":"Milenkovic","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2018.2827942","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1612.01361"},{"id":"doi:10.1017/CBO9780511804090","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set. The book starts with a broad introduction to the field and progresses to advanced results. Contents include: definition of Turing machines and basic time and space complexity classes, probabilistic algorithms, interactive proofs, cryptography, quantum computation, lower bounds for concrete computational models (decision trees, communication complexity, constant depth, algebraic and monotone circuits, proof complexity), average-case complexity and hardness amplification, derandomization and pseudorandom constructions, and the PCP theorem.</jats:p>","DOI":"10.1017/cbo9780511804090","source":"Crossref","title":"Computational Complexity","author":[{"given":"Sanjeev","family":"Arora","sequence":"first","affiliation":[]},{"given":"Boaz","family":"Barak","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2009,4,20]]},"ISBN":["9780521424264","9780511804090"],"URL":"http://dx.doi.org/10.1017/CBO9780511804090","note":"edition-number:1"},{"id":"doi:10.1145/301250.301397","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/301250.301397","source":"Crossref","title":"Pseudorandom generators without the XOR Lemma (extended abstract)","author":[{"given":"Madhu","family":"Sudan","sequence":"first","affiliation":[{"name":"Laboratory for Computer Science, 545 Technology Square, MIT, Cambridge, MA"}]},{"given":"Luca","family":"Trevisan","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Columbia University, 500W 120th St., New York, NY"}]},{"given":"Salil","family":"Vadhan","sequence":"additional","affiliation":[{"name":"Laboratory for Computer Science, 542 Technology Square, MIT, Cambridge, MA"}]}],"event":"STOC99: ACM Symposium on Theory of Computing","container-title":"Proceedings of the thirty-first annual ACM symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[1999,5]]},"URL":"http://dx.doi.org/10.1145/301250.301397","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/301250.301397\n10.1145/301250"},{"id":"doi:10.1145/2993749.2993761","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"3","abstract":"<jats:p>We survey the state of the art in constructions of locally testable codes, locally decodable codes and locally correctable codes of high rate.</jats:p>","DOI":"10.1145/2993749.2993761","page":"46-66","source":"Crossref","title":"Guest Column","volume":"47","author":[{"given":"Swastik","family":"Kopparty","sequence":"first","affiliation":[{"name":"Rutgers University, Piscataway, NJ"}]},{"given":"Shubhangi","family":"Saraf","sequence":"additional","affiliation":[{"name":"Rutgers University, Piscataway, NJ"}]}],"container-title":"ACM SIGACT News","original-title":[],"language":"en","issued":{"date-parts":[[2016,8,31]]},"URL":"http://dx.doi.org/10.1145/2993749.2993761","ISSN":["0163-5700"],"container-title-short":"SIGACT News","note":"alternative-id:10.1145/2993749.2993761"},{"id":"doi:10.1145/335305.335315","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/335305.335315","source":"Crossref","title":"On the efficiency of local decoding procedures for error-correcting codes","author":[{"given":"Jonathan","family":"Katz","sequence":"first","affiliation":[{"name":"Department of Computer Science, Columbia University"}]},{"given":"Luca","family":"Trevisan","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Columbia University"}]}],"event":"STOC00: The 32nd Annual ACM Symposium on Theory of Computing","container-title":"Proceedings of the thirty-second annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2000,5]]},"URL":"http://dx.doi.org/10.1145/335305.335315","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/335305.335315\n10.1145/335305"},{"id":"arxiv:0705.3806","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/focs.2008.45","page":"477-486","source":"Crossref","title":"A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs","author":[{"given":"Avraham","family":"Ben-Aroya","sequence":"first","affiliation":[]},{"given":"Oded","family":"Regev","sequence":"additional","affiliation":[]},{"given":"Ronald","family":"de Wolf","sequence":"additional","affiliation":[]}],"event":"2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS)","container-title":"2008 49th Annual IEEE Symposium on Foundations of Computer Science","original-title":[],"issued":{"date-parts":[[2008,10]]},"URL":"http://dx.doi.org/10.1109/FOCS.2008.45","note":"arxivid:0705.3806"},{"id":"doi:10.1109/CCC.2002.1004353","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/ccc.2002.1004353","page":"175-183","source":"Crossref","title":"Lower bounds for linear locally decodable codes and private information retrieval","author":[{"given":"O.","family":"Goldreich","sequence":"first","affiliation":[]},{"given":"H.","family":"Karloff","sequence":"additional","affiliation":[]},{"given":"L.J.","family":"Schulman","sequence":"additional","affiliation":[]},{"given":"L.","family":"Trevisan","sequence":"additional","affiliation":[]}],"event":"17th IEEE Annual Conference on Computational Complexity","container-title":"Proceedings 17th IEEE Annual Conference on Computational Complexity","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/CCC.2002.1004353"},{"id":"doi:10.1007/978-3-642-15369-3_50","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-15369-3_50","page":"670-682","source":"Crossref","title":"Locally Testable vs. Locally Decodable Codes","author":[{"given":"Tali","family":"Kaufman","sequence":"first","affiliation":[]},{"given":"Michael","family":"Viderman","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2010]]},"ISBN":["9783642153686","9783642153693"],"URL":"http://dx.doi.org/10.1007/978-3-642-15369-3_50","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:0806.2101","type":"article-journal","author":[{"given":"Jop","family":"Briët"},{"given":"Ronald","family":"de Wolf"}],"title":"Locally Decodable Quantum Codes","issued":{"date-parts":[[2008,6,12]]},"note":"arxivid:0806.2101\narxiv_version_number:1"},{"id":"arxiv:1611.06980","type":"article-journal","author":[{"given":"Arnab","family":"Bhattacharyya"},{"given":"Sivakanth","family":"Gopi"},{"given":"Avishay","family":"Tal"}],"title":"Lower bounds for 2-query LCCs over large alphabet","issued":{"date-parts":[[2017,4,28]]},"note":"arxivid:1611.06980\narxiv_version_number:2"},{"id":"doi:10.1145/1007352.1007396","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/1007352.1007396","page":"262-271","source":"Crossref","title":"Batch codes and their applications","author":[{"given":"Yuval","family":"Ishai","sequence":"first","affiliation":[{"name":"Technion"}]},{"given":"Eyal","family":"Kushilevitz","sequence":"additional","affiliation":[{"name":"Technion"}]},{"given":"Rafail","family":"Ostrovsky","sequence":"additional","affiliation":[{"name":"UCLA"}]},{"given":"Amit","family":"Sahai","sequence":"additional","affiliation":[{"name":"Princeton University, Princeton, NJ"}]}],"event":"STOC04: Symposium of Theory of Computing 2004","container-title":"Proceedings of the thirty-sixth annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2004,6,13]]},"URL":"http://dx.doi.org/10.1145/1007352.1007396","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/1007352.1007396\n10.1145/1007352"},{"id":"doi:10.1515/popets-2016-0036","type":"journal-article","publisher":"Privacy Enhancing Technologies Symposium Advisory Board","issue":"4","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p> Private information retrieval (PIR) is a way for clients to query a remote database without the database holder learning the clients’ query terms or the responses they generate. Compelling applications for PIR are abound in the cryptographic and privacy research literature, yet existing PIR techniques are notoriously inefficient. Consequently, no such PIRbased application to date has seen real-world at-scale deployment. This paper proposes new “batch coding” techniques to help address PIR’s efficiency problem. The new techniques exploit the connection between ramp secret sharing schemes and efficient information-theoretically secure PIR (IT-PIR) protocols. This connection was previously observed by Henry, Huang, and Goldberg (NDSS 2013), who used ramp schemes to construct efficient “batch queries” with which clients can fetch several database records for the same cost as fetching a single record using a standard, non-batch query. The new techniques in this paper generalize and extend those of Henry et al. to construct “batch codes” with which clients can fetch several records for only a fraction the cost of fetching a single record using a standard non-batch query over an unencoded database. The batch codes are highly tuneable, providing a means to trade off (i) lower server-side computation cost, (ii) lower server-side storage cost, and/or (iii) lower uni- or bi-directional communication cost, in exchange for a comparatively modest decrease in resilience to Byzantine database servers.</jats:p>","DOI":"10.1515/popets-2016-0036","page":"202-218","source":"Crossref","title":"Polynomial Batch Codes for Efficient IT-PIR","volume":"2016","author":[{"given":"Ryan","family":"Henry","sequence":"first","affiliation":[{"name":"Indiana University"}]}],"container-title":"Proceedings on Privacy Enhancing Technologies","original-title":[],"language":"en","issued":{"date-parts":[[2016,7,14]]},"URL":"http://dx.doi.org/10.1515/popets-2016-0036","ISSN":["2299-0984"],"note":"alternative-id:10.1515/popets-2016-0036"},{"id":"arxiv:1701.02708","type":"article-journal","author":[{"given":"Hui","family":"Zhang"},{"given":"Eitan","family":"Yaakobi"},{"given":"Natalia","family":"Silberstein"}],"title":"Multiset Combinatorial Batch Codes","issued":{"date-parts":[[2017,1,10]]},"note":"arxivid:1701.02708\narxiv_version_number:1"},{"id":"doi:10.1007/978-3-642-14358-8","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-14358-8","source":"Crossref","title":"Locally Decodable Codes and Private Information Retrieval Schemes","author":[{"given":"Sergey","family":"Yekhanin","sequence":"first","affiliation":[]}],"container-title":"Information Security and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2010]]},"ISBN":["9783642143571","9783642143588"],"URL":"http://dx.doi.org/10.1007/978-3-642-14358-8","ISSN":["1619-7100"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:1202.2414","type":"article-journal","author":[{"given":"N.","family":"Prakash"},{"given":"Govinda M.","family":"Kamath"},{"given":"V.","family":"Lalitha"},{"given":"P. Vijay","family":"Kumar"}],"title":"Optimal Linear Codes with a Local-Error-Correction Property","issued":{"date-parts":[[2012,2,11]]},"note":"arxivid:1202.2414\narxiv_version_number:1"},{"id":"doi:10.4086/toc.2020.v016a018","type":"journal-article","publisher":"Theory of Computing Exchange","issue":"1","DOI":"10.4086/toc.2020.v016a018","page":"1-68","source":"Crossref","title":"","volume":"16","author":[{"given":"Tom","family":"Gur","sequence":"first","affiliation":[]},{"given":"Govind","family":"Ramnarayan","sequence":"additional","affiliation":[]},{"given":"Ron","family":"Rothblum","sequence":"additional","affiliation":[]}],"container-title":"Theory of Computing","original-title":[],"language":"en","issued":{"date-parts":[[2020]]},"URL":"http://dx.doi.org/10.4086/toc.2020.v016a018","ISSN":["1557-2862"],"container-title-short":"Theory of Comput."},{"id":"arxiv:2311.00558","type":"article-journal","author":[{"given":"Pravesh K.","family":"Kothari"},{"given":"Peter","family":"Manohar"}],"title":"An Exponential Lower Bound for Linear 3-Query Locally Correctable Codes","issued":{"date-parts":[[2023,11,1]]},"note":"arxivid:2311.00558\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.2016.2518663","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2016.2518663","page":"3070-3083","source":"Crossref","title":"Bounds on the Parameters of Locally Recoverable Codes","volume":"62","author":[{"given":"Itzhak","family":"Tamo","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6734-0179","authenticated-orcid":false,"given":"Alexey","family":"Frolov","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2016,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2016.2518663","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1408.4862","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2015.2472521","page":"5810-5821","source":"Crossref","title":"Storage Capacity of Repairable Networks","volume":"61","author":[{"given":"Arya","family":"Mazumdar","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2015,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2015.2472521","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1408.4862"},{"id":"arxiv:1603.08876","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2015.7282656","page":"1252-1256","source":"Crossref","title":"Locally recoverable codes on algebraic curves","author":[{"given":"Alexander","family":"Barg","sequence":"first","affiliation":[{"name":"Dept. of ECE and ISR, University of Maryland, College Park, 20742, USA"}]},{"given":"Itzhak","family":"Tamo","sequence":"additional","affiliation":[{"name":"Dept. of EE-Systems, Tel Aviv University, Israel"}]},{"given":"Serge","family":"Vlăduţ","sequence":"additional","affiliation":[{"name":"Institut de Math&#x00E9;matiques de Marseille, Aix-Marseille Universit&#x00E9;, IML, Luminy case 907, 13288, France"}]}],"event":"2015 IEEE International Symposium on Information Theory (ISIT)","container-title":"2015 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2015,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2015.7282656","note":"arxivid:1603.08876"},{"id":"doi:10.1109/TIT.2015.2477406","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2015.2477406","page":"5787-5794","source":"Crossref","title":"Bounds on the Size of Locally Recoverable Codes","volume":"61","author":[{"given":"Viveck R.","family":"Cadambe","sequence":"first","affiliation":[]},{"given":"Arya","family":"Mazumdar","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2015,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2015.2477406","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1702.02685","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.2018.2800042","page":"3481-3492","source":"Crossref","title":"Combinatorial Alphabet-Dependent Bounds for Locally Recoverable Codes","volume":"64","author":[{"given":"Abhishek","family":"Agarwal","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8972-4413","authenticated-orcid":false,"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2910-3025","authenticated-orcid":false,"given":"Sihuang","family":"Hu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4605-7996","authenticated-orcid":false,"given":"Arya","family":"Mazumdar","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8000-0419","authenticated-orcid":false,"given":"Itzhak","family":"Tamo","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,5]]},"URL":"http://dx.doi.org/10.1109/TIT.2018.2800042","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1702.02685"},{"id":"manual:-C.-Huang-H.-Simitci-Y.-Xu-A.","type":"paper-conference","author":[{"family":"Huang","given":"C."},{"family":"Simitci","given":"H."},{"family":"Xu","given":"Y."},{"family":"Ogus","given":"A."},{"family":"Calder","given":"B."},{"family":"Gopalan","given":"P."},{"family":"Li","given":"J."},{"family":"Yekhanin","given":"S."}],"title":"Erasure coding in Windows Azure Storage","container-title":"Proc. USENIX Annual Technical Conference (ATC","issued":"2012-06","page":"15–26,","publisher-place":"Boston, Massachusetts"},{"id":"manual:-S.-Muralidhar-W.-Lloyd-S.-Ro","type":"paper-conference","author":[{"family":"Muralidhar","given":"S."},{"family":"Lloyd","given":"W."},{"family":"Roy","given":"S."},{"family":"Hill","given":"C."},{"family":"Lin","given":"E."},{"family":"Liu","given":"W."},{"family":"Pan","given":"S."},{"family":"Shankar","given":"S."},{"family":"Sivakumar","given":"V."},{"family":"Tang","given":"L."},{"family":"Kumar","given":"S."}],"title":"f4: Facebook’s warm {BLOB} storage system","container-title":"11th USENIX Symposium on Operating Systems Design and Implementation (OSDI 14","issued":"2014","page":"383–398"},{"id":"doi:10.1007/978-3-319-70293-3_16","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-70293-3_16","page":"427-442","source":"Crossref","title":"Batch and PIR Codes and Their Connections to Locally Repairable Codes","author":[{"given":"Vitaly","family":"Skachek","sequence":"first","affiliation":[]}],"container-title":"Signals and Communication Technology","original-title":[],"issued":{"date-parts":[[2018]]},"ISBN":["9783319702926","9783319702933"],"URL":"http://dx.doi.org/10.1007/978-3-319-70293-3_16","ISSN":["1860-4862","1860-4870"],"note":"publisher-location:Cham"},{"id":"doi:10.1109/TIT.2021.3131620","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2021.3131620","page":"1545-1559","source":"Crossref","title":"Batch Codes for Asynchronous Recovery of Data","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0002-8310-6809","authenticated-orcid":false,"given":"Ago-Erik","family":"Riet","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0626-2437","authenticated-orcid":false,"given":"Vitaly","family":"Skachek","sequence":"additional","affiliation":[]},{"given":"Eldho K.","family":"Thomas","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3131620","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1505.06241","type":"article-journal","author":[{"given":"Arman","family":"Fazeli"},{"given":"Alexander","family":"Vardy"},{"given":"Eitan","family":"Yaakobi"}],"title":"PIR with Low Storage Overhead: Coding instead of Replication","issued":{"date-parts":[[2015,5,22]]},"note":"arxivid:1505.06241\narxiv_version_number:1"},{"id":"arxiv:1611.09914","type":"article-journal","author":[{"given":"Vitaly","family":"Skachek"}],"title":"Batch and PIR Codes and Their Connections to Locally Repairable Codes","issued":{"date-parts":[[2017,9,5]]},"note":"arxivid:1611.09914\narxiv_version_number:4"},{"id":"arxiv:1205.0997","type":"article-journal","author":[{"given":"Mario","family":"Blaum"},{"given":"James Lee","family":"Hafner"},{"given":"Steven","family":"Hetzler"}],"title":"Partial-MDS Codes and their Application to RAID Type of Architectures","issued":{"date-parts":[[2014,9,11]]},"note":"arxivid:1205.0997\narxiv_version_number:2"},{"id":"arxiv:1307.4150","type":"article-journal","author":[{"given":"Parikshit","family":"Gopalan"},{"given":"Cheng","family":"Huang"},{"given":"Bob","family":"Jenkins"},{"given":"Sergey","family":"Yekhanin"}],"title":"Explicit Maximally Recoverable Codes with Locality","issued":{"date-parts":[[2013,7,20]]},"note":"arxivid:1307.4150\narxiv_version_number:2"},{"id":"doi:10.1145/2560013","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"1","abstract":"<jats:p>Traditionally, when storage systems employ erasure codes, they are designed to tolerate the failures of entire disks. However, the most common types of failures are latent sector failures, which only affect individual disk sectors, and block failures which arise through wear on SSD’s. This article introduces SD codes, which are designed to tolerate combinations of disk and sector failures. As such, they consume far less storage resources than traditional erasure codes. We specify the codes with enough detail for the storage practitioner to employ them, discuss their practical properties, and detail an open-source implementation.</jats:p>","DOI":"10.1145/2560013","page":"1-17","source":"Crossref","title":"Sector-Disk (SD) Erasure Codes for Mixed Failure Modes in RAID Systems","volume":"10","author":[{"given":"James S.","family":"Plank","sequence":"first","affiliation":[{"name":"University of Tennessee"}]},{"given":"Mario","family":"Blaum","sequence":"additional","affiliation":[{"name":"IBM Almaden Research Center"}]}],"container-title":"ACM Transactions on Storage","original-title":[],"language":"en","issued":{"date-parts":[[2014,1]]},"URL":"http://dx.doi.org/10.1145/2560013","ISSN":["1553-3077","1553-3093"],"container-title-short":"ACM Trans. Storage","note":"alternative-id:10.1145/2560013"},{"id":"arxiv:1106.3625","type":"article-journal","author":[{"given":"Parikshit","family":"Gopalan"},{"given":"Cheng","family":"Huang"},{"given":"Huseyin","family":"Simitci"},{"given":"Sergey","family":"Yekhanin"}],"title":"On the Locality of Codeword Symbols","issued":{"date-parts":[[2011,6,18]]},"note":"arxivid:1106.3625\narxiv_version_number:1"},{"id":"arxiv:1206.3804","type":"article-journal","author":[{"given":"Dimitris S.","family":"Papailiopoulos"},{"given":"Alexandros G.","family":"Dimakis"}],"title":"Locally Repairable Codes","issued":{"date-parts":[[2014,5,3]]},"note":"arxivid:1206.3804\narxiv_version_number:2"},{"id":"doi:10.1007/978-3-642-16367-8_6","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-16367-8_6","page":"65-104","source":"Crossref","title":"Short Locally Testable Codes and Proofs: A Survey in Two Parts","author":[{"given":"Oded","family":"Goldreich","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2010]]},"ISBN":["9783642163661","9783642163678"],"URL":"http://dx.doi.org/10.1007/978-3-642-16367-8_6","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1145/103418.103428","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/103418.103428","page":"21-32","source":"Crossref","title":"Checking computations in polylogarithmic time","author":[{"given":"László","family":"Babai","sequence":"first","affiliation":[]},{"given":"Lance","family":"Fortnow","sequence":"additional","affiliation":[]},{"given":"Leonid A.","family":"Levin","sequence":"additional","affiliation":[]},{"given":"Mario","family":"Szegedy","sequence":"additional","affiliation":[]}],"event":"the twenty-third annual ACM symposium","container-title":"Proceedings of the twenty-third annual ACM symposium on Theory of computing  - STOC '91","original-title":[],"issued":{"date-parts":[[1991]]},"URL":"http://dx.doi.org/10.1145/103418.103428","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"preset:Arora94","type":"book","author":[{"family":"Arora","given":"S."}],"title":"Probabilistic checking of proofs and hardness of approximation problems","publisher":"UC Berkeley","issued":"1994"},{"id":"doi:10.1137/S0097539793255151","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","DOI":"10.1137/s0097539793255151","page":"252-271","source":"Crossref","title":"Robust Characterizations of Polynomials with Applications to Program Testing","volume":"25","author":[{"given":"Ronitt","family":"Rubinfeld","sequence":"first","affiliation":[]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[1996,4]]},"URL":"http://dx.doi.org/10.1137/S0097539793255151","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/S0097539793255151"},{"id":"arxiv:1307.3975","type":"article-journal","author":[{"given":"Katalin","family":"Friedl"},{"given":"Madhu","family":"Sudan"}],"title":"Some Improvements to Total Degree Tests","issued":{"date-parts":[[2013,7,15]]},"note":"arxivid:1307.3975\narxiv_version_number:1"},{"id":"doi:10.1007/BF01200056","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01200056","page":"3-40","source":"Crossref","title":"Non-deterministic exponential time has two-prover interactive protocols","volume":"1","author":[{"given":"L�szl�","family":"Babai","sequence":"first","affiliation":[]},{"given":"Lance","family":"Fortnow","sequence":"additional","affiliation":[]},{"given":"Carsten","family":"Lund","sequence":"additional","affiliation":[]}],"container-title":"Computational Complexity","original-title":[],"language":"en","issued":{"date-parts":[[1991,3]]},"URL":"http://dx.doi.org/10.1007/BF01200056","ISSN":["1016-3328","1420-8954"],"container-title-short":"Comput Complexity","note":"alternative-id:BF01200056"},{"id":"doi:10.1145/146585.146605","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"4","abstract":"<jats:p>A new algebraic technique for the construction of interactive proof systems is presented. Our technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system. This technique played a pivotal role in the recent proofs that IP = PSPACE [28] and that MIP = NEXP [4].</jats:p>","DOI":"10.1145/146585.146605","page":"859-868","source":"Crossref","title":"Algebraic methods for interactive proof systems","volume":"39","author":[{"given":"Carsten","family":"Lund","sequence":"first","affiliation":[]},{"given":"Lance","family":"Fortnow","sequence":"additional","affiliation":[]},{"given":"Howard","family":"Karloff","sequence":"additional","affiliation":[]},{"given":"Noam","family":"Nisan","sequence":"additional","affiliation":[]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[1992,10]]},"URL":"http://dx.doi.org/10.1145/146585.146605","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/146585.146605"},{"id":"doi:10.1145/273865.273901","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"1","abstract":"<jats:p>\n            We give a new characterization of NP: the class NP contains exactly those languages\n            <jats:italic>L</jats:italic>\n            for which membership proofs (a proof that an input\n            <jats:italic>x</jats:italic>\n            is in\n            <jats:italic>L</jats:italic>\n            ) can be verified probabilistically in polynomial time using\n            <jats:italic>logarithmic</jats:italic>\n            number of random bits and by reading\n            <jats:italic>sublogarithmic</jats:italic>\n            number of bits from the proof.\n          </jats:p>\n          <jats:p>We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NP-hard.</jats:p>","DOI":"10.1145/273865.273901","page":"70-122","source":"Crossref","title":"Probabilistic checking of proofs","volume":"45","author":[{"given":"Sanjeev","family":"Arora","sequence":"first","affiliation":[{"name":"Princeton Univ., Princeton, NJ"}]},{"given":"Shmuel","family":"Safra","sequence":"additional","affiliation":[{"name":"Tel-Aviv Univ., Tel-Aviv, Israel"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[1998,1]]},"URL":"http://dx.doi.org/10.1145/273865.273901","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/273865.273901"},{"id":"doi:10.1145/278298.278306","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"3","abstract":"<jats:p>\n            We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a\n            <jats:italic>constant</jats:italic>\n            number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof” with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [1998] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length).\n          </jats:p>\n          <jats:p>\n            As a consequence, we prove that no MAX SNP-hard problem has a polynomial  time approximation scheme, unless NP = P. The class MAX SNP was defined by Papadimitriou and Yannakakis [1991] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige et al. [1996] and Arora and Safra [1998] and show that there exists a positive ε such that approximating the maximum clique size in an\n            <jats:italic>N</jats:italic>\n            -vertex graph to within a factor of\n            <jats:italic>N</jats:italic>\n            <jats:sup>ε</jats:sup>\n            is NP-hard.\n          </jats:p>","DOI":"10.1145/278298.278306","page":"501-555","source":"Crossref","title":"Proof verification and the hardness of approximation problems","volume":"45","author":[{"given":"Sanjeev","family":"Arora","sequence":"first","affiliation":[{"name":"Princeton Univ., Princeton, NJ"}]},{"given":"Carsten","family":"Lund","sequence":"additional","affiliation":[{"name":"AT&amp;T Bell Labs, Murray Hill, NJ"}]},{"given":"Rajeev","family":"Motwani","sequence":"additional","affiliation":[{"name":"Stanford Univ., Stanford, CA"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge"}]},{"given":"Mario","family":"Szegedy","sequence":"additional","affiliation":[{"name":"AT&amp;T Bell Labs, Murray Hill, NJ"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[1998,5]]},"URL":"http://dx.doi.org/10.1145/278298.278306","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/278298.278306"},{"id":"doi:10.4230/LIPIcs.CCC.2022.14","type":"document","categories":["local testing","multiplicity codes","Reed Muller codes","Theory of computation → Error-correcting codes"],"language":"en","author":[{"family":"Karliner","given":"Dan"},{"family":"Salama","given":"Roie"},{"family":"Ta-Shma","given":"Amnon"}],"contributor":[{"family":"Lovett","given":"Shachar"}],"editor":[{"family":"Lovett","given":"Shachar"}],"issued":{"date-parts":[[2022]]},"abstract":"Multiplicity codes are a generalization of RS and RM codes where for each evaluation point we output the evaluation of a low-degree polynomial and all of its directional derivatives up to order s. Multi-variate multiplicity codes are locally decodable with the natural local decoding algorithm that reads values on a random line and corrects to the closest uni-variate multiplicity code. However, it was not known whether multiplicity codes are locally testable, and this question has been posed since the introduction of these codes with no progress up to date. In fact, it has been also open whether multiplicity codes can be characterized by local constraints, i.e., if there exists a probabilistic algorithm that queries few symbols of a word c, accepts every c in the code with probability 1, and rejects every c not in the code with nonzero probability. \r\nWe begin by giving a simple example showing the line test does not give local characterization when d > q. Surprisingly, we then show the plane test is a local characterization when s < q and d < qs-1 for prime q. In addition, we show the s-dimensional test is a local tester for multiplicity codes, when s < q. Combining the two results, we show our main result that the plane test is a local tester for multiplicity codes of degree d < qs-1, with constant rejection probability for constant q, s.\r\nOur technique is new. We represent the given input as a possibly very high-degree polynomial, and we show that for some choice of plane, the restriction of the polynomial to the plane is a high-degree bi-variate polynomial. The argument has to work modulo the appropriate kernels, and for that we use Grobner theory, the Combinatorial Nullstellensatz theorem and its generalization to multiplicities. Even given that, the argument is delicate and requires choosing a non-standard monomial order for the argument to work.","container-title":"LIPIcs, Volume 234, CCC 2022","DOI":"10.4230/LIPICS.CCC.2022.14","volume":"234","number":"14","page":"14:1-14:33","page-first":"14:1","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"The Plane Test Is a Local Tester for Multiplicity Codes","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.14","note":"copyright:Creative Commons Attribution 4.0 International license"},{"id":"doi:10.4230/LIPIcs.APPROX/RANDOM.202","type":"document","categories":["local testing","multiplicity codes","Reed Muller codes","Theory of computation → Error-correcting codes"],"language":"en","author":[{"family":"Karliner","given":"Dan"},{"family":"Ta-Shma","given":"Amnon"}],"contributor":[{"family":"Chakrabarti","given":"Amit"},{"family":"Swamy","given":"Chaitanya"}],"editor":[{"family":"Chakrabarti","given":"Amit"},{"family":"Swamy","given":"Chaitanya"}],"issued":{"date-parts":[[2022]]},"abstract":"Multiplicity codes are a generalization of Reed-Muller codes which include derivatives as well as the values of low degree polynomials, evaluated in every point in 𝔽_p^m. Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing. Recently, [Karliner et al., 2022] showed that the plane test, which tests the degree of the codeword on a random plane, is a good local tester for small enough degrees. In this work we simplify and extend the analysis of local testing for multiplicity codes, giving a more general and tight analysis. In particular, we show that multiplicity codes MRM_p(m, d, s) over prime fields with arbitrary d are locally testable by an appropriate k-flat test, which tests the degree of the codeword on a random k-dimensional affine subspace. The relationship between the degree parameter d and the required dimension k is shown to be nearly optimal, and improves on [Karliner et al., 2022] in the case of planes.\r\nOur analysis relies on a generalization of the technique of canonincal monomials introduced in [Haramaty et al., 2013]. Generalizing canonical monomials to the multiplicity case requires substantially different proofs which exploit the algebraic structure of multiplicity codes.","container-title":"LIPIcs, Volume 245, APPROX/RANDOM 2022","DOI":"10.4230/LIPICS.APPROX/RANDOM.2022.11","volume":"245","number":"11","page":"11:1-11:19","page-first":"11:1","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"Improved Local Testing for Multiplicity Codes","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.11","note":"copyright:Creative Commons Attribution 4.0 International license"},{"id":"arxiv:2403.03651","type":"article-journal","author":[{"given":"Pavel","family":"Panteleev"},{"given":"Gleb","family":"Kalachev"}],"title":"Maximally Extendable Sheaf Codes","issued":{"date-parts":[[2024,3,6]]},"note":"arxivid:2403.03651\narxiv_version_number:1"},{"id":"arxiv:2403.19388","type":"article-journal","author":[{"given":"Uriya A.","family":"First"},{"given":"Tali","family":"Kaufman"}],"title":"Cosystolic Expansion of Sheaves on Posets with Applications to Good 2-Query Locally Testable Codes and Lifted Codes","issued":{"date-parts":[[2024,5,13]]},"note":"arxivid:2403.19388\narxiv_version_number:3"},{"id":"doi:10.1145/1806689.1806748","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/1806689.1806748","page":"417-426","source":"Crossref","title":"Local list-decoding and testing of random linear codes from high error","author":[{"given":"Swastik","family":"Kopparty","sequence":"first","affiliation":[{"name":"MIT, Cambridge, MA, USA"}]},{"given":"Shubhangi","family":"Saraf","sequence":"additional","affiliation":[{"name":"MIT, Cambridge, MA, USA"}]}],"event":"STOC'10: Symposium on Theory of Computing","container-title":"Proceedings of the forty-second ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2010,6,5]]},"URL":"http://dx.doi.org/10.1145/1806689.1806748","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/1806689.1806748\n10.1145/1806689"},{"id":"arxiv:2201.11369","type":"article-journal","author":[{"given":"Ting-Chun","family":"Lin"},{"given":"Min-Hsiu","family":"Hsieh"}],"title":"$c^3$-Locally Testable Codes from Lossless Expanders","issued":{"date-parts":[[2022,1,28]]},"note":"arxivid:2201.11369\narxiv_version_number:2"},{"id":"arxiv:1510.02082","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/focs.2017.46","page":"427-438","source":"Crossref","title":"Local Hamiltonians Whose Ground States Are Hard to Approximate","author":[{"given":"Lior","family":"Eldar","sequence":"first","affiliation":[]},{"given":"Aram W.","family":"Harrow","sequence":"additional","affiliation":[]}],"event":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","container-title":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","original-title":[],"issued":{"date-parts":[[2017,10]]},"URL":"http://dx.doi.org/10.1109/FOCS.2017.46","note":"arxivid:1510.02082"},{"id":"doi:10.1109/TIT.2016.2524007","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2016.2524007","page":"1592-1604","source":"Crossref","title":"Batch Codes Through Dense Graphs Without Short Cycles","volume":"62","author":[{"given":"Ankit Singh","family":"Rawat","sequence":"first","affiliation":[]},{"given":"Zhao","family":"Song","sequence":"additional","affiliation":[]},{"given":"Alexandros G.","family":"Dimakis","sequence":"additional","affiliation":[]},{"given":"Anna","family":"Gal","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2016,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2016.2524007","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1701.07206","type":"article-journal","author":[{"given":"Hilal","family":"Asi"},{"given":"Eitan","family":"Yaakobi"}],"title":"Nearly Optimal Constructions of PIR and Batch Codes","issued":{"date-parts":[[2017,6,5]]},"note":"arxivid:1701.07206\narxiv_version_number:2"},{"id":"doi:10.1145/293347.293350","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"6","abstract":"<jats:p>\n            Publicly accessible databases are an indispensable resource for retrieving up-to-date information. But they also pose a significant risk to the privacy of the user, since a curious database operator can follow the user's queries and infer what the user is after. Indeed, in cases where the users' intentions are to be kept secret, users are often cautious about accessing the database. It can be shown that when accessing a single database, to completely guarantee the privacy of the user, the whole database should be down-loaded; namely\n            <jats:italic>n</jats:italic>\n            bits should be communicated (where\n            <jats:italic>n</jats:italic>\n            is the number of bits in the database).\n          </jats:p>\n          <jats:p>\n            In this work, we investigate whether by replicating the database, more efficient solutions to the private retrieval problem can be  obtained. We describe schemes that enable a user to access\n            <jats:italic>k</jats:italic>\n            replicated copies of a database (\n            <jats:italic>k</jats:italic>\n            ≥2) and\n            <jats:italic>privately</jats:italic>\n            retrieve information stored in the database. This means that each individual server (holding a replicated copy of the database) gets no information on the identity of the item retrieved by the user. Our schemes use the replication to gain substantial saving. In particular, we present a two-server scheme with communication complexity\n            <jats:italic>O(n</jats:italic>\n            <jats:sup>1/3</jats:sup>\n            ).\n          </jats:p>","DOI":"10.1145/293347.293350","page":"965-981","source":"Crossref","title":"Private information retrieval","volume":"45","author":[{"given":"Benny","family":"Chor","sequence":"first","affiliation":[{"name":"Technion, Haifa, Israel"}]},{"given":"Eyal","family":"Kushilevitz","sequence":"additional","affiliation":[{"name":"Technion, Haifa, Israel"}]},{"given":"Oded","family":"Goldreich","sequence":"additional","affiliation":[{"name":"Weizmann Institute of Science, Rehovot, Israel"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachusetts  Institute of Technology, Cambridge"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[1998,11]]},"URL":"http://dx.doi.org/10.1145/293347.293350","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/293347.293350"},{"id":"doi:10.1109/SFCS.1995.492461","type":"proceedings-article","publisher":"IEEE Comput. Soc. Press","DOI":"10.1109/sfcs.1995.492461","source":"Crossref","title":"Private information retrieval","author":[{"given":"B.","family":"Chor","sequence":"first","affiliation":[]},{"given":"O.","family":"Goldreich","sequence":"additional","affiliation":[]},{"given":"E.","family":"Kushilevitz","sequence":"additional","affiliation":[]},{"given":"M.","family":"Sudan","sequence":"additional","affiliation":[]}],"event":"IEEE 36th Annual Foundations of Computer Science","container-title":"Proceedings of IEEE 36th Annual Foundations of Computer Science","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.1995.492461"},{"id":"doi:10.1109/ISIT45174.2021.9517900","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit45174.2021.9517900","page":"2684-2689","source":"Crossref","title":"Private Data Access in Blockchain Systems Employing Coded Sharding","author":[{"given":"Birenjith","family":"Sasidharan","sequence":"first","affiliation":[]},{"given":"Emanuele","family":"Viterbo","sequence":"additional","affiliation":[]}],"event":"2021 IEEE International Symposium on Information Theory (ISIT)","container-title":"2021 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2021,7,12]]},"URL":"http://dx.doi.org/10.1109/ISIT45174.2021.9517900"},{"id":"doi:10.1006/jtbi.1996.0142","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1006/jtbi.1996.0142","page":"45-58","source":"Crossref","title":"A Complementary Circular Code in the Protein Coding Genes","volume":"182","author":[{"given":"Didier G.","family":"Arquès","sequence":"first","affiliation":[]},{"given":"Christian J.","family":"Michel","sequence":"additional","affiliation":[]}],"container-title":"Journal of Theoretical Biology","original-title":[],"language":"en","issued":{"date-parts":[[1996,9]]},"URL":"http://dx.doi.org/10.1006/jtbi.1996.0142","ISSN":["0022-5193"],"container-title-short":"Journal of Theoretical Biology","note":"alternative-id:S0022519396901423"},{"id":"doi:10.1007/s00285-014-0806-7","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7","DOI":"10.1007/s00285-014-0806-7","page":"1623-1644","source":"Crossref","title":"Circular codes, symmetries and transformations","volume":"70","author":[{"given":"Elena","family":"Fimmel","sequence":"first","affiliation":[]},{"given":"Simone","family":"Giannerini","sequence":"additional","affiliation":[]},{"given":"Diego Luis","family":"Gonzalez","sequence":"additional","affiliation":[]},{"given":"Lutz","family":"Strüngmann","sequence":"additional","affiliation":[]}],"container-title":"Journal of Mathematical Biology","original-title":[],"language":"en","issued":{"date-parts":[[2014,7,10]]},"URL":"http://dx.doi.org/10.1007/s00285-014-0806-7","ISSN":["0303-6812","1432-1416"],"container-title-short":"J. Math. Biol.","note":"alternative-id:806"},{"id":"arxiv:1410.8837","type":"article-journal","author":[{"given":"Han Mao","family":"Kiah"},{"given":"Gregory J.","family":"Puleo"},{"given":"Olgica","family":"Milenkovic"}],"title":"Codes for DNA Storage Channels","issued":{"date-parts":[[2015,11,3]]},"note":"arxivid:1410.8837\narxiv_version_number:2"},{"id":"arxiv:1505.02199","type":"article-journal","author":[{"given":"S. M. Hossein Tabatabaei","family":"Yazdi"},{"given":"Yongbo","family":"Yuan"},{"given":"Jian","family":"Ma"},{"given":"Huimin","family":"Zhao"},{"given":"Olgica","family":"Milenkovic"}],"title":"A Rewritable, Random-Access DNA-Based Storage System","issued":{"date-parts":[[2015,5,8]]},"note":"arxivid:1505.02199\narxiv_version_number:1"},{"id":"doi:10.1038/s41598-017-05188-1","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>DNA-based data storage is an emerging nonvolatile memory technology of potentially unprecedented density, durability, and replication efficiency. The basic system implementation steps include synthesizing DNA strings that contain user information and subsequently retrieving them via high-throughput sequencing technologies. Existing architectures enable reading and writing but do not offer random-access and error-free data recovery from low-cost, portable devices, which is crucial for making the storage technology competitive with classical recorders. Here we show for the first time that a portable, random-access platform may be implemented in practice using nanopore sequencers. The novelty of our approach is to design an integrated processing pipeline that encodes data to avoid costly synthesis and sequencing errors, enables random access through addressing, and leverages efficient portable sequencing via new iterative alignment and deletion error-correcting codes. Our work represents the only known random access DNA-based data storage system that uses error-prone nanopore sequencers, while still producing error-free readouts with the highest reported information rate. As such, it represents a crucial step towards practical employment of DNA molecules as storage media.</jats:p>","DOI":"10.1038/s41598-017-05188-1","source":"Crossref","title":"Portable and Error-Free DNA-Based Data Storage","volume":"7","author":[{"given":"S. M. Hossein Tabatabaei","family":"Yazdi","sequence":"first","affiliation":[]},{"given":"Ryan","family":"Gabrys","sequence":"additional","affiliation":[]},{"given":"Olgica","family":"Milenkovic","sequence":"additional","affiliation":[]}],"container-title":"Scientific Reports","original-title":[],"language":"en","issued":{"date-parts":[[2017,7,10]]},"URL":"http://dx.doi.org/10.1038/s41598-017-05188-1","ISSN":["2045-2322"],"container-title-short":"Sci Rep","page":"5011","note":"updated-by:{\"updated\":{\"date-parts\":[[2020,4,22]],\"date-time\":\"2020-04-22T00:00:00Z\",\"timestamp\":1587513600000},\"DOI\":\"10.1038/s41598-020-60080-9\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nalternative-id:5188"},{"id":"doi:10.1073/pnas.2004821117","type":"journal-article","publisher":"Proceedings of the National Academy of Sciences","issue":"31","abstract":"<jats:title>Significance</jats:title>\n          <jats:p>\n            This paper constructs an error-correcting code for the\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                <mml:mrow>\n                  <mml:mo>{</mml:mo>\n                  <mml:mrow>\n                    <mml:mi>A</mml:mi>\n                    <mml:mo>,</mml:mo>\n                    <mml:mi>C</mml:mi>\n                    <mml:mo>,</mml:mo>\n                    <mml:mi>G</mml:mi>\n                    <mml:mo>,</mml:mo>\n                    <mml:mi>T</mml:mi>\n                  </mml:mrow>\n                  <mml:mo>}</mml:mo>\n                </mml:mrow>\n              </mml:math>\n            </jats:inline-formula>\n            alphabet of DNA. By contrast with previous work, the code corrects insertions and deletions directly, in a single strand of DNA, without the need for multiple alignment of strands. This code, when coupled to a standard outer code, can achieve error-free storage of petabyte-scale data even when\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                <mml:mo>∼</mml:mo>\n              </mml:math>\n            </jats:inline-formula>\n            10% of all nucleotides are erroneous.\n          </jats:p>","DOI":"10.1073/pnas.2004821117","page":"18489-18496","source":"Crossref","title":"HEDGES error-correcting code for DNA storage corrects indels and allows sequence constraints","volume":"117","author":[{"ORCID":"https://orcid.org/0000-0003-0771-0841","authenticated-orcid":false,"given":"William H.","family":"Press","sequence":"first","affiliation":[{"name":"Department of Computer Science, The University of Texas at Austin, Austin, TX 78712;"},{"name":"Department of Integrative Biology, The University of Texas at Austin, Austin, TX 78712;"}]},{"ORCID":"https://orcid.org/0000-0002-2943-9397","authenticated-orcid":false,"given":"John A.","family":"Hawkins","sequence":"additional","affiliation":[{"name":"Oden Institute of Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712;"},{"name":"Department of Molecular Biosciences, The University of Texas at Austin, Austin, TX 78712;"},{"name":"Institute for Cellular and Molecular Biology, The University of Texas at Austin, Austin, TX 78712"}]},{"ORCID":"https://orcid.org/0000-0001-8387-1917","authenticated-orcid":false,"suffix":"Jr","given":"Stephen K.","family":"Jones","sequence":"additional","affiliation":[{"name":"Department of Molecular Biosciences, The University of Texas at Austin, Austin, TX 78712;"},{"name":"Institute for Cellular and Molecular Biology, The University of Texas at Austin, Austin, TX 78712"}]},{"ORCID":"https://orcid.org/0000-0002-1130-2675","authenticated-orcid":false,"given":"Jeffrey M.","family":"Schaub","sequence":"additional","affiliation":[{"name":"Department of Molecular Biosciences, The University of Texas at Austin, Austin, TX 78712;"},{"name":"Institute for Cellular and Molecular Biology, The University of Texas at Austin, Austin, TX 78712"}]},{"ORCID":"https://orcid.org/0000-0002-9371-2431","authenticated-orcid":false,"given":"Ilya J.","family":"Finkelstein","sequence":"additional","affiliation":[{"name":"Department of Molecular Biosciences, The University of Texas at Austin, Austin, TX 78712;"},{"name":"Institute for Cellular and Molecular Biology, The University of Texas at Austin, Austin, TX 78712"}]}],"container-title":"Proceedings of the National Academy of Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,16]]},"URL":"http://dx.doi.org/10.1073/pnas.2004821117","ISSN":["0027-8424","1091-6490"],"container-title-short":"Proc. Natl. Acad. Sci. U.S.A.","note":"alternative-id:10.1073/pnas.2004821117"},{"id":"doi:10.1109/TIT.2021.3066430","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2021.3066430","page":"5602-5613","source":"Crossref","title":"Capacity-Approaching Constrained Codes With Error Correction for DNA-Based Data Storage","volume":"67","author":[{"ORCID":"https://orcid.org/0000-0002-3179-9471","authenticated-orcid":false,"given":"Tuan Thanh","family":"Nguyen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2059-0071","authenticated-orcid":false,"given":"Kui","family":"Cai","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6747-9261","authenticated-orcid":false,"given":"Kees A.","family":"Schouhamer Immink","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5611-0848","authenticated-orcid":false,"given":"Han Mao","family":"Kiah","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2021,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3066430","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1803.03322","type":"article-journal","author":[{"given":"Reinhard","family":"Heckel"},{"given":"Gediminas","family":"Mikutis"},{"given":"Robert N.","family":"Grass"}],"title":"A Characterization of the DNA Data Storage Channel","issued":{"date-parts":[[2018,3,8]]},"note":"arxivid:1803.03322\narxiv_version_number:1"},{"id":"arxiv:1502.00517","type":"article-journal","author":[{"given":"Han Mao","family":"Kiah"},{"given":"Gregory J.","family":"Puleo"},{"given":"Olgica","family":"Milenkovic"}],"title":"Codes for DNA Sequence Profiles","issued":{"date-parts":[[2015,2,2]]},"note":"arxivid:1502.00517\narxiv_version_number:1"},{"id":"arxiv:1507.01611","type":"article-journal","author":[{"given":"S. M. Hossein Tabatabaei","family":"Yazdi"},{"given":"Han Mao","family":"Kiah"},{"given":"Eva Ruiz","family":"Garcia"},{"given":"Jian","family":"Ma"},{"given":"Huimin","family":"Zhao"},{"given":"Olgica","family":"Milenkovic"}],"title":"DNA-Based Storage: Trends and Methods","issued":{"date-parts":[[2015,7,6]]},"note":"arxivid:1507.01611\narxiv_version_number:1"},{"id":"manual:-T.-A.-Terlep.-A-Mathematical","type":"thesis","author":[{"family":"Terlep","given":"T.A."}],"title":"A Mathematical Theory of Communication with Graphs and Symbols","genre":"PhD thesis,","publisher":"Purdue University","issued":"2024"},{"id":"doi:10.1002/anie.201411378","type":"journal-article","publisher":"Wiley","issue":"8","abstract":"<jats:title>Abstract</jats:title><jats:p>Information, such as text printed on paper or images projected onto microfilm, can survive for over 500 years. However, the storage of digital information for time frames exceeding 50 years is challenging. Here we show that digital information can be stored on DNA and recovered without errors for considerably longer time frames. To allow for the perfect recovery of the information, we encapsulate the DNA in an inorganic matrix, and employ error‐correcting codes to correct storage‐related errors. Specifically, we translated 83 kB of information to 4991 DNA segments, each 158 nucleotides long, which were encapsulated in silica. Accelerated aging experiments were performed to measure DNA decay kinetics, which show that data can be archived on DNA for millennia under a wide range of conditions. The original information could be recovered error free, even after treating the DNA in silica at 70 °C for one week. This is thermally equivalent to storing information on DNA in central Europe for 2000 years.</jats:p>","DOI":"10.1002/anie.201411378","page":"2552-2555","source":"Crossref","title":"Robust Chemical Preservation of Digital Information on DNA in Silica with Error‐Correcting Codes","volume":"54","author":[{"given":"Robert N.","family":"Grass","sequence":"first","affiliation":[]},{"given":"Reinhard","family":"Heckel","sequence":"additional","affiliation":[]},{"given":"Michela","family":"Puddu","sequence":"additional","affiliation":[]},{"given":"Daniela","family":"Paunescu","sequence":"additional","affiliation":[]},{"given":"Wendelin J.","family":"Stark","sequence":"additional","affiliation":[]}],"container-title":"Angewandte Chemie International Edition","original-title":[],"language":"en","issued":{"date-parts":[[2015,2,4]]},"URL":"http://dx.doi.org/10.1002/anie.201411378","ISSN":["1433-7851","1521-3773"],"container-title-short":"Angew Chem Int Ed","note":"alternative-id:10.1002/anie.201411378"},{"id":"manual:-V.-I.-Levenshtein-Binary-cod","type":"article-journal","author":[{"family":"Levenshtein","given":"V.I."}],"title":"Binary codes capable of correcting deletions, insertions, and reversals","volume":"10","container-title":"Soviet Physics Doklady","issue":"8","issued":"1966"},{"id":"arxiv:2007.09075","type":"article-journal","author":[{"given":"Kuan","family":"Cheng"},{"given":"Venkatesan","family":"Guruswami"},{"given":"Bernhard","family":"Haeupler"},{"given":"Xin","family":"Li"}],"title":"Efficient Linear and Affine Codes for Correcting Insertions/Deletions","issued":{"date-parts":[[2022,7,21]]},"note":"arxivid:2007.09075\narxiv_version_number:4"},{"id":"doi:10.1109/TIT.2021.3056317","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2021.3056317","page":"3190-3206","source":"Crossref","title":"Synchronization Strings and Codes for Insertions and Deletions—A Survey","volume":"67","author":[{"given":"Bernhard","family":"Haeupler","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4013-9096","authenticated-orcid":false,"given":"Amirbehshad","family":"Shahrasbi","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2021,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3056317","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.651060","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.651060","page":"328-333","source":"Crossref","title":"On the constructions of constant-weight codes","volume":"44","author":[{"family":"Fang-Wei Fu","sequence":"first","affiliation":[]},{"given":"A.J.","family":"Han Vinck","sequence":"additional","affiliation":[]},{"family":"Shi-Yi Shen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.651060","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1090/S0002-9947-01-02905-1","type":"journal-article","publisher":"American Mathematical Society (AMS)","issue":"3","DOI":"10.1090/s0002-9947-01-02905-1","page":"1007-1026","source":"Crossref","title":"The structure of linear codes of constant weight","volume":"354","author":[{"given":"Jay A.","family":"Wood","sequence":"first","affiliation":[]}],"container-title":"Transactions of the American Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[2001,10,26]]},"URL":"http://dx.doi.org/10.1090/S0002-9947-01-02905-1","ISSN":["0002-9947","1088-6850"],"container-title-short":"Trans. Amer. Math. Soc.","note":"alternative-id:S0002994701029051"},{"id":"arxiv:1812.08562","type":"article-journal","author":[{"given":"Mustafa Cemil","family":"Coşkun"},{"given":"Giuseppe","family":"Durisi"},{"given":"Thomas","family":"Jerkovits"},{"given":"Gianluigi","family":"Liva"},{"given":"William","family":"Ryan"},{"given":"Brian","family":"Stein"},{"given":"Fabian","family":"Steiner"}],"title":"Efficient Error-Correcting Codes in the Short Blocklength Regime","issued":{"date-parts":[[2019,3,10]]},"note":"arxivid:1812.08562\narxiv_version_number:2"},{"id":"manual:-E.-Prange-Cyclic-Error-Corre","type":"document","author":[{"family":"Prange","given":"E."}],"title":"Cyclic Error-Correcting Codes in Two Symbols","volume":"TN-57-/03","issued":"1957-09"},{"id":"preset:Prange58","type":"article-journal","author":[{"family":"Prange","given":"E."}],"title":"Some cyclic error-correcting codes with simple decoding algorithms","container-title":"AFCRC-TN-58-156","issued":"1958-04"},{"id":"manual:-E.-Prange-The-use-of-coset-e","type":"document","author":[{"family":"Prange","given":"E."}],"title":"The use of coset equivalence in the analysis and decoding of group codes","issued":"1959","page":"–59– 64,"},{"id":"manual:-E.-Prange-An-algorithm-for-f","type":"chapter","author":[{"family":"Prange","given":"E."}],"title":"An algorithm for factoring xn - I over a finite field","container-title":"TN-59-/75","issued":"1959-10"},{"id":"manual:-T.-A.-Gulliver-href-https://","type":"thesis","author":[{"family":"Gulliver","given":"T.A."}],"title":"\\href{https://www.ece.uvic.ca/~agullive/gulliverthesis.pdf}{Construction of quasi-cyclic codes}","genre":"PhD thesis,","publisher":"University of New Brunswick","issued":"1989"},{"id":"doi:10.1109/TIT.1967.1053974","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1967.1053974","page":"183-195","source":"Crossref","title":"Self-orthogonal quasi-cyclic codes","volume":"13","author":[{"given":"R.","family":"Townsend","sequence":"first","affiliation":[]},{"given":"E.","family":"Weldon","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1967,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1967.1053974","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-G.-D.-Forney-Jr.-Why-quasi-c","type":"manuscript","author":[{"family":"Forney","given":"G.D.","suffix":"Jr"}],"title":"Why quasi cyclic codes are interesting","note":"unpublished note,","issued":"1970"},{"id":"doi:10.1137/0137027","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","DOI":"10.1137/0137027","page":"358-369","source":"Crossref","title":"A Connection Between Block and Convolutional Codes","volume":"37","author":[{"given":"G.","family":"Solomon","sequence":"first","affiliation":[]},{"given":"H. C. A.","family":"Tilborg","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1979,10]]},"URL":"http://dx.doi.org/10.1137/0137027","ISSN":["0036-1399","1095-712X"],"container-title-short":"SIAM J. Appl. Math.","note":"alternative-id:10.1137/0137027"},{"id":"manual:-R.-M.-Tanner-ERROR-CORRECTIN","type":"patent","author":[{"family":"Tanner","given":"R.M."}],"title":"ERROR-CORRECTING CODING SYSTEM, U.S","genre":"Patent 4295218,","issued":"1981"},{"id":"manual:-R.-M.-Tanner.-Convolutional-","type":"book","author":[{"family":"Tanner","given":"R.M."}],"title":"Convolutional codes from quasi-cyclic codes: A link between the theories of block and convolutional codes","publisher":"University of California","issued":"1987","publisher-place":"Santa Cruz"},{"id":"manual:-H.-H.-Ma-Generalized-tail-bi","type":"thesis","author":[{"family":"Ma","given":"H.H."}],"title":"Generalized tail-biting convolutional codes","genre":"PhD thesis,","publisher":"Univ. of Massachusetts","issued":"1985","publisher-place":"Amherst"},{"id":"manual:-Y.-Levy-and-J.-Costello-Jr.-","type":"article-journal","author":[{"family":"Levy","given":"Y."},{"family":"Costello","given":"J.","suffix":"Jr"}],"title":"An algebraic approach to constructing convolutional codes from quasi-cyclic codes","volume":"14","container-title":"DIMACS Ser. Discr. Math. and Theor. Comp. Sci","issued":"1993","page":"189–198,"},{"id":"doi:10.1109/18.651076","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.651076","page":"431-435","source":"Crossref","title":"A link between quasi-cyclic codes and convolutional codes","volume":"44","author":[{"given":"M.","family":"Esmaeili","sequence":"first","affiliation":[]},{"given":"T.A.","family":"Gulliver","sequence":"additional","affiliation":[]},{"given":"N.P.","family":"Secord","sequence":"additional","affiliation":[]},{"given":"S.A.","family":"Mahmoud","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.651076","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1102.3181","type":"article-journal","author":[{"given":"Manabu","family":"Hagiwara"},{"given":"Kenta","family":"Kasai"},{"given":"Hideki","family":"Imai"},{"given":"Kohichi","family":"Sakaniwa"}],"title":"Spatially Coupled Quasi-Cyclic Quantum LDPC Codes","issued":{"date-parts":[[2011,2,15]]},"note":"arxivid:1102.3181\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9708021","type":"journal-article","publisher":"Wiley","issue":"4-5","DOI":"10.1002/(sici)1521-3978(199806)46:4/5<443::aid-prop443>3.0.co;2-8","page":"443-457","source":"Crossref","title":"Space, Time, Parallelism and Noise Requirements for Reliable Quantum Computing","volume":"46","author":[{"given":"A.M.","family":"Steane","sequence":"first","affiliation":[]}],"container-title":"Fortschritte der Physik","original-title":[],"language":"de","issued":{"date-parts":[[1998,6]]},"URL":"http://dx.doi.org/10.1002/(SICI)1521-3978(199806)46:4/5<443::AID-PROP443>3.0.CO;2-8","ISSN":["0015-8208","1521-3978"],"container-title-short":"Fortschr. Phys.","note":"arxivid:quant-ph/9708021"},{"id":"doi:10.1504/IJICOT.2011.044674","type":"journal-article","publisher":"Inderscience Publishers","issue":"1","DOI":"10.1504/ijicot.2011.044674","page":"10","source":"Crossref","title":"Skew cyclic codes of arbitrary length","volume":"2","author":[{"given":"Irfan","family":"Siap","sequence":"first","affiliation":[]},{"given":"Taher","family":"Abualrub","sequence":"additional","affiliation":[]},{"given":"Nuh","family":"Aydin","sequence":"additional","affiliation":[]},{"given":"Padmapani","family":"Seneviratne","sequence":"additional","affiliation":[]}],"container-title":"International Journal of Information and Coding Theory","original-title":[],"language":"en","issued":{"date-parts":[[2011]]},"URL":"http://dx.doi.org/10.1504/IJICOT.2011.044674","ISSN":["1753-7703","1753-7711"],"container-title-short":"IJICOT"},{"id":"arxiv:1605.09598","type":"article-journal","author":[{"given":"Jihao","family":"Fan"},{"given":"Yonghui","family":"Li"},{"given":"Min-Hsiu","family":"Hsieh"},{"given":"Hanwu","family":"Chen"}],"title":"On Quantum Tensor Product Codes","issued":{"date-parts":[[2017,10,25]]},"note":"arxivid:1605.09598\narxiv_version_number:4"},{"id":"arxiv:math/0604603","type":"article-journal","author":[{"given":"Delphine","family":"Boucher"},{"given":"Willi","family":"Geiselmann"},{"given":"Félix","family":"Ulmer"}],"title":"Skew-cyclic codes","issued":{"date-parts":[[2006,4,27]]},"note":"arxivid:math/0604603\narxiv_version_number:1"},{"id":"doi:10.1142/9407","type":"monograph","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/9407","source":"Crossref","title":"Algebraic Coding Theory","author":[{"given":"Elwyn R","family":"Berlekamp","sequence":"first","affiliation":[{"name":"UC Berkeley"}]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2014,10,30]]},"ISBN":["9789814635899","9789814635905"],"URL":"http://dx.doi.org/10.1142/9407","note":"alternative-id:10.1142/9407"},{"id":"arxiv:quant-ph/0312164","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"01","abstract":"<jats:p> We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛<jats:sub>q</jats:sub> exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q<jats:sup>2</jats:sup>, q<jats:sup>2</jats:sup>-2d+2, d〛<jats:sub>q</jats:sub> for 1≤d≤q which additionally give rise to shortened codes 〚q<jats:sup>2</jats:sup>-s, q<jats:sup>2</jats:sup>-2d+2-s, d〛<jats:sub>q</jats:sub> for some s. </jats:p>","DOI":"10.1142/s0219749904000079","page":"55-64","source":"Crossref","title":"ON OPTIMAL QUANTUM CODES","volume":"02","author":[{"given":"MARKUS","family":"GRASSL","sequence":"first","affiliation":[{"name":"Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, 76 128 Karlsruhe, Germany"}]},{"given":"THOMAS","family":"BETH","sequence":"additional","affiliation":[{"name":"Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, 76 128 Karlsruhe, Germany"}]},{"given":"MARTIN","family":"RÖTTELER","sequence":"additional","affiliation":[{"name":"Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1, Canada"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2004,3]]},"URL":"http://dx.doi.org/10.1142/S0219749904000079","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749904000079\narxivid:quant-ph/0312164"},{"id":"arxiv:0906.2509","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.82.052316","source":"Crossref","title":"Construction of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mrow/><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>quantum codes for odd prime power<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>","volume":"82","author":[{"given":"Ruihu","family":"Li","sequence":"first","affiliation":[]},{"given":"Zongben","family":"Xu","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,11,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.82.052316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052316","note":"arxivid:0906.2509"},{"id":"arxiv:1507.08355","type":"article-journal","author":[{"given":"Xianmang","family":"He"},{"given":"Liqing","family":"Xu"},{"given":"Hao","family":"Chen"}],"title":"New $q$-ary Quantum MDS Codes with Distances Bigger than $\\frac{q}{2}$","issued":{"date-parts":[[2015,10,7]]},"note":"arxivid:1507.08355\narxiv_version_number:2"},{"id":"arxiv:1803.07927","type":"article-journal","author":[{"given":"Liangdong","family":"Lu"},{"given":"Wenping","family":"Ma"},{"given":"Ruihu","family":"Li"},{"given":"Yuena","family":"Ma"},{"given":"Luobin","family":"Guo"}],"title":"New Quantum MDS codes constructed from Constacyclic codes","issued":{"date-parts":[[2018,3,20]]},"note":"arxivid:1803.07927\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.2011.2159039","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2011.2159039","page":"5551-5554","source":"Crossref","title":"New Quantum MDS Codes","volume":"57","author":[{"given":"Giuliano G.","family":"La Guardia","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2011,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2011.2159039","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2014.2308180","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2014.2308180","page":"2080-2086","source":"Crossref","title":"Constacyclic Codes and Some New Quantum MDS Codes","volume":"60","author":[{"given":"Xiaoshan","family":"Kai","sequence":"first","affiliation":[]},{"given":"Shixin","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Ping","family":"Li","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2308180","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2015.2388576","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2015.2388576","page":"1474-1484","source":"Crossref","title":"Application of Constacyclic Codes to Quantum MDS Codes","volume":"61","author":[{"given":"Bocong","family":"Chen","sequence":"first","affiliation":[]},{"given":"San","family":"Ling","sequence":"additional","affiliation":[]},{"given":"Guanghui","family":"Zhang","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2015,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2015.2388576","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2012.2220519","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2012.2220519","page":"1193-1197","source":"Crossref","title":"New Quantum MDS Codes From Negacyclic Codes","volume":"59","author":[{"given":"Xiaoshan","family":"Kai","sequence":"first","affiliation":[]},{"given":"Shixin","family":"Zhu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2012.2220519","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2312.06504","type":"article-journal","author":[{"given":"Reza","family":"Dastbasteh"},{"given":"Josu Etxezarreta","family":"Martinez"},{"given":"Andrew","family":"Nemec"},{"given":"Antonio deMarti","family":"iOlius"},{"given":"Pedro Crespo","family":"Bofill"}],"title":"An infinite class of quantum codes derived from duadic constacyclic codes","issued":{"date-parts":[[2024,5,27]]},"note":"arxivid:2312.06504\narxiv_version_number:2"},{"id":"preset:HKSconstacyclicrings","type":"chapter","author":[{"family":"Dinh","given":"H.Q."},{"family":"López-Permouth","given":"S.R."}],"title":"Constacyclic Codes over Finite Commutative Chain Rings","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"doi:10.1007/s10623-016-0207-6","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-016-0207-6","page":"115-143","source":"Crossref","title":"On linear codes admitting large automorphism groups","volume":"83","author":[{"given":"Nicola","family":"Pace","sequence":"first","affiliation":[]},{"given":"Angelo","family":"Sonnino","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2016,4,23]]},"URL":"http://dx.doi.org/10.1007/s10623-016-0207-6","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:207"},{"id":"doi:10.1109/18.904536","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.904536","page":"361-365","source":"Crossref","title":"Concatenated codes with fixed inner code and random outer code","volume":"47","author":[{"given":"A.","family":"Barg","sequence":"first","affiliation":[]},{"given":"J.","family":"Justesen","sequence":"additional","affiliation":[]},{"given":"C.","family":"Thommesen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.904536","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2405.08584","type":"article-journal","author":[{"given":"Dean","family":"Doron"},{"given":"Jonathan","family":"Mosheiff"},{"given":"Mary","family":"Wootters"}],"title":"When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?","issued":{"date-parts":[[2024,7,10]]},"note":"arxivid:2405.08584\narxiv_version_number:3"},{"id":"arxiv:0905.0428","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2009.5205592","page":"953-957","source":"Crossref","title":"Generalized concatenation for quantum codes","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"event":"2009 IEEE International Symposium on Information Theory - ISIT","container-title":"2009 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2009,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2009.5205592","note":"arxivid:0905.0428"},{"id":"doi:10.1109/TIT.1983.1056765","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1983.1056765","page":"850-853","source":"Crossref","title":"The existence of binary linear concatenated codes with Reed - Solomon outer codes which asymptotically meet the Gilbert- Varshamov bound","volume":"29","author":[{"given":"C.","family":"Thommesen","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1983,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1983.1056765","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-V.-A.-Zinoviev-Generalized-C","type":"article-journal","author":[{"family":"Zinoviev","given":"V.A."}],"title":"Generalized Cascade Codes","volume":"12","container-title":"Problemy Peredachi Informatsii","issue":"1","issued":"1976","page":"5–15"},{"id":"manual:-E-.-L.-Blokh-and-V.-V.-Zyabl","type":"article-journal","author":[{"family":"Blokh","given":"È.L."},{"family":"Zyablov","given":"V.V."}],"title":"Coding of Generalized Concatenated Codes","volume":"10","container-title":"Problemy Peredachi Informatsii","issue":"3","issued":"1974","page":"45–50"},{"id":"doi:10.1002/ett.4460100606","type":"journal-article","publisher":"Wiley","issue":"6","abstract":"<jats:title>Abstract</jats:title><jats:p>This paper presents an introduction to the theory of generalized concatenated codes. We consider the encoding and decoding procedures of both ordinary concatenated codes, and generalized concatenated codes where block codes are used as inner codes. We give the main parameters of these codes, and by means of examples, show that under the same conditions the latter outperforms the former with respect to minimum code distance. We also heuristically describe the generalized concatenated decoding procedure. Then we show how the generalized concatenated coding ideas can be applied to encoded memoryless modulation. Such construction is often referred to as a multilevel code in literature. We consider the multilevel codes from the standpoint of generalized concatenated codes and with the help of a simple example, show how encoding and decoding procedures can be carried out. Ordinary concatenated and generalized concatenated coding schemes are considered next, using inner convolutional codes or inner modulation with memory. With the help of examples, we analyze the distance properties of such constructions.</jats:p>","DOI":"10.1002/ett.4460100606","page":"609-622","source":"Crossref","title":"An Introduction to Generalized Concatenated Codes","volume":"10","author":[{"given":"Victor","family":"Zyablov","sequence":"first","affiliation":[]},{"given":"Sergo","family":"Shavgulidze","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Bossert","sequence":"additional","affiliation":[]}],"container-title":"European Transactions on Telecommunications","original-title":[],"language":"en","issued":{"date-parts":[[1999,11]]},"URL":"http://dx.doi.org/10.1002/ett.4460100606","ISSN":["1124-318X","1541-8251"],"container-title-short":"Trans Emerging Tel Tech","note":"alternative-id:10.1002/ett.4460100606"},{"id":"manual:-I.-I.-Dumer-Concatenated-cod","type":"chapter","author":[{"family":"Dumer","given":"I.I."}],"title":"Concatenated codes and their multilevel generalizations","container-title":"\\emph{Handbook of Coding Theory}","volume":"II, Part 3","editor":[{"family":"V. S. Pless","particle":"eds."},{"family":"Huffman","given":"W.C."}],"publisher":"Elsevier","issued":"1998","page":"1911–1988"},{"id":"arxiv:1802.07010","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2019.2896110","page":"4023-4040","source":"Crossref","title":"Capacity-Achieving Guessing Random Additive Noise Decoding","volume":"65","author":[{"ORCID":"https://orcid.org/0000-0001-5587-9356","authenticated-orcid":false,"given":"Ken R.","family":"Duffy","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5201-4338","authenticated-orcid":false,"given":"Jiange","family":"Li","sequence":"additional","affiliation":[]},{"given":"Muriel","family":"Medard","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2019,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2019.2896110","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1802.07010"},{"id":"doi:10.1109/ISIT.2018.8437648","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2018.8437648","page":"671-675","source":"Crossref","title":"Guessing noise, not code-words","author":[{"given":"Ken R.","family":"Duffy","sequence":"first","affiliation":[]},{"given":"Jiange","family":"Li","sequence":"additional","affiliation":[]},{"given":"Muriel","family":"Medard","sequence":"additional","affiliation":[]}],"event":"2018 IEEE International Symposium on Information Theory (ISIT)","container-title":"2018 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2018,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2018.8437648"},{"id":"arxiv:2303.14448","type":"article-journal","author":[{"given":"Trevor","family":"McCourt"},{"given":"Ila R.","family":"Fiete"},{"given":"Isaac L.","family":"Chuang"}],"title":"Noisy dynamical systems evolve error correcting codes and modularity","issued":{"date-parts":[[2023,4,12]]},"note":"arxivid:2303.14448\narxiv_version_number:2"},{"id":"doi:10.1109/TIT.2002.800480","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2002.800480","page":"2568-2573","source":"Crossref","title":"Random codes: minimum distances and error exponents","volume":"48","author":[{"given":"A.","family":"Barg","sequence":"first","affiliation":[]},{"given":"G.D.","family":"Forney","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2002,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.800480","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/978-3-642-22792-9_42","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-22792-9_42","page":"743-760","source":"Crossref","title":"Smaller Decoding Exponents: Ball-Collision Decoding","author":[{"given":"Daniel J.","family":"Bernstein","sequence":"first","affiliation":[]},{"given":"Tanja","family":"Lange","sequence":"additional","affiliation":[]},{"given":"Christiane","family":"Peters","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2011]]},"ISBN":["9783642227912","9783642227929"],"URL":"http://dx.doi.org/10.1007/978-3-642-22792-9_42","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1007/978-3-642-29011-4_31","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-29011-4_31","page":"520-536","source":"Crossref","title":"Decoding Random Binary Linear Codes in 2 n/20: How 1 + 1 = 0 Improves Information Set Decoding","author":[{"given":"Anja","family":"Becker","sequence":"first","affiliation":[]},{"given":"Antoine","family":"Joux","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"May","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Meurer","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2012]]},"ISBN":["9783642290107","9783642290114"],"URL":"http://dx.doi.org/10.1007/978-3-642-29011-4_31","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1007/978-3-642-10366-7_6","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-10366-7_6","page":"88-105","source":"Crossref","title":"Security Bounds for the Design of Code-Based Cryptosystems","author":[{"given":"Matthieu","family":"Finiasz","sequence":"first","affiliation":[]},{"given":"Nicolas","family":"Sendrier","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2009]]},"ISBN":["9783642103650","9783642103667"],"URL":"http://dx.doi.org/10.1007/978-3-642-10366-7_6","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/TPDS.2008.251","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tpds.2008.251","page":"1653-1667","source":"Crossref","title":"Priority Random Linear Codes in Distributed Storage Systems","volume":"20","author":[{"family":"Yunfeng Lin","sequence":"first","affiliation":[]},{"family":"Ben Liang","sequence":"additional","affiliation":[]},{"family":"Baochun Li","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Parallel and Distributed Systems","original-title":[],"issued":{"date-parts":[[2009,11]]},"URL":"http://dx.doi.org/10.1109/TPDS.2008.251","ISSN":["1045-9219"],"container-title-short":"IEEE Trans. Parallel Distrib. Syst."},{"id":"doi:10.1145/1568318.1568324","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"6","abstract":"<jats:p>\n            Our main result is a reduction from worst-case lattice problems such as GapSVP and SIVP to a certain learning problem. This learning problem is a natural extension of the “learning from parity with error” problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quantum. Hence, an efficient solution to the learning problem implies a\n            <jats:italic>quantum</jats:italic>\n            algorithm for GapSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., nonquantum).\n          </jats:p>\n          <jats:p>\n            We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GapSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ) and encrypting a message increases its size by a factor of Õ(\n            <jats:italic>n</jats:italic>\n            ) (in previous cryptosystems these values are Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>4</jats:sup>\n            ) and Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ), respectively). In fact, under the assumption that all parties share a random bit string of length Õ(\n            <jats:italic>n</jats:italic>\n            <jats:sup>2</jats:sup>\n            ), the size of the public key can be reduced to Õ(\n            <jats:italic>n</jats:italic>\n            ).\n          </jats:p>","DOI":"10.1145/1568318.1568324","page":"1-40","source":"Crossref","title":"On lattices, learning with errors, random linear codes, and cryptography","volume":"56","author":[{"given":"Oded","family":"Regev","sequence":"first","affiliation":[{"name":"Tel Aviv University, Tel Aviv, Israel"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[2009,9]]},"URL":"http://dx.doi.org/10.1145/1568318.1568324","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/1568318.1568324"},{"id":"doi:10.1007/978-3-540-72540-4_17","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-540-72540-4_17","page":"291-310","source":"Crossref","title":"Secure Computation from Random Error Correcting Codes","author":[{"given":"Hao","family":"Chen","sequence":"first","affiliation":[]},{"given":"Ronald","family":"Cramer","sequence":"additional","affiliation":[]},{"given":"Shafi","family":"Goldwasser","sequence":"additional","affiliation":[]},{"given":"Robbert","family":"de Haan","sequence":"additional","affiliation":[]},{"given":"Vinod","family":"Vaikuntanathan","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2007]]},"ISBN":["9783540725398","9783540725404"],"URL":"http://dx.doi.org/10.1007/978-3-540-72540-4_17","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:1603.05202","type":"article-journal","author":[{"given":"Henry","family":"Cohn"}],"title":"Packing, coding, and ground states","issued":{"date-parts":[[2016,3,16]]},"note":"arxivid:1603.05202\narxiv_version_number:1"},{"id":"doi:10.1109/18.841185","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.841185","page":"1056-1059","source":"Crossref","title":"On cyclic reversible self-dual additive codes with odd length over Z/sub 2//sup 2/","volume":"46","author":[{"given":"M.","family":"Ran","sequence":"first","affiliation":[]},{"given":"J.","family":"Snyders","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000,5]]},"URL":"http://dx.doi.org/10.1109/18.841185","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2501.15465","type":"article-journal","author":[{"given":"Ruihu","family":"Li"},{"given":"Yuezhen","family":"Ren"},{"given":"Chaofeng","family":"Guan"},{"given":"Yang","family":"Liu"}],"title":"Geometry of the symplectic group and optimal EAQECC codes","issued":{"date-parts":[[2025,8,12]]},"note":"arxivid:2501.15465\narxiv_version_number:2"},{"id":"doi:10.1109/18.568704","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.568704","page":"953-968","source":"Crossref","title":"New code parameters from Reed-Solomon subfield codes","volume":"43","author":[{"given":"J.","family":"Bierbrauer","sequence":"first","affiliation":[]},{"given":"Y.","family":"Edel","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1997,5]]},"URL":"http://dx.doi.org/10.1109/18.568704","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1006/ffta.1997.0188","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1006/ffta.1997.0188","page":"314-333","source":"Crossref","title":"Extending and LengtheningBCHCodes","volume":"3","author":[{"given":"Jürgen","family":"Bierbrauer","sequence":"first","affiliation":[]},{"given":"Yves","family":"Edel","sequence":"additional","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[1997,10]]},"URL":"http://dx.doi.org/10.1006/ffta.1997.0188","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S107157979790188X"},{"id":"doi:10.1002/(SICI)1520-6610(1997)5:5","type":"journal-article","publisher":"Wiley","issue":"5","DOI":"10.1002/(sici)1520-6610(1997)5:5<377::aid-jcd6>3.0.co;2-c","page":"377-389","source":"Crossref","title":"Twisted BCH-codes","volume":"5","author":[{"given":"Yves","family":"Edel","sequence":"first","affiliation":[]},{"given":"J�rgen","family":"Bierbauer","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Designs","original-title":[],"language":"en","issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1002/(SICI)1520-6610(1997)5:5<377::AID-JCD6>3.0.CO;2-C","ISSN":["1063-8539","1520-6610"],"container-title-short":"J. Combin. Designs"},{"id":"manual:-V.-D.-Goppa-Codes-Associated","type":"article-journal","author":[{"family":"Goppa","given":"V.D."}],"title":"Codes Associated with Divisors","volume":"13","container-title":"Problemy Peredachi Informatsii","issue":"1","issued":"1977","page":"33–39"},{"id":"manual:-V.-D.-Goppa-Codes-on-algebra","type":"article-journal","author":[{"family":"Goppa","given":"V.D."}],"title":"Codes on algebraic curves","volume":"259","container-title":"Doklady Akademii Nauk SSSR","issue":"6","issued":"1981","page":"1289–1290"},{"id":"manual:-V.-D.-Goppa-Algebraico-geome","type":"article-journal","author":[{"family":"Goppa","given":"V.D."}],"title":"Algebraico-geometric codes","volume":"46","container-title":"Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya","issue":"4","issued":"1982","page":"762–781"},{"id":"doi:10.1002/mana.19821090103","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1002/mana.19821090103","page":"21-28","source":"Crossref","title":"Modular curves, Shimura curves, and Goppa codes, better than Varshamov‐Gilbert bound","volume":"109","author":[{"given":"M. A.","family":"Tsfasman","sequence":"first","affiliation":[]},{"given":"S. G.","family":"Vlădutx","sequence":"additional","affiliation":[]},{"given":"Th.","family":"Zink","sequence":"additional","affiliation":[]}],"container-title":"Mathematische Nachrichten","original-title":[],"language":"en","issued":{"date-parts":[[1982,1]]},"URL":"http://dx.doi.org/10.1002/mana.19821090103","ISSN":["0025-584X","1522-2616"],"container-title-short":"Mathematische Nachrichten","note":"alternative-id:10.1002/mana.19821090103"},{"id":"doi:10.1007/BF01884295","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01884295","page":"211-222","source":"Crossref","title":"A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound","volume":"121","author":[{"given":"Arnaldo","family":"Garcia","sequence":"first","affiliation":[]},{"given":"Henning","family":"Stichtenoth","sequence":"additional","affiliation":[]}],"container-title":"Inventiones Mathematicae","original-title":[],"language":"en","issued":{"date-parts":[[1995,12]]},"URL":"http://dx.doi.org/10.1007/BF01884295","ISSN":["0020-9910","1432-1297"],"container-title-short":"Invent Math","note":"alternative-id:BF01884295"},{"id":"doi:10.1006/jnth.1996.0147","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1006/jnth.1996.0147","page":"248-273","source":"Crossref","title":"On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields","volume":"61","author":[{"given":"Arnaldo","family":"Garcia","sequence":"first","affiliation":[]},{"given":"Henning","family":"Stichtenoth","sequence":"additional","affiliation":[]}],"container-title":"Journal of Number Theory","original-title":[],"language":"en","issued":{"date-parts":[[1996,12]]},"URL":"http://dx.doi.org/10.1006/jnth.1996.0147","ISSN":["0022-314X"],"container-title-short":"Journal of Number Theory","note":"alternative-id:S0022314X9690147X"},{"id":"preset:HPAlgCodes","type":"chapter","author":[{"family":"Høholdt","given":"T."},{"family":"Lint","given":"J.H.","particle":"van"},{"family":"Pellikaan","given":"R."}],"title":"Algebraic geometry codes","container-title":"\\emph{Handbook of Coding Theory}","volume":"I, Part 1","editor":[{"family":"V. S. Pless","particle":"eds."},{"family":"Huffman","given":"W.C."}],"publisher":"Elsevier","issued":"1998","page":"871–961"},{"id":"manual:-S.-G.-Vla-dut-and-V.-G.-Drin","type":"article-journal","author":[{"family":"Vlăduţ","given":"S.G."},{"family":"Drinfeld","given":"V.G."}],"title":"Number of points of an algebraic curve","volume":"17","container-title":"Funktsional'nyi Analiz i ego Prilozheniya","issue":"1","issued":"1983","page":"68–69"},{"id":"doi:10.1007/978-94-015-6870-8","type":"book","publisher":"Springer Netherlands","DOI":"10.1007/978-94-015-6870-8","source":"Crossref","title":"Geometry and Codes","author":[{"given":"V. D.","family":"Goppa","sequence":"first","affiliation":[]}],"container-title":"Mathematics and Its Applications","original-title":[],"language":"en","issued":{"date-parts":[[1988]]},"ISBN":["9781402003028","9789401568708"],"URL":"http://dx.doi.org/10.1007/978-94-015-6870-8","ISSN":["0169-6378"],"note":"publisher-location:Dordrecht"},{"id":"arxiv:2101.06461","type":"article-journal","author":[{"given":"Lin","family":"Sok"}],"title":"On linear codes with one-dimensional Euclidean hull and their applications to EAQECCs","issued":{"date-parts":[[2021,1,16]]},"note":"arxivid:2101.06461\narxiv_version_number:1"},{"id":"doi:10.1007/978-3-319-51103-0","type":"book","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-51103-0","source":"Crossref","title":"Error-Correction Coding and Decoding","author":[{"given":"Martin","family":"Tomlinson","sequence":"first","affiliation":[]},{"given":"Cen Jung","family":"Tjhai","sequence":"additional","affiliation":[]},{"given":"Marcel A.","family":"Ambroze","sequence":"additional","affiliation":[]},{"given":"Mohammed","family":"Ahmed","sequence":"additional","affiliation":[]},{"given":"Mubarak","family":"Jibril","sequence":"additional","affiliation":[]}],"container-title":"Signals and Communication Technology","original-title":[],"issued":{"date-parts":[[2017]]},"ISBN":["9783319511023","9783319511030"],"URL":"http://dx.doi.org/10.1007/978-3-319-51103-0","ISSN":["1860-4862","1860-4870"],"note":"publisher-location:Cham"},{"id":"doi:10.1109/18.179340","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.179340","page":"37-45","source":"Crossref","title":"Decoding algebraic-geometric codes up to the designed minimum distance","volume":"39","author":[{"given":"G.-L.","family":"Feng","sequence":"first","affiliation":[]},{"given":"T.R.N.","family":"Rao","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1993]]},"URL":"http://dx.doi.org/10.1109/18.179340","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1515/9783110811056.155","type":"book-chapter","publisher":"DE GRUYTER","DOI":"10.1515/9783110811056.155","source":"Crossref","title":"The shift bound for cyclic, Reed-Muller and geometric Goppa codes","author":[{"given":"Ruud","family":"Pellikaan","sequence":"first","affiliation":[]}],"container-title":"Arithmetic, Geometry, and Coding Theory","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1515/9783110811056.155","note":"publisher-location:Berlin, New York"},{"id":"doi:10.1016/j.ffa.2006.09.006","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/j.ffa.2006.09.006","page":"665-680","source":"Crossref","title":"The order bound for general algebraic geometric codes","volume":"13","author":[{"given":"Peter","family":"Beelen","sequence":"first","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2007,7]]},"URL":"http://dx.doi.org/10.1016/j.ffa.2006.09.006","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S1071579706000724"},{"id":"doi:10.1109/18.333893","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.333893","page":"1678-1681","source":"Crossref","title":"A determination of the parameters of a large class of Goppa codes","volume":"40","author":[{"given":"T.","family":"Johnsen","sequence":"first","affiliation":[]},{"given":"S.","family":"Manshadi","sequence":"additional","affiliation":[]},{"given":"N.","family":"Monzavi","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1994]]},"URL":"http://dx.doi.org/10.1109/18.333893","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.32157","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/18.32157","page":"811-821","source":"Crossref","title":"Construction and decoding of a class of algebraic geometry codes","volume":"35","author":[{"given":"J.","family":"Justesen","sequence":"first","affiliation":[]},{"given":"K.J.","family":"Larsen","sequence":"additional","affiliation":[]},{"given":"H.E.","family":"Jensen","sequence":"additional","affiliation":[]},{"given":"A.","family":"Havemose","sequence":"additional","affiliation":[]},{"given":"T.","family":"Hoholdt","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1989,7]]},"URL":"http://dx.doi.org/10.1109/18.32157","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.165441","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.165441","page":"1663-1676","source":"Crossref","title":"Decoding geometric Goppa codes using an extra place","volume":"38","author":[{"given":"S.C.","family":"Porter","sequence":"first","affiliation":[]},{"given":"B.-Z.","family":"Shen","sequence":"additional","affiliation":[]},{"given":"R.","family":"Pellikaan","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1992]]},"URL":"http://dx.doi.org/10.1109/18.165441","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/BFb0087989","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/bfb0087989","page":"18-25","source":"Crossref","title":"Decoding Algebraic-Geometric Codes by solving a key equation","author":[{"given":"Dirk","family":"Ehrhard","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Mathematics","original-title":[],"issued":{"date-parts":[[1992]]},"ISBN":["9783540556510","9783540472674"],"URL":"http://dx.doi.org/10.1007/BFb0087989","ISSN":["0075-8434","1617-9692"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/18.45279","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.45279","page":"1228-1232","source":"Crossref","title":"On a decoding algorithm for codes on maximal curves","volume":"35","author":[{"given":"R.","family":"Pellikaan","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1989]]},"URL":"http://dx.doi.org/10.1109/18.45279","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.59945","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.59945","page":"1461-1463","source":"Crossref","title":"On the decoding of algebraic-geometric codes over F/sub q/ for q&amp;lt;or=16","volume":"36","author":[{"given":"S.","family":"Vladut","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1990]]},"URL":"http://dx.doi.org/10.1109/18.59945","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/S0747-7171(88)80033-6","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s0747-7171(88)80033-6","page":"321-337","source":"Crossref","title":"Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array","volume":"5","author":[{"given":"Shojiro","family":"Sakata","sequence":"first","affiliation":[]}],"container-title":"Journal of Symbolic Computation","original-title":[],"language":"en","issued":{"date-parts":[[1988,6]]},"URL":"http://dx.doi.org/10.1016/S0747-7171(88)80033-6","ISSN":["0747-7171"],"container-title-short":"Journal of Symbolic Computation","note":"alternative-id:S0747717188800336"},{"id":"doi:10.1016/0890-5401(90)90039-K","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0890-5401(90)90039-k","page":"207-239","source":"Crossref","title":"Extension of the Berlekamp-Massey algorithm to N dimensions","volume":"84","author":[{"given":"Shojiro","family":"Sakata","sequence":"first","affiliation":[]}],"container-title":"Information and Computation","original-title":[],"language":"en","issued":{"date-parts":[[1990,2]]},"URL":"http://dx.doi.org/10.1016/0890-5401(90)90039-K","ISSN":["0890-5401"],"container-title-short":"Information and Computation","note":"alternative-id:089054019090039K"},{"id":"doi:10.1109/18.86974","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/18.86974","page":"1200-1203","source":"Crossref","title":"Decoding binary 2-D cyclic codes by the 2-D Berlekamp-Massey algorithm","volume":"37","author":[{"given":"S.","family":"Sakata","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1991,7]]},"URL":"http://dx.doi.org/10.1109/18.86974","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.256485","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.256485","page":"743-751","source":"Crossref","title":"Achieving the designed error capacity in decoding algebraic-geometric codes","volume":"39","author":[{"given":"D.","family":"Ehrhard","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1993,5]]},"URL":"http://dx.doi.org/10.1109/18.256485","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.748993","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.748993","page":"432-437","source":"Crossref","title":"List decoding of algebraic-geometric codes","volume":"45","author":[{"given":"M.A.","family":"Shokrollahi","sequence":"first","affiliation":[]},{"given":"H.","family":"Wasserman","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999,3]]},"URL":"http://dx.doi.org/10.1109/18.748993","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.476214","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.476214","page":"1589-1614","source":"Crossref","title":"On the decoding of algebraic-geometric codes","volume":"41","author":[{"given":"T.","family":"Hoholdt","sequence":"first","affiliation":[]},{"given":"R.","family":"Pellikaan","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1995]]},"URL":"http://dx.doi.org/10.1109/18.476214","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1142/9789812794017_0002","type":"book-chapter","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/9789812794017_0002","page":"49-98","source":"Crossref","title":"The Decoding of Algebraic Geometry Codes","author":[{"given":"Peter","family":"Beelen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303S, DK 2800, Kgs. Lyngby, Denmark"}]},{"given":"Tom","family":"Høholdt","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303S, DK 2800, Kgs. Lyngby, Denmark"}]}],"container-title":"Series on Coding Theory and Cryptology","original-title":[],"issued":{"date-parts":[[2008,10]]},"URL":"http://dx.doi.org/10.1142/9789812794017_0002","ISSN":["1793-2238"],"note":"alternative-id:10.1142/9789812794017_0002\n10.1142/SCTC\n10.1142/6767"},{"id":"doi:10.1017/CBO9780511543401","type":"monograph","publisher":"Cambridge University Press","DOI":"10.1017/cbo9780511543401","source":"Crossref","title":"Algebraic Codes on Lines, Planes, and Curves","author":[{"given":"Richard E.","family":"Blahut","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2001,1,1]]},"ISBN":["9780521771948","9780511543401"],"URL":"http://dx.doi.org/10.1017/CBO9780511543401","note":"edition-number:1"},{"id":"doi:10.1109/18.79915","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.79915","page":"583-602","source":"Crossref","title":"Which linear codes are algebraic-geometric?","volume":"37","author":[{"given":"R.","family":"Pellikan","sequence":"first","affiliation":[{"name":"Dept. of Math. & Comput. Sci., Eindhoven Univ. of Technol., Netherlands"}]},{"given":"B.-Z.","family":"Shen","sequence":"additional","affiliation":[{"name":"Dept. of Math. & Comput. Sci., Eindhoven Univ. of Technol., Netherlands"}]},{"given":"G.J.M.","family":"van Wee","sequence":"additional","affiliation":[{"name":"Dept. of Math. & Comput. Sci., Eindhoven Univ. of Technol., Netherlands"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1991,5]]},"URL":"http://dx.doi.org/10.1109/18.79915","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/S1071-5797(03)00010-8","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s1071-5797(03)00010-8","page":"351-371","source":"Crossref","title":"On codes from norm–trace curves","volume":"9","author":[{"given":"Olav","family":"Geil","sequence":"first","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2003,7]]},"URL":"http://dx.doi.org/10.1016/S1071-5797(03)00010-8","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S1071579703000108"},{"id":"arxiv:2110.00769","type":"article-journal","author":[{"given":"Lin","family":"Sok"}],"title":"New families of quantum stabilizer codes from Hermitian self-orthogonal algebraic geometry codes","issued":{"date-parts":[[2021,12,12]]},"note":"arxivid:2110.00769\narxiv_version_number:2"},{"id":"doi:10.1016/j.ffa.2016.11.003","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.ffa.2016.11.003","page":"1-17","source":"Crossref","title":"Hermitian codes with automorphism group isomorphic to PGL(2,q) with q odd","volume":"44","author":[{"given":"Gábor","family":"Korchmáros","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4914-1168","authenticated-orcid":false,"given":"Pietro","family":"Speziali","sequence":"additional","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2017,3]]},"URL":"http://dx.doi.org/10.1016/j.ffa.2016.11.003","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"special_numbering:C\nalternative-id:S1071579716300703"},{"id":"doi:10.1109/TIT.1987.1057327","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1987.1057327","page":"605-609","source":"Crossref","title":"Remarks on codes from Hermitian curves (Corresp.)","volume":"33","author":[{"given":"H.","family":"Tiersma","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1987,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1987.1057327","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/BFb0087995","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/bfb0087995","page":"99-107","source":"Crossref","title":"On the true minimum distance of Hermitian codes","author":[{"given":"Kyeongcheol","family":"Yang","sequence":"first","affiliation":[]},{"given":"P. Vijay","family":"Kumar","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Mathematics","original-title":[],"issued":{"date-parts":[[1992]]},"ISBN":["9783540556510","9783540472674"],"URL":"http://dx.doi.org/10.1007/BFb0087995","ISSN":["0075-8434","1617-9692"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:2510.06603","type":"article-journal","author":[{"given":"Andi","family":"Gu"},{"given":"Stephen P.","family":"Jordan"}],"title":"Algebraic Geometry Codes and Decoded Quantum Interferometry","issued":{"date-parts":[[2025,10,8]]},"note":"arxivid:2510.06603\narxiv_version_number:1"},{"id":"doi:10.1007/BF00124208","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf00124208","page":"5-17","source":"Crossref","title":"Hermitian codes as generalized Reed-Solomon codes","volume":"2","author":[{"given":"Tomik","family":"Yaghoobian","sequence":"first","affiliation":[]},{"given":"Ian F.","family":"Blake","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1992,3]]},"URL":"http://dx.doi.org/10.1007/BF00124208","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:BF00124208"},{"id":"manual:-J.-P.-Hansen.-Group-codes-on","type":"article-journal","author":[{"family":"Hansen","given":"J.P."}],"title":"Group codes on algebraic curves. Universität zu Göttingen","container-title":"SFB Geometrie und Analysis","issued":"1987"},{"id":"doi:10.1109/TIT.1987.1057365","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1987.1057365","page":"923-925","source":"Crossref","title":"Codes on the Klein quartic, ideals, and decoding (Corresp.)","volume":"33","author":[{"given":"J.","family":"Hansen","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1987,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1987.1057365","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-A.-M.-Barg-G.-L.-Katsman-and","type":"article-journal","author":[{"family":"Barg","given":"A.M."},{"family":"Katsman","given":"G.L."},{"family":"Tsfasman","given":"M.A."}],"title":"Algebraic-Geometric Codes from Curves of Small Genus","volume":"23","container-title":"Problemy Peredachi Informatsii","issue":"1","issued":"1987","page":"42–46"},{"id":"arxiv:0905.2345","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.jalgebra.2011.09.030","page":"84-107","source":"Crossref","title":"The dual minimum distance of arbitrary-dimensional algebraic–geometric codes","volume":"350","author":[{"given":"Alain","family":"Couvreur","sequence":"first","affiliation":[]}],"container-title":"Journal of Algebra","original-title":[],"language":"en","issued":{"date-parts":[[2012,1]]},"URL":"http://dx.doi.org/10.1016/j.jalgebra.2011.09.030","ISSN":["0021-8693"],"container-title-short":"Journal of Algebra","note":"alternative-id:S0021869311005576\narxivid:0905.2345"},{"id":"doi:10.1109/18.57204","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.57204","page":"1051-1060","source":"Crossref","title":"On the decoding of algebraic-geometric codes","volume":"36","author":[{"given":"A.N.","family":"Skorobogatov","sequence":"first","affiliation":[]},{"given":"S.G.","family":"Vladut","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1990]]},"URL":"http://dx.doi.org/10.1109/18.57204","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-V.-Yu.-Krachkovskii-Decoding","type":"paper-conference","author":[{"family":"Krachkovskii","given":"V.Yu"}],"title":"Decoding of codes on algebraic curves","container-title":"Russian), Conference Odessa","issued":"1988"},{"id":"doi:10.1007/BF01810849","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01810849","page":"67-77","source":"Crossref","title":"Group codes on certain algebraic curves with many rational points","volume":"1","author":[{"given":"Johan P.","family":"Hansen","sequence":"first","affiliation":[]},{"given":"Henning","family":"Stichtenoth","sequence":"additional","affiliation":[]}],"container-title":"Applicable Algebra in Engineering, Communication and Computing","original-title":[],"language":"en","issued":{"date-parts":[[1990,3]]},"URL":"http://dx.doi.org/10.1007/BF01810849","ISSN":["0938-1279","1432-0622"],"container-title-short":"AAECC","note":"alternative-id:BF01810849"},{"id":"arxiv:0802.2349","type":"article-journal","author":[{"given":"John B.","family":"Little"}],"title":"Algebraic geometry codes from higher dimensional varieties","issued":{"date-parts":[[2008,2,16]]},"note":"arxivid:0802.2349\narxiv_version_number:1"},{"id":"arxiv:1907.13217","type":"article-journal","author":[{"given":"Delio","family":"Jaramillo"},{"given":"Maria Vaz","family":"Pinto"},{"given":"Rafael H.","family":"Villarreal"}],"title":"Evaluation codes and their basic parameters","issued":{"date-parts":[[2020,5,19]]},"note":"arxivid:1907.13217\narxiv_version_number:5"},{"id":"doi:10.1007/BF02249124","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1007/bf02249124","page":"2611-2643","source":"Crossref","title":"Linear codes and modular curves","volume":"30","author":[{"given":"S. G.","family":"Vléduts","sequence":"first","affiliation":[]},{"given":"Yu. I.","family":"Manin","sequence":"additional","affiliation":[]}],"container-title":"Journal of Soviet Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1985,9]]},"URL":"http://dx.doi.org/10.1007/BF02249124","ISSN":["0090-4104","1573-8795"],"container-title-short":"J Math Sci","note":"alternative-id:BF02249124"},{"id":"doi:10.1142/9789812794017_0004","type":"book-chapter","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/9789812794017_0004","page":"153-180","source":"Crossref","title":"Evaluation Codes from an Affine Variety Code Perspective","author":[{"given":"Olav","family":"Geil","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Aalborg University, Fr. Bajersvej 7G, 9220 Aalborg Ø, Denmark"}]}],"container-title":"Series on Coding Theory and Cryptology","original-title":[],"issued":{"date-parts":[[2008,10]]},"URL":"http://dx.doi.org/10.1142/9789812794017_0004","ISSN":["1793-2238"],"note":"alternative-id:10.1142/9789812794017_0004\n10.1142/SCTC\n10.1142/6767"},{"id":"doi:10.1007/978-3-540-76878-4","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-540-76878-4","source":"Crossref","title":"Algebraic Function Fields and Codes","author":[{"given":"Henning","family":"Stichtenoth","sequence":"first","affiliation":[]}],"container-title":"Graduate Texts in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2009]]},"ISBN":["9783540768777","9783540768784"],"URL":"http://dx.doi.org/10.1007/978-3-540-76878-4","ISSN":["0072-5285"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"manual:-G.-Lachaud-(1985).-Les-codes","type":"article-journal","author":[{"family":"Lachaud","given":"G."}],"title":"Les codes géométriques de Goppa","volume":"37","container-title":"Sem. Bourbaki","issued":"1985","page":"1984–85"},{"id":"preset:Rosenbloom97","type":"article-journal","author":[{"family":"Rosenbloom","given":"M.Yu"},{"family":"Tsfasman","given":"M.A."}],"title":"Codes for the m-Metric","volume":"33","container-title":"Problemy Peredachi Informatsii","issue":"1","issued":"1997","page":"55–63"},{"id":"manual:-R.-R.-Nielsen.-List-decoding","type":"thesis","author":[{"family":"Nielsen","given":"R.R."}],"title":"List decoding of linear block codes","genre":"PhD thesis,","publisher":"Technical University of Denmark","issued":"2001"},{"id":"doi:10.1145/2629416","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"5","abstract":"<jats:p>\n            Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been well studied, and it has widely been suspected that nontrivial locality must come at the price of low rate. A particular setting of potential interest in practice is codes of constant rate. For such codes, decoding algorithms with locality\n            <jats:italic>O</jats:italic>\n            (\n            <jats:italic>k∈</jats:italic>\n            ) were known only for codes of rate\n            <jats:italic>∈</jats:italic>\n            Ω(1/\n            <jats:italic>∈</jats:italic>\n            ), where\n            <jats:italic>k</jats:italic>\n            is the length of the message. Furthermore, for codes of rate &gt; 1/2, no nontrivial locality had been achieved.\n          </jats:p>\n          <jats:p>\n            In this article, we construct a new family of locally decodable codes that have very efficient local decoding algorithms, and at the same time have rate approaching 1. We show that for every\n            <jats:italic>∈</jats:italic>\n            &gt; 0 and\n            <jats:italic>α</jats:italic>\n            &gt; 0, for infinitely many\n            <jats:italic>k</jats:italic>\n            , there exists a code\n            <jats:italic>C</jats:italic>\n            which encodes messages of length\n            <jats:italic>k</jats:italic>\n            with rate 1 −\n            <jats:italic>α</jats:italic>\n            , and is locally decodable from a constant fraction of errors using\n            <jats:italic>O</jats:italic>\n            (\n            <jats:italic>k∈</jats:italic>\n            ) queries and time.\n          </jats:p>\n          <jats:p>These codes, which we call multiplicity codes, are based on evaluating multivariate polynomials and their derivatives. Multiplicity codes extend traditional multivariate polynomial codes; they inherit the local-decodability of these codes, and at the same time achieve better tradeoffs and flexibility in the rate and minimum distance.</jats:p>","DOI":"10.1145/2629416","page":"1-20","source":"Crossref","title":"High-rate codes with sublinear-time decoding","volume":"61","author":[{"given":"Swastik","family":"Kopparty","sequence":"first","affiliation":[{"name":"Institute for Advanced Study"}]},{"given":"Shubhangi","family":"Saraf","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology"}]},{"given":"Sergey","family":"Yekhanin","sequence":"additional","affiliation":[{"name":"Microsoft Research Silicon Valley"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,8]]},"URL":"http://dx.doi.org/10.1145/2629416","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/2629416"},{"id":"arxiv:0901.2529","type":"article-journal","author":[{"given":"Zeev","family":"Dvir"},{"given":"Swastik","family":"Kopparty"},{"given":"Shubhangi","family":"Saraf"},{"given":"Madhu","family":"Sudan"}],"title":"Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers","issued":{"date-parts":[[2009,5,13]]},"note":"arxivid:0901.2529\narxiv_version_number:2"},{"id":"arxiv:2408.15925","type":"article-journal","author":[{"given":"Yeyuan","family":"Chen"},{"given":"Zihan","family":"Zhang"}],"title":"Explicit Folded Reed-Solomon and Multiplicity Codes Achieve Relaxed Generalized Singleton Bounds","issued":{"date-parts":[[2025,4,14]]},"note":"arxivid:2408.15925\narxiv_version_number:3"},{"id":"doi:10.4086/toc.2015.v011a005","type":"journal-article","publisher":"Theory of Computing Exchange","issue":"1","DOI":"10.4086/toc.2015.v011a005","page":"149-182","source":"Crossref","title":"","volume":"11","author":[{"given":"Swastik","family":"Kopparty","sequence":"first","affiliation":[]}],"container-title":"Theory of Computing","original-title":[],"language":"en","issued":{"date-parts":[[2015]]},"URL":"http://dx.doi.org/10.4086/toc.2015.v011a005","ISSN":["1557-2862"],"container-title-short":"Theory of Comput."},{"id":"arxiv:2012.01530","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3406325.3451027","page":"1489-1501","source":"Crossref","title":"Decoding multivariate multiplicity codes on product sets","author":[{"given":"Siddharth","family":"Bhandari","sequence":"first","affiliation":[{"name":"Tata Institute of Fundamental Research, India"}]},{"given":"Prahladh","family":"Harsha","sequence":"additional","affiliation":[{"name":"Tata Institute of Fundamental Research, India"}]},{"given":"Mrinal","family":"Kumar","sequence":"additional","affiliation":[{"name":"IIT Bombay, India"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Harvard University, USA"}]}],"event":"STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing","container-title":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[2021,6,15]]},"URL":"http://dx.doi.org/10.1145/3406325.3451027","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3406325.3451027\n10.1145/3406325\narxivid:2012.01530"},{"id":"arxiv:1106.3951","type":"article-journal","author":[{"given":"Venkatesan","family":"Guruswami"},{"given":"Carol","family":"Wang"}],"title":"Optimal rate list decoding via derivative codes","issued":{"date-parts":[[2011,6,20]]},"note":"arxivid:1106.3951\narxiv_version_number:1"},{"id":"arxiv:1505.07547","type":"article-journal","author":[{"given":"Swastik","family":"Kopparty"}],"title":"Some remarks on multiplicity codes","issued":{"date-parts":[[2015,5,28]]},"note":"arxivid:1505.07547\narxiv_version_number:1"},{"id":"arxiv:2103.07930","type":"document","categories":["List Decodability","List Decoding Capacity","Polynomial Ideal Codes","Multiplicity Codes","Folded Reed-Solomon Codes","Mathematics of computing → Coding theory"],"language":"en","author":[{"family":"Bhandari","given":"Siddharth"},{"family":"Harsha","given":"Prahladh"},{"family":"Kumar","given":"Mrinal"},{"family":"Sudan","given":"Madhu"}],"contributor":[{"family":"Wootters","given":"Mary"},{"family":"Sanità","given":"Laura"}],"editor":[{"family":"Wootters","given":"Mary"},{"family":"Sanità","given":"Laura"}],"issued":{"date-parts":[[2021]]},"abstract":"In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their encoding is the residue modulo the ideals. We present an alternate way of viewing this class of codes in terms of linear operators, and show that this alternate view makes their algorithmic list-decodability amenable to analysis.\r\nOur framework leads to a new class of codes that we call affine Folded Reed-Solomon codes (which are themselves a special case of the broader class we explore). These codes are common generalizations of the well-studied Folded Reed-Solomon codes and Univariate Multiplicity codes, while also capturing the less-studied Additive Folded Reed-Solomon codes as well as a large family of codes that were not previously known/studied. \r\nMore significantly our framework also captures the algorithmic list-decodability of the constituent codes. Specifically, we present a unified view of the decoding algorithm for ideal-theoretic codes and show that the decodability reduces to the analysis of the distance of some related codes. We show that good bounds on this distance lead to capacity-achieving performance of the underlying code, providing a unifying explanation of known capacity-achieving results. In the specific case of affine Folded Reed-Solomon codes, our framework shows that they are list-decodable up to capacity (for appropriate setting of the parameters), thereby unifying the previous results for Folded Reed-Solomon, Multiplicity and Additive Folded Reed-Solomon codes.","container-title":"LIPIcs, Volume 207, APPROX/RANDOM 2021","DOI":"10.4230/LIPICS.APPROX/RANDOM.2021.56","volume":"207","number":"56","page":"56:1-56:21","page-first":"56:1","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"Ideal-Theoretic Explanation of Capacity-Achieving Decoding","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.56","note":"copyright:Creative Commons Attribution 4.0 International license\narxivid:2103.07930"},{"id":"arxiv:math/0104115","type":"article-journal","author":[{"given":"Noam D.","family":"Elkies"}],"title":"Excellent nonlinear codes from modular curves","issued":{"date-parts":[[2001,4,10]]},"note":"arxivid:math/0104115\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.2003.813559","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2003.813559","page":"1653-1657","source":"Crossref","title":"Nonlinear codes from algebraic curves improving the Tsfasman-Vladut-Zink bound","volume":"49","author":[{"family":"Chaoping Xing","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2003,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2003.813559","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:math/0308046","type":"article-journal","author":[{"given":"Noam D.","family":"Elkies"}],"title":"Still better nonlinear codes from modular curves","issued":{"date-parts":[[2003,8,5]]},"note":"arxivid:math/0308046\narxiv_version_number:1"},{"id":"doi:10.1007/978-3-0348-7865-4_18","type":"book-chapter","publisher":"Birkhäuser Basel","DOI":"10.1007/978-3-0348-7865-4_18","page":"259-275","source":"Crossref","title":"Constructive Asymptotic Codes with an Improvement on the Tsfasman-Vlăduţ-Zink and Xing Bounds","author":[{"given":"Harald","family":"Niederreiter","sequence":"first","affiliation":[]},{"given":"Ferruh","family":"Özbudak","sequence":"additional","affiliation":[]}],"container-title":"Coding, Cryptography and Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2004]]},"ISBN":["9783034896023","9783034878654"],"URL":"http://dx.doi.org/10.1007/978-3-0348-7865-4_18","note":"publisher-location:Basel"},{"id":"doi:10.1109/TIT.2005.856977","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2005.856977","page":"4044-4046","source":"Crossref","title":"Excellent Nonlinear Codes From Algebraic Function Fields","volume":"51","author":[{"given":"H.","family":"Stichtenoth","sequence":"first","affiliation":[]},{"given":"C.","family":"Xing","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2005,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.856977","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2003.819333","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2003.819333","page":"2975-2984","source":"Crossref","title":"Reed-solomon codes for correcting phased error bursts","volume":"49","author":[{"given":"V.Y.","family":"Krachkovsky","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2003,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2003.819333","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:cs/0511072","type":"article-journal","author":[{"given":"Venkatesan","family":"Guruswami"},{"given":"Atri","family":"Rudra"}],"title":"Explicit Codes Achieving List Decoding Capacity: Error-correction with Optimal Redundancy","issued":{"date-parts":[[2007,10,8]]},"note":"arxivid:cs/0511072\narxiv_version_number:2"},{"id":"manual:-A.-Rudra.-List-Decoding-and-","type":"thesis","author":[{"family":"Rudra","given":"A."}],"title":"List Decoding and Property Testing of Error Correcting Codes","genre":"PhD thesis,","publisher":"University of Washington","issued":"2007"},{"id":"doi:10.1109/SFCS.2005.29","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/sfcs.2005.29","page":"285-294","source":"Crossref","title":"Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time","author":[{"given":"F.","family":"Parvaresh","sequence":"first","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]}],"event":"46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)","container-title":"46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.2005.29"},{"id":"arxiv:1106.0436","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/ccc.2011.22","page":"77-85","source":"Crossref","title":"Linear-Algebraic List Decoding of Folded Reed-Solomon Codes","author":[{"given":"Venkatesan","family":"Guruswami","sequence":"first","affiliation":[]}],"event":"2011 IEEE Annual Conference on Computational Complexity (CCC)","container-title":"2011 IEEE 26th Annual Conference on Computational Complexity","original-title":[],"issued":{"date-parts":[[2011,6]]},"URL":"http://dx.doi.org/10.1109/CCC.2011.22","note":"arxivid:1106.0436"},{"id":"doi:10.1109/TIT.2008.2008124","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2008.2008124","page":"19-26","source":"Crossref","title":"Better Binary List Decodable Codes Via Multilevel Concatenation","volume":"55","author":[{"given":"Venkatesan","family":"Guruswami","sequence":"first","affiliation":[]},{"given":"Atri","family":"Rudra","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2009,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.2008124","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-V.-Guruswami-A.-Rudra-and-M.","type":"document","author":[{"family":"Guruswami","given":"V."},{"family":"Rudra","given":"A."},{"family":"Sudan","given":"M."}],"title":"\\emph{Essential coding theory}","note":"Draft available at \\href{https://cse.buffalo.edu/faculty/atri/courses/coding-theory/book/}{this URL} (2012"},{"id":"arxiv:2204.02063","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"3","abstract":"<jats:p>\n            We show the following hold, unconditionally unless otherwise stated, relative to a random oracle:\n            <jats:list list-type=\"simple\">\n              <jats:list-item>\n                <jats:label>—</jats:label>\n                <jats:p>\n                  There are NP\n                  <jats:italic>search</jats:italic>\n                  problems solvable by quantum polynomial-time (QPT) machines but not classical probabilistic polynomial-time (PPT) machines.\n                </jats:p>\n              </jats:list-item>\n              <jats:list-item>\n                <jats:label>—</jats:label>\n                <jats:p>There exist functions that are one-way, and even collision resistant, against classical adversaries but are easily inverted quantumly. Similar counterexamples exist for digital signatures and CPA-secure public key encryption (the latter requiring the assumption of a classically CPA-secure encryption scheme). Interestingly, the counterexample does not necessarily extend to the case of other cryptographic objects such as PRGs.</jats:p>\n              </jats:list-item>\n              <jats:list-item>\n                <jats:label>—</jats:label>\n                <jats:p>There are unconditional publicly verifiable proofs of quantumness with the minimal rounds of interaction: for uniform adversaries, the proofs are non-interactive, whereas for non-uniform adversaries the proofs are two message public coin.</jats:p>\n              </jats:list-item>\n              <jats:list-item>\n                <jats:label>—</jats:label>\n                <jats:p>Our results do not appear to contradict the Aaronson-Ambanis conjecture. Assuming this conjecture, there exist publicly verifiable certifiable randomness, again with the minimal rounds of interaction.</jats:p>\n              </jats:list-item>\n            </jats:list>\n            By replacing the random oracle with a concrete cryptographic hash function such as SHA2, we obtain plausible Minicrypt instantiations of the above results. Previous analogous results all required substantial structure, either in terms of highly structured oracles and/or algebraic assumptions in Cryptomania and beyond.\n          </jats:p>","DOI":"10.1145/3658665","page":"1-50","source":"Crossref","title":"Verifiable Quantum Advantage without Structure","volume":"71","author":[{"ORCID":"https://orcid.org/0000-0003-1712-3026","authenticated-orcid":false,"given":"Takashi","family":"Yamakawa","sequence":"first","affiliation":[{"name":"Social Informatics Laboratories, Nippon Telegraph and Telephone Corporation, Musashino-shi, Japan"}]},{"ORCID":"https://orcid.org/0000-0001-7071-6272","authenticated-orcid":false,"given":"Mark","family":"Zhandry","sequence":"additional","affiliation":[{"name":"NTT Research Inc, Sunnyvale, United States"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,11]]},"URL":"http://dx.doi.org/10.1145/3658665","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/3658665\narxivid:2204.02063"},{"id":"doi:10.1109/TIT.1968.1054109","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1968.1054109","page":"242-242","source":"Crossref","title":"Nonbinary BCH decoding (Abstr.)","volume":"14","author":[{"given":"E.","family":"Berlekamp","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1968,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1968.1054109","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2003.819332","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2003.819332","page":"2809-2825","source":"Crossref","title":"Algebraic soft-decision decoding of reed-solomon codes","volume":"49","author":[{"given":"R.","family":"Koetter","sequence":"first","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2003,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2003.819332","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-A.-Berman-A.-Dor-Y.-Shany-I.","type":"article-journal","author":[{"family":"Berman","given":"A."},{"family":"Dor","given":"A."},{"family":"Shany","given":"Y."},{"family":"Shapir","given":"I."},{"family":"Doubchak","given":"A."}],"title":"U.S","volume":null,"publisher":"U.S. Patent and Trademark Office","container-title":"Patent","issue":"11,855,658","issued":"2023","publisher-place":"Washington, DC"},{"id":"preset:McEliece78","type":"report","author":[{"family":"McEliece","given":"R.J."}],"title":"A public-key cryptosystem based on algebraic coding theory","genre":"Technical report, Jet Propulsion Lab. DSN Progress Report","issued":"1978"},{"id":"doi:10.1007/BF00173300","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf00173300","source":"Crossref","title":"McEliece public key cryptosystems using algebraic-geometric codes","volume":"8","author":[{"given":"Heeralal","family":"Janwa","sequence":"first","affiliation":[]},{"given":"Oscar","family":"Moreno","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1996,6]]},"URL":"http://dx.doi.org/10.1007/BF00173300","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:BF00173300"},{"id":"manual:-H.-Niederreiter-(1986)-Knaps","type":"article-journal","author":[{"family":"Niederreiter","given":"H."}],"title":"Knapsack-type cryptosystems and algebraic coding theory. Problems of Control and Information Theory","volume":"15","container-title":"Problemy Upravlenija I Teorii Informacii","issued":"1986","page":"159–166"},{"id":"doi:10.1515/dma.1992.2.4.439","type":"journal-article","publisher":"Walter de Gruyter GmbH","issue":"4","DOI":"10.1515/dma.1992.2.4.439","source":"Crossref","title":"On insecurity of cryptosystems based on generalized Reed-Solomon codes","volume":"2","author":[{"given":"V. M.","family":"SIDELNIKOV","sequence":"first","affiliation":[]},{"given":"S. O.","family":"SHESTAKOV","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics and Applications","original-title":[],"issued":{"date-parts":[[1992]]},"URL":"http://dx.doi.org/10.1515/dma.1992.2.4.439","ISSN":["0924-9265","1569-3929"]},{"id":"arxiv:1307.6458","type":"article-journal","author":[{"given":"Alain","family":"Couvreur"},{"given":"Philippe","family":"Gaborit"},{"given":"Valérie","family":"Gauthier-Umaña"},{"given":"Ayoub","family":"Otmani"},{"given":"Jean-Pierre","family":"Tillich"}],"title":"Distinguisher-Based Attacks on Public-Key Cryptosystems Using Reed-Solomon Codes","issued":{"date-parts":[[2014,3,28]]},"note":"arxivid:1307.6458\narxiv_version_number:2"},{"id":"arxiv:1108.2462","type":"article-journal","author":[{"given":"Marco","family":"Baldi"},{"given":"Marco","family":"Bianchi"},{"given":"Franco","family":"Chiaraluce"},{"given":"Joachim","family":"Rosenthal"},{"given":"Davide","family":"Schipani"}],"title":"Enhanced public key security for the McEliece cryptosystem","issued":{"date-parts":[[2014,5,20]]},"note":"arxivid:1108.2462\narxiv_version_number:4"},{"id":"doi:10.1109/TIT.1975.1055435","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1975.1055435","page":"575-576","source":"Crossref","title":"On subfield subcodes of modified Reed-Solomon codes (Corresp.)","volume":"21","author":[{"given":"P.","family":"Delsarte","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1975,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1975.1055435","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-R.-F.-Babindamana-and-C.-T.-","type":"article-journal","author":[{"family":"Babindamana","given":"R.F."},{"family":"Gueye","given":"C.T."}],"title":"Gabidulin codes that are Generalized Reed Solomon Codes","volume":"4","container-title":"Int. J. Algebra","issue":"3","issued":"2010","page":"119–142"},{"id":"arxiv:1502.05267","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2015.7282626","page":"1104-1108","source":"Crossref","title":"Quantum MDS codes over small fields","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Martin","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"2015 IEEE International Symposium on Information Theory (ISIT)","container-title":"2015 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2015,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2015.7282626","note":"arxivid:1502.05267"},{"id":"doi:10.1007/s10623-022-01174-5","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","DOI":"10.1007/s10623-022-01174-5","page":"1685-1694","source":"Crossref","title":"Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes","volume":"91","author":[{"ORCID":"https://orcid.org/0000-0003-4845-2084","authenticated-orcid":false,"given":"Simeon","family":"Ball","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,31]]},"URL":"http://dx.doi.org/10.1007/s10623-022-01174-5","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:1174"},{"id":"doi:10.1007/s10623-023-01294-6","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","DOI":"10.1007/s10623-023-01294-6","page":"4143-4151","source":"Crossref","title":"A class of constacyclic codes are generalized Reed–Solomon codes","volume":"91","author":[{"given":"Hongwei","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6827-9886","authenticated-orcid":false,"given":"Shengwei","family":"Liu","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2023,9,4]]},"URL":"http://dx.doi.org/10.1007/s10623-023-01294-6","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:1294"},{"id":"arxiv:2212.09935","type":"article-journal","author":[{"given":"Thiago","family":"Bergamaschi"},{"given":"Louis","family":"Golowich"},{"given":"Sam","family":"Gunn"}],"title":"Approaching the Quantum Singleton Bound with Approximate Error Correction","issued":{"date-parts":[[2022,12,20]]},"note":"arxivid:2212.09935\narxiv_version_number:1"},{"id":"manual:-L.-B.-Vries-and-K.-Odaka-198","type":"paper-conference","author":[{"family":"Vries","given":"L.B."},{"family":"Odaka","given":"K."}],"title":"CIRC-the error-correcting code for the compact disc digital audio system","container-title":"Audio Engineering Society Conference: 1st International Conference: Digital Audio","publisher":"Audio Engineering Society","issued":"1982-06"},{"id":"manual:-K.-Odaka-Y.-Sako-I.-Iwamoto-","type":"patent","author":[{"family":"Odaka","given":"K."},{"family":"Sako","given":"Y."},{"family":"Iwamoto","given":"I."},{"family":"Doi","given":"T."},{"family":"Vries","given":"L.B."}],"title":"SONY: ERROR CORRECTABLE DATA TRANSMISSION METHOD","genre":"(Patent US4413340)","issued":"1980-05-21"},{"id":"doi:10.1109/9780470546345","type":"edited-book","publisher":"IEEE","DOI":"10.1109/9780470546345","source":"Crossref","title":"Reed-Solomon Codes and Their Applications","author":[{"given":"Stephen B.","family":"Wicker","sequence":"first","affiliation":[]},{"given":"Vijay K.","family":"Bhargava","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[1999]]},"ISBN":["9780470546345","0470546344"],"URL":"http://dx.doi.org/10.1109/9780470546345"},{"id":"doi:10.1016/j.tcs.2007.02.043","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/j.tcs.2007.02.043","page":"348-360","source":"Crossref","title":"Decoding interleaved Reed–Solomon codes over noisy channels","volume":"379","author":[{"given":"Daniel","family":"Bleichenbacher","sequence":"first","affiliation":[]},{"given":"Aggelos","family":"Kiayias","sequence":"additional","affiliation":[]},{"given":"Moti","family":"Yung","sequence":"additional","affiliation":[]}],"container-title":"Theoretical Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2007,6]]},"URL":"http://dx.doi.org/10.1016/j.tcs.2007.02.043","ISSN":["0304-3975"],"container-title-short":"Theoretical Computer Science","note":"alternative-id:S0304397507001351"},{"id":"doi:10.1145/780542.780563","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/780542.780563","page":"136-142","source":"Crossref","title":"Reconstructing curves in three (and higher) dimensional space from noisy data","author":[{"given":"Don","family":"Coppersmith","sequence":"first","affiliation":[{"name":"IBM Thomas J. Watson Research Center, Yorktown Heights, New York"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]}],"event":"STOC03: The 35th Annual ACM Symposium on Theory of Computing","container-title":"Proceedings of the thirty-fifth annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2003,6,9]]},"URL":"http://dx.doi.org/10.1145/780542.780563","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/780542.780563\n10.1145/780542"},{"id":"doi:10.1016/0097-3165(87)90060-4","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(87)90060-4","page":"69-82","source":"Crossref","title":"The automorphism groups of Reed-Solomon codes","volume":"44","author":[{"given":"Arne","family":"Dür","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1987,1]]},"URL":"http://dx.doi.org/10.1016/0097-3165(87)90060-4","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316587900604"},{"id":"preset:Schurer15","type":"document","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Extended Reed–Solomon Code. From MinT—the database of optimal net, code, OA, and OOA parameters","note":"Version: 2015-09-03.","URL":"https://web.archive.org/web/20240420202309/https://mint.sbg.ac.at/desc_CReedSolomon-extended.html"},{"id":"doi:10.4171/JEMS/316","type":"journal-article","publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"3","abstract":"<jats:p>\n                    It is shown that the maximum size of a set\n                    <jats:inline-formula>\n                      <jats:tex-math>{ S}</jats:tex-math>\n                    </jats:inline-formula>\n                    of vectors of a\n                    <jats:inline-formula>\n                      <jats:tex-math>k</jats:tex-math>\n                    </jats:inline-formula>\n                    -dimensional vector space over\n                    <jats:inline-formula>\n                      <jats:tex-math>{\\mathbb F}_q</jats:tex-math>\n                    </jats:inline-formula>\n                    , with the property that every subset of size\n                    <jats:inline-formula>\n                      <jats:tex-math>k</jats:tex-math>\n                    </jats:inline-formula>\n                    is a basis, is at most\n                    <jats:inline-formula>\n                      <jats:tex-math>q+1</jats:tex-math>\n                    </jats:inline-formula>\n                    , if\n                    <jats:inline-formula>\n                      <jats:tex-math>k \\leq p</jats:tex-math>\n                    </jats:inline-formula>\n                    , and at most\n                    <jats:inline-formula>\n                      <jats:tex-math>q+k-p</jats:tex-math>\n                    </jats:inline-formula>\n                    , if\n                    <jats:inline-formula>\n                      <jats:tex-math>q \\geq k \\geq p+1 \\geq 4</jats:tex-math>\n                    </jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:tex-math>q=p^h</jats:tex-math>\n                    </jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>p</jats:tex-math>\n                    </jats:inline-formula>\n                    is prime. Moreover, for\n                    <jats:inline-formula>\n                      <jats:tex-math>k\\leq p</jats:tex-math>\n                    </jats:inline-formula>\n                    , the sets\n                    <jats:inline-formula>\n                      <jats:tex-math>S</jats:tex-math>\n                    </jats:inline-formula>\n                    of maximum size are classified, generalising Beniamino Segre's “arc is a conic'' theorem.\n                  </jats:p>\n                  <jats:p>\n                    These results have various implications. One such implication is that a\n                    <jats:inline-formula>\n                      <jats:tex-math>k\\times (p+2)</jats:tex-math>\n                    </jats:inline-formula>\n                    matrix, with\n                    <jats:inline-formula>\n                      <jats:tex-math>k \\leq p</jats:tex-math>\n                    </jats:inline-formula>\n                    and entries from\n                    <jats:inline-formula>\n                      <jats:tex-math>{\\mathbb F}_p</jats:tex-math>\n                    </jats:inline-formula>\n                    , has\n                    <jats:inline-formula>\n                      <jats:tex-math>k</jats:tex-math>\n                    </jats:inline-formula>\n                    columns which are linearly dependent. Another is that the uniform matroid of rank\n                    <jats:inline-formula>\n                      <jats:tex-math>r</jats:tex-math>\n                    </jats:inline-formula>\n                    that has a base set of size\n                    <jats:inline-formula>\n                      <jats:tex-math>n \\geq r+2</jats:tex-math>\n                    </jats:inline-formula>\n                    is representable over\n                    <jats:inline-formula>\n                      <jats:tex-math>{\\mathbb F}_p</jats:tex-math>\n                    </jats:inline-formula>\n                    if and only if\n                    <jats:inline-formula>\n                      <jats:tex-math>n \\leq p+1</jats:tex-math>\n                    </jats:inline-formula>\n                    . It also implies that the main conjecture for maximum distance separable codes is true for prime fields; that there are no maximum distance separable linear codes over\n                    <jats:inline-formula>\n                      <jats:tex-math>{\\mathbb F}_p</jats:tex-math>\n                    </jats:inline-formula>\n                    , of dimension at most\n                    <jats:inline-formula>\n                      <jats:tex-math>p</jats:tex-math>\n                    </jats:inline-formula>\n                    , longer than the longest Reed-Solomon codes. The classification implies that the longest maximum distance separable linear codes, whose dimension is bounded above by the characteristic of the field, are Reed–Solomon codes.\n                  </jats:p>","DOI":"10.4171/jems/316","page":"733-748","source":"Crossref","title":"On sets of vectors of a finite vector space in which every subset of basis size is a basis","volume":"14","author":[{"given":"Simeon","family":"Ball","sequence":"first","affiliation":[{"name":"Universitat Politécnica de Catalunya, Barcelona, Spain"}]}],"container-title":"Journal of the European Mathematical Society","original-title":[],"issued":{"date-parts":[[2012,3,7]]},"URL":"http://dx.doi.org/10.4171/JEMS/316","ISSN":["1435-9855","1435-9863"],"container-title-short":"J. Eur. Math. Soc."},{"id":"preset:Bush52","type":"article-journal","author":[{"family":"Bush","given":"K.A."}],"title":"Orthogonal Arrays of Index Unity","volume":"23","container-title":"The Annals of Mathematical Statistics","issue":"3","issued":"1952","page":"426–34"},{"id":"doi:10.1214/aoms/1177729387","type":"journal-article","publisher":"Institute of Mathematical Statistics","issue":"3","DOI":"10.1214/aoms/1177729387","page":"426-434","source":"Crossref","title":"Orthogonal Arrays of Index Unity","volume":"23","author":[{"given":"K. A.","family":"Bush","sequence":"first","affiliation":[]}],"container-title":"The Annals of Mathematical Statistics","original-title":[],"language":"en","issued":{"date-parts":[[1952,9]]},"URL":"http://dx.doi.org/10.1214/aoms/1177729387","ISSN":["0003-4851"],"container-title-short":"Ann. Math. Statist."},{"id":"doi:10.1137/0108018","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","DOI":"10.1137/0108018","page":"300-304","source":"Crossref","title":"Polynomial Codes Over Certain Finite Fields","volume":"8","author":[{"given":"I. S.","family":"Reed","sequence":"first","affiliation":[]},{"given":"G.","family":"Solomon","sequence":"additional","affiliation":[]}],"container-title":"Journal of the Society for Industrial and Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1960,6]]},"URL":"http://dx.doi.org/10.1137/0108018","ISSN":["0368-4245","2168-3484"],"container-title-short":"Journal of the Society for Industrial and Applied Mathematics","note":"alternative-id:10.1137/0108018"},{"id":"doi:10.1017/CBO9780511808968","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This 2006 book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed-Solomon codes and their derivative codes. After reviewing linear codes and finite fields, the author describes Reed-Solomon codes and various decoding algorithms. Cyclic codes are presented, as are MDS codes, graph codes, and codes in the Lee metric. Concatenated, trellis, and convolutional codes are also discussed in detail. Homework exercises introduce additional concepts such as Reed-Muller codes, and burst error correction. The end-of-chapter notes often deal with algorithmic issues, such as the time complexity of computational problems. While mathematical rigor is maintained, the text is designed to be accessible to a broad readership, including students of computer science, electrical engineering, and mathematics, from senior-undergraduate to graduate level.</jats:p>","DOI":"10.1017/cbo9780511808968","source":"Crossref","title":"Introduction to Coding Theory","author":[{"given":"Ron","family":"Roth","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2006,2,23]]},"ISBN":["9780521845045","9780511808968"],"URL":"http://dx.doi.org/10.1017/CBO9780511808968","note":"edition-number:1"},{"id":"doi:10.1109/TIT.1982.1056591","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1982.1056591","page":"869-874","source":"Crossref","title":"Bit-serial Reed - Solomon encoders","volume":"28","author":[{"given":"E.","family":"Berlekamp","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1982,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1982.1056591","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/49.1926","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/49.1926","page":"572-577","source":"Crossref","title":"A fast algorithm for the Fourier transform over finite fields and its VLSI implementation","volume":"6","author":[{"given":"Y.","family":"Wang","sequence":"first","affiliation":[]},{"given":"X.","family":"Zhu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[1988,4]]},"URL":"http://dx.doi.org/10.1109/49.1926","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1016/0097-3165(89)90020-4","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0097-3165(89)90020-4","page":"285-300","source":"Crossref","title":"On arithmetical algorithms over finite fields","volume":"50","author":[{"given":"David G","family":"Cantor","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1989,3]]},"URL":"http://dx.doi.org/10.1016/0097-3165(89)90020-4","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316589900204"},{"id":"doi:10.1017/CBO9781139856065","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.</jats:p>","DOI":"10.1017/cbo9781139856065","source":"Crossref","title":"Modern Computer Algebra","author":[{"given":"Joachim","family":"von zur Gathen","sequence":"first","affiliation":[]},{"given":"Jürgen","family":"Gerhard","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2013,4,25]]},"ISBN":["9781107039032","9781139856065"],"URL":"http://dx.doi.org/10.1017/CBO9781139856065","note":"edition-number:3"},{"id":"arxiv:cs/0405005","type":"article-journal","author":[{"given":"Venkatesan","family":"Guruswami"},{"given":"Alexander","family":"Vardy"}],"title":"Maximum-likelihood decoding of Reed-Solomon Codes is NP-hard","issued":{"date-parts":[[2004,5,4]]},"note":"arxivid:cs/0405005\narxiv_version_number:1"},{"id":"doi:10.1137/0109020","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","DOI":"10.1137/0109020","page":"207-214","source":"Crossref","title":"A Class of Error-Correcting Codes in $p^m $ Symbols","volume":"9","author":[{"given":"Daniel","family":"Gorenstein","sequence":"first","affiliation":[]},{"given":"Neal","family":"Zierler","sequence":"additional","affiliation":[]}],"container-title":"Journal of the Society for Industrial and Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1961,6]]},"URL":"http://dx.doi.org/10.1137/0109020","ISSN":["0368-4245","2168-3484"],"container-title-short":"Journal of the Society for Industrial and Applied Mathematics","note":"alternative-id:10.1137/0109020"},{"id":"manual:-E.-R.-Berlekamp-and-L.-Welch","type":"document","author":[{"family":"Berlekamp","given":"E.R."},{"family":"Welch","given":"L."},{"family":"Patent","given":"E.R.R.O.R.C.O.R.R.E.C.T.I.O.N.O.F.A.L.G.E.B.R.A.I.C.B.L.O.C.K.C.O.D.E.S.U.S."}],"volume":"Number 4,633,470","issued":"1986"},{"id":"doi:10.1016/0020-0190(92)90195-2","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/0020-0190(92)90195-2","page":"169-174","source":"Crossref","title":"Highly resilient correctors for polynomials","volume":"43","author":[{"given":"Peter","family":"Gemmell","sequence":"first","affiliation":[]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[]}],"container-title":"Information Processing Letters","original-title":[],"language":"en","issued":{"date-parts":[[1992,9]]},"URL":"http://dx.doi.org/10.1016/0020-0190(92)90195-2","ISSN":["0020-0190"],"container-title-short":"Information Processing Letters","note":"alternative-id:0020019092901952"},{"id":"arxiv:0906.3778","type":"article-journal","author":[{"given":"Dilip V.","family":"Sarwate"},{"given":"Zhiyuan","family":"Yan"}],"title":"Modified Euclidean Algorithms for Decoding Reed-Solomon Codes","issued":{"date-parts":[[2009,6,22]]},"note":"arxivid:0906.3778\narxiv_version_number:1"},{"id":"doi:10.1007/978-1-4757-3789-9_5","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-1-4757-3789-9_5","page":"55-68","source":"Crossref","title":"A New Algorithm for Decoding Reed-Solomon Codes","author":[{"given":"Shuhong","family":"Gao","sequence":"first","affiliation":[]}],"container-title":"Communications, Information and Network Security","original-title":[],"issued":{"date-parts":[[2003]]},"ISBN":["9781441953186","9781475737899"],"URL":"http://dx.doi.org/10.1007/978-1-4757-3789-9_5","note":"publisher-location:Boston, MA"},{"id":"doi:10.1109/TIT.1978.1055816","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1978.1055816","page":"100-106","source":"Crossref","title":"The fast decoding of Reed-Solomon codes using Fermat theoretic transforms and continued fractions","volume":"24","author":[{"given":"I.","family":"Reed","sequence":"first","affiliation":[]},{"given":"R.","family":"Scholtz","sequence":"additional","affiliation":[]},{"family":"Treiu-Kien Truong","sequence":"additional","affiliation":[]},{"given":"L.","family":"Welch","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1978,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1978.1055816","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1006/JCOM.1997.0439","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1006/jcom.1997.0439","page":"180-193","source":"Crossref","title":"Decoding of Reed Solomon Codes beyond the Error-Correction Bound","volume":"13","author":[{"given":"Madhu","family":"Sudan","sequence":"first","affiliation":[]}],"container-title":"Journal of Complexity","original-title":[],"language":"en","issued":{"date-parts":[[1997,3]]},"URL":"http://dx.doi.org/10.1006/JCOM.1997.0439","ISSN":["0885-064X"],"container-title-short":"Journal of Complexity","note":"alternative-id:S0885064X97904398"},{"id":"doi:10.1109/18.817522","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.817522","page":"246-257","source":"Crossref","title":"Efficient decoding of Reed-Solomon codes beyond half the minimum distance","volume":"46","author":[{"given":"R.M.","family":"Roth","sequence":"first","affiliation":[]},{"given":"G.","family":"Ruckenstein","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000]]},"URL":"http://dx.doi.org/10.1109/18.817522","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2006.878164","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2006.878164","page":"3642-3649","source":"Crossref","title":"Limits to List Decoding Reed–Solomon Codes","volume":"52","author":[{"given":"V.","family":"Guruswami","sequence":"first","affiliation":[]},{"given":"A.","family":"Rudra","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2006,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2006.878164","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2206.05256","type":"article-journal","author":[{"given":"Joshua","family":"Brakensiek"},{"given":"Sivakanth","family":"Gopi"},{"given":"Visu","family":"Makam"}],"title":"Generic Reed-Solomon Codes Achieve List-decoding Capacity","issued":{"date-parts":[[2024,8,28]]},"note":"arxivid:2206.05256\narxiv_version_number:4"},{"id":"doi:10.1109/26.79287","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/26.79287","page":"440-444","source":"Crossref","title":"Bit-level soft-decision decoding of Reed-Solomon codes","volume":"39","author":[{"given":"A.","family":"Vardy","sequence":"first","affiliation":[]},{"given":"Y.","family":"Be'ery","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[1991,3]]},"URL":"http://dx.doi.org/10.1109/26.79287","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1145/301250.301312","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/301250.301312","page":"245-254","source":"Crossref","title":"Oblivious transfer and polynomial evaluation","author":[{"given":"Moni","family":"Naor","sequence":"first","affiliation":[{"name":"Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel"}]},{"given":"Benny","family":"Pinkas","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel"}]}],"event":"STOC99: ACM Symposium on Theory of Computing","container-title":"Proceedings of the thirty-first annual ACM symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[1999,5]]},"URL":"http://dx.doi.org/10.1145/301250.301312","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/301250.301312\n10.1145/301250"},{"id":"doi:10.1109/92.953498","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/92.953498","page":"641-655","source":"Crossref","title":"High-speed architectures for Reed-Solomon decoders","volume":"9","author":[{"given":"D.V.","family":"Sarwate","sequence":"first","affiliation":[]},{"given":"N.R.","family":"Shanbhag","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Very Large Scale Integration (VLSI) Systems","original-title":[],"issued":{"date-parts":[[2001,10]]},"URL":"http://dx.doi.org/10.1109/92.953498","ISSN":["1063-8210","1557-9999"],"container-title-short":"IEEE Trans. VLSI Syst."},{"id":"doi:10.1109/ICC.2012.6364040","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.2012.6364040","page":"3160-3164","source":"Crossref","title":"Impulse noise detection techniques for retransmission to reduce delay in DSL systems","author":[{"given":"Dan","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Khuong","family":"Ho-Van","sequence":"additional","affiliation":[]},{"given":"Tho","family":"Le-Ngoc","sequence":"additional","affiliation":[]}],"event":"ICC 2012 - 2012 IEEE International Conference on Communications","container-title":"2012 IEEE International Conference on Communications (ICC)","original-title":[],"issued":{"date-parts":[[2012,6]]},"URL":"http://dx.doi.org/10.1109/ICC.2012.6364040"},{"id":"doi:10.1007/3-540-45465-9_21","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-45465-9_21","page":"232-243","source":"Crossref","title":"Cryptographic Hardness Based on the Decoding of Reed-Solomon Codes","author":[{"given":"Aggelos","family":"Kiayias","sequence":"first","affiliation":[]},{"given":"Moti","family":"Yung","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2002]]},"ISBN":["9783540438649","9783540454656"],"URL":"http://dx.doi.org/10.1007/3-540-45465-9_21","ISSN":["0302-9743"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:1809.10361","type":"article-journal","author":[{"given":"Songze","family":"Li"},{"given":"Mingchao","family":"Yu"},{"given":"Chien-Sheng","family":"Yang"},{"given":"A. Salman","family":"Avestimehr"},{"given":"Sreeram","family":"Kannan"},{"given":"Pramod","family":"Viswanath"}],"title":"PolyShard: Coded Sharding Achieves Linearly Scaling Efficiency and Security Simultaneously","issued":{"date-parts":[[2020,1,24]]},"note":"arxivid:1809.10361\narxiv_version_number:2"},{"id":"manual:-International-Organization-f","type":"document","title":"International Organization for Standardization, Information Technology: Automatic Identification and Data Capture Techniques-QR Code 2005 Bar Code Symbology Specification","edition":"2nd","note":"IEC18004 (ISO, 2006"},{"id":"doi:10.1109/PIMRC.2010.5672091","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/pimrc.2010.5672091","page":"1021-1026","source":"Crossref","title":"Reed-Solomon coding for cooperative wireless communication","author":[{"given":"Ismail","family":"Shakeel","sequence":"first","affiliation":[]},{"given":"Mazen O.","family":"Hasna","sequence":"additional","affiliation":[]},{"given":"Adnan","family":"Abu-Dayya","sequence":"additional","affiliation":[]},{"given":"Ali","family":"Ghrayeb","sequence":"additional","affiliation":[]}],"event":"2010 IEEE 21st International Symposium on Personal, Indoor and Mobile Radio Communications - (PIMRC 2010)","container-title":"21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications","original-title":[],"issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1109/PIMRC.2010.5672091"},{"id":"doi:10.1126/sciadv.abc5961","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"37","abstract":"<jats:p>Single-stage group testing uses compressed sensing for efficient high-throughput SARS-CoV-2 testing.</jats:p>","DOI":"10.1126/sciadv.abc5961","source":"Crossref","title":"Efficient high-throughput SARS-CoV-2 testing to detect asymptomatic carriers","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-9645-3773","authenticated-orcid":true,"given":"Noam","family":"Shental","sequence":"first","affiliation":[{"name":"Department of Computer Science, The Open University of Israel, Ra’anana, Israel."}]},{"given":"Shlomia","family":"Levy","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"given":"Vered","family":"Wuvshet","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"given":"Shosh","family":"Skorniakov","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"given":"Bar","family":"Shalem","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel."}]},{"ORCID":"https://orcid.org/0000-0001-7582-0466","authenticated-orcid":true,"given":"Aner","family":"Ottolenghi","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"given":"Yariv","family":"Greenshpan","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"given":"Rachel","family":"Steinberg","sequence":"additional","affiliation":[{"name":"Soroka University Medical Center, Beer-Sheva, Israel."}]},{"ORCID":"https://orcid.org/0000-0002-9232-6635","authenticated-orcid":true,"given":"Avishay","family":"Edri","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"ORCID":"https://orcid.org/0000-0001-6226-2573","authenticated-orcid":true,"given":"Roni","family":"Gillis","sequence":"additional","affiliation":[{"name":"Goldman Medical School, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"given":"Michal","family":"Goldhirsh","sequence":"additional","affiliation":[{"name":"Goldman Medical School, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"ORCID":"https://orcid.org/0000-0003-1409-4047","authenticated-orcid":true,"given":"Khen","family":"Moscovici","sequence":"additional","affiliation":[{"name":"Goldman Medical School, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"ORCID":"https://orcid.org/0000-0002-6340-880X","authenticated-orcid":true,"given":"Sinai","family":"Sachren","sequence":"additional","affiliation":[{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"ORCID":"https://orcid.org/0000-0002-5399-7356","authenticated-orcid":true,"given":"Lilach M.","family":"Friedman","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"ORCID":"https://orcid.org/0000-0002-1230-0428","authenticated-orcid":true,"given":"Lior","family":"Nesher","sequence":"additional","affiliation":[{"name":"Soroka University Medical Center, Beer-Sheva, Israel."}]},{"given":"Yonat","family":"Shemer-Avni","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"Soroka University Medical Center, Beer-Sheva, Israel."}]},{"ORCID":"https://orcid.org/0000-0003-1540-4449","authenticated-orcid":true,"given":"Angel","family":"Porgador","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."}]},{"ORCID":"https://orcid.org/0000-0002-0561-1578","authenticated-orcid":true,"given":"Tomer","family":"Hertz","sequence":"additional","affiliation":[{"name":"Department of Microbiology and Immunology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"National Institute of Biotechnology in the Negev, Ben-Gurion University of the Negev, Beer-Sheva, Israel."},{"name":"Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, Seattle, WA, USA."}]}],"container-title":"Science Advances","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,11]]},"URL":"http://dx.doi.org/10.1126/sciadv.abc5961","ISSN":["2375-2548"],"container-title-short":"Sci. Adv.","page":"eabc5961","note":"alternative-id:10.1126/sciadv.abc5961"},{"id":"manual:-R.-Sch-rer-and-W.-Ch.-Schmid","type":"article-journal","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Reed–Solomon Code. From MinT—the database of optimal net, code, OA, and OOA parameters","container-title":"Version","issued":"2015-09-03","URL":"https://web.archive.org/web/20240228195739/http://mint.sbg.ac.at/desc_CReedSolomon.html","note":"Available at \\url{https://web.archive.org/web/20240228195739/http://mint.sbg.ac.at/desc_CReedSolomon.html}"},{"id":"arxiv:2408.08292","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8086","DOI":"10.1038/s41586-025-09527-5","page":"831-836","source":"Crossref","title":"Optimization by decoded quantum interferometry","volume":"646","author":[{"ORCID":"https://orcid.org/0000-0003-1035-1827","authenticated-orcid":false,"given":"Stephen P.","family":"Jordan","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6035-2812","authenticated-orcid":false,"given":"Noah","family":"Shutty","sequence":"additional","affiliation":[]},{"given":"Mary","family":"Wootters","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2585-2424","authenticated-orcid":false,"given":"Adam","family":"Zalcman","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Schmidhuber","sequence":"additional","affiliation":[]},{"given":"Robbie","family":"King","sequence":"additional","affiliation":[]},{"given":"Sergei V.","family":"Isakov","sequence":"additional","affiliation":[]},{"given":"Tanuj","family":"Khattar","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6979-9533","authenticated-orcid":false,"given":"Ryan","family":"Babbush","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,22]]},"URL":"http://dx.doi.org/10.1038/s41586-025-09527-5","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:9527\narxivid:2408.08292"},{"id":"arxiv:2411.12553","type":"article-journal","author":[{"given":"André","family":"Chailloux"},{"given":"Jean-Pierre","family":"Tillich"}],"title":"Quantum advantage from soft decoders","issued":{"date-parts":[[2026,3,9]]},"note":"arxivid:2411.12553\narxiv_version_number:2"},{"id":"arxiv:2510.10967","type":"article-journal","author":[{"given":"Tanuj","family":"Khattar"},{"given":"Noah","family":"Shutty"},{"given":"Craig","family":"Gidney"},{"given":"Adam","family":"Zalcman"},{"given":"Noureldin","family":"Yosri"},{"given":"Dmitri","family":"Maslov"},{"given":"Ryan","family":"Babbush"},{"given":"Stephen P.","family":"Jordan"}],"title":"Verifiable Quantum Advantage via Optimized DQI Circuits","issued":{"date-parts":[[2025,10,13]]},"note":"arxivid:2510.10967\narxiv_version_number:1"},{"id":"preset:Schurer24","type":"document","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Reed-Solomon Codes for OOAs. From MinT—the database of optimal net, code, OA, and OOA parameters","note":"Version: 2024-09-05.","URL":"https://web.archive.org/web/20240420202309/https://mint.sbg.ac.at/desc_OReedSolomon.html"},{"id":"doi:10.1137/050646445","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","DOI":"10.1137/050646445","page":"551-607","source":"Crossref","title":"Short PCPs with Polylog Query Complexity","volume":"38","author":[{"given":"Eli","family":"Ben-Sasson","sequence":"first","affiliation":[]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[2008,1]]},"URL":"http://dx.doi.org/10.1137/050646445","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/050646445"},{"id":"doi:10.1007/3-540-56686-4_51","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-56686-4_51","page":"291-303","source":"Crossref","title":"On hyperbolic cascaded Reed-Solomon codes","author":[{"given":"Keith","family":"Saints","sequence":"first","affiliation":[]},{"given":"Chris","family":"Heegard","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[1993]]},"ISBN":["9783540566861","9783540476306"],"URL":"http://dx.doi.org/10.1007/3-540-56686-4_51","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/18.476241","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.476241","page":"1678-1693","source":"Crossref","title":"Improved geometric Goppa codes. I. Basic theory","volume":"41","author":[{"family":"Gui-Liang Feng","sequence":"first","affiliation":[]},{"given":"T.R.N.","family":"Rao","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1995]]},"URL":"http://dx.doi.org/10.1109/18.476241","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-P.-Piret.-Convolutional-code","type":"book","author":[{"family":"Piret","given":"P."}],"title":"Convolutional codes: an algebraic approach","publisher":"MIT press","issued":"1988"},{"id":"doi:10.1002/0470866969","type":"edited-book","publisher":"Wiley","DOI":"10.1002/0470866969","source":"Crossref","title":"GSM, GPRS and EDGE Performance","container-title":[],"original-title":[],"language":"en","editor":[{"given":"Timo","family":"Halonen","sequence":"first","affiliation":[]},{"given":"Javier","family":"Romero","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Melero","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2003,9,15]]},"ISBN":["9780470866948","9780470866962"],"URL":"http://dx.doi.org/10.1002/0470866969","note":"edition-number:1\nalternative-id:10.1002/0470866969"},{"id":"manual:-T.-Baicheva-I.-Bouyukliev-S.","type":"article-journal","author":[{"family":"Baicheva","given":"T."},{"family":"Bouyukliev","given":"I."},{"family":"Dodunekov","given":"S."},{"family":"Willems","given":"W."}],"title":"On the [10; 5; 6] Reed-Solomon and Glynn codes","volume":"18","container-title":"Mathematica Balkanica, New Series","issued":"2004","page":"67–78,"},{"id":"doi:10.1109/tit.2008.928297","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2008.928297","page":"4354-4360","source":"Crossref","title":"New MDS or Near-MDS Self-Dual Codes","volume":"54","author":[{"given":"T. Aaron","family":"Gulliver","sequence":"first","affiliation":[]},{"given":"Jon-Lark","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Yoonjin","family":"Lee","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,9]]},"URL":"http://dx.doi.org/10.1109/tit.2008.928297","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s10623-009-9267-1","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-009-9267-1","page":"57-81","source":"Crossref","title":"On circulant self-dual codes over small fields","volume":"52","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"T. Aaron","family":"Gulliver","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2009,2,6]]},"URL":"http://dx.doi.org/10.1007/s10623-009-9267-1","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9267"},{"id":"arxiv:quant-ph/0511016","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2006.890698","page":"865-880","source":"Crossref","title":"Convolutional and Tail-Biting Quantum Error-Correcting Codes","volume":"53","author":[{"given":"G. David","family":"Forney","sequence":"first","affiliation":[]},{"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2007,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2006.890698","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0511016"},{"id":"arxiv:quant-ph/0610153","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"},{"given":"Andreas","family":"Klappenecker"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Subsystem Codes","issued":{"date-parts":[[2006,10,18]]},"note":"arxivid:quant-ph/0610153\narxiv_version_number:1"},{"id":"arxiv:1206.4728","type":"article-journal","author":[{"given":"Alain","family":"Couvreur"}],"title":"Codes and the Cartier Operator","issued":{"date-parts":[[2012,9,10]]},"note":"arxivid:1206.4728\narxiv_version_number:2"},{"id":"manual:-V.-D.-Goppa-A-new-class-of-l","type":"article-journal","author":[{"family":"Goppa","given":"V.D."}],"title":"A new class of linear error-correcting codes","volume":"6","container-title":"Problemy Peredachi Informatsii","issue":"3","issued":"1970-09","page":"24–30,"},{"id":"manual:-V.-D.-Goppa-Rational-represe","type":"article-journal","author":[{"family":"Goppa","given":"V.D."}],"title":"Rational representation of codes and (Lg) codes","volume":"7","container-title":"Problemy Peredachi Informatsii","issue":"3","issued":"1971-09","page":"41–49,"},{"id":"doi:10.1109/TIT.1973.1055088","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1973.1055088","page":"590-592","source":"Crossref","title":"Goppa codes","volume":"19","author":[{"given":"E.","family":"Berlekamp","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1973,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1973.1055088","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1975.1055350","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1975.1055350","page":"203-207","source":"Crossref","title":"The algebraic decoding of Goppa codes","volume":"21","author":[{"given":"N.","family":"Patterson","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1975,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1975.1055350","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-D.-J.-Bernstein-Understandin","type":"article-journal","author":[{"family":"Bernstein","given":"D.J."}],"title":"Understanding binary-Goppa decoding","container-title":"Cryptology ePrint Archive","issued":"2022"},{"id":"doi:10.1109/TIT.2013.2243800","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2013.2243800","page":"3269-3281","source":"Crossref","title":"On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes","volume":"59","author":[{"given":"Peter","family":"Beelen","sequence":"first","affiliation":[]},{"given":"Tom","family":"Hoholdt","sequence":"additional","affiliation":[]},{"given":"Johan S. R.","family":"Nielsen","sequence":"additional","affiliation":[]},{"given":"Yingquan","family":"Wu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2243800","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2407.15740","type":"article-journal","author":[{"given":"Hugues","family":"Randriambololona"}],"title":"The syzygy distinguisher","issued":{"date-parts":[[2025,10,16]]},"note":"arxivid:2407.15740\narxiv_version_number:6"},{"id":"doi:10.1023/A:1027351723034","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1023/a:1027351723034","page":"293-307","source":"Crossref","title":"McEliece Public Key Cryptosystems Using Algebraic-Geometric Codes","volume":"8","author":[{"given":"Heeralal","family":"Janwa","sequence":"first","affiliation":[]},{"given":"Oscar","family":"Moreno","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1996,6]]},"URL":"http://dx.doi.org/10.1023/A:1027351723034","ISSN":["0925-1022","1573-7586"],"container-title-short":"Designs, Codes and Cryptography","note":"alternative-id:116717"},{"id":"doi:10.1007/11818175_31","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/11818175_31","page":"521-536","source":"Crossref","title":"Algebraic Geometric Secret Sharing Schemes and Secure Multi-Party Computations over Small Fields","author":[{"given":"Hao","family":"Chen","sequence":"first","affiliation":[]},{"given":"Ronald","family":"Cramer","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2006]]},"ISBN":["9783540374329","9783540374336"],"URL":"http://dx.doi.org/10.1007/11818175_31","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"manual:-Y.-Ihara.-Some-remarks-on-th","type":"article-journal","author":[{"family":"Ihara","given":"Y."}],"title":"Some remarks on the number of rational points of algebraic curves over finite fields","volume":"28","container-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math","issued":"1982","page":"721–724"},{"id":"arxiv:2408.09254","type":"article-journal","author":[{"given":"Louis","family":"Golowich"},{"given":"Venkatesan","family":"Guruswami"}],"title":"Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates","issued":{"date-parts":[[2024,8,17]]},"note":"arxivid:2408.09254\narxiv_version_number:1"},{"id":"doi:10.1109/18.825832","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.825832","page":"635-641","source":"Crossref","title":"Footprints or generalized Bezout's theorem","volume":"46","author":[{"given":"O.","family":"Geil","sequence":"first","affiliation":[]},{"given":"T.","family":"Hoholdt","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000,3]]},"URL":"http://dx.doi.org/10.1109/18.825832","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/0012-365X(93)90321-J","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(93)90321-j","page":"1-17","source":"Crossref","title":"The automorphism group of Generalized Reed-Muller codes","volume":"117","author":[{"given":"Thierry","family":"Berger","sequence":"first","affiliation":[]},{"given":"Pascale","family":"Charpin","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1993,7]]},"URL":"http://dx.doi.org/10.1016/0012-365X(93)90321-J","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9390321J"},{"id":"doi:10.1109/TIT.1970.1054554","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1970.1054554","page":"760-769","source":"Crossref","title":"On cyclic codes that are invariant under the general linear group","volume":"16","author":[{"given":"P.","family":"Delsarte","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1970,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1970.1054554","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1968.1054127","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1968.1054127","page":"189-199","source":"Crossref","title":"New generalizations of the Reed-Muller codes--I: Primitive codes","volume":"14","author":[{"given":"T.","family":"Kasami","sequence":"first","affiliation":[]},{"family":"Shu Lin","sequence":"additional","affiliation":[]},{"given":"W.","family":"Peterson","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1968,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1968.1054127","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1968.1054128","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1968.1054128","page":"199-205","source":"Crossref","title":"New generalizations of the Reed-Muller codes--II: Nonprimitive codes","volume":"14","author":[{"given":"E.","family":"Weldon","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1968,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1968.1054128","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/S0019-9958(70)90214-7","type":"journal-article","publisher":"Elsevier BV","issue":"5","DOI":"10.1016/s0019-9958(70)90214-7","page":"403-442","source":"Crossref","title":"On generalized ReedMuller codes and their relatives","volume":"16","author":[{"given":"P.","family":"Delsarte","sequence":"first","affiliation":[]},{"given":"J.M.","family":"Goethals","sequence":"additional","affiliation":[]},{"given":"F.J.","family":"Mac Williams","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1970,7]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(70)90214-7","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995870902147"},{"id":"arxiv:2305.07779","type":"article-journal","author":[{"given":"Galen","family":"Reeves"},{"given":"Henry D.","family":"Pfister"}],"title":"Achieving Capacity on Non-Binary Channels with Generalized Reed-Muller Codes","issued":{"date-parts":[[2023,5,12]]},"note":"arxivid:2305.07779\narxiv_version_number:1"},{"id":"manual:-E.-F.-Assmus-Jr.-and-J.-D.-K","type":"chapter","author":[{"family":"Assmus","given":"E.F.","suffix":"Jr."},{"family":"Key","given":"J.D."}],"title":"Polynomial codes and finite geometries","container-title":"\\emph{Handbook of Coding Theory}","volume":"II, Part 2","editor":[{"family":"V. S. Pless","particle":"eds."},{"family":"Huffman","given":"W.C."}],"publisher":"Elsevier","issued":"1998","page":"1269–1343"},{"id":"doi:10.1007/PL00004226","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1007/pl00004226","page":"477-500","source":"Crossref","title":"Matrix-Product Codes over ? q","volume":"12","author":[{"given":"Tim","family":"Blackmore","sequence":"first","affiliation":[]},{"given":"Graham H.","family":"Norton","sequence":"additional","affiliation":[]}],"container-title":"Applicable Algebra in Engineering, Communication and Computing","original-title":[],"language":"en","issued":{"date-parts":[[2001,12]]},"URL":"http://dx.doi.org/10.1007/PL00004226","ISSN":["0938-1279"],"container-title-short":"AAECC","note":"alternative-id:10120477.200"},{"id":"doi:10.1007/BF00141972","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf00141972","page":"273-285","source":"Crossref","title":"Classical codes as ideals in group algebras","volume":"2","author":[{"given":"Peter","family":"Landrock","sequence":"first","affiliation":[]},{"given":"Olaf","family":"Manz","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1992,9]]},"URL":"http://dx.doi.org/10.1007/BF00141972","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:BF00141972"},{"id":"doi:10.1145/258533.258641","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/258533.258641","page":"475-484","source":"Crossref","title":"A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP","author":[{"given":"Ran","family":"Raz","sequence":"first","affiliation":[]},{"given":"Shmuel","family":"Safra","sequence":"additional","affiliation":[]}],"event":"the twenty-ninth annual ACM symposium","container-title":"Proceedings of the twenty-ninth annual ACM symposium on Theory of computing  - STOC '97","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1145/258533.258641","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"doi:10.1145/258533.258642","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/258533.258642","page":"485-495","source":"Crossref","title":"Improved low-degree testing and its applications","author":[{"given":"Sanjeev","family":"Arora","sequence":"first","affiliation":[]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[]}],"event":"the twenty-ninth annual ACM symposium","container-title":"Proceedings of the twenty-ninth annual ACM symposium on Theory of computing  - STOC '97","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1145/258533.258642","note":"publisher-location:New York, New York, USA\nproceedings-subject:Theory of computing"},{"id":"arxiv:2311.04681","type":"article-journal","author":[{"given":"Michael","family":"Chapman"},{"given":"Thomas","family":"Vidick"},{"given":"Henry","family":"Yuen"}],"title":"Efficiently stable presentations from error-correcting codes","issued":{"date-parts":[[2023,11,8]]},"note":"arxivid:2311.04681\narxiv_version_number:1"},{"id":"arxiv:1204.5467","type":"article-journal","author":[{"given":"Noga","family":"Ron-Zewi"},{"given":"Madhu","family":"Sudan"}],"title":"A new upper bound on the query complexity for testing generalized Reed-Muller codes","issued":{"date-parts":[[2012,4,24]]},"note":"arxivid:1204.5467\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0604102","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"},{"given":"Andreas","family":"Klappenecker"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"On Quantum and Classical BCH Codes","issued":{"date-parts":[[2006,4,14]]},"note":"arxivid:quant-ph/0604102\narxiv_version_number:1"},{"id":"doi:10.1201/9781584889007-18","type":"book-chapter","publisher":"Chapman and Hall/CRC","DOI":"10.1201/9781584889007-18","page":"305-326","source":"Crossref","title":"Nonbinary Stabilizer Codes","container-title":"Mathematics of Quantum Computation and Quantum Technology","original-title":[],"language":"en","issued":{"date-parts":[[2007,9,19]]},"ISBN":["9780429143410"],"URL":"http://dx.doi.org/10.1201/9781584889007-18","note":"edition-number:0"},{"id":"arxiv:quant-ph/0502001","type":"article-journal","author":[{"given":"Pradeep Kiran","family":"Sarvepalli"},{"given":"Andreas","family":"Klappenecker"}],"title":"Nonbinary Quantum Reed-Muller Codes","issued":{"date-parts":[[2005,1,31]]},"note":"arxivid:quant-ph/0502001\narxiv_version_number:1"},{"id":"arxiv:0712.0103","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2011.2165812","page":"7163-7179","source":"Crossref","title":"A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices","volume":"57","author":[{"given":"Ching-Yi","family":"Lai","sequence":"first","affiliation":[]},{"given":"Chung-Chin","family":"Lu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2011,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2011.2165812","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0712.0103"},{"id":"doi:10.1515/9783110811056.77","type":"book-chapter","publisher":"DE GRUYTER","DOI":"10.1515/9783110811056.77","source":"Crossref","title":"Number of points of plane sections and linear codes defined on algebraic varieties","author":[{"given":"Gilles","family":"Lachaud","sequence":"first","affiliation":[]}],"container-title":"Arithmetic, Geometry, and Coding Theory","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1515/9783110811056.77","note":"publisher-location:Berlin, New York"},{"id":"doi:10.1109/18.104317","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.104317","page":"1567-1576","source":"Crossref","title":"Projective Reed-Muller codes","volume":"37","author":[{"given":"A.B.","family":"Sorensen","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1991]]},"URL":"http://dx.doi.org/10.1109/18.104317","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2406.04757","type":"article-journal","author":[{"given":"Nathan","family":"Kaplan"},{"given":"Jon-Lark","family":"Kim"}],"title":"Hulls of Projective Reed-Muller Codes","issued":{"date-parts":[[2024,6,7]]},"note":"arxivid:2406.04757\narxiv_version_number:1"},{"id":"doi:10.1016/0012-365X(90)90155-B","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0012-365x(90)90155-b","page":"217-221","source":"Crossref","title":"The parameters of projective Reed–Müller codes","volume":"81","author":[{"given":"Gilles","family":"Lachaud","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1990,4]]},"URL":"http://dx.doi.org/10.1016/0012-365X(90)90155-B","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9090155B"},{"id":"doi:10.1007/s002000000047","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1007/s002000000047","page":"455-462","source":"Crossref","title":"Reed-Muller Codes on Complete Intersections","volume":"11","author":[{"given":"I. M.","family":"Duursma","sequence":"first","affiliation":[]},{"given":"C.","family":"Rentería","sequence":"additional","affiliation":[]},{"given":"H.","family":"Tapia-Recillas","sequence":"additional","affiliation":[]}],"container-title":"Applicable Algebra in Engineering, Communication and Computing","original-title":[],"issued":{"date-parts":[[2001,5,1]]},"URL":"http://dx.doi.org/10.1007/s002000000047","ISSN":["0938-1279","1432-0622"],"container-title-short":"Applicable Algebra in Engineering, Communication and Computing","note":"alternative-id:23BCW79F3H0FP38J"},{"id":"doi:10.1016/j.jpaa.2004.08.015","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.jpaa.2004.08.015","page":"91-99","source":"Crossref","title":"Cayley–Bacharach and evaluation codes on complete intersections","volume":"196","author":[{"given":"Leah","family":"Gold","sequence":"first","affiliation":[]},{"given":"John","family":"Little","sequence":"additional","affiliation":[]},{"given":"Hal","family":"Schenck","sequence":"additional","affiliation":[]}],"container-title":"Journal of Pure and Applied Algebra","original-title":[],"language":"en","issued":{"date-parts":[[2005,3]]},"URL":"http://dx.doi.org/10.1016/j.jpaa.2004.08.015","ISSN":["0022-4049"],"container-title-short":"Journal of Pure and Applied Algebra","note":"alternative-id:S0022404904001628"},{"id":"doi:10.1006/ffta.2001.0313","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1006/ffta.2001.0313","page":"530-552","source":"Crossref","title":"Error-Correcting Codes from Higher-Dimensional Varieties","volume":"7","author":[{"given":"Søren Have","family":"Hansen","sequence":"first","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2001,10]]},"URL":"http://dx.doi.org/10.1006/ffta.2001.0313","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S1071579701903132"},{"id":"manual:-S.-H.-Hansen-The-geometry-of","type":"thesis","author":[{"family":"Hansen","given":"S.H."}],"title":"The geometry of Deligne-Lusztig varieties: Higher dimensional AG codes","genre":"PhD thesis,","publisher":"University of Aarhus","issued":"1999"},{"id":"doi:10.1007/s002290050146","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s002290050146","page":"363-375","source":"Crossref","title":"Canonical bundles of Deligne-Lusztig varieties","volume":"98","author":[{"given":"S&#x000F8;ren Have","family":"Hansen","sequence":"first","affiliation":[]}],"container-title":"manuscripta mathematica","original-title":[],"issued":{"date-parts":[[1999,3,1]]},"URL":"http://dx.doi.org/10.1007/s002290050146","ISSN":["0025-2611","1432-1785"],"container-title-short":"manuscripta mathematica","note":"alternative-id:CFEXPJLRW836FT50"},{"id":"doi:10.1007/BFb0087993","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/bfb0087993","page":"63-81","source":"Crossref","title":"Deligne-Lusztig varieties and group codes","author":[{"given":"Johan P.","family":"Hansen","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Mathematics","original-title":[],"issued":{"date-parts":[[1992]]},"ISBN":["9783540556510","9783540472674"],"URL":"http://dx.doi.org/10.1007/BFb0087993","ISSN":["0075-8434","1617-9692"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1016/S0022-4049(02)00188-3","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0022-4049(02)00188-3","page":"203-214","source":"Crossref","title":"Codes from flag varieties over a finite field","volume":"178","author":[{"given":"F.","family":"Rodier","sequence":"first","affiliation":[]}],"container-title":"Journal of Pure and Applied Algebra","original-title":[],"language":"en","issued":{"date-parts":[[2003,3]]},"URL":"http://dx.doi.org/10.1016/S0022-4049(02)00188-3","ISSN":["0022-4049"],"container-title-short":"Journal of Pure and Applied Algebra","note":"alternative-id:S0022404902001883"},{"id":"manual:-C.-T.-Ryan-An-application-of","type":"article-journal","author":[{"family":"Ryan","given":"C.T."}],"title":"An application of Grassmannian varieties to coding theory","volume":"57","container-title":"Congr. Numer","issued":"1987","page":"257–271"},{"id":"manual:-C.-T.-Ryan-Projective-codes-","type":"article-journal","author":[{"family":"Ryan","given":"C.T."}],"title":"Projective codes based on Grassmann varieties, Congr","volume":"57","container-title":"Numer","issued":"1987","page":"273–279"},{"id":"doi:10.1016/0166-218X(90)90112-P","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0166-218x(90)90112-p","page":"149-156","source":"Crossref","title":"The minimum weight of the Grassmann codes C(k,n),","volume":"28","author":[{"given":"Charles T.","family":"Ryan","sequence":"first","affiliation":[]},{"given":"Kevin M.","family":"Ryan","sequence":"additional","affiliation":[]}],"container-title":"Discrete Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1990,8]]},"URL":"http://dx.doi.org/10.1016/0166-218X(90)90112-P","ISSN":["0166-218X"],"container-title-short":"Discrete Applied Mathematics","note":"alternative-id:0166218X9090112P"},{"id":"doi:10.1007/978-1-4613-8994-1_4","type":"book-chapter","publisher":"Springer New York","DOI":"10.1007/978-1-4613-8994-1_4","page":"35-50","source":"Crossref","title":"Families of Codes with Few Distinct Weights from Singular and Non-Singular Hermitian Varieties and Quadrics in Projective Geometries and Hadamard Difference Sets and Designs Associated with Two-Weight Codes","author":[{"given":"I. M.","family":"Chakravarti","sequence":"first","affiliation":[]}],"container-title":"The IMA Volumes in Mathematics and Its Applications","original-title":[],"language":"en","issued":{"date-parts":[[1990]]},"ISBN":["9781461389965","9781461389941"],"URL":"http://dx.doi.org/10.1007/978-1-4613-8994-1_4","ISSN":["0940-6573"],"note":"publisher-location:New York, NY"},{"id":"doi:10.1016/0012-365X(75)90017-5","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0012-365x(75)90017-5","page":"185-211","source":"Crossref","title":"Codes projectifs a deux ou trois poids associfs aux hyperquadriques d'une geometrie finie","volume":"13","author":[{"given":"J.","family":"Wolfmann","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"fr","issued":{"date-parts":[[1975]]},"URL":"http://dx.doi.org/10.1016/0012-365X(75)90017-5","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X75900175"},{"id":"doi:10.1007/BFb0087988","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/bfb0087988","page":"4-17","source":"Crossref","title":"Reed-Muller codes associated to projective algebraic varieties","author":[{"given":"Yves","family":"Aubry","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Mathematics","original-title":[],"issued":{"date-parts":[[1992]]},"ISBN":["9783540556510","9783540472674"],"URL":"http://dx.doi.org/10.1007/BFb0087988","ISSN":["0075-8434","1617-9692"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:math/0309123","type":"article-journal","author":[{"given":"Chris","family":"Lomont"}],"title":"Error Correcting Codes on Algebraic Surfaces","issued":{"date-parts":[[2003,9,7]]},"note":"arxivid:math/0309123\narxiv_version_number:1"},{"id":"doi:10.1007/978-3-642-57189-3_11","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-57189-3_11","page":"122-131","source":"Crossref","title":"Higher Weights of Grassmann Codes","author":[{"given":"Sudhir R.","family":"Ghorpade","sequence":"first","affiliation":[]},{"given":"Gilles","family":"Lachaud","sequence":"additional","affiliation":[]}],"container-title":"Coding Theory, Cryptography and Related Areas","original-title":[],"language":"en","issued":{"date-parts":[[2000]]},"ISBN":["9783540662488","9783642571893"],"URL":"http://dx.doi.org/10.1007/978-3-642-57189-3_11","note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/18.850689","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/18.850689","page":"1535-1538","source":"Crossref","title":"On the minimum distances of Schubert codes","volume":"46","author":[{"family":"Hao Chen","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000,7]]},"URL":"http://dx.doi.org/10.1109/18.850689","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1023/B:DESI.0000035470.05639.","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1023/b:desi.0000035470.05639.2b","page":"173-180","source":"Crossref","title":"On the Linear Codes Arising from Schubert Varieties","volume":"33","author":[{"given":"Lucio","family":"Guerra","sequence":"first","affiliation":[]},{"given":"Rita","family":"Vincenti","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2004,9]]},"URL":"http://dx.doi.org/10.1023/B:DESI.0000035470.05639.2b","ISSN":["0925-1022","1573-7586"],"container-title-short":"Designs, Codes and Cryptography","note":"alternative-id:5273656"},{"id":"doi:10.1016/j.ffa.2004.09.002","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/j.ffa.2004.09.002","page":"684-699","source":"Crossref","title":"Schubert varieties, linear codes and enumerative combinatorics","volume":"11","author":[{"given":"Sudhir R.","family":"Ghorpade","sequence":"first","affiliation":[]},{"given":"Michael A.","family":"Tsfasman","sequence":"additional","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2005,11]]},"URL":"http://dx.doi.org/10.1016/j.ffa.2004.09.002","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S1071579704000589"},{"id":"doi:10.1006/ffta.2002.0360","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1006/ffta.2002.0360","page":"511-518","source":"Crossref","title":"Reed-Muller-Type Codes Over the Segre Variety","volume":"8","author":[{"given":"M.","family":"González-Sarabia","sequence":"first","affiliation":[]},{"given":"C.","family":"Renterı́a","sequence":"additional","affiliation":[]},{"given":"H.","family":"Tapia-Recillas","sequence":"additional","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2002,10]]},"URL":"http://dx.doi.org/10.1006/ffta.2002.0360","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S1071579702903606"},{"id":"doi:10.1007/978-3-642-57189-3_12","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-57189-3_12","page":"132-142","source":"Crossref","title":"Toric Surfaces and Error-correcting Codes","author":[{"given":"Johan P.","family":"Hansen","sequence":"first","affiliation":[]}],"container-title":"Coding Theory, Cryptography and Related Areas","original-title":[],"language":"en","issued":{"date-parts":[[2000]]},"ISBN":["9783540662488","9783642571893"],"URL":"http://dx.doi.org/10.1007/978-3-642-57189-3_12","note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:math/0208155","type":"article-journal","author":[{"given":"David","family":"Joyner"}],"title":"Toric codes over finite fields","issued":{"date-parts":[[2003,7,30]]},"note":"arxivid:math/0208155\narxiv_version_number:2"},{"id":"arxiv:math/0512285","type":"article-journal","author":[{"given":"Diego","family":"Ruano"}],"title":"On the Parameters of r-dimensional Toric Codes","issued":{"date-parts":[[2005,12,13]]},"note":"arxivid:math/0512285\narxiv_version_number:1"},{"id":"arxiv:math/0507598","type":"article-journal","author":[{"given":"John","family":"Little"},{"given":"Hal","family":"Schenck"}],"title":"Toric surface codes and Minkowski sums","issued":{"date-parts":[[2006,6,19]]},"note":"arxivid:math/0507598\narxiv_version_number:2"},{"id":"arxiv:1009.4966","type":"article-journal","author":[{"given":"Eliseo","family":"Sarmiento"},{"given":"Maria Vaz","family":"Pinto"},{"given":"Rafael H.","family":"Villarreal"}],"title":"The minimum distance of parameterized codes on projective tori","issued":{"date-parts":[[2011,7,21]]},"note":"arxivid:1009.4966\narxiv_version_number:3"},{"id":"arxiv:1202.0085","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-012-9714-2","page":"5-19","source":"Crossref","title":"Affine cartesian codes","volume":"71","author":[{"given":"Hiram H.","family":"López","sequence":"first","affiliation":[]},{"given":"Carlos","family":"Rentería-Márquez","sequence":"additional","affiliation":[]},{"given":"Rafael H.","family":"Villarreal","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2012,6,17]]},"URL":"http://dx.doi.org/10.1007/s10623-012-9714-2","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9714\narxivid:1202.0085"},{"id":"arxiv:1706.02114","type":"article-journal","author":[{"given":"Peter","family":"Beelen"},{"given":"Mrinmoy","family":"Datta"}],"title":"Generalized Hamming weights of affine cartesian codes","issued":{"date-parts":[[2017,8,1]]},"note":"arxivid:1706.02114\narxiv_version_number:2"},{"id":"arxiv:1702.06569","type":"article-journal","author":[{"given":"Johan P.","family":"Hansen"}],"title":"Toric Codes, Multiplicative Structure and Decoding","issued":{"date-parts":[[2017,2,20]]},"note":"arxivid:1702.06569\narxiv_version_number:1"},{"id":"doi:10.1016/S0019-9958(72)80007-X","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s0019-9958(72)80007-x","page":"280-290","source":"Crossref","title":"Noncyclic generalizations of BCH and srivastava codes","volume":"21","author":[{"given":"H.J.","family":"Helgert","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1972,10]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(72)80007-X","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S001999587280007X"},{"id":"doi:10.1016/S0019-9958(74)80005-7","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/s0019-9958(74)80005-7","page":"369-380","source":"Crossref","title":"Alternant codes","volume":"26","author":[{"given":"H.J.","family":"Helgert","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1974,12]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(74)80005-7","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995874800057"},{"id":"doi:10.1016/S0019-9958(75)90099-6","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0019-9958(75)90099-6","page":"101-108","source":"Crossref","title":"Binary primitive alternant codes","volume":"27","author":[{"given":"H.J.","family":"Helgert","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1975,2]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(75)90099-6","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995875900996"},{"id":"doi:10.1109/TIT.1977.1055730","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1977.1055730","page":"513-514","source":"Crossref","title":"Decoding of alternant codes (Corresp.)","volume":"23","author":[{"given":"H.","family":"Helgert","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1977,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1977.1055730","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2022.3201239","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2022.3201239","page":"262-272","source":"Crossref","title":"Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically","volume":"69","author":[{"ORCID":"https://orcid.org/0000-0003-4466-2025","authenticated-orcid":false,"given":"Jihao","family":"Fan","sequence":"first","affiliation":[{"name":"School of Cyber Science and Engineering, Nanjing University of Science and Technology, Nanjing, China"}]},{"ORCID":"https://orcid.org/0000-0002-6239-2922","authenticated-orcid":false,"given":"Jun","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, China"}]},{"given":"Ya","family":"Wang","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Microscale Magnetic Resonance and the Department of Modern Physics, University of Science and Technology of China, Hefei, China"}]},{"ORCID":"https://orcid.org/0000-0001-7702-1123","authenticated-orcid":false,"given":"Yonghui","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electrical and Information Engineering, The University of Sydney, Sydney, NSW, Australia"}]},{"ORCID":"https://orcid.org/0000-0002-3396-8427","authenticated-orcid":false,"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[{"name":"Quantum Computing Research Center, Hon Hai Research Institute, Taipei, Taiwan"}]},{"given":"Jiangfeng","family":"Du","sequence":"additional","affiliation":[{"name":"CAS Key Laboratory of Microscale Magnetic Resonance and the Department of Modern Physics, University of Science and Technology of China, Hefei, China"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2023,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2022.3201239","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1975.1055336","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1975.1055336","page":"70-79","source":"Crossref","title":"Algebraic generalization of BCH-Goppa-Helgert codes","volume":"21","author":[{"given":"R.","family":"Chien","sequence":"first","affiliation":[]},{"given":"D.","family":"Choy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1975,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1975.1055336","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2311.16044","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit57864.2024.10619479","page":"2281-2286","source":"Crossref","title":"Robust Syndrome Extraction via BCH Encoding","author":[{"given":"Eren","family":"Guttentag","sequence":"first","affiliation":[{"name":"Duke University,Duke Quantum Center,Durham,NC,USA,27701"}]},{"given":"Andrew","family":"Nemec","sequence":"additional","affiliation":[{"name":"Duke University,Duke Quantum Center,Durham,NC,USA,27701"}]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Duke University,Duke Quantum Center,Durham,NC,USA,27701"}]}],"event":"2024 IEEE International Symposium on Information Theory (ISIT)","container-title":"2024 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2024,7,7]]},"URL":"http://dx.doi.org/10.1109/ISIT57864.2024.10619479","note":"arxivid:2311.16044"},{"id":"arxiv:1409.2559","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.90.062304","source":"Crossref","title":"Ability of stabilizer quantum error correction to protect itself from its own imperfection","volume":"90","author":[{"given":"Yuichiro","family":"Fujiwara","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.062304","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062304","note":"arxivid:1409.2559"},{"id":"doi:10.1109/TIT.2019.2913389","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2019.2913389","page":"4699-4714","source":"Crossref","title":"A Class of Narrow-Sense BCH Codes","volume":"65","author":[{"given":"Shixin","family":"Zhu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8975-1163","authenticated-orcid":false,"given":"Zhonghua","family":"Sun","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4384-1470","authenticated-orcid":false,"given":"Xiaoshan","family":"Kai","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2019,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2019.2913389","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.335966","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/18.335966","page":"1083-1096","source":"Crossref","title":"Lee-metric BCH codes and their application to constrained and partial-response channels","volume":"40","author":[{"given":"R.M.","family":"Roth","sequence":"first","affiliation":[]},{"given":"P.H.","family":"Siegel","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1994,7]]},"URL":"http://dx.doi.org/10.1109/18.335966","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1972.1054760","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1972.1054760","page":"292-297","source":"Crossref","title":"Srivastava codes","volume":"18","author":[{"given":"H.","family":"Helgert","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1972,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1972.1054760","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-P.-Elias.-Coding-for-noisy-c","type":"article-journal","author":[{"family":"Elias","given":"P."}],"title":"Coding for noisy channels","volume":"3","container-title":"IRE Convention Records","issue":"4","issued":"1955","page":"37–46"},{"id":"doi:10.1109/TIT.1979.1056004","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1979.1056004","page":"85-90","source":"Crossref","title":"Linear programming bounds for tree codes (Corresp.)","volume":"25","author":[{"given":"M.","family":"Aaltonen","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1979,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1979.1056004","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.12928/telkomnika.v10i4.171","type":"journal-article","publisher":"Universitas Ahmad Dahlan","issue":"4","DOI":"10.12928/telkomnika.v10i4.171","source":"Crossref","title":"Effects of Puncturing Patterns on Punctured Convolutional Codes","volume":"10","author":[{"given":"Lydia","family":"Sari","sequence":"first","affiliation":[]}],"container-title":"TELKOMNIKA (Telecommunication, Computing, Electronics and Control)","original-title":[],"issued":{"date-parts":[[2012,12,1]]},"URL":"http://dx.doi.org/10.12928/telkomnika.v10i4.171","ISSN":["2302-9293","1693-6930"],"container-title-short":"TELKOMNIKA"},{"id":"doi:10.1109/TIT.1967.1054010","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1967.1054010","page":"260-269","source":"Crossref","title":"Error bounds for convolutional codes and an asymptotically optimum decoding algorithm","volume":"13","author":[{"given":"A.","family":"Viterbi","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1967,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1967.1054010","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1974.1055186","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1974.1055186","page":"284-287","source":"Crossref","title":"Optimal decoding of linear codes for minimizing symbol error rate (Corresp.)","volume":"20","author":[{"given":"L.","family":"Bahl","sequence":"first","affiliation":[]},{"given":"J.","family":"Cocke","sequence":"additional","affiliation":[]},{"given":"F.","family":"Jelinek","sequence":"additional","affiliation":[]},{"given":"J.","family":"Raviv","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1974,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1974.1055186","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.5815/ijcnis.2014.07.03","type":"journal-article","publisher":"MECS Publisher","issue":"7","DOI":"10.5815/ijcnis.2014.07.03","page":"19-27","source":"Crossref","title":"Forward Error Correction Convolutional Codes for RTAs' Networks: An Overview","volume":"6","author":[{"given":"Salehe","family":"I. Mrutu","sequence":"first","affiliation":[]},{"given":"Anael","family":"Sam","sequence":"additional","affiliation":[]},{"given":"Nerey","family":"H. Mvungi","sequence":"additional","affiliation":[]}],"container-title":"International Journal of Computer Network and Information Security","original-title":[],"issued":{"date-parts":[[2014,6,8]]},"URL":"http://dx.doi.org/10.5815/ijcnis.2014.07.03","ISSN":["2074-9090","2074-9104"],"container-title-short":"IJCNIS"},{"id":"manual:-Butman-Deutsch-and-Miller.-P","type":"document","author":[{"family":"Butman","given":"Deutsch"},{"given":"Miller"}],"title":"Performance of concatenated codes for deep space missions","issued":"1981"},{"id":"doi:10.1103/PhysRevLett.70.2968","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevlett.70.2968","page":"2968-2971","source":"Crossref","title":"Finite temperature error-correcting codes","volume":"70","author":[{"given":"Pál","family":"Ruján","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1993,5,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.70.2968","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"manual:-S.-Lin-An-introduction-to-er","type":"book","author":[{"family":"Lin","given":"S."}],"title":"An introduction to error-correcting codes","publisher":"Prentice Hall","issued":"1970"},{"id":"manual:-M.-Tuchler-and-J.-Hagenauer-","type":"paper-conference","author":[{"family":"Tuchler","given":"M."},{"family":"Hagenauer","given":"J."}],"title":"EXIT charts of irregular codes","container-title":"Proc. 36th Annual Conference on Information Sciences and Systems (CDROM","issued":"2002-03","publisher-place":"Princeton, NJ, USA"},{"id":"doi:10.1109/TCOMM.2003.822697","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tcomm.2003.822697","page":"209-218","source":"Crossref","title":"Design of Serially Concatenated Systems Depending on the Block Length","volume":"52","author":[{"given":"M.","family":"Tuchler","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"language":"en","issued":{"date-parts":[[2004,2]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2003.822697","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/ICC.1995.525138","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.1995.525138","page":"54-59","source":"Crossref","title":"Turbo codes for PCS applications","volume":"1","author":[{"given":"D.","family":"Divsalar","sequence":"first","affiliation":[]},{"given":"F.","family":"Pollara","sequence":"additional","affiliation":[]}],"event":"IEEE International Conference on Communications ICC '95","container-title":"Proceedings IEEE International Conference on Communications ICC '95","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ICC.1995.525138"},{"id":"doi:10.1109/49.924867","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/49.924867","page":"831-837","source":"Crossref","title":"Interleaver design for turbo codes","volume":"19","author":[{"given":"H.R.","family":"Sadjadpour","sequence":"first","affiliation":[]},{"given":"N.J.A.","family":"Sloane","sequence":"additional","affiliation":[]},{"given":"M.","family":"Salehi","sequence":"additional","affiliation":[]},{"given":"G.","family":"Nebe","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[2001,5]]},"URL":"http://dx.doi.org/10.1109/49.924867","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1109/26.539767","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/26.539767","page":"1261-1271","source":"Crossref","title":"Near optimum error correcting coding and decoding: turbo-codes","volume":"44","author":[{"given":"C.","family":"Berrou","sequence":"first","affiliation":[]},{"given":"A.","family":"Glavieux","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[1996]]},"URL":"http://dx.doi.org/10.1109/26.539767","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/ICC.1993.397441","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.1993.397441","page":"1064-1070","source":"Crossref","title":"Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1","volume":"2","author":[{"given":"C.","family":"Berrou","sequence":"first","affiliation":[]},{"given":"A.","family":"Glavieux","sequence":"additional","affiliation":[]},{"given":"P.","family":"Thitimajshima","sequence":"additional","affiliation":[]}],"event":"ICC '93 - IEEE International Conference on Communications","container-title":"Proceedings of ICC '93 - IEEE International Conference on Communications","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ICC.1993.397441"},{"id":"doi:10.1109/ISIT.2002.1023356","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2002.1023356","page":"84","source":"Crossref","title":"On the minimum distance of turbo codes","author":[{"given":"D.","family":"Truhachev","sequence":"first","affiliation":[]},{"given":"M.","family":"Lentmaier","sequence":"additional","affiliation":[]},{"given":"O.","family":"Wintzell","sequence":"additional","affiliation":[]},{"given":"K.Sh.","family":"Zigangirov","sequence":"additional","affiliation":[]}],"event":"IEEE International Symposium on Information Theory","container-title":"Proceedings IEEE International Symposium on Information Theory,","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2002.1023356"},{"id":"doi:10.1109/TIT.2002.1003833","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2002.1003833","page":"1451-1461","source":"Crossref","title":"Coding theorems for turbo code ensembles","volume":"48","author":[{"family":"Hui Jin","sequence":"first","affiliation":[]},{"given":"R.J.","family":"McEliece","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2002,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.1003833","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/26.681398","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/26.681398","page":"717-723","source":"Crossref","title":"New performance bounds for turbo codes","volume":"46","author":[{"given":"T.M.","family":"Duman","sequence":"first","affiliation":[]},{"given":"M.","family":"Salehi","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[1998,6]]},"URL":"http://dx.doi.org/10.1109/26.681398","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/49.661103","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/49.661103","page":"140-152","source":"Crossref","title":"Turbo decoding as an instance of Pearl's \"belief propagation\" algorithm","volume":"16","author":[{"given":"R.J.","family":"McEliece","sequence":"first","affiliation":[]},{"given":"D.J.C.","family":"MacKay","sequence":"additional","affiliation":[]},{"family":"Jung-Fu Cheng","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/49.661103","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1117/12.394081","type":"proceedings-article","publisher":"SPIE","DOI":"10.1117/12.394081","page":"73-83","source":"Crossref","title":"&lt;title&gt;Maximum a posteriori decoding algorithms for turbo codes&lt;/title&gt;","volume":"4045","author":[{"given":"Hamid R.","family":"Sadjadpour","sequence":"first","affiliation":[]}],"event":"AeroSense 2000","container-title":"SPIE Proceedings","original-title":[],"editor":[{"given":"Raghuveer M.","family":"Rao","sequence":"first","affiliation":[]},{"given":"Soheil A.","family":"Dianat","sequence":"additional","affiliation":[]},{"given":"Michael D.","family":"Zoltowski","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2000,7,26]]},"URL":"http://dx.doi.org/10.1117/12.394081","ISSN":["0277-786X"]},{"id":"doi:10.1016/j.protcy.2012.10.080","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.protcy.2012.10.080","page":"666-673","source":"Crossref","title":"Soft Output Decoding Algorithm for Turbo Codes Implementation in Mobile Wi-Max Environment","volume":"6","author":[{"given":"Jagdish D.","family":"Kene","sequence":"first","affiliation":[]},{"given":"Kishor D.","family":"Kulat","sequence":"additional","affiliation":[]}],"container-title":"Procedia Technology","original-title":[],"language":"en","issued":{"date-parts":[[2012]]},"URL":"http://dx.doi.org/10.1016/j.protcy.2012.10.080","ISSN":["2212-0173"],"container-title-short":"Procedia Technology","note":"special_numbering:C\nalternative-id:S2212017312006251"},{"id":"doi:10.1109/35.642838","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/35.642838","page":"94-102","source":"Crossref","title":"A primer on turbo code concepts","volume":"35","author":[{"given":"B.","family":"Sklar","sequence":"first","affiliation":[]}],"container-title":"IEEE Communications Magazine","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1109/35.642838","ISSN":["0163-6804"],"container-title-short":"IEEE Commun. Mag."},{"id":"doi:10.1109/26.681410","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/26.681410","page":"754-762","source":"Crossref","title":"List decoding of turbo codes","volume":"46","author":[{"given":"K.R.","family":"Narayanan","sequence":"first","affiliation":[]},{"given":"G.L.","family":"Stuber","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[1998,6]]},"URL":"http://dx.doi.org/10.1109/26.681410","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/92.784098","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/92.784098","page":"369-379","source":"Crossref","title":"VLSI architectures for turbo codes","volume":"7","author":[{"given":"G.","family":"Masera","sequence":"first","affiliation":[]},{"given":"G.","family":"Piccinini","sequence":"additional","affiliation":[]},{"given":"M.R.","family":"Roch","sequence":"additional","affiliation":[]},{"given":"M.","family":"Zamboni","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Very Large Scale Integration (VLSI) Systems","original-title":[],"issued":{"date-parts":[[1999,9]]},"URL":"http://dx.doi.org/10.1109/92.784098","ISSN":["1063-8210","1557-9999"],"container-title-short":"IEEE Trans. VLSI Syst."},{"id":"arxiv:1909.12120","type":"article-journal","author":[{"given":"Eren","family":"Balevi"},{"given":"Jeffrey G.","family":"Andrews"}],"title":"Autoencoder-Based Error Correction Coding for One-Bit Quantization","issued":{"date-parts":[[2019,9,24]]},"note":"arxivid:1909.12120\narxiv_version_number:1"},{"id":"manual:-D.-Divsalar-and-F.-Pollara-T","type":"chapter","author":[{"family":"Divsalar","given":"D."},{"family":"Pollara","given":"F."}],"title":"Turbo codes for deep-space communications","container-title":"The Telecommunications and Data Acquisition Report 29-39 (SEE","issued":"1995","page":"95–21532 06–32"},{"id":"doi:10.2514/6.2002-T5-11","type":"proceedings-article","publisher":"American Institute of Aeronautics and Astronautics","DOI":"10.2514/6.2002-t5-11","source":"Crossref","title":"CCSDS Telemetry/Tele-command Standards Restructured as Communications Protocols","author":[{"given":"Takahiro","family":"Yamada","sequence":"first","affiliation":[{"name":"ISAS, Japan"}]}],"event":"SpaceOps 2002 Conference","container-title":"SpaceOps 2002 Conference","original-title":[],"issued":{"date-parts":[[2002,3,27]]},"URL":"http://dx.doi.org/10.2514/6.2002-T5-11","note":"publisher-location:Reston, Virigina\nalternative-id:10.2514/6.2002-T5-11\n10.2514/MSPOPS02"},{"id":"doi:10.1109/JPROC.2007.895197","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/jproc.2007.895197","page":"1228-1254","source":"Crossref","title":"Turbo and Turbo-Like Codes: Principles and Applications in Telecommunications","volume":"95","author":[{"given":"Ken","family":"Gracie","sequence":"first","affiliation":[]},{"given":"Marie-Helene","family":"Hamon","sequence":"additional","affiliation":[]}],"container-title":"Proceedings of the IEEE","original-title":[],"issued":{"date-parts":[[2007,6]]},"URL":"http://dx.doi.org/10.1109/JPROC.2007.895197","ISSN":["0018-9219"],"container-title-short":"Proc. IEEE"},{"id":"doi:10.1002/047085474X","type":"monograph","publisher":"Wiley","DOI":"10.1002/047085474x","source":"Crossref","title":"Turbo Coding, Turbo Equalisation and Space‐Time Coding","author":[{"given":"L.","family":"Hanzo","sequence":"first","affiliation":[]},{"given":"T.H.","family":"Liew","sequence":"additional","affiliation":[]},{"given":"B.L.","family":"Yeap","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2002,7,11]]},"ISBN":["9780470847268","9780470854747"],"URL":"http://dx.doi.org/10.1002/047085474X","note":"edition-number:1\nalternative-id:10.1002/047085474X"},{"id":"doi:10.1007/978-1-4615-4469-2","type":"book","publisher":"Springer US","DOI":"10.1007/978-1-4615-4469-2","source":"Crossref","title":"Turbo Codes","author":[{"given":"Branka","family":"Vucetic","sequence":"first","affiliation":[]},{"given":"Jinhong","family":"Yuan","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2000]]},"ISBN":["9781461370130","9781461544692"],"URL":"http://dx.doi.org/10.1007/978-1-4615-4469-2","note":"publisher-location:Boston, MA"},{"id":"doi:10.1007/978-1-4757-2999-3","type":"book","publisher":"Springer US","DOI":"10.1007/978-1-4757-2999-3","source":"Crossref","title":"Turbo Coding","author":[{"given":"Chris","family":"Heegard","sequence":"first","affiliation":[]},{"given":"Stephen B.","family":"Wicker","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1999]]},"ISBN":["9781441950697","9781475729993"],"URL":"http://dx.doi.org/10.1007/978-1-4757-2999-3","note":"publisher-location:Boston, MA"},{"id":"doi:10.1002/9780470978481","type":"monograph","publisher":"Wiley","DOI":"10.1002/9780470978481","source":"Crossref","title":"Turbo Coding, Turbo Equalisation and Space–Time Coding","author":[{"given":"L.","family":"Hanzo","sequence":"first","affiliation":[]},{"given":"T. H.","family":"Liew","sequence":"additional","affiliation":[]},{"given":"B. L.","family":"Yeap","sequence":"additional","affiliation":[]},{"given":"R. Y. S.","family":"Tee","sequence":"additional","affiliation":[]},{"given":"S. X.","family":"Ng","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2011,3,4]]},"ISBN":["9780470972908","9780470978481"],"URL":"http://dx.doi.org/10.1002/9780470978481","note":"edition-number:1\nalternative-id:10.1002/9780470978481"},{"id":"manual:-M.-Helmling-S.-Scholl-F.-Gen","type":"book","author":[{"family":"Helmling","given":"M."},{"family":"Scholl","given":"S."},{"family":"Gensheimer","given":"F."},{"family":"Dietz","given":"T."},{"family":"Kraft","given":"K."},{"family":"Ruzika","given":"S."},{"family":"Wehn","given":"N."}],"title":"Database of Channel Codes and ML Simulation Results","publisher":"www.uni-kl.de/channel-codes","issued":"2022"},{"id":"doi:10.1109/49.661105","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/49.661105","page":"160-174","source":"Crossref","title":"Design and analysis of turbo codes on Rayleigh fading channels","volume":"16","author":[{"given":"E.K.","family":"Hall","sequence":"first","affiliation":[]},{"given":"S.G.","family":"Wilson","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/49.661105","ISSN":["0733-8716"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"doi:10.1145/2435204.2435207","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"1","abstract":"<jats:p>We design flexible schemes to explore the tradeoffs between storage space and access efficiency in reliable data storage systems. Aiming at this goal, two new classes of erasure-resilient codes are introduced -- Basic Pyramid Codes (BPC) and Generalized Pyramid Codes (GPC). Both schemes require slightly more storage space than conventional schemes, but significantly improve the critical performance of read during failures and unavailability.</jats:p>\n          <jats:p>\n            As a by-product, we establish a necessary matching condition to characterize the limit of failure recovery, that is, unless the matching condition is satisfied, a failure case is impossible to recover. In addition, we define a maximally recoverable (MR) property. For all ERC schemes holding the MR property, the matching condition becomes sufficient, that is, all failure cases satisfying the matching condition are indeed recoverable. We show that GPC is the\n            <jats:italic>first</jats:italic>\n            class of non-MDS schemes holding the MR property.\n          </jats:p>","DOI":"10.1145/2435204.2435207","page":"1-28","source":"Crossref","title":"Pyramid Codes","volume":"9","author":[{"given":"Cheng","family":"Huang","sequence":"first","affiliation":[{"name":"Microsoft Research"}]},{"given":"Minghua","family":"Chen","sequence":"additional","affiliation":[{"name":"Chinese University of Hong Kong"}]},{"given":"Jin","family":"Li","sequence":"additional","affiliation":[{"name":"Microsoft Research"}]}],"container-title":"ACM Transactions on Storage","original-title":[],"language":"en","issued":{"date-parts":[[2013,3]]},"URL":"http://dx.doi.org/10.1145/2435204.2435207","ISSN":["1553-3077","1553-3093"],"container-title-short":"ACM Trans. Storage","note":"alternative-id:10.1145/2435204.2435207"},{"id":"arxiv:1311.3284","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2014.2321280","page":"4661-4676","source":"Crossref","title":"A Family of Optimal Locally Recoverable Codes","volume":"60","author":[{"given":"Itzhak","family":"Tamo","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2321280","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1311.3284"},{"id":"arxiv:1501.04904","type":"article-journal","author":[{"given":"Alexander","family":"Barg"},{"given":"Itzhak","family":"Tamo"},{"given":"Serge","family":"Vladut"}],"title":"Locally recoverable codes on algebraic curves","issued":{"date-parts":[[2015,5,10]]},"note":"arxivid:1501.04904\narxiv_version_number:2"},{"id":"doi:10.1145/1060590.1060631","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/1060590.1060631","page":"266-275","source":"Crossref","title":"Simple PCPs with poly-log rate and query complexity","author":[{"given":"Eli","family":"Ben-Sasson","sequence":"first","affiliation":[{"name":"Israel Institute of Technology, Haifa, Israel and Toyota Technological Institute, Chicago, IL"}]},{"given":"Madhu","family":"Sudan","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, Cambridge, MA"}]}],"event":"STOC05: Symposium on Theory of Computing","container-title":"Proceedings of the thirty-seventh annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2005,5,22]]},"URL":"http://dx.doi.org/10.1145/1060590.1060631","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/1060590.1060631\n10.1145/1060590"},{"id":"doi:10.1145/1374376.1374419","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/1374376.1374419","page":"285-294","source":"Crossref","title":"Combinatorial construction of locally testable codes","author":[{"given":"Or","family":"Meir","sequence":"first","affiliation":[{"name":"Weizmann Institute of Science, Rehovot, Israel"}]}],"event":"STOC '08: Symposium on Theory of Computing","container-title":"Proceedings of the fortieth annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2008,5,17]]},"URL":"http://dx.doi.org/10.1145/1374376.1374419","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/1374376.1374419\n10.1145/1374376"},{"id":"doi:10.1109/TIT.1964.1053661","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1964.1053661","page":"116-118","source":"Crossref","title":"Maximum distanceq-nary codes","volume":"10","author":[{"given":"R.","family":"Singleton","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1964,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1964.1053661","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/j.ffa.2014.10.006","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.ffa.2014.10.006","page":"44-81","source":"Crossref","title":"Open problems in finite projective spaces","volume":"32","author":[{"ORCID":"https://orcid.org/0000-0003-2635-9268","authenticated-orcid":false,"given":"J.W.P.","family":"Hirschfeld","sequence":"first","affiliation":[]},{"given":"J.A.","family":"Thas","sequence":"additional","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2015,3]]},"URL":"http://dx.doi.org/10.1016/j.ffa.2014.10.006","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"special_numbering:C\nalternative-id:S1071579714001191"},{"id":"doi:10.1109/18.748994","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.748994","page":"438-452","source":"Crossref","title":"Binomial moments of the distance distribution: bounds and applications","volume":"45","author":[{"given":"A.","family":"Ashikhmin","sequence":"first","affiliation":[]},{"given":"A.","family":"Barg","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999,3]]},"URL":"http://dx.doi.org/10.1109/18.748994","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:Bose47","type":"article-journal","author":[{"family":"Bose","given":"R.C."}],"title":"Mathematical theory of the symmetrical factorial design","container-title":"Sankhyā: The Indian Journal of Statistics","issued":"1947","page":"107–166"},{"id":"doi:10.1109/ISIT.1994.395042","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.1994.395042","page":"427","source":"Crossref","title":"On near-MDS codes","author":[{"given":"S.M.","family":"Dodunekov","sequence":"first","affiliation":[]},{"given":"I.N.","family":"Landgev","sequence":"additional","affiliation":[]}],"event":"1994 IEEE International Symposium on Information Theory","container-title":"Proceedings of 1994 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.1994.395042"},{"id":"doi:10.1007/978-1-4613-0283-4_13","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-1-4613-0283-4_13","page":"201-246","source":"Crossref","title":"The Packing Problem in Statistics, Coding Theory and Finite Projective Spaces: Update 2001","author":[{"given":"J. W. P.","family":"Hirschfeld","sequence":"first","affiliation":[]},{"given":"L.","family":"Storme","sequence":"additional","affiliation":[]}],"container-title":"Developments in Mathematics","original-title":[],"issued":{"date-parts":[[2001]]},"ISBN":["9781461379775","9781461302834"],"URL":"http://dx.doi.org/10.1007/978-1-4613-0283-4_13","ISSN":["1389-2177"],"note":"publisher-location:Boston, MA"},{"id":"manual:-A.-Beutelspacher-and-U.-Rose","type":"book","author":[{"family":"Beutelspacher","given":"A."},{"family":"Rosenbaum","given":"U."}],"title":"Projective geometry: from foundations to applications","publisher":"Cambridge University Press","issued":"1998"},{"id":"doi:10.1109/18.476239","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.476239","page":"1667-1671","source":"Crossref","title":"Formula for the number of [9,3] MDS codes","volume":"41","author":[{"given":"A.V.","family":"Iampolskaia","sequence":"first","affiliation":[]},{"given":"A.N.","family":"Skorobogatov","sequence":"additional","affiliation":[]},{"given":"E.A.","family":"Sorokin","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1995]]},"URL":"http://dx.doi.org/10.1109/18.476239","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-O.-Taussky-and-J.-Todd-Cover","type":"article-journal","author":[{"family":"Taussky","given":"O."},{"family":"Todd","given":"J."}],"title":"Covering theorems for groups","volume":"54","container-title":"CHARLES ST","publisher":"AMER MATHEMATICAL SOC","issue":"3","issued":"1948","page":"201","publisher-place":"PROVIDENCE, RI 02940-2213"},{"id":"doi:10.1112/jlms/s1-44.1.60","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1112/jlms/s1-44.1.60","page":"60-64","source":"Crossref","title":"A Combinatorial Problem in Matching","volume":"s1-44","author":[{"given":"J. G.","family":"Kalbfleisch","sequence":"first","affiliation":[]},{"given":"R. G.","family":"Stanton","sequence":"additional","affiliation":[]}],"container-title":"Journal of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1969]]},"URL":"http://dx.doi.org/10.1112/jlms/s1-44.1.60","ISSN":["0024-6107"],"container-title-short":"Journal of the London Mathematical Society"},{"id":"manual:-A.-V.-Thapliyal-Multipartite","type":"document","author":[{"family":"Thapliyal","given":"A.V."}],"title":"Multipartite maximally entangled states, minimal entanglement generating states and entropic inequalities unpublished presentation","issued":"2003"},{"id":"arxiv:1306.2536","type":"article-journal","author":[{"given":"Wolfram","family":"Helwig"},{"given":"Wei","family":"Cui"}],"title":"Absolutely Maximally Entangled States: Existence and Applications","issued":{"date-parts":[[2013,6,11]]},"note":"arxivid:1306.2536\narxiv_version_number:1"},{"id":"arxiv:1306.2879","type":"article-journal","author":[{"given":"Wolfram","family":"Helwig"}],"title":"Absolutely Maximally Entangled Qudit Graph States","issued":{"date-parts":[[2013,6,12]]},"note":"arxivid:1306.2879\narxiv_version_number:1"},{"id":"arxiv:1701.03359","type":"journal-article","publisher":"IOP Publishing","issue":"7","DOI":"10.1088/1751-8121/aaa151","page":"075301","source":"Crossref","title":"Optimal quantum error correcting codes from absolutely maximally entangled states","volume":"51","author":[{"ORCID":"https://orcid.org/0000-0002-9168-8212","authenticated-orcid":false,"given":"Zahra","family":"Raissi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0290-4698","authenticated-orcid":false,"given":"Christian","family":"Gogolin","sequence":"additional","affiliation":[]},{"given":"Arnau","family":"Riera","sequence":"additional","affiliation":[]},{"given":"Antonio","family":"Acín","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2018,1,15]]},"URL":"http://dx.doi.org/10.1088/1751-8121/aaa151","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1701.03359"},{"id":"arxiv:1706.08318","type":"article-journal","author":[{"given":"Daniel","family":"Alsina"}],"title":"PhD thesis: Multipartite entanglement and quantum algorithms","issued":{"date-parts":[[2017,7,11]]},"note":"arxivid:1706.08318\narxiv_version_number:2"},{"id":"arxiv:1602.02235","type":"article-journal","author":[{"given":"Jihao","family":"Fan"},{"given":"Hanwu","family":"Chen"},{"given":"Juan","family":"Xu"}],"title":"Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q + 1","issued":{"date-parts":[[2016,2,6]]},"note":"arxivid:1602.02235\narxiv_version_number:1"},{"id":"arxiv:1306.4774","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2014.2351404","page":"6979-6987","source":"Crossref","title":"Repair Locality With Multiple Erasure Tolerance","volume":"60","author":[{"ORCID":"https://orcid.org/0000-0003-2100-4521","authenticated-orcid":false,"given":"Anyu","family":"Wang","sequence":"first","affiliation":[]},{"given":"Zhifang","family":"Zhang","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2351404","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1306.4774"},{"id":"arxiv:1501.04264","type":"article-journal","author":[{"given":"Anyu","family":"Wang"},{"given":"Zhifang","family":"Zhang"}],"title":"Achieving Arbitrary Locality and Availability in Binary Codes","issued":{"date-parts":[[2015,1,18]]},"note":"arxivid:1501.04264\narxiv_version_number:1"},{"id":"arxiv:1302.5518","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2013.6620355","page":"892-896","source":"Crossref","title":"Locally repairable codes with multiple repair alternatives","author":[{"given":"Lluis","family":"Pamies-Juarez","sequence":"first","affiliation":[]},{"given":"Henk D.L.","family":"Hollmann","sequence":"additional","affiliation":[]},{"given":"Frederique","family":"Oggier","sequence":"additional","affiliation":[]}],"event":"2013 IEEE International Symposium on Information Theory (ISIT)","container-title":"2013 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2013,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2013.6620355","note":"arxivid:1302.5518"},{"id":"arxiv:1401.2422","type":"article-journal","author":[{"given":"N.","family":"Prakash"},{"given":"V.","family":"Lalitha"},{"given":"P. Vijay","family":"Kumar"}],"title":"Codes with Locality for Two Erasures","issued":{"date-parts":[[2014,1,27]]},"note":"arxivid:1401.2422\narxiv_version_number:2"},{"id":"arxiv:1812.02502","type":"article-journal","author":[{"given":"S. B.","family":"Balaji"},{"given":"Ganesh R.","family":"Kini"},{"given":"P. Vijay","family":"Kumar"}],"title":"A Tight Rate Bound and Matching Construction for Locally Recoverable Codes with Sequential Recovery From Any Number of Multiple Erasures","issued":{"date-parts":[[2018,12,6]]},"note":"arxivid:1812.02502\narxiv_version_number:1"},{"id":"doi:10.1109/18.30988","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.30988","page":"655-657","source":"Crossref","title":"A construction of non-Reed-Solomon type MDS codes","volume":"35","author":[{"given":"R.M.","family":"Roth","sequence":"first","affiliation":[]},{"given":"A.","family":"Lempel","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1989,5]]},"URL":"http://dx.doi.org/10.1109/18.30988","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s10623-016-0282-8","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s10623-016-0282-8","page":"473-492","source":"Crossref","title":"Galois self-dual constacyclic codes","volume":"84","author":[{"given":"Yun","family":"Fan","sequence":"first","affiliation":[]},{"given":"Liang","family":"Zhang","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2016,9,29]]},"URL":"http://dx.doi.org/10.1007/s10623-016-0282-8","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:282"},{"id":"doi:10.1088/1751-8113/41/14/145304","type":"journal-article","publisher":"IOP Publishing","issue":"14","DOI":"10.1088/1751-8113/41/14/145304","page":"145304","source":"Crossref","title":"Stabilizer quantum codes over the Clifford algebra","volume":"41","author":[{"given":"Guihua","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Yuan","family":"Li","sequence":"additional","affiliation":[]},{"given":"Ying","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Moon Ho","family":"Lee","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2008,3,26]]},"URL":"http://dx.doi.org/10.1088/1751-8113/41/14/145304","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(08)57182-4"},{"id":"arxiv:2503.08736","type":"article-journal","author":[{"given":"Sanchayan","family":"Dutta"}],"title":"A Note on Clifford Stabilizer Codes for Ising Anyons","issued":{"date-parts":[[2025,3,11]]},"note":"arxivid:2503.08736\narxiv_version_number:1"},{"id":"doi:10.1109/18.959288","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/18.959288","page":"3065-3072","source":"Crossref","title":"Nonbinary quantum stabilizer codes","volume":"47","author":[{"given":"A.","family":"Ashikhmin","sequence":"first","affiliation":[]},{"given":"E.","family":"Knill","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001]]},"URL":"http://dx.doi.org/10.1109/18.959288","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1016/0012-365X(92)90563-U","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/0012-365x(92)90563-u","page":"337-342","source":"Crossref","title":"Linear codes with complementary duals","volume":"106-107","author":[{"given":"James L.","family":"Massey","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1992,9]]},"URL":"http://dx.doi.org/10.1016/0012-365X(92)90563-U","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"special_numbering:C\nalternative-id:0012365X9290563U"},{"id":"arxiv:1908.03294","type":"article-journal","author":[{"given":"Makoto","family":"Araya"},{"given":"Masaaki","family":"Harada"},{"given":"Ken","family":"Saito"}],"title":"Characterization and classification of optimal LCD codes","issued":{"date-parts":[[2021,1,4]]},"note":"arxivid:1908.03294\narxiv_version_number:3"},{"id":"arxiv:1506.01955","type":"article-journal","author":[{"given":"Steven T.","family":"Dougherty"},{"given":"Jon-Lark","family":"Kim"},{"given":"Buket","family":"Ozkaya"},{"given":"Lin","family":"Sok"},{"given":"Patrick","family":"Solé"}],"title":"The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices","issued":{"date-parts":[[2015,6,5]]},"note":"arxivid:1506.01955\narxiv_version_number:1"},{"id":"arxiv:1606.00134","type":"article-journal","author":[{"given":"Kenza","family":"Guenda"},{"given":"Somphong","family":"Jitman"},{"given":"T. Aaron","family":"Gulliver"}],"title":"Constructions of Good Entanglement-Assisted Quantum Error Correcting Codes","issued":{"date-parts":[[2016,6,1]]},"note":"arxivid:1606.00134\narxiv_version_number:1"},{"id":"manual:-V.-Pless-On-a-new-family-of-","type":"article-journal","author":[{"family":"Pless","given":"V."}],"title":"On a new family of symmetry codes and related new five-designs","volume":"75","container-title":"BAMS","issued":"1969","page":"1339–1342"},{"id":"arxiv:2010.15453","type":"article-journal","author":[{"given":"Kirill","family":"Ivanov"},{"given":"Rüdiger L.","family":"Urbanke"}],"title":"Capacity-achieving codes: a review on double transitivity","issued":{"date-parts":[[2020,10,29]]},"note":"arxivid:2010.15453\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.1973.1054975","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1973.1054975","page":"251-251","source":"Crossref","title":"Is there a (72,36) d = 16 self-dual code? (Corresp.)","volume":"19","author":[{"given":"N.","family":"Sloane","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1973,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1973.1054975","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:math/0208001","type":"article-journal","author":[{"given":"E. M.","family":"Rains"},{"given":"N. J. A.","family":"Sloane"}],"title":"Self-Dual Codes","issued":{"date-parts":[[2002,8,1]]},"note":"arxivid:math/0208001\narxiv_version_number:1"},{"id":"arxiv:1104.3727","type":"article-journal","author":[{"given":"Koichi","family":"Betsumiya"},{"given":"Masaaki","family":"Harada"},{"given":"Akihiro","family":"Munemasa"}],"title":"A complete classification of doubly even self-dual codes of length 40","issued":{"date-parts":[[2012,11,12]]},"note":"arxivid:1104.3727\narxiv_version_number:4"},{"id":"preset:HPRainsSloane","type":"chapter","author":[{"family":"Rains","given":"E.M."},{"family":"Sloane","given":"N.J.A."}],"title":"Self-dual codes","container-title":"\\emph{Handbook of Coding Theory}","volume":"I, Part 1","editor":[{"family":"V. S. Pless","particle":"eds."},{"family":"Huffman","given":"W.C."}],"publisher":"Elsevier","issued":"1998","page":"177–294"},{"id":"doi:10.1016/j.ffa.2005.05.012","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/j.ffa.2005.05.012","page":"451-490","source":"Crossref","title":"On the classification and enumeration of self-dual codes","volume":"11","author":[{"given":"W.","family":"Cary Huffman","sequence":"first","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2005,8]]},"URL":"http://dx.doi.org/10.1016/j.ffa.2005.05.012","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S1071579705000444"},{"id":"doi:10.1016/j.jcta.2008.11.011","type":"journal-article","publisher":"Elsevier BV","issue":"5","DOI":"10.1016/j.jcta.2008.11.011","page":"1063-1072","source":"Crossref","title":"A complete classification of ternary self-dual codes of length 24","volume":"116","author":[{"given":"Masaaki","family":"Harada","sequence":"first","affiliation":[]},{"given":"Akihiro","family":"Munemasa","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[2009,7]]},"URL":"http://dx.doi.org/10.1016/j.jcta.2008.11.011","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S0097316509000259"},{"id":"doi:10.1090/S0025-5718-08-02194-7","type":"journal-article","publisher":"American Mathematical Society (AMS)","issue":"267","abstract":"<p>\n                    All\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"28\">\n                        <mml:semantics>\n                          <mml:mn>28</mml:mn>\n                          <mml:annotation encoding=\"application/x-tex\">28</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    -dimensional unimodular lattices with minimum norm\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\n                        <mml:semantics>\n                          <mml:mn>3</mml:mn>\n                          <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    are known. Using this classification, we give a classification of ternary extremal self-dual codes of length\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"28\">\n                        <mml:semantics>\n                          <mml:mn>28</mml:mn>\n                          <mml:annotation encoding=\"application/x-tex\">28</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    . Up to equivalence, there are 6,931 such codes.\n                  </p>","DOI":"10.1090/s0025-5718-08-02194-7","page":"1787-1796","source":"Crossref","title":"Classification of ternary extremal self-dual codes of length 28","volume":"78","author":[{"given":"Masaaki","family":"Harada","sequence":"first","affiliation":[]},{"given":"Akihiro","family":"Munemasa","sequence":"additional","affiliation":[]},{"given":"Boris","family":"Venkov","sequence":"additional","affiliation":[]}],"container-title":"Mathematics of Computation","original-title":[],"language":"en","issued":{"date-parts":[[2008,10,24]]},"URL":"http://dx.doi.org/10.1090/S0025-5718-08-02194-7","ISSN":["1088-6842","0025-5718"],"container-title-short":"Math. Comp.","note":"alternative-id:S0025-5718-08-02194-7"},{"id":"doi:10.1016/j.disc.2004.06.010","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/j.disc.2004.06.010","page":"1-10","source":"Crossref","title":"Some optimal self-orthogonal and self-dual codes","volume":"287","author":[{"given":"Stefka","family":"Bouyuklieva","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2004,10]]},"URL":"http://dx.doi.org/10.1016/j.disc.2004.06.010","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X04002912"},{"id":"arxiv:1706.01694","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","DOI":"10.1007/s10623-017-0380-2","page":"1085-1094","source":"Crossref","title":"Binary extremal self-dual codes of length 60 and related codes","volume":"86","author":[{"given":"Masaaki","family":"Harada","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2017,6,24]]},"URL":"http://dx.doi.org/10.1007/s10623-017-0380-2","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:380\narxivid:1706.01694"},{"id":"doi:10.1016/S1071-5797(03)00011-X","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s1071-5797(03)00011-x","page":"372-394","source":"Crossref","title":"Experimental constructions of self-dual codes","volume":"9","author":[{"given":"Philippe","family":"Gaborit","sequence":"first","affiliation":[]},{"given":"Ayoub","family":"Otmani","sequence":"additional","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2003,7]]},"URL":"http://dx.doi.org/10.1016/S1071-5797(03)00011-X","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"alternative-id:S107157970300011X"},{"id":"preset:HKSquantum","type":"chapter","author":[{"family":"Ezerman","given":"M.F."}],"title":"Quantum Error-Control Codes","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"arxiv:1811.00589","type":"journal-article","publisher":"Canadian Mathematical Society","issue":"2","abstract":"<jats:title>Abstract</jats:title><jats:p>We observe that every self-dual ternary code determines a holomorphic <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline1.png\" /><jats:tex-math>\n$\\mathcal N=1$\n</jats:tex-math></jats:alternatives></jats:inline-formula> superconformal field theory. This provides ternary constructions of some well-known holomorphic <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline2.png\" /><jats:tex-math>\n$\\mathcal N=1$\n</jats:tex-math></jats:alternatives></jats:inline-formula> superconformal field theories (SCFTs), including Duncan’s “supermoonshine” model and the fermionic “beauty and the beast” model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline3.png\" /><jats:tex-math>\n$24$\n</jats:tex-math></jats:alternatives></jats:inline-formula>; we are able to prove this conjecture except in the case when the code has length <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline4.png\" /><jats:tex-math>\n$12$\n</jats:tex-math></jats:alternatives></jats:inline-formula> mod <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline5.png\" /><jats:tex-math>\n$24$\n</jats:tex-math></jats:alternatives></jats:inline-formula>. Lastly, we discuss a conjecture of Stolz and Teichner relating <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline6.png\" /><jats:tex-math>\n$\\mathcal N=1$\n</jats:tex-math></jats:alternatives></jats:inline-formula> SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each <jats:italic>k</jats:italic>, a holomorphic <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline7.png\" /><jats:tex-math>\n$\\mathcal N=1$\n</jats:tex-math></jats:alternatives></jats:inline-formula> SCFT of central charge <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline8.png\" /><jats:tex-math>\n$12k$\n</jats:tex-math></jats:alternatives></jats:inline-formula> and index <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline9.png\" /><jats:tex-math>\n$24/\\gcd (k,24)$\n</jats:tex-math></jats:alternatives></jats:inline-formula>. We give ternary code constructions of SCFTs realizing this suggestion for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0008414X2100002X_inline10.png\" /><jats:tex-math>\n$k\\leq 5$\n</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>","DOI":"10.4153/s0008414x2100002x","page":"573-601","source":"Crossref","title":"Holomorphic SCFTs with small index","volume":"74","author":[{"given":"Davide","family":"Gaiotto","sequence":"first","affiliation":[]},{"given":"Theo","family":"Johnson-Freyd","sequence":"additional","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2021,1,18]]},"URL":"http://dx.doi.org/10.4153/S0008414X2100002X","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. J. Math.-J. Can. Math.","note":"alternative-id:S0008414X2100002X\narxivid:1811.00589"},{"id":"doi:10.1109/TIT.2010.2092415","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2010.2092415","page":"2243-2251","source":"Crossref","title":"On Self-Dual Cyclic Codes Over Finite Fields","volume":"57","author":[{"family":"Yan Jia","sequence":"first","affiliation":[]},{"family":"San Ling","sequence":"additional","affiliation":[]},{"family":"Chaoping Xing","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2011,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2092415","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.2012.2236383","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.2012.2236383","page":"3046-3058","source":"Crossref","title":"Abelian Codes in Principal Ideal Group Algebras","volume":"59","author":[{"given":"Somphong","family":"Jitman","sequence":"first","affiliation":[]},{"given":"San","family":"Ling","sequence":"additional","affiliation":[]},{"given":"Hongwei","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Xiaoli","family":"Xie","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,5]]},"URL":"http://dx.doi.org/10.1109/TIT.2012.2236383","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2501.10163","type":"article-journal","author":[{"given":"Amolak Ratan","family":"Kalra"},{"given":"Shiroman","family":"Prakash"}],"title":"Invariant Theory, Magic State Distillation, and Bounds on Classical Codes","issued":{"date-parts":[[2026,3,23]]},"note":"arxivid:2501.10163\narxiv_version_number:3"},{"id":"arxiv:2408.09685","type":"article-journal","author":[{"given":"Minjia","family":"Shi"},{"given":"Haodong","family":"Lu"},{"given":"Jon-Lark","family":"Kim"},{"given":"Patrick","family":"Sole"}],"title":"Triorthogonal Codes and Self-dual Codes","issued":{"date-parts":[[2024,8,19]]},"note":"arxivid:2408.09685\narxiv_version_number:1"},{"id":"manual:-P.-Gaborit-W.-C.-Huffman-J.-","type":"chapter","author":[{"family":"Gaborit","given":"P."},{"family":"Huffman","given":"W.C."},{"family":"Kim","given":"J.-L."},{"family":"Pless","given":"V."}],"title":"On additive GF(4) codes","container-title":"Codes and Association Schemes (DIMACS Workshop","editor":[{"family":"A. Barg","particle":"eds."},{"family":"Litsyn","given":"S."}],"publisher":"American Mathematical Society","issued":"November 9–12, 1999","page":"135–149","publisher-place":"Providence, RI"},{"id":"arxiv:math/0504522","type":"journal-article","publisher":"Elsevier BV","issue":"7","DOI":"10.1016/j.jcta.2005.12.004","page":"1351-1367","source":"Crossref","title":"On the classification of all self-dual additive codes over GF(4) of length up to 12","volume":"113","author":[{"given":"Lars Eirik","family":"Danielsen","sequence":"first","affiliation":[]},{"given":"Matthew G.","family":"Parker","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[2006,10]]},"URL":"http://dx.doi.org/10.1016/j.jcta.2005.12.004","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S0097316505002347\narxivid:math/0504522"},{"id":"doi:10.1109/ISIT.2019.8849731","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2019.8849731","page":"1912-1916","source":"Crossref","title":"The punctured Dodecacode is uniformly packed","author":[{"given":"Denis","family":"Krotov","sequence":"first","affiliation":[]},{"given":"Patrick","family":"Sole","sequence":"additional","affiliation":[]}],"event":"2019 IEEE International Symposium on Information Theory (ISIT)","container-title":"2019 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2019,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2019.8849731"},{"id":"arxiv:quant-ph/0406063","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","DOI":"10.1007/s11128-005-0002-1","page":"399-431","source":"Crossref","title":"Probabilities of Failure for Quantum Error Correction","volume":"4","author":[{"given":"A. J.","family":"Scott","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2005,10,12]]},"URL":"http://dx.doi.org/10.1007/s11128-005-0002-1","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:2\narxivid:quant-ph/0406063"},{"id":"doi:10.1016/0012-365X(86)90067-1","type":"journal-article","publisher":"Elsevier BV","issue":"1-2","DOI":"10.1016/0012-365x(86)90067-1","page":"43-51","source":"Crossref","title":"The non-classical 10-arc of PG(4, 9)","volume":"59","author":[{"given":"David G","family":"Glynn","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1986,4]]},"URL":"http://dx.doi.org/10.1016/0012-365X(86)90067-1","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X86900671"},{"id":"arxiv:1907.04391","type":"article-journal","author":[{"given":"Simeon","family":"Ball"}],"title":"Some constructions of quantum MDS codes","issued":{"date-parts":[[2021,1,14]]},"note":"arxivid:1907.04391\narxiv_version_number:5"},{"id":"arxiv:2108.08088","type":"article-journal","author":[{"given":"Simeon","family":"Ball"},{"given":"Ricard","family":"Vilar"}],"title":"The geometry of Hermitian self-orthogonal codes","issued":{"date-parts":[[2021,8,18]]},"note":"arxivid:2108.08088\narxiv_version_number:1"},{"id":"manual:-R.-Sch-rer-and-W.-Ch.-Schmid","type":"article-journal","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Hexacode. From MinT—the database of optimal net, code, OA, and OOA parameters","container-title":"Version","issued":"2015-09-03","URL":"https://web.archive.org/web/20240418205700/http://mint.sbg.ac.at/desc_CHexa.html","note":"Available at \\url{https://web.archive.org/web/20240418205700/http://mint.sbg.ac.at/desc_CHexa.html}"},{"id":"preset:HKStwoweight","type":"chapter","author":[{"family":"Brouwer","given":"A.E."}],"title":"Two-weight Codes","container-title":"Concise Encyclopedia of Coding Theory","note":"Chapman and Hall/CRC, 2021","URL":"https://doi.org/10.1201/9781315147901","DOI":"10.1201/9781315147901"},{"id":"manual:-R.-A.-Wilson-On-lexicographi","type":"article-journal","author":[{"family":"Wilson","given":"R.A."}],"title":"On lexicographic codes of minimal distance 4, Atti Sem","volume":"33","container-title":"Mat. Fis. Univ. Modena","issued":"1984"},{"id":"arxiv:quant-ph/0310137","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.69.052330","source":"Crossref","title":"Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions","volume":"69","author":[{"given":"A. J.","family":"Scott","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,5,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.69.052330","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052330","note":"arxivid:quant-ph/0310137"},{"id":"arxiv:quant-ph/9704043","type":"article-journal","author":[{"given":"Eric M.","family":"Rains"}],"title":"Quantum codes of minimum distance two","issued":{"date-parts":[[1997,4,24]]},"note":"arxivid:quant-ph/9704043\narxiv_version_number:1"},{"id":"manual:-V.-V.-Albert-private-communi","type":"document","author":[{"family":"Albert","given":"V.V."}],"genre":"private communication,","issued":"2024"},{"id":"doi:10.1017/CBO9781316257449","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>The projective and polar geometries that arise from a vector space over a finite field are particularly useful in the construction of combinatorial objects, such as latin squares, designs, codes and graphs. This book provides an introduction to these geometries and their many applications to other areas of combinatorics. Coverage includes a detailed treatment of the forbidden subgraph problem from a geometrical point of view, and a chapter on maximum distance separable codes, which includes a proof that such codes over prime fields are short. The author also provides more than 100 exercises (complete with detailed solutions), which show the diversity of applications of finite fields and their geometries. Finite Geometry and Combinatorial Applications is ideal for anyone, from a third-year undergraduate to a researcher, who wishes to familiarise themselves with and gain an appreciation of finite geometry.</jats:p>","DOI":"10.1017/cbo9781316257449","source":"Crossref","title":"Finite Geometry and Combinatorial Applications","author":[{"given":"Simeon","family":"Ball","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2015,4,30]]},"ISBN":["9781107107991","9781107518438","9781316257449"],"URL":"http://dx.doi.org/10.1017/CBO9781316257449","note":"edition-number:1"},{"id":"manual:-J.-W.-P.-Hirschfeld.-Rationa","type":"article-journal","author":[{"family":"Hirschfeld","given":"J.W.P."}],"title":"Rational curves on quadrics over finite fields of characteristic two","volume":"4","container-title":"Rend. Mat","issued":"1971","page":"773–795"},{"id":"doi:10.1016/j.patcog.2007.05.020","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.patcog.2007.05.020","page":"67-76","source":"Crossref","title":"Data-driven decomposition for multi-class classification","volume":"41","author":[{"given":"Jie","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Hanchuan","family":"Peng","sequence":"additional","affiliation":[]},{"given":"Ching Y.","family":"Suen","sequence":"additional","affiliation":[]}],"container-title":"Pattern Recognition","original-title":[],"language":"en","issued":{"date-parts":[[2008,1]]},"URL":"http://dx.doi.org/10.1016/j.patcog.2007.05.020","ISSN":["0031-3203"],"container-title-short":"Pattern Recognition","note":"alternative-id:S0031320307002610"},{"id":"manual:-E.-L.-Allwein-R.-E.-Schapire","type":"article-journal","author":[{"family":"Allwein","given":"E.L."},{"family":"Schapire","given":"R.E."},{"family":"Singer","given":"Y."}],"title":"Reducing multiclass to binary: A unifying approach for margin classifiers","volume":"1","container-title":"Journal of machine learning research","issue":"Dec","issued":"2000","page":"113–141"},{"id":"doi:10.1016/S1566-2535(02)00101-X","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s1566-2535(02)00101-x","page":"11-21","source":"Crossref","title":"Coding and decoding strategies for multi-class learning problems","volume":"4","author":[{"given":"Terry","family":"Windeatt","sequence":"first","affiliation":[]},{"given":"Reza","family":"Ghaderi","sequence":"additional","affiliation":[]}],"container-title":"Information Fusion","original-title":[],"language":"en","issued":{"date-parts":[[2003,3]]},"URL":"http://dx.doi.org/10.1016/S1566-2535(02)00101-X","ISSN":["1566-2535"],"container-title-short":"Information Fusion","note":"alternative-id:S156625350200101X"},{"id":"doi:10.1016/j.patcog.2007.04.008","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/j.patcog.2007.04.008","page":"713-725","source":"Crossref","title":"An incremental node embedding technique for error correcting output codes","volume":"41","author":[{"given":"Oriol","family":"Pujol","sequence":"first","affiliation":[]},{"given":"Sergio","family":"Escalera","sequence":"additional","affiliation":[]},{"given":"Petia","family":"Radeva","sequence":"additional","affiliation":[]}],"container-title":"Pattern Recognition","original-title":[],"language":"en","issued":{"date-parts":[[2008,2]]},"URL":"http://dx.doi.org/10.1016/j.patcog.2007.04.008","ISSN":["0031-3203"],"container-title-short":"Pattern Recognition","note":"alternative-id:S0031320307002014"},{"id":"doi:10.1109/TNN.2003.820841","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tnn.2003.820841","page":"45-54","source":"Crossref","title":"New Results on Error Correcting Output Codes of Kernel Machines","volume":"15","author":[{"given":"A.","family":"Passerini","sequence":"first","affiliation":[]},{"given":"M.","family":"Pontil","sequence":"additional","affiliation":[]},{"given":"P.","family":"Frasconi","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Neural Networks","original-title":[],"language":"en","issued":{"date-parts":[[2004,1]]},"URL":"http://dx.doi.org/10.1109/TNN.2003.820841","ISSN":["1045-9227"],"container-title-short":"IEEE Trans. Neural Netw."},{"id":"doi:10.1049/cp:20030513","type":"proceedings-article","publisher":"IEE","DOI":"10.1049/cp:20030513","page":"165-168","source":"Crossref","title":"Boosted ECOC ensembles for face recognition","volume":"2003","author":[{"given":"T.","family":"Windeatt","sequence":"first","affiliation":[]}],"event":"International Conference on Visual Information Engineering (VIE 2003). Ideas, Applications, Experience","container-title":"International Conference on Visual Information Engineering (VIE 2003). Ideas, Applications, Experience","original-title":[],"issued":{"date-parts":[[2003]]},"URL":"http://dx.doi.org/10.1049/cp:20030513"},{"id":"doi:10.1109/CVPR.2001.990552","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/cvpr.2001.990552","page":"I-755-I-760","source":"Crossref","title":"Face verification using error correcting output codes","volume":"1","author":[{"given":"J.","family":"Kittler","sequence":"first","affiliation":[]},{"given":"R.","family":"Ghaderi","sequence":"additional","affiliation":[]},{"given":"T.","family":"Windeatt","sequence":"additional","affiliation":[]},{"given":"J.","family":"Matas","sequence":"additional","affiliation":[]}],"event":"2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001","container-title":"Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/CVPR.2001.990552"},{"id":"doi:10.1109/ICDM.2001.989574","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/icdm.2001.989574","page":"597-598","source":"Crossref","title":"Combining labeled and unlabeled data for text classification with a large number of categories","author":[{"given":"R.","family":"Ghani","sequence":"first","affiliation":[]}],"event":"2001 IEEE International Conference on Data Mining","container-title":"Proceedings 2001 IEEE International Conference on Data Mining","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ICDM.2001.989574"},{"id":"doi:10.1109/ICDAR.2005.246","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icdar.2005.246","page":"484-488 Vol. 1","source":"Crossref","title":"Unconstrained numeral pair recognition using enhanced error correcting output coding: a holistic approach","author":[{"given":"J.","family":"Zhou","sequence":"first","affiliation":[]},{"given":"C.Y.","family":"Suen","sequence":"additional","affiliation":[]}],"event":"Eighth International Conference on Document Analysis and Recognition (ICDAR'05)","container-title":"Eighth International Conference on Document Analysis and Recognition (ICDAR'05)","original-title":[],"issued":{"date-parts":[[2005]]},"URL":"http://dx.doi.org/10.1109/ICDAR.2005.246"},{"id":"doi:10.1109/TPAMI.2008.266","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tpami.2008.266","page":"120-134","source":"Crossref","title":"On the Decoding Process in Ternary Error-Correcting Output Codes","volume":"32","author":[{"given":"S.","family":"Escalera","sequence":"first","affiliation":[]},{"given":"O.","family":"Pujol","sequence":"additional","affiliation":[]},{"given":"P.","family":"Radeva","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence","original-title":[],"issued":{"date-parts":[[2010,1]]},"URL":"http://dx.doi.org/10.1109/TPAMI.2008.266","ISSN":["0162-8828"],"container-title-short":"IEEE Trans. Pattern Anal. Mach. Intell."},{"id":"manual:-S.-Escalera-O.-Pujol-and-P.-","type":"article-journal","author":[{"family":"Escalera","given":"S."},{"family":"Pujol","given":"O."},{"family":"Radeva","given":"P."}],"title":"Error-correcting output codes library","volume":"11","container-title":"The Journal of Machine Learning Research","issued":"2010","page":"661–664"},{"id":"doi:10.1109/72.165597","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/72.165597","page":"962-968","source":"Crossref","title":"Handwritten digit recognition by neural networks with single-layer training","volume":"3","author":[{"given":"S.","family":"Knerr","sequence":"first","affiliation":[]},{"given":"L.","family":"Personnaz","sequence":"additional","affiliation":[]},{"given":"G.","family":"Dreyfus","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Neural Networks","original-title":[],"issued":{"date-parts":[[1992]]},"URL":"http://dx.doi.org/10.1109/72.165597","ISSN":["1045-9227"],"container-title-short":"IEEE Trans. Neural Netw."},{"id":"doi:10.1214/aos/1028144844","type":"journal-article","publisher":"Institute of Mathematical Statistics","issue":"2","DOI":"10.1214/aos/1028144844","source":"Crossref","title":"Classification by pairwise coupling","volume":"26","author":[{"given":"Trevor","family":"Hastie","sequence":"first","affiliation":[]},{"given":"Robert","family":"Tibshirani","sequence":"additional","affiliation":[]}],"container-title":"The Annals of Statistics","original-title":[],"issued":{"date-parts":[[1998,4,1]]},"URL":"http://dx.doi.org/10.1214/aos/1028144844","ISSN":["0090-5364"],"container-title-short":"Ann. Statist."},{"id":"doi:10.1134/S0032946009040024","type":"journal-article","publisher":"Pleiades Publishing Ltd","issue":"4","DOI":"10.1134/s0032946009040024","page":"309-316","source":"Crossref","title":"The group of permutation automorphisms of a q-ary hamming code","volume":"45","author":[{"given":"E. V.","family":"Gorkunov","sequence":"first","affiliation":[]}],"container-title":"Problems of Information Transmission","original-title":[],"language":"en","issued":{"date-parts":[[2009,12]]},"URL":"http://dx.doi.org/10.1134/S0032946009040024","ISSN":["0032-9460","1608-3253"],"container-title-short":"Probl Inf Transm","note":"alternative-id:4057"},{"id":"doi:10.1111/j.1469-1809.1943.tb02332","type":"journal-article","publisher":"Wiley","issue":"1","abstract":"<jats:p>The articles published by the Annals of Eugenics (1925–1954) have been made available online as an historical archive intended for scholarly use. The work of eugenicists was often pervaded by prejudice against racial, ethnic and disabled groups. The online publication of this material for scholarly research purposes is not an endorsement of those views nor a promotion of eugenics in any way.</jats:p>","DOI":"10.1111/j.1469-1809.1943.tb02332.x","page":"283-290","source":"Crossref","title":"A SYSTEM OF CONFOUNDING FOR FACTORS WITH MORE THAN TWO ALTERNATIVES, GIVING COMPLETELY ORTHOGONAL CUBES AND HIGHER POWERS","volume":"12","author":[{"given":"R. A.","family":"FISHER","sequence":"first","affiliation":[]}],"container-title":"Annals of Eugenics","original-title":[],"language":"en","issued":{"date-parts":[[1943,1]]},"URL":"http://dx.doi.org/10.1111/j.1469-1809.1943.tb02332.x","ISSN":["2050-1420","2050-1439"],"container-title-short":"Annals of Eugenics","note":"alternative-id:10.1111/j.1469-1809.1943.tb02332.x"},{"id":"arxiv:quant-ph/9607027","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"Pasting Quantum Codes","issued":{"date-parts":[[1996,7,31]]},"note":"arxivid:quant-ph/9607027\narxiv_version_number:1"},{"id":"doi:10.2140/iig.2008.6.53","type":"journal-article","publisher":"Mathematical Sciences Publishers","issue":"1","DOI":"10.2140/iig.2008.6.53","page":"53-71","source":"Crossref","title":"The geometry of quantum codes","volume":"6","author":[{"given":"Jürgen","family":"Bierbrauer","sequence":"first","affiliation":[]},{"given":"Giorgio","family":"Faina","sequence":"additional","affiliation":[]},{"given":"Massimo","family":"Giulietti","sequence":"additional","affiliation":[]},{"given":"Stefano","family":"Marcugini","sequence":"additional","affiliation":[]},{"given":"Fernanda","family":"Pambianco","sequence":"additional","affiliation":[]}],"container-title":"Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial","original-title":[],"language":"en","issued":{"date-parts":[[2008]]},"URL":"http://dx.doi.org/10.2140/iig.2008.6.53","ISSN":["2640-7345","2640-7337"],"container-title-short":"Innov. Incidence Geom."},{"id":"arxiv:1503.06237","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1007/jhep06(2015)149","source":"Crossref","title":"Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence","volume":"2015","author":[{"given":"Fernando","family":"Pastawski","sequence":"first","affiliation":[]},{"given":"Beni","family":"Yoshida","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Harlow","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2015,6]]},"URL":"http://dx.doi.org/10.1007/JHEP06(2015)149","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"149","note":"alternative-id:1540\narxivid:1503.06237"},{"id":"manual:-Veikkaus-Lotto-(Veikkaaja)-m","type":"book","title":"Veikkaus-Lotto (Veikkaaja) magazine, issues 27, 28, and 33","publisher":"August-September","issued":"1947"},{"id":"doi:10.1016/0022-314X(87)90099-0","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0022-314x(87)90099-0","page":"96-116","source":"Crossref","title":"The diophantine equation x2 = 4qa2 + 4q + 1, with an application to coding theory","volume":"26","author":[{"given":"Nikos","family":"Tzanakis","sequence":"first","affiliation":[]},{"given":"John","family":"Wolfskill","sequence":"additional","affiliation":[]}],"container-title":"Journal of Number Theory","original-title":[],"language":"en","issued":{"date-parts":[[1987,5]]},"URL":"http://dx.doi.org/10.1016/0022-314X(87)90099-0","ISSN":["0022-314X"],"container-title-short":"Journal of Number Theory","note":"alternative-id:0022314X87900990"},{"id":"doi:10.1017/9781009057226","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Strongly regular graphs lie at the intersection of statistical design, group theory, finite geometry, information and coding theory, and extremal combinatorics. This monograph collects all the major known results together for the first time in book form, creating an invaluable text that researchers in algebraic combinatorics and related areas will refer to for years to come. The book covers the theory of strongly regular graphs, polar graphs, rank 3 graphs associated to buildings and Fischer groups, cyclotomic graphs, two-weight codes and graphs related to combinatorial configurations such as Latin squares, quasi-symmetric designs and spherical designs. It gives the complete classification of rank 3 graphs, including some new constructions. More than 100 graphs are treated individually. Some unified and streamlined proofs are featured, along with original material including a new approach to the (affine) half spin graphs of rank 5 hyperbolic polar spaces.</jats:p>","DOI":"10.1017/9781009057226","source":"Crossref","title":"Strongly Regular Graphs","author":[{"given":"Andries E.","family":"Brouwer","sequence":"first","affiliation":[]},{"given":"H.","family":"Van Maldeghem","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2022,1,6]]},"ISBN":["9781009057226","9781316512036"],"URL":"http://dx.doi.org/10.1017/9781009057226","note":"edition-number:1"},{"id":"doi:10.1016/0097-3165(83)90002-X","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0097-3165(83)90002-x","page":"126-144","source":"Crossref","title":"The packing problem for projective geometries over GF(3) with dimension greater than five","volume":"35","author":[{"given":"Richard A","family":"Games","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1983,9]]},"URL":"http://dx.doi.org/10.1016/0097-3165(83)90002-X","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:009731658390002X"},{"id":"manual:-R.-Sch-rer-and-W.-Ch.-Schmid","type":"article-journal","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Tetracode. From MinT—the database of optimal net, code, OA, and OOA parameters","container-title":"Version","issued":"2015-09-03","URL":"https://web.archive.org/web/20240525080503/http://mint.sbg.ac.at/desc_CTetra.html","note":"Available at \\url{https://web.archive.org/web/20240525080503/http://mint.sbg.ac.at/desc_CTetra.html}"},{"id":"arxiv:1608.06674","type":"article-journal","author":[{"given":"Yongsheng","family":"Tang"},{"given":"Shixin","family":"Zhu"},{"given":"Xiaoshan","family":"Kai"},{"given":"Jian","family":"Ding"}],"title":"New quantum codes from dual-containing cyclic codes over finite rings","issued":{"date-parts":[[2016,8,24]]},"note":"arxivid:1608.06674\narxiv_version_number:1"},{"id":"doi:10.1016/0097-3165(86)90066-X","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0097-3165(86)90066-x","page":"258-276","source":"Crossref","title":"Q-codes","volume":"43","author":[{"given":"Vera","family":"Pless","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1986,11]]},"URL":"http://dx.doi.org/10.1016/0097-3165(86)90066-X","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:009731658690066X"},{"id":"doi:10.7907/CZA5-PX53","type":"thesis","categories":["Mathematics","FOS: Mathematics","Duadic codes","Smith normal form"],"language":"en","author":[{"family":"Rushanan","given":"Joseph John"}],"issued":{"date-parts":[[1986]]},"abstract":"This thesis consists of two independent chapters.\n\nThe first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix A to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation.\n\nThe second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring F[G], where G is a finite abelian group of odd order n and F is a finite field with characteristic relatively prime to n. We define generalized Q-codes from a pair of idempotents of F[G] and an automorphism of G which together obey two simple equations. These codes are (n, (n+1)/2) and (n, (n-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group G. When F has characteristic 2, we give simple numerical conditions on G and F which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.","DOI":"10.7907/CZA5-PX53","publisher":"California Institute of Technology","title":"Topics in Integral Matrices and Abelian Group Codes","URL":"https://resolver.caltech.edu/CaltechETD:etd-05022003-113743","version":"Final","note":"copyright:No commercial reproduction, distribution, display or performance rights in this work are provided."},{"id":"doi:10.1109/TIT.1987.1057300","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1987.1057300","page":"432-433","source":"Crossref","title":"Duadic codes (Corresp.)","volume":"33","author":[{"given":"M.","family":"Smid","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1987,5]]},"URL":"http://dx.doi.org/10.1109/TIT.1987.1057300","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1978.1055965","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1978.1055965","page":"730-737","source":"Crossref","title":"Generalized quadratic residue codes","volume":"24","author":[{"given":"J.","family":"van Lint","sequence":"first","affiliation":[]},{"given":"F.","family":"MacWilliams","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1978,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1978.1055965","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/978-3-662-39641-4_6","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-662-39641-4_6","page":"285-310","source":"Crossref","title":"Generalized quadratic-residue codes","author":[{"given":"J. H.","family":"Lint","sequence":"first","affiliation":[]}],"container-title":"Algebraic Coding Theory and Applications","original-title":[],"issued":{"date-parts":[[1979]]},"ISBN":["9783662387528","9783662396414"],"URL":"http://dx.doi.org/10.1007/978-3-662-39641-4_6","note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:1704.01199","type":"article-journal","author":[{"given":"Cunsheng","family":"Ding"},{"given":"Hao","family":"Liu"},{"given":"Vladimir D.","family":"Tonchev"}],"title":"All binary linear codes that are invariant under $\\PSL_2(n)$","issued":{"date-parts":[[2017,4,4]]},"note":"arxivid:1704.01199\narxiv_version_number:1"},{"id":"arxiv:0810.3787","type":"journal-article","publisher":"Wiley","issue":"5","DOI":"10.1112/blms/bdp026","page":"769-778","source":"Crossref","title":"Automorphisms of doubly even self-dual binary codes","volume":"41","author":[{"given":"Annika","family":"Günther","sequence":"first","affiliation":[]},{"given":"Gabriele","family":"Nebe","sequence":"additional","affiliation":[]}],"container-title":"Bulletin of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[2009,5,27]]},"URL":"http://dx.doi.org/10.1112/blms/bdp026","ISSN":["0024-6093"],"container-title-short":"Bulletin of the London Mathematical Society","note":"arxivid:0810.3787"},{"id":"manual:-A.-G-nther.-Automorphism-gro","type":"thesis","author":[{"family":"Günther","given":"A."}],"title":"Automorphism groups of self-dual codes","genre":"PhD thesis,","publisher":"Aachen, Techn. Hochsch","issued":"2009"},{"id":"doi:10.1515/9783110870138","type":"edited-book","publisher":"DE GRUYTER","DOI":"10.1515/9783110870138","source":"Crossref","title":"Finite Soluble Groups","author":[{"given":"Klaus","family":"Doerk","sequence":"first","affiliation":[]},{"given":"Trevor O.","family":"Hawkes","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1992,12,31]]},"ISBN":["9783110128925","9783110870138"],"URL":"http://dx.doi.org/10.1515/9783110870138","note":"alternative-id:10.1515/9783110870138"},{"id":"arxiv:2412.09695","type":"article-journal","author":[{"given":"Miguel","family":"Sales-Cabrera"},{"given":"Xaro","family":"Soler-Escrivà"},{"given":"Víctor","family":"Sotomayor"}],"title":"Codes in algebras of direct products of groups","issued":{"date-parts":[[2025,12,15]]},"note":"arxivid:2412.09695\narxiv_version_number:3"},{"id":"doi:10.1007/s10623-008-9261-z","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s10623-008-9261-z","page":"289-300","source":"Crossref","title":"An intrinsical description of group codes","volume":"51","author":[{"given":"José Joaquín","family":"Bernal","sequence":"first","affiliation":[]},{"given":"Ángel","family":"del Río","sequence":"additional","affiliation":[]},{"given":"Juan Jacobo","family":"Simón","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2009,1,6]]},"URL":"http://dx.doi.org/10.1007/s10623-008-9261-z","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9261"},{"id":"arxiv:2306.16400","type":"article-journal","author":[{"given":"Hsiang-Ku","family":"Lin"},{"given":"Leonid P.","family":"Pryadko"}],"title":"Quantum two-block group algebra codes","issued":{"date-parts":[[2023,6,28]]},"note":"arxivid:2306.16400\narxiv_version_number:1"},{"id":"manual:-V.-I.-Levenshtein-(1960).-A-","type":"article-journal","author":[{"family":"Levenshtein","given":"V.I."}],"title":"A class of systematic codes","container-title":"Doklady Akademii Nauk","volume":"131","publisher":"Russian Academy of Sciences","issue":"5","issued":"1960","page":"1011–1014"},{"id":"manual:-C.-R.-Rao-Hypercubes-of-stre","type":"article-journal","author":[{"family":"Rao","given":"C.R."}],"title":"Hypercubes of strength d leading to confounded designs in factorial experiments","volume":"38","container-title":"Bull. Calcutta Math. Soc","page":"67–78"},{"id":"doi:10.2307/2983576","type":"journal-article","publisher":"Oxford University Press (OUP)","issue":"1","DOI":"10.2307/2983576","page":"128-139","source":"Crossref","title":"Factorial Experiments Derivable from Combinatorial Arrangements of Arrays","volume":"9","author":[{"given":"C. Radhakrishna","family":"Rao","sequence":"first","affiliation":[{"name":"King’s College , Cambridge"}]}],"container-title":"Journal of the Royal Statistical Society Series B: Statistical Methodology","original-title":[],"language":"en","issued":{"date-parts":[[1947,1,1]]},"URL":"http://dx.doi.org/10.2307/2983576","ISSN":["1369-7412","1467-9868"]},{"id":"doi:10.1017/S0013091500002650","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"3","abstract":"<jats:p>In this paper, I introduce a class of arrangements called <jats:italic>arrays of strength d</jats:italic> and discuss methods of constructing them with the help of finite geometrical configurations and algebraic groups involving elements of a Galois field. The definitions of arrays of strength <jats:italic>d</jats:italic> and other configurations that are used are given below.</jats:p>","DOI":"10.1017/s0013091500002650","page":"119-125","source":"Crossref","title":"On a Class of Arrangements","volume":"8","author":[{"given":"C. Radhakrishna","family":"Rao","sequence":"first","affiliation":[]}],"container-title":"Proceedings of the Edinburgh Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1949,12]]},"URL":"http://dx.doi.org/10.1017/S0013091500002650","ISSN":["0013-0915","1464-3839"],"container-title-short":"Proceedings of the Edinburgh Mathematical Society","note":"alternative-id:S0013091500002650"},{"id":"arxiv:1302.4295","type":"article-journal","author":[{"given":"Greg","family":"Kuperberg"},{"given":"Shachar","family":"Lovett"},{"given":"Ron","family":"Peled"}],"title":"Probabilistic existence of regular combinatorial structures","issued":{"date-parts":[[2017,4,6]]},"note":"arxivid:1302.4295\narxiv_version_number:3"},{"id":"arxiv:1404.3586","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.90.022316","source":"Crossref","title":"Genuinely multipartite entangled states and orthogonal arrays","volume":"90","author":[{"given":"Dardo","family":"Goyeneche","sequence":"first","affiliation":[]},{"given":"Karol","family":"Życzkowski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,8,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.022316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022316","note":"arxivid:1404.3586"},{"id":"arxiv:2303.15001","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s10773-023-05316-w","source":"Crossref","title":"Quantum k-Uniform States From Quantum Orthogonal Arrays","volume":"62","author":[{"given":"Yajuan","family":"Zang","sequence":"first","affiliation":[]},{"given":"Zihong","family":"Tian","sequence":"additional","affiliation":[]},{"given":"Shao-Ming","family":"Fei","sequence":"additional","affiliation":[]},{"given":"Hui-Juan","family":"Zuo","sequence":"additional","affiliation":[]}],"container-title":"International Journal of Theoretical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,23]]},"URL":"http://dx.doi.org/10.1007/s10773-023-05316-w","ISSN":["1572-9575"],"container-title-short":"Int J Theor Phys","page":"73","note":"alternative-id:5316\narxivid:2303.15001"},{"id":"doi:10.1038/s41534-019-0165-8","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>A pure quantum state of <jats:italic>N</jats:italic> subsystems, each with <jats:italic>d</jats:italic> levels, is said to be <jats:italic>k</jats:italic>-uniform if all of its reductions to <jats:italic>k</jats:italic> qudits are maximally mixed. Only the uniform states obtained from orthogonal arrays (OAs) are considered throughout this work. The Hamming distances of OAs are specially applied to the theory of quantum information. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant orthogonal arrays (IrOAs), then answer the open questions of whether there exist 3-uniform states of <jats:italic>N</jats:italic> qubits and 2-uniform states of <jats:italic>N</jats:italic> qutrits, and whether 3-uniform states of qudits (<jats:italic>d</jats:italic> &gt; 2) for high values of <jats:italic>N</jats:italic> can be explicitly constructed. In fact, we obtain 3-uniform states for an arbitrary number of <jats:italic>N</jats:italic> ≥ 8 qubits and 2-uniform states of <jats:italic>N</jats:italic> qutrits for every <jats:italic>N</jats:italic> ≥ 4. Additionally, we provide explicit constructions of the 3-uniform states of <jats:italic>N</jats:italic> ≥ 8 qutrits, <jats:italic>N</jats:italic> = 6 and <jats:italic>N</jats:italic> ≥ 8 ququarts and ququints, <jats:italic>N</jats:italic> ≥ 6 qudits having <jats:italic>d</jats:italic> levels for any prime power <jats:italic>d</jats:italic> &gt; 6, and <jats:italic>N</jats:italic> = 8 and <jats:italic>N</jats:italic> ≥ 12 qudits having <jats:italic>d</jats:italic> levels for non-prime-power <jats:italic>d</jats:italic> ≥ 6. Moreover, we describe an explicit construction scheme for the 2-uniform states of qudits having <jats:italic>d</jats:italic> ≥ 4 levels. The proofs of existence of the 2-uniform states of <jats:italic>N</jats:italic> ≥ 6 qubits are simplified by using a class of OAs. Two special 3-uniform states are obtained from IrOA(32, 10, 2, 3) and IrOA(32, 11, 2, 3) using the interaction column property of OAs.</jats:p>","DOI":"10.1038/s41534-019-0165-8","source":"Crossref","title":"Two and three-uniform states from irredundant orthogonal arrays","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0001-9000-802X","authenticated-orcid":false,"given":"Shan-Qi","family":"Pang","sequence":"first","affiliation":[]},{"given":"Xiao","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Xiao","family":"Lin","sequence":"additional","affiliation":[]},{"given":"Qing-Juan","family":"Zhang","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2019,6,17]]},"URL":"http://dx.doi.org/10.1038/s41534-019-0165-8","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"52","note":"updated-by:{\"DOI\":\"10.1038/s41534-019-0178-3\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2019,8,2]],\"date-time\":\"2019-08-02T00:00:00Z\",\"timestamp\":1564704000000}}\nalternative-id:165"},{"id":"doi:10.1103/PhysRevA.99.042332","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.99.042332","source":"Crossref","title":"<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>\n-uniform quantum states arising from orthogonal arrays","volume":"99","author":[{"given":"Mao-Sheng","family":"Li","sequence":"first","affiliation":[]},{"given":"Yan-Ling","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,4,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.042332","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042332"},{"id":"doi:10.3390/e25040680","type":"journal-article","publisher":"MDPI AG","issue":"4","abstract":"<jats:p>In this paper, by using the Hamming distance, we establish a relation between quantum error-correcting codes ((N,K,d+1))s and orthogonal arrays with orthogonal partitions. Therefore, this is a generalization of the relation between quantum error-correcting codes ((N,1,d+1))s and irredundant orthogonal arrays. This relation is used for the construction of pure quantum error-correcting codes. As applications of this method, numerous infinite families of optimal quantum codes can be constructed explicitly such as ((3,s,2))s for all si≥3, ((4,s2,2))s for all si≥5, ((5,s,3))s for all si≥4, ((6,s2,3))s for all si≥5, ((7,s3,3))s for all si≥7, ((8,s2,4))s for all si≥9, ((9,s3,4))s for all si≥11, ((9,s,5))s for all si≥9, ((10,s2,5))s for all si≥11, ((11,s,6))s for all si≥11, and ((12,s2,6))s for all si≥13, where s=s1⋯sn and s1,…,sn are all prime powers. The advantages of our approach over existing methods lie in the facts that these results are not just existence results, but constructive results, the codes constructed are pure, and each basis state of these codes has far less terms. Moreover, the above method developed can be extended to construction of quantum error-correcting codes over mixed alphabets.</jats:p>","DOI":"10.3390/e25040680","page":"680","source":"Crossref","title":"Quantum Error-Correcting Codes Based on Orthogonal Arrays","volume":"25","author":[{"given":"Rong","family":"Yan","sequence":"first","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}]},{"ORCID":"https://orcid.org/0000-0001-9000-802X","authenticated-orcid":false,"given":"Shanqi","family":"Pang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}]},{"given":"Mengqian","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}]},{"given":"Fuyuan","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}]}],"container-title":"Entropy","original-title":[],"language":"en","issued":{"date-parts":[[2023,4,19]]},"URL":"http://dx.doi.org/10.3390/e25040680","ISSN":["1099-4300"],"container-title-short":"Entropy","note":"alternative-id:e25040680"},{"id":"doi:10.1109/TIT.1984.1056978","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1984.1056978","page":"800-804","source":"Crossref","title":"The complexity of computing the covering radius of a code","volume":"30","author":[{"given":"A.","family":"McLoughlin","sequence":"first","affiliation":[{"name":"IEEE IT"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1984,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1984.1056978","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-G.-Cohen-I.-Honkala-S.-Litsy","type":"book","author":[{"family":"Cohen","given":"G."},{"family":"Honkala","given":"I."},{"family":"Litsyn","given":"S."},{"family":"Lobstein","given":"A."}],"title":"Covering Codes","publisher":"Elsevier","issued":"1997"},{"id":"doi:10.2307/2974552","type":"journal-article","publisher":"JSTOR","issue":"7","DOI":"10.2307/2974552","page":"579","source":"Crossref","title":"Football Pools--A Game for Mathematicians","volume":"102","author":[{"given":"Heikki","family":"Hamalainen","sequence":"first","affiliation":[]},{"given":"Iiro","family":"Honkala","sequence":"additional","affiliation":[]},{"given":"Simon","family":"Litsyn","sequence":"additional","affiliation":[]},{"given":"Patric","family":"Ostergard","sequence":"additional","affiliation":[]}],"container-title":"The American Mathematical Monthly","original-title":[],"issued":{"date-parts":[[1995,8]]},"URL":"http://dx.doi.org/10.2307/2974552","ISSN":["0002-9890"],"container-title-short":"The American Mathematical Monthly"},{"id":"preset:CoveringBook","type":"book","author":[{"family":"Cohen","given":"G."},{"family":"Honkala","given":"I."},{"family":"Litsyn","given":"S."},{"family":"Lobstein","given":"A."}],"title":"\\emph{Covering Codes}","publisher":"Elsevier","issued":"1997"},{"id":"doi:10.1147/rd.31.0025","type":"journal-article","publisher":"IBM","issue":"1","DOI":"10.1147/rd.31.0025","page":"25-34","source":"Crossref","title":"On the Mathematical Theory of Error-Correcting Codes","volume":"3","author":[{"given":"H. S.","family":"Shapiro","sequence":"first","affiliation":[]},{"given":"D. L.","family":"Slotnick","sequence":"additional","affiliation":[]}],"container-title":"IBM Journal of Research and Development","original-title":[],"issued":{"date-parts":[[1959,1]]},"URL":"http://dx.doi.org/10.1147/rd.31.0025","ISSN":["0018-8646","0018-8646"],"container-title-short":"IBM J. Res. &amp; Dev."},{"id":"doi:10.1016/S0019-9958(68)90167-8","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0019-9958(68)90167-8","page":"23-26","source":"Crossref","title":"On linear and nonlinear single-error-correcting q-nary perfect codes","volume":"12","author":[{"given":"J.","family":"Schnheim","sequence":"first","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1968,1]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(68)90167-8","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S0019995868901678"},{"id":"manual:-B.-Lindstr-m-On-group-and-no","type":"article-journal","author":[{"family":"Lindström","given":"B."}],"title":"On group and nongroup perfect codes in q symbols","volume":"25","container-title":"Mathematica Scandinavica","issue":"2","issued":"1969","page":"149–158"},{"id":"doi:10.1016/0012-365X(84)90121-3","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0012-365x(84)90121-3","page":"209-212","source":"Crossref","title":"Une nouvelle famille de 3-codes parfaits sur GF(q)","volume":"49","author":[{"given":"M.","family":"Mollard","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"fr","issued":{"date-parts":[[1984,4]]},"URL":"http://dx.doi.org/10.1016/0012-365X(84)90121-3","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X84901213"},{"id":"doi:10.1137/0607013","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","DOI":"10.1137/0607013","page":"113-115","source":"Crossref","title":"A Generalized Parity Function and Its Use in the Construction of Perfect Codes","volume":"7","author":[{"given":"M.","family":"Mollard","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Algebraic Discrete Methods","original-title":[],"language":"en","issued":{"date-parts":[[1986,1]]},"URL":"http://dx.doi.org/10.1137/0607013","ISSN":["0196-5212","2168-345X"],"container-title-short":"SIAM. J. on Algebraic and Discrete Methods","note":"alternative-id:10.1137/0607013"},{"id":"doi:10.1007/BF02579339","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf02579339","page":"21-33","source":"Crossref","title":"Algebraic techniques for nonlinear codes","volume":"3","author":[{"given":"H.","family":"Bauer","sequence":"first","affiliation":[]},{"given":"B.","family":"Ganter","sequence":"additional","affiliation":[]},{"given":"F.","family":"Hergert","sequence":"additional","affiliation":[]}],"container-title":"Combinatorica","original-title":[],"language":"en","issued":{"date-parts":[[1983,3]]},"URL":"http://dx.doi.org/10.1007/BF02579339","ISSN":["0209-9683","1439-6912"],"container-title-short":"Combinatorica","note":"alternative-id:BF02579339"},{"id":"manual:-J.-M.-Laborde-and-M.-Sch-tze","type":"article-journal","author":[{"family":"Laborde","given":"J.M."},{"family":"Schützenberger","given":"M."}],"title":"Une nouvelle famille de codes binaires, parfaits, non linéaires. Comptes rendus des séances de l'Académie des sciences","volume":"297","container-title":"Série","issue":"1","issued":"1983","page":"67–70"},{"id":"doi:10.1142/12823","type":"monograph","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/12823","source":"Crossref","title":"Perfect Codes and Related Structures","author":[{"given":"Tuvi","family":"Etzion","sequence":"first","affiliation":[{"name":"TechnionIsrael"}]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2022,1,27]]},"ISBN":["9789811255878","9789811255885"],"URL":"http://dx.doi.org/10.1142/12823","note":"alternative-id:10.1142/12823"},{"id":"arxiv:1703.08684","type":"article-journal","author":[{"given":"J.","family":"Borges"},{"given":"J.","family":"Rifà"},{"given":"V. A.","family":"Zinoviev"}],"title":"On Completely Regular Codes","issued":{"date-parts":[[2017,3,25]]},"note":"arxivid:1703.08684\narxiv_version_number:1"},{"id":"manual:-J.-M.-Goethals-and-H.-C.-A.-","type":"article-journal","author":[{"family":"Goethals","given":"J.M."},{"family":"Tilborg","given":"H.C.A.","particle":"Van"}],"title":"Uniformly packed codes","volume":"30","container-title":"Philips Research Reports","issue":"1","issued":"1975"},{"id":"manual:-H.-C.-A.-van-Tilborg.-Unifor","type":"book","author":[{"family":"Tilborg","given":"H.C.A.","particle":"van"}],"title":"Uniformly packed codes","publisher":"Technische Hogeschool Eindhoven","issued":"1976"},{"id":"doi:10.1007/BF01294651","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/bf01294651","page":"273-337","source":"Crossref","title":"Point sets and sequences with small discrepancy","volume":"104","author":[{"given":"Harald","family":"Niederreiter","sequence":"first","affiliation":[]}],"container-title":"Monatshefte für Mathematik","original-title":[],"language":"en","issued":{"date-parts":[[1987,12]]},"URL":"http://dx.doi.org/10.1007/BF01294651","ISSN":["0026-9255","1436-5081"],"container-title-short":"Monatshefte für Mathematik","note":"alternative-id:BF01294651"},{"id":"doi:10.1016/0012-365X(91)90315-S","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0012-365x(91)90315-s","page":"221-228","source":"Crossref","title":"A combinatorial problem for vector spaces over finite fields","volume":"96","author":[{"given":"Harald","family":"Niederreiter","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1991,12]]},"URL":"http://dx.doi.org/10.1016/0012-365X(91)90315-S","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9190315S"},{"id":"doi:10.1016/0012-365X(92)90566-X","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/0012-365x(92)90566-x","page":"361-367","source":"Crossref","title":"Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes","volume":"106-107","author":[{"given":"Harald","family":"Niederreiter","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1992,9]]},"URL":"http://dx.doi.org/10.1016/0012-365X(92)90566-X","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"special_numbering:C\nalternative-id:0012365X9290566X"},{"id":"doi:10.1007/s10623-006-9025-6","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s10623-006-9025-6","page":"127-143","source":"Crossref","title":"A direct approach to linear programming bounds for codes and tms-nets","volume":"42","author":[{"given":"Jürgen","family":"Bierbrauer","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2006,12,8]]},"URL":"http://dx.doi.org/10.1007/s10623-006-9025-6","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:9025"},{"id":"doi:10.1016/0012-365X(94)00228-B","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(94)00228-b","page":"57-72","source":"Crossref","title":"Codes with a poset metric","volume":"147","author":[{"given":"Richard A.","family":"Brualdi","sequence":"first","affiliation":[]},{"given":"Janine Smolin","family":"Graves","sequence":"additional","affiliation":[]},{"given":"K.Mark","family":"Lawrence","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1995,12]]},"URL":"http://dx.doi.org/10.1016/0012-365X(94)00228-B","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9400228B"},{"id":"arxiv:1205.1090","type":"article-journal","author":[{"given":"Soohak","family":"Choi"},{"given":"Jong Yoon","family":"Hyun"},{"given":"Hyun Kwang","family":"Kim"},{"given":"Dong Yeol","family":"Oh"}],"title":"MacWilliams-type equivalence relations","issued":{"date-parts":[[2013,1,1]]},"note":"arxivid:1205.1090\narxiv_version_number:2"},{"id":"doi:10.1007/978-3-319-93821-9","type":"book","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-93821-9","source":"Crossref","title":"Poset Codes: Partial Orders, Metrics and Coding Theory","author":[{"given":"Marcelo","family":"Firer","sequence":"first","affiliation":[]},{"given":"Marcelo Muniz","family":"S. Alves","sequence":"additional","affiliation":[]},{"given":"Jerry Anderson","family":"Pinheiro","sequence":"additional","affiliation":[]},{"given":"Luciano","family":"Panek","sequence":"additional","affiliation":[]}],"container-title":"SpringerBriefs in Mathematics","original-title":[],"issued":{"date-parts":[[2018]]},"ISBN":["9783319938202","9783319938219"],"URL":"http://dx.doi.org/10.1007/978-3-319-93821-9","ISSN":["2191-8198","2191-8201"],"note":"publisher-location:Cham"},{"id":"doi:10.1007/s00200-009-0113-5","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5-6","DOI":"10.1007/s00200-009-0113-5","page":"497-507","source":"Crossref","title":"Construction and decoding of matrix-product codes from nested codes","volume":"20","author":[{"given":"Fernando","family":"Hernando","sequence":"first","affiliation":[]},{"given":"Kristine","family":"Lally","sequence":"additional","affiliation":[]},{"given":"Diego","family":"Ruano","sequence":"additional","affiliation":[]}],"container-title":"Applicable Algebra in Engineering, Communication and Computing","original-title":[],"language":"en","issued":{"date-parts":[[2009,10,9]]},"URL":"http://dx.doi.org/10.1007/s00200-009-0113-5","ISSN":["0938-1279","1432-0622"],"container-title-short":"AAECC","note":"alternative-id:113"},{"id":"doi:10.1007/s11128-020-02921-0","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","DOI":"10.1007/s11128-020-02921-0","source":"Crossref","title":"Construction of new quantum codes via Hermitian dual-containing matrix-product codes","volume":"19","author":[{"given":"Meng","family":"Cao","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4964-7959","authenticated-orcid":false,"given":"Jianlian","family":"Cui","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2020,11,24]]},"URL":"http://dx.doi.org/10.1007/s11128-020-02921-0","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","page":"427","note":"alternative-id:2921"},{"id":"arxiv:1604.05823","type":"article-journal","author":[{"given":"Xiusheng","family":"Liu"},{"given":"Hualu","family":"Liu"},{"given":"Long","family":"Yu"}],"title":"On New Quantum Codes From Matrix Product Codes","issued":{"date-parts":[[2021,5,25]]},"note":"arxivid:1604.05823\narxiv_version_number:2"},{"id":"arxiv:2209.13474","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Several notions of code products are known in quantum error correction, such as hypergraph products, homological products, lifted products, balanced products, to name a few. In this paper we introduce a new product code construction which is a natural generalization of classical product codes to quantum codes: starting from a set of component Calderbank-Shor-Steane (CSS) codes, a larger CSS code is obtained where both <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math> parity checks and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> parity checks are associated to classical product codes. We deduce several properties of product CSS codes from the properties of the component codes, including bounds to the code distance, and show that built-in redundancies in the parity checks result in so-called meta-checks which can be exploited to correct syndrome read-out errors. We then specialize to the case of single-parity-check (SPC) product codes which in the classical domain are a common choice for constructing product codes. Logical error rate simulations of a SPC <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>3</mml:mn></mml:math>-fold product CSS code having parameters <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>512</mml:mn><mml:mo>,</mml:mo><mml:mn>174</mml:mn><mml:mo>,</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> are shown under both a maximum likelihood decoder for the erasure channel and belief propagation decoding for depolarizing noise. We compare the results with other codes of comparable length and dimension, including a code from the family of asymptotically good Tanner codes. We observe that our reference product CSS code outperforms all the other examined codes.</jats:p>","DOI":"10.22331/q-2024-07-22-1420","page":"1420","source":"Crossref","title":"Classical product code constructions for quantum Calderbank-Shor-Steane codes","volume":"8","author":[{"given":"Dimiter","family":"Ostrev","sequence":"first","affiliation":[{"name":"Institute of Communications and Navigation, German Aerospace Center (DLR), 82234 Weßling, Germany"}]},{"given":"Davide","family":"Orsucci","sequence":"additional","affiliation":[{"name":"Institute of Communications and Navigation, German Aerospace Center (DLR), 82234 Weßling, Germany"}]},{"given":"Francisco","family":"Lázaro","sequence":"additional","affiliation":[{"name":"Institute of Communications and Navigation, German Aerospace Center (DLR), 82234 Weßling, Germany"}]},{"given":"Balazs","family":"Matuz","sequence":"additional","affiliation":[{"name":"Institute of Communications and Navigation, German Aerospace Center (DLR), 82234 Weßling, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,22]]},"URL":"http://dx.doi.org/10.22331/q-2024-07-22-1420","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2209.13474"},{"id":"doi:10.1109/TIT.1965.1053771","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1965.1053771","page":"281-284","source":"Crossref","title":"On codes derivable from the tensor product of check matrices","volume":"11","author":[{"given":"J.","family":"Wolf","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1965,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1965.1053771","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ITW2.2006.323741","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw2.2006.323741","page":"6-10","source":"Crossref","title":"An Introduction to Tensor Product Codes and Applications to Digital Storage Systems","author":[{"given":"Jack Keil","family":"Wolf","sequence":"first","affiliation":[]}],"event":"2006 IEEE Information Theory Workshop - ITW '06 Chengdu","container-title":"2006 IEEE Information Theory Workshop - ITW '06 Chengdu","original-title":[],"issued":{"date-parts":[[2006,10]]},"URL":"http://dx.doi.org/10.1109/ITW2.2006.323741"},{"id":"doi:10.1109/TIT.1960.1057584","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1960.1057584","page":"445-450","source":"Crossref","title":"Binary codes with specified minimum distance","volume":"6","author":[{"given":"M.","family":"Plotkin","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1960,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1960.1057584","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-S.-Dodunekov-Optimal-linear-","type":"thesis","author":[{"family":"Dodunekov","given":"S."}],"title":"Optimal linear codes","genre":"PhD thesis,","publisher":"Sofia","issued":"1985"},{"id":"doi:10.1147/rd.45.0532","type":"journal-article","publisher":"IBM","issue":"5","DOI":"10.1147/rd.45.0532","page":"532-542","source":"Crossref","title":"A Bound for Error-Correcting Codes","volume":"4","author":[{"given":"J. H.","family":"Griesmer","sequence":"first","affiliation":[]}],"container-title":"IBM Journal of Research and Development","original-title":[],"issued":{"date-parts":[[1960,11]]},"URL":"http://dx.doi.org/10.1147/rd.45.0532","ISSN":["0018-8646","0018-8646"],"container-title-short":"IBM J. Res. &amp; Dev."},{"id":"doi:10.1016/S0019-9958(65)90080-X","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0019-9958(65)90080-x","page":"170-179","source":"Crossref","title":"Algebraically punctured cyclic codes","volume":"8","author":[{"given":"G.","family":"Solomon","sequence":"first","affiliation":[]},{"given":"J.J.","family":"Stiffler","sequence":"additional","affiliation":[]}],"container-title":"Information and Control","original-title":[],"language":"en","issued":{"date-parts":[[1965,4]]},"URL":"http://dx.doi.org/10.1016/S0019-9958(65)90080-X","ISSN":["0019-9958"],"container-title-short":"Information and Control","note":"alternative-id:S001999586590080X"},{"id":"manual:-R.-R.-Varshamov-On-the-gener","type":"thesis","author":[{"family":"Varshamov","given":"R.R."}],"title":"On the general theory of linear coding","genre":"PhD thesis,","publisher":"Moscow State University","issued":"1959"},{"id":"doi:10.1006/jcta.1997.2864","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1006/jcta.1997.2864","page":"79-93","source":"Crossref","title":"Divisibility of Codes Meeting the Griesmer Bound","volume":"83","author":[{"given":"Harold N.","family":"Ward","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1998,7]]},"URL":"http://dx.doi.org/10.1006/jcta.1997.2864","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S0097316597928648"},{"id":"doi:10.1007/978-1-4613-0283-4_14","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-1-4613-0283-4_14","page":"247-256","source":"Crossref","title":"The Geometric Approach to Linear Codes","author":[{"given":"I. N.","family":"Landjev","sequence":"first","affiliation":[]}],"container-title":"Developments in Mathematics","original-title":[],"issued":{"date-parts":[[2001]]},"ISBN":["9781461379775","9781461302834"],"URL":"http://dx.doi.org/10.1007/978-1-4613-0283-4_14","ISSN":["1389-2177"],"note":"publisher-location:Boston, MA"},{"id":"doi:10.1016/0378-3758(89)90098-0","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0378-3758(89)90098-0","page":"323-336","source":"Crossref","title":"A characterization of {νμ + 1 + ε, νμ; t, q}-min.hypers and its applications to error-correcting codes and factorial designs","volume":"22","author":[{"given":"Noboru","family":"Hamada","sequence":"first","affiliation":[]},{"given":"Michel","family":"Deza","sequence":"additional","affiliation":[]}],"container-title":"Journal of Statistical Planning and Inference","original-title":[],"language":"en","issued":{"date-parts":[[1989,7]]},"URL":"http://dx.doi.org/10.1016/0378-3758(89)90098-0","ISSN":["0378-3758"],"container-title-short":"Journal of Statistical Planning and Inference","note":"alternative-id:0378375889900980"},{"id":"manual:-N.-Hamada-Characterization-o","type":"article-journal","author":[{"family":"Hamada","given":"N."}],"title":"Characterization of minihypers in a finite projective geometry and its applications to error-correcting codes, Bull","volume":"24","container-title":"Osaka Women's Univ","issued":"1987","page":"1–24"},{"id":"arxiv:0812.2674","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.81.032318","source":"Crossref","title":"Degenerate quantum codes and the quantum Hamming bound","volume":"81","author":[{"given":"Pradeep","family":"Sarvepalli","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,3,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.032318","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032318","note":"arxivid:0812.2674"},{"id":"doi:10.1109/JSAIT.2025.3562287","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/jsait.2025.3562287","page":"74-84","source":"Crossref","title":"Puncturing Quantum Stabilizer Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-0882-4621","authenticated-orcid":false,"given":"Jaron Skovsted","family":"Gundersen","sequence":"first","affiliation":[{"name":"Department of Electronic Systems, Aalborg University, Aalborg, Denmark"}]},{"ORCID":"https://orcid.org/0000-0002-9209-3739","authenticated-orcid":false,"given":"René Bødker","family":"Christensen","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences and the Department of Electronic Systems, Aalborg University, Aalborg, Denmark"}]},{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gdansk, Gda&#x0144;sk, Poland"}]},{"ORCID":"https://orcid.org/0000-0001-6195-4797","authenticated-orcid":false,"given":"Petar","family":"Popovski","sequence":"additional","affiliation":[{"name":"Department of Electronic Systems, Aalborg University, Aalborg, Denmark"}]},{"ORCID":"https://orcid.org/0000-0001-6719-8427","authenticated-orcid":false,"given":"Rafał","family":"Wisniewski","sequence":"additional","affiliation":[{"name":"Department of Electronic Systems, Aalborg University, Aalborg, Denmark"}]}],"container-title":"IEEE Journal on Selected Areas in Information Theory","original-title":[],"issued":{"date-parts":[[2025]]},"URL":"http://dx.doi.org/10.1109/JSAIT.2025.3562287","ISSN":["2641-8770"],"container-title-short":"IEEE J. Sel. Areas Inf. Theory"},{"id":"manual:-R.-Sch-rer-and-W.-Ch.-Schmid","type":"article-journal","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Denniston Codes. From MinT—the database of optimal net, code, OA, and OOA parameters","container-title":"Version","issued":"2015-09-03","URL":"https://web.archive.org/web/20240422025950/http://mint.sbg.ac.at/desc_CDenniston.html","note":"Available at \\url{https://web.archive.org/web/20240422025950/http://mint.sbg.ac.at/desc_CDenniston.html}"},{"id":"doi:10.1006/jcta.2002.3265","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1006/jcta.2002.3265","page":"128-142","source":"Crossref","title":"Projective Geometric Codes","volume":"99","author":[{"given":"Bhaskar","family":"Bagchi","sequence":"first","affiliation":[]},{"given":"S.P.","family":"Inamdar","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[2002,7]]},"URL":"http://dx.doi.org/10.1006/jcta.2002.3265","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S0097316502932656"},{"id":"manual:-M.-Lavrauw-L.-Storme-and-G.-","type":"chapter","author":[{"family":"Lavrauw","given":"M."},{"family":"Storme","given":"L."},{"family":"Voorde","given":"G.","particle":"Van de"}],"title":"Linear codes from projective spaces","editor":[{"family":"Bruen","given":"A."},{"family":"Wehlau","given":"D."}],"container-title":"Contemporary Mathematics","volume":"523","publisher":"American Mathematical Society (AMS","issued":"2010","page":"185–202","publisher-place":"Providence, RI, USA"},{"id":"manual:-E.-Prange-The-use-of-coset-e","type":"chapter","author":[{"family":"Prange","given":"E."}],"title":"The use of coset equivalence in the analysis and decoding of group codes","container-title":"AIR FORCE CAMBRIDGE RESEARCH LABS HANSCOM AFB MA","issued":"1959"},{"id":"doi:10.1109/TIT.1967.1053994","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1967.1053994","page":"305-307","source":"Crossref","title":"A class of majority logic decodable codes (Corresp.)","volume":"13","author":[{"given":"L.","family":"Rudolph","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1967,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1967.1053994","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1002/sapm1976552119","type":"journal-article","publisher":"Wiley","issue":"2","abstract":"<jats:p>The problem of computing the weight distribution of a linear code is proved to be a special case of the problem of finding the Tutte polynomial (or generalized dichromatic polynomial) of a combinatorial geometry. Many other connections between the two fields are discussed.</jats:p>","DOI":"10.1002/sapm1976552119","page":"119-128","source":"Crossref","title":"Weight Enumeration and the Geometry of Linear Codes","volume":"55","author":[{"given":"Curtis","family":"Greene","sequence":"first","affiliation":[]}],"container-title":"Studies in Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1976,6]]},"URL":"http://dx.doi.org/10.1002/sapm1976552119","ISSN":["0022-2526","1467-9590"],"container-title-short":"Stud Appl Math","note":"alternative-id:10.1002/sapm1976552119"},{"id":"doi:10.1023/A:1008294128110","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1023/a:1008294128110","page":"261-266","source":"Crossref","title":[],"volume":"11","author":[{"given":"A. E.","family":"Brouwer","sequence":"first","affiliation":[]},{"given":"M. van","family":"Eupen","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1023/A:1008294128110","ISSN":["0925-1022"],"note":"alternative-id:131334"},{"id":"doi:10.1112/blms/18.2.97","type":"journal-article","publisher":"Wiley","issue":"2","DOI":"10.1112/blms/18.2.97","page":"97-122","source":"Crossref","title":"The Geometry of Two-Weight Codes","volume":"18","author":[{"given":"R.","family":"Calderbank","sequence":"first","affiliation":[]},{"given":"W. M.","family":"Kantor","sequence":"additional","affiliation":[]}],"container-title":"Bulletin of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1986,3]]},"URL":"http://dx.doi.org/10.1112/blms/18.2.97","ISSN":["0024-6093"],"container-title-short":"Bulletin of the London Mathematical Society"},{"id":"doi:10.1016/j.ffa.2017.12.010","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.ffa.2017.12.010","page":"372-381","source":"Crossref","title":"The classification of antipodal two-weight linear codes","volume":"50","author":[{"given":"Dieter","family":"Jungnickel","sequence":"first","affiliation":[]},{"given":"Vladimir D.","family":"Tonchev","sequence":"additional","affiliation":[]}],"container-title":"Finite Fields and Their Applications","original-title":[],"language":"en","issued":{"date-parts":[[2018,3]]},"URL":"http://dx.doi.org/10.1016/j.ffa.2017.12.010","ISSN":["1071-5797"],"container-title-short":"Finite Fields and Their Applications","note":"special_numbering:C\nalternative-id:S1071579717301600"},{"id":"doi:10.1142/9789812837691_0003","type":"proceedings-article","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/9789812837691_0003","page":"60-123","source":"Crossref","title":"Linear Codes over Finite Chain Rings and Projective Hjelmslev Geometries","author":[{"given":"Thomas","family":"Honold","sequence":"first","affiliation":[{"name":"Department of Information and Electronic Engineering, Zhejiang University, 38 Zheda Road, 310027 Hangzhou, China"}]},{"given":"Ivan","family":"Landjev","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Informatics, BAS, 8 Acad. G. Bonchev str., 1113 Sofia, Bulgaria"},{"name":"New Bulgarian University, 21 Montevideo str., 1618 Sofia, Bulgaria"}]}],"event":"Codes Over Rings - The CIMPA Summer School","container-title":"Codes Over Rings","original-title":[],"issued":{"date-parts":[[2009,7]]},"URL":"http://dx.doi.org/10.1142/9789812837691_0003","note":"alternative-id:10.1142/9789812837691_0003\n10.1142/7140"},{"id":"arxiv:2007.05992","type":"journal-article","publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"2","abstract":"<jats:p>This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum error-correcting codes allow the negation of these effects in order to successfully restore the original quantum information. We briefly describe the necessary quantum-mechanical background to be able to understand how quantum error correction works. We go on to construct quantum codes: firstly qubit stabilizer codes, then qubit non-stabilizer codes, and finally codes with a higher local dimension. We will delve into the geometry of these codes. This allows one to deduce the parameters of the code efficiently, deduce the inequivalence between codes that have the same parameters, and presents a useful tool in deducing the feasibility of certain parameters. We also include sections on quantum maximum distance separable codes and the quantum MacWilliams identities.</jats:p>","DOI":"10.4171/aihpd/160","page":"337-405","source":"Crossref","title":"Quantum error-correcting codes and their geometries","volume":"10","author":[{"given":"Simeon","family":"Ball","sequence":"first","affiliation":[{"name":"Universitat Politécnica de Catalunya, Barcelona, Spain"}]},{"given":"Aina","family":"Centelles","sequence":"additional","affiliation":[{"name":"Universitat Politècnica de Catalunya, Barcelona, Spain"}]},{"given":"Felix","family":"Huber","sequence":"additional","affiliation":[{"name":"ICFO - The Institute of Photonic Sciences, Barcelona, Spain"},{"name":"Jagiellonian University, Kraków, Poland"}]}],"container-title":"Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions","original-title":[],"issued":{"date-parts":[[2023,2,9]]},"URL":"http://dx.doi.org/10.4171/aihpd/160","ISSN":["2308-5827","2308-5835"],"container-title-short":"Ann. Inst. Henri Poincaré Comb. Phys. Interact.","note":"arxivid:2007.05992"},{"id":"preset:Glynn04","type":"document","author":[{"family":"Glynn","given":"D.G."},{"family":"Gulliver","given":"T.A."},{"family":"Maks","given":"J.G."},{"family":"Gupta","given":"M.K."}],"title":"The geometry of additive quantum codes","note":"submitted to Springer-Verlag (2004"},{"id":"arxiv:2107.11281","type":"article-journal","author":[{"given":"Simeon","family":"Ball"},{"given":"Pablo","family":"Puig"}],"title":"The geometry of non-additive stabiliser codes","issued":{"date-parts":[[2021,7,23]]},"note":"arxivid:2107.11281\narxiv_version_number:1"},{"id":"manual:-S.-Ball-Stabiliser-codes-and","type":"paper-conference","author":[{"family":"Ball","given":"S."}],"title":"Stabiliser codes and quantum sets of lines","container-title":"Finite Geometries Sixth Irsee Conference","issued":"2022"},{"id":"manual:-B.-Qvist.-Some-remarks-conce","type":"article-journal","author":[{"family":"Qvist","given":"B."}],"title":"Some remarks concerning curves of the second degree in a finite plane","container-title":"Suomalainen tiedeakatemia","issued":"1952"},{"id":"doi:10.1016/S0021-9800(69)80095-5","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s0021-9800(69)80095-5","page":"317-319","source":"Crossref","title":"Some maximal arcs in finite projective planes","volume":"6","author":[{"given":"R.H.F.","family":"Denniston","sequence":"first","affiliation":[]}],"container-title":"Journal of Combinatorial Theory","original-title":[],"language":"en","issued":{"date-parts":[[1969,4]]},"URL":"http://dx.doi.org/10.1016/S0021-9800(69)80095-5","ISSN":["0021-9800"],"container-title-short":"Journal of Combinatorial Theory","note":"alternative-id:S0021980069800955"},{"id":"manual:-R.-Hill-(1973).-On-the-large","type":"article-journal","author":[{"family":"Hill","given":"R."}],"title":"On the largest size of cap in s53. Atti della Accademia Nazionale dei Lincei","volume":"54","container-title":"Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti","issue":"3","issued":"1973","page":"378–384"},{"id":"manual:-R.-Hill-Caps-and-groups-pp.-","type":"paper-conference","author":[{"family":"Hill","given":"R."}],"title":"Caps and groups","container-title":"Proc. Rome 1973, Atti dei Convegni Lincei","issued":"1976","page":"389–394"},{"id":"manual:-R.-Hill-(1973).-Caps-and-gro","type":"document","author":[{"family":"Hill","given":"R."}],"title":"Caps and groups. Colloquio Internazionale sulle Teorie Combinatorie","volume":"2","issued":"1973","page":"389–394","publisher-place":"Rome"},{"id":"doi:10.1016/0012-365X(72)90024-6","type":"journal-article","publisher":"Elsevier BV","issue":"1-3","DOI":"10.1016/0012-365x(72)90024-6","page":"47-64","source":"Crossref","title":"Weights of linear codes and strongly regular normed spaces","volume":"3","author":[{"given":"Ph.","family":"Delsarte","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1972]]},"URL":"http://dx.doi.org/10.1016/0012-365X(72)90024-6","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X72900246"},{"id":"doi:10.1016/S0012-365X(02)00390-4","type":"journal-article","publisher":"Elsevier BV","issue":"1-2","DOI":"10.1016/s0012-365x(02)00390-4","page":"465-469","source":"Crossref","title":"Strongly regular graphs from differences of quadrics","volume":"256","author":[{"given":"Nicholas","family":"Hamilton","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2002,9]]},"URL":"http://dx.doi.org/10.1016/S0012-365X(02)00390-4","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X02003904"},{"id":"doi:10.3934/amc.2012.6.505","type":"journal-article","publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"4","DOI":"10.3934/amc.2012.6.505","page":"505-516","source":"Crossref","title":"Partial permutation decoding for simplex codes","volume":"6","author":[{"given":"Washiela","family":"Fish","sequence":"first","affiliation":[]},{"name":",Department of Mathematics and Applied Mathematics, University of the Western Cape, 7535 Bellville","sequence":"first","affiliation":[]},{"given":"Jennifer","family":"D. Key","sequence":"additional","affiliation":[]},{"given":"Eric","family":"Mwambene","sequence":"additional","affiliation":[]}],"container-title":"Advances in Mathematics of Communications","original-title":[],"language":"en","issued":{"date-parts":[[2012]]},"URL":"http://dx.doi.org/10.3934/amc.2012.6.505","ISSN":["1930-5338"]},{"id":"doi:10.1007/s00200-016-0286-7","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","DOI":"10.1007/s00200-016-0286-7","page":"399-412","source":"Crossref","title":"Partial permutation decoding for MacDonald codes","volume":"27","author":[{"given":"Jennifer D","family":"Key","sequence":"first","affiliation":[]},{"given":"Padmapani","family":"Seneviratne","sequence":"additional","affiliation":[]}],"container-title":"Applicable Algebra in Engineering, Communication and Computing","original-title":[],"language":"en","issued":{"date-parts":[[2016,2,9]]},"URL":"http://dx.doi.org/10.1007/s00200-016-0286-7","ISSN":["0938-1279","1432-0622"],"container-title-short":"AAECC","note":"alternative-id:286"},{"id":"manual:-R.-Sch-rer-and-W.-Ch.-Schmid","type":"article-journal","author":[{"family":"Schürer","given":"R."},{"family":"Schmid","given":"W.Ch"}],"title":"Simplex Code. From MinT—the database of optimal net, code, OA, and OOA parameters","container-title":"Version","issued":"2015-09-03","URL":"https://web.archive.org/web/20240228194024/http://mint.sbg.ac.at/desc_CSimplex.html","note":"Available at \\url{https://web.archive.org/web/20240228194024/http://mint.sbg.ac.at/desc_CSimplex.html}"},{"id":"doi:10.1142/6400","type":"monograph","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/6400","source":"Crossref","title":"Codes for Error Detection","author":[{"given":"Torleiv","family":"Kløve","sequence":"first","affiliation":[{"name":"University of BergenNorway"}]}],"container-title":"Series on Coding Theory and Cryptology","original-title":[],"issued":{"date-parts":[[2007,4]]},"ISBN":["9789812705860","9789812770516"],"URL":"http://dx.doi.org/10.1142/6400","ISSN":["1793-2238"],"note":"alternative-id:10.1142/6400\n10.1142/SCTC"},{"id":"doi:10.1109/TCOMM.2024.3420721","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tcomm.2024.3420721","page":"12-21","source":"Crossref","title":"Correcting a Substring Edit Error of Bounded Length","volume":"73","author":[{"ORCID":"https://orcid.org/0000-0003-2946-7782","authenticated-orcid":false,"given":"Yuanyuan","family":"Tang","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, USA"}]},{"ORCID":"https://orcid.org/0009-0005-0928-3910","authenticated-orcid":false,"given":"Sarvin","family":"Motamen","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, USA"}]},{"ORCID":"https://orcid.org/0000-0002-6133-2987","authenticated-orcid":false,"given":"Hao","family":"Lou","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, USA"}]},{"given":"Kallie","family":"Whritenour","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Virginia, Charlottesville, VA, USA"}]},{"ORCID":"https://orcid.org/0000-0003-1582-4873","authenticated-orcid":false,"given":"Shuche","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Operations Research and Analytics, National University of Singapore, Queenstown, Singapore"}]},{"ORCID":"https://orcid.org/0000-0002-9197-3371","authenticated-orcid":false,"given":"Ryan","family":"Gabrys","sequence":"additional","affiliation":[{"name":"Calit2, University of California at San Diego, La Jolla, CA, USA"}]},{"ORCID":"https://orcid.org/0000-0002-8684-4487","authenticated-orcid":false,"given":"Farzad","family":"Farnoud","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering and the Department of Computer Science, University of Virginia, Charlottesville, VA, USA"}]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2025,1]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2024.3420721","ISSN":["0090-6778","1558-0857"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1002/j.1538-7305.1963.tb04003","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1002/j.1538-7305.1963.tb04003.x","page":"79-94","source":"Crossref","title":"A Theorem on the Distribution of Weights in a Systematic Code†","volume":"42","author":[{"given":"Jessie","family":"Macwilliams","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1963,1]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1963.tb04003.x","ISSN":["0005-8580"]},{"id":"doi:10.1002/j.1538-7305.1972.tb01947","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1002/j.1538-7305.1972.tb01947.x","page":"803-819","source":"Crossref","title":"The MacWilliams Identities for Nonlinear Codes","volume":"51","author":[{"given":"F. J.","family":"MacWilliams","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]},{"given":"J.-M.","family":"Goethals","sequence":"additional","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1972,4]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1972.tb01947.x","ISSN":["0005-8580"]},{"id":"doi:10.1002/j.1538-7305.1952.tb01393","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1002/j.1538-7305.1952.tb01393.x","page":"504-522","source":"Crossref","title":"A Comparison of Signalling Alphabets","volume":"31","author":[{"given":"E. N.","family":"Gilbert","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1952,5]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1952.tb01393.x","ISSN":["0005-8580"]},{"id":"manual:-R.-R.-Varshamov-Estimate-of-","type":"article-journal","author":[{"family":"Varshamov","given":"R.R."}],"title":"Estimate of the number of signals in error correcting codes","volume":"117","container-title":"Doklady Akademii Nauk SSSR Soviet Math. - Doklady","issued":"1957","page":"739–741"},{"id":"doi:10.1016/0012-365X(90)90002-Y","type":"journal-article","publisher":"Elsevier BV","issue":"2-3","DOI":"10.1016/0012-365x(90)90002-y","page":"139-160","source":"Crossref","title":"A new upper bound on nonbinary block codes","volume":"83","author":[{"given":"Matti","family":"Aaltonen","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1990,8]]},"URL":"http://dx.doi.org/10.1016/0012-365X(90)90002-Y","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:0012365X9090002Y"},{"id":"arxiv:2306.13262","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2024.3486278","page":"9152-9164","source":"Crossref","title":"Reliable Computation by Large-Alphabet Formulas in the Presence of Noise","volume":"70","author":[{"ORCID":"https://orcid.org/0009-0006-8183-5203","authenticated-orcid":false,"given":"Andrew K.","family":"Tan","sequence":"first","affiliation":[{"name":"Department of Physics and the Co-Design Center for Quantum Advantage, Massachusetts Institute of Technology, Cambridge, MA, USA"}]},{"ORCID":"https://orcid.org/0000-0001-8662-0873","authenticated-orcid":false,"given":"Matthew H.","family":"Ho","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA"}]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[{"name":"Department of Physics and the Institute for Artificial Intelligence and Fundamental Interactions, Massachusetts Institute of Technology, Cambridge, MA, USA"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2024,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3486278","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2306.13262"},{"id":"manual:-H.-W.-Lenstra-Codes-from-alg","type":"chapter","author":[{"family":"Lenstra","given":"H.W."}],"title":"Codes from algebraic number fields","container-title":"Mathematics and computer science II, Fundamental contributions in the Netherlands since","volume":"4","issued":"1945","page":"94–104"},{"id":"manual:-M.-A.-Tsfasman-Global-fields","type":"chapter","author":[{"family":"Tsfasman","given":"M.A."}],"title":"Global fields, codes and sphere packings","container-title":"Journees Arithmetiques 1989,-\" Asterisque","volume":"198","issued":"1992","page":"199–200"},{"id":"doi:10.1109/TIT.2002.808131","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2002.808131","page":"594-603","source":"Crossref","title":"Constructions of codes from number fields","volume":"49","author":[{"given":"V.","family":"Guruswami","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2003,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.808131","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ISIT.1997.613333","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.1997.613333","page":"396","source":"Crossref","title":"Maximum likelihood decoding with reduced complexity","author":[{"given":"I.","family":"Dumer","sequence":"first","affiliation":[]}],"event":"IEEE International Symposium on Information Theory","container-title":"Proceedings of IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.1997.613333"},{"id":"doi:10.1109/TIT.1976.1055617","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.1976.1055617","page":"514-517","source":"Crossref","title":"An optimum symbol-by-symbol decoding rule for linear codes","volume":"22","author":[{"given":"C.","family":"Hartmann","sequence":"first","affiliation":[]},{"given":"L.","family":"Rudolph","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1976,9]]},"URL":"http://dx.doi.org/10.1109/TIT.1976.1055617","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/978-3-642-12929-2_7","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-12929-2_7","page":"81-94","source":"Crossref","title":"Information-Set Decoding for Linear Codes over F q","author":[{"given":"Christiane","family":"Peters","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2010]]},"ISBN":["9783642129285","9783642129292"],"URL":"http://dx.doi.org/10.1007/978-3-642-12929-2_7","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/TIT.1978.1055821","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1978.1055821","page":"76-80","source":"Crossref","title":"Efficient maximum likelihood decoding of linear block codes using a trellis","volume":"24","author":[{"given":"J.","family":"Wolf","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1978,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1978.1055821","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1704.02420","type":"article-journal","author":[{"given":"Atri","family":"Rudra"},{"given":"Mary","family":"Wootters"}],"title":"Average-radius list-recovery of random linear codes: it really ties the room together","issued":{"date-parts":[[2017,4,8]]},"note":"arxivid:1704.02420\narxiv_version_number:1"},{"id":"manual:-R.-K-tter.-A-unified-descrip","type":"paper-conference","author":[{"family":"Kötter","given":"R."}],"title":"A unified description of an error locating procedure for linear codes","editor":[{"family":"Yorgov","given":"D."}],"container-title":"Proc. 3rd International Workshop on Algebraic and Combinatorial Coding Theory","publisher":"Voneshta Voda","note":"Hermes","issued":"1992-06","page":"113–117,","publisher-place":"Bulgaria"},{"id":"doi:10.1016/0012-365X(92)90567-Y","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/0012-365x(92)90567-y","page":"369-381","source":"Crossref","title":"On decoding by error location and dependent sets of error positions","volume":"106-107","author":[{"given":"Ruud","family":"Pellikaan","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1992,9]]},"URL":"http://dx.doi.org/10.1016/0012-365X(92)90567-Y","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"special_numbering:C\nalternative-id:0012365X9290567Y"},{"id":"preset:Sabo","type":"document","author":[{"family":"Sabo","given":"E."}],"title":"CodingTheory","URL":"https://github.com/esabo/CodingTheory"},{"id":"arxiv:1111.3275","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2013.07.009","page":"134-166","source":"Crossref","title":"Information storage capacity of discrete spin systems","volume":"338","author":[{"given":"Beni","family":"Yoshida","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2013,11]]},"URL":"http://dx.doi.org/10.1016/j.aop.2013.07.009","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491613001693\narxivid:1111.3275"},{"id":"doi:10.1063/1.1665111","type":"journal-article","publisher":"AIP Publishing","issue":"11","abstract":"<jats:p>Models of 2-dimensional hydrogen bonded crystals obeying the ice rule, which previously have been solved exactly, are generalized by removing the ice rule. Many of the peculiar and unique properties of the solutions for the constrained models are now explained by showing that these models, above critical temperature, are equivalent to new unconstrained models at critical temperature. In addition to locating the critical temperature for the general but unsolved models, we locate the singularities of the ground state energy of a related ring of interacting spins.</jats:p>","DOI":"10.1063/1.1665111","page":"3183-3186","source":"Crossref","title":"Two-Dimensional Hydrogen Bonded Crystals without the Ice Rule","volume":"11","author":[{"given":"Bill","family":"Sutherland","sequence":"first","affiliation":[{"name":"Physics Department, University of California, Berkeley, California 94720"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1970,11,1]]},"URL":"http://dx.doi.org/10.1063/1.1665111","ISSN":["0022-2488","1089-7658"]},{"id":"doi:10.1103/PhysRevLett.26.832","type":"journal-article","publisher":"American Physical Society (APS)","issue":"14","DOI":"10.1103/physrevlett.26.832","page":"832-833","source":"Crossref","title":"Eight-Vertex Model in Lattice Statistics","volume":"26","author":[{"given":"R. J.","family":"Baxter","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1971,4,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.26.832","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"doi:10.1016/0003-4916(72)90335-1","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0003-4916(72)90335-1","page":"193-228","source":"Crossref","title":"Partition function of the Eight-Vertex lattice model","volume":"70","author":[{"given":"Rodney J","family":"Baxter","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[1972,3]]},"URL":"http://dx.doi.org/10.1016/0003-4916(72)90335-1","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:0003491672903351"},{"id":"doi:10.1109/ITW.1998.706440","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.1998.706440","source":"Crossref","title":"Low density parity check codes over GF(q)","author":[{"given":"M.C.","family":"Davey","sequence":"first","affiliation":[]},{"given":"D.J.C.","family":"MacKay","sequence":"additional","affiliation":[]}],"event":"1998 Information Theory Workshop","container-title":"1998 Information Theory Workshop (Cat. No.98EX131)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ITW.1998.706440"},{"id":"doi:10.1109/ICC.2004.1312545","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icc.2004.1312545","page":"528-532 Vol.1","source":"Crossref","title":"Binary representation of cycle Tanner-graph GF(2/sup b/) codes","author":[{"given":"X.-Yu.","family":"Hu","sequence":"first","affiliation":[]},{"given":"E.","family":"Eleftheriou","sequence":"additional","affiliation":[]}],"event":"2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577)","container-title":"2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577)","original-title":[],"issued":{"date-parts":[[2004]]},"URL":"http://dx.doi.org/10.1109/ICC.2004.1312545"},{"id":"doi:10.1109/ITW.2007.4313042","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.2007.4313042","page":"13-18","source":"Crossref","title":"Design of Nonbinary Quasi-Cyclic LDPC Cycle Codes","author":[{"given":"Rong-Hui","family":"Peng","sequence":"first","affiliation":[]},{"given":"Rong-Rong","family":"Chen","sequence":"additional","affiliation":[]}],"event":"2007 IEEE Information Theory Workshop","container-title":"2007 IEEE Information Theory Workshop","original-title":[],"issued":{"date-parts":[[2007,9]]},"URL":"http://dx.doi.org/10.1109/ITW.2007.4313042"},{"id":"doi:10.1109/TCOMM.2008.060527","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tcomm.2008.060527","page":"1626-1635","source":"Crossref","title":"Design of regular (2,d/sub c/)-LDPC codes over GF(q) using their binary images","volume":"56","author":[{"given":"Charly","family":"Poulliat","sequence":"first","affiliation":[]},{"given":"Marc","family":"Fossorier","sequence":"additional","affiliation":[]},{"given":"David","family":"Declercq","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2008,10]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2008.060527","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"doi:10.1109/ISTC.2010.5613806","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/istc.2010.5613806","page":"68-72","source":"Crossref","title":"Protograph-based q-ary LDPC codes for higher-order modulation","author":[{"given":"Andrea","family":"Marinoni","sequence":"first","affiliation":[]},{"given":"Pietro","family":"Savazzi","sequence":"additional","affiliation":[]},{"given":"Richard D.","family":"Wesel","sequence":"additional","affiliation":[]}],"event":"Iterative Information Processing (ISTC)","container-title":"2010 6th International Symposium on Turbo Codes &amp; Iterative Information Processing","original-title":[],"issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1109/ISTC.2010.5613806"},{"id":"doi:10.1109/ISIT.2011.6034271","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2011.6034271","page":"913-917","source":"Crossref","title":"Enumerators for protograph-based ensembles of nonbinary LDPC codes","author":[{"given":"Dariush","family":"Divsalar","sequence":"first","affiliation":[]},{"given":"Lara","family":"Dolecek","sequence":"additional","affiliation":[]}],"event":"2011 IEEE International Symposium on Information Theory - ISIT","container-title":"2011 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2011,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2011.6034271"},{"id":"doi:10.1109/ISIT.2014.6874958","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2014.6874958","page":"876-880","source":"Crossref","title":"Performance comparison of non-binary LDPC block and spatially coupled codes","author":[{"given":"Kechao","family":"Huang","sequence":"first","affiliation":[]},{"given":"David G. M.","family":"Mitchell","sequence":"additional","affiliation":[]},{"given":"Lai","family":"Wei","sequence":"additional","affiliation":[]},{"given":"Xiao","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Daniel J.","family":"Costello","sequence":"additional","affiliation":[]}],"event":"2014 IEEE International Symposium on Information Theory (ISIT)","container-title":"2014 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2014.6874958"},{"id":"doi:10.1109/TIT.2014.2316215","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2014.2316215","page":"3913-3941","source":"Crossref","title":"Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs","volume":"60","author":[{"given":"Lara","family":"Dolecek","sequence":"first","affiliation":[]},{"given":"Dariush","family":"Divsalar","sequence":"additional","affiliation":[]},{"given":"Yizeng","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Behzad","family":"Amiri","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2316215","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ITA.2012.6181802","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/ita.2012.6181802","page":"192-198","source":"Crossref","title":"On the typical minimum distance of protograph-based non-binary LDPC codes","author":[{"given":"Dariush","family":"Divsalar","sequence":"first","affiliation":[]},{"given":"Lara","family":"Dolecek","sequence":"additional","affiliation":[]}],"event":"2012 Information Theory and Applications Workshop (ITA)","container-title":"2012 Information Theory and Applications Workshop","original-title":[],"issued":{"date-parts":[[2012,2]]},"URL":"http://dx.doi.org/10.1109/ITA.2012.6181802"},{"id":"doi:10.1002/ett.4460060507","type":"journal-article","publisher":"Wiley","issue":"5","abstract":"<jats:title>Abstract</jats:title><jats:p>A general framework, based on ideas of Tanner, for the description of codes and iterative decoding (“turbo coding”) is developed. Just like trellis‐based code descriptions are naturally matched to Viterbi decoding, code descriptions based on Tanner graphs (which may be viewed as generalized trellises) are naturally matched to iterative decoding. Two basic iterative decoding algorithms (which are versions of the algorithms of Berrou et al. and of Hagenauer, respectively) are shown to be natural generalizations of the forward‐backward algorithm (Bahl et al.) and the Viterbi algorithm, respectively, to arbitrary Tanner graphs. The careful derivation of these algorithms clarifies, in particular, which a priori probabilities are admissible and how they are properly dealt with. For cycle codes (a class of binary linear block codes), a complete characterization is given of the error patterns that are corrected by the generalized Viterbi algorithm after infinitely many iterations.</jats:p>","DOI":"10.1002/ett.4460060507","page":"513-525","source":"Crossref","title":"Codes and iterative decoding on general graphs","volume":"6","author":[{"given":"Niclas","family":"Wiberg","sequence":"first","affiliation":[]},{"given":"Hans‐Andrea","family":"Loeliger","sequence":"additional","affiliation":[]},{"given":"Ralf","family":"Kotter","sequence":"additional","affiliation":[]}],"container-title":"European Transactions on Telecommunications","original-title":[],"language":"en","issued":{"date-parts":[[1995,9]]},"URL":"http://dx.doi.org/10.1002/ett.4460060507","ISSN":["1124-318X","1541-8251"],"container-title-short":"Trans Emerging Tel Tech","note":"alternative-id:10.1002/ett.4460060507"},{"id":"manual:-N.-Wiberg-Codes-and-decoding","type":"thesis","author":[{"family":"Wiberg","given":"N."}],"title":"Codes and decoding on general graphs","genre":"PhD thesis,","publisher":"Linköping University","issued":"1996","publisher-place":"Linköping, Sweden"},{"id":"manual:-B.-J.-Frey.-Graphical-models","type":"book","author":[{"family":"Frey","given":"B.J."}],"title":"Graphical models for machine learning and digital communication","publisher":"MIT press","issued":"1998"},{"id":"doi:10.1109/PACRIM.2001.953537","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/pacrim.2001.953537","page":"119-123","source":"Crossref","title":"Decoding low-density parity-check codes with probabilistic schedule","volume":"1","author":[{"family":"Yongyi Mao","sequence":"first","affiliation":[]},{"given":"A.H.","family":"Banihashemi","sequence":"additional","affiliation":[]}],"event":"2001 IEEE Pacific Rim Conference on Communications, Computers and Signal Processing","container-title":"2001 IEEE Pacific Rim Conference on Communications, Computers and Signal Processing (IEEE Cat. No.01CH37233)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/PACRIM.2001.953537"},{"id":"doi:10.1109/ISIT.2000.866299","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2000.866299","page":"9","source":"Crossref","title":"Codes on graphs: normal realizations","author":[{"given":"G.D.","family":"Forney","sequence":"first","affiliation":[]}],"event":"2000 IEEE International Symposium on Information Theory","container-title":"2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2000.866299"},{"id":"doi:10.1016/S1571-0653(04)00166-0","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/s1571-0653(04)00166-0","page":"148-157","source":"Crossref","title":"LDPC codes: a group algebra formulation","volume":"6","author":[{"given":"E.","family":"Soljanin","sequence":"first","affiliation":[]},{"given":"E.","family":"Offer","sequence":"additional","affiliation":[]}],"container-title":"Electronic Notes in Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2001,4]]},"URL":"http://dx.doi.org/10.1016/S1571-0653(04)00166-0","ISSN":["1571-0653"],"container-title-short":"Electronic Notes in Discrete Mathematics","note":"special_numbering:C\nalternative-id:S1571065304001660"},{"id":"doi:10.1109/18.782170","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.782170","page":"2173-2181","source":"Crossref","title":"Which codes have cycle-free Tanner graphs?","volume":"45","author":[{"given":"T.","family":"Etzion","sequence":"first","affiliation":[]},{"given":"A.","family":"Trachtenberg","sequence":"additional","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1999,9]]},"URL":"http://dx.doi.org/10.1109/18.782170","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1201/b11707","type":"reference-book","publisher":"CRC Press","DOI":"10.1201/b11707","source":"Crossref","title":"Algebraic and Stochastic Coding Theory","author":[{"given":"Dave K.","family":"Kythe","sequence":"first","affiliation":[]},{"given":"Prem K.","family":"Kythe","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2017,7,28]]},"ISBN":["9781466505629"],"URL":"http://dx.doi.org/10.1201/b11707","note":"edition-number:0"},{"id":"manual:-N.-V.-Semakov-and-V.-A.-Zino","type":"article-journal","author":[{"family":"Semakov","given":"N.V."},{"family":"Zinoviev","given":"V.A."}],"title":"Equidistant q-ary Codes with Maximal Distance and Resolvable Balanced Incomplete Block Designs","volume":"4","container-title":"Problemy Peredachi Informatsii","issue":"2","issued":"1968","page":"3–10"},{"id":"manual:-N.-V.-Semakov-V.-A.-Zinoviev","type":"article-journal","author":[{"family":"Semakov","given":"N.V."},{"family":"Zinoviev","given":"V.A."},{"family":"Zaitsev","given":"G.V."}],"title":"A Class of Maximum Equidistant Codes","volume":"5","container-title":"Problemy Peredachi Informatsii","issue":"2","issued":"1969","page":"84–87"},{"id":"doi:10.1109/18.412678","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.412678","page":"1303-1321","source":"Crossref","title":"Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces","volume":"41","author":[{"given":"V.I.","family":"Levenshtein","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1995]]},"URL":"http://dx.doi.org/10.1109/18.412678","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s10623-016-0286-4","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1-2","DOI":"10.1007/s10623-016-0286-4","page":"411-433","source":"Crossref","title":"Energy bounds for codes and designs in Hamming spaces","volume":"82","author":[{"given":"P. G.","family":"Boyvalenkov","sequence":"first","affiliation":[]},{"given":"P. D.","family":"Dragnev","sequence":"additional","affiliation":[]},{"given":"D. P.","family":"Hardin","sequence":"additional","affiliation":[]},{"given":"E. B.","family":"Saff","sequence":"additional","affiliation":[]},{"given":"M. M.","family":"Stoyanova","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2016,10,14]]},"URL":"http://dx.doi.org/10.1007/s10623-016-0286-4","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:286"},{"id":"doi:10.1109/18.915662","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.915662","page":"1050-1061","source":"Crossref","title":"Estimates of the distance distribution of codes and designs","volume":"47","author":[{"given":"A.","family":"Askikhmin","sequence":"first","affiliation":[]},{"given":"A.","family":"Barg","sequence":"additional","affiliation":[]},{"given":"S.","family":"Litsyn","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001,3]]},"URL":"http://dx.doi.org/10.1109/18.915662","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1986.1057136","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1986.1057136","page":"51-53","source":"Crossref","title":"Efficient balanced codes","volume":"32","author":[{"given":"D.","family":"Knuth","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1986,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1986.1057136","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.52490","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.52490","page":"406-408","source":"Crossref","title":"On balanced codes","volume":"36","author":[{"given":"S.","family":"Al-Bassam","sequence":"first","affiliation":[]},{"given":"B.","family":"Bose","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1990,3]]},"URL":"http://dx.doi.org/10.1109/18.52490","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/jsac.2010.100207","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/jsac.2010.100207","page":"188-192","source":"Crossref","title":"Very Efficient Balanced Codes","volume":"28","author":[{"given":"Kees A.","family":"Schouhamer Immink","sequence":"first","affiliation":[]},{"given":"Jos H.","family":"Weber","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[2010,2]]},"URL":"http://dx.doi.org/10.1109/jsac.2010.100207","ISSN":["0733-8716","1558-0008"],"container-title-short":"IEEE J. Select. Areas Commun."},{"id":"manual:-K.-A.-S.-Immink.-emph-Codes-","type":"book","author":[{"family":"Immink","given":"K.A.S."}],"title":"\\emph{Codes for mass data storage systems}","publisher":"Shannon Foundation Publisher","issued":"2004"},{"id":"arxiv:2112.11763","type":"article-journal","author":[{"given":"Sascha","family":"Kurz"}],"title":"Divisible Codes","issued":{"date-parts":[[2025,12,20]]},"note":"arxivid:2112.11763\narxiv_version_number:4"},{"id":"doi:10.1007/BF01223730","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01223730","page":"485-494","source":"Crossref","title":"Divisible codes","volume":"36","author":[{"given":"Harold N.","family":"Ward","sequence":"first","affiliation":[]}],"container-title":"Archiv der Mathematik","original-title":[],"language":"en","issued":{"date-parts":[[1981,12]]},"URL":"http://dx.doi.org/10.1007/BF01223730","ISSN":["0003-889X","1420-8938"],"container-title-short":"Arch. Math","note":"alternative-id:BF01223730"},{"id":"arxiv:1012.4134","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1112/jlms/jdr054","page":"1-16","source":"Crossref","title":"On triply even binary codes","volume":"86","author":[{"given":"Koichi","family":"Betsumiya","sequence":"first","affiliation":[{"name":"Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan"}]},{"given":"Akihiro","family":"Munemasa","sequence":"additional","affiliation":[{"name":"Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan,"}]}],"container-title":"Journal of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[2012,2,9]]},"URL":"http://dx.doi.org/10.1112/jlms/jdr054","ISSN":["0024-6107","1469-7750"],"container-title-short":"Journal of the London Mathematical Society","note":"alternative-id:10.1112/jlms/jdr054\narxivid:1012.4134"},{"id":"arxiv:1209.2426","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.86.052329","source":"Crossref","title":"Magic-state distillation with low overhead","volume":"86","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,11,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.052329","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052329","note":"arxivid:1209.2426"},{"id":"arxiv:2005.13623","type":"article-journal","author":[{"given":"P. G.","family":"Boyvalenkov"},{"given":"K. V.","family":"Delchev"},{"given":"D. V.","family":"Zinoviev"},{"given":"V. A.","family":"Zinoviev"}],"title":"On two-weight codes","issued":{"date-parts":[[2020,11,30]]},"note":"arxivid:2005.13623\narxiv_version_number:3"},{"id":"arxiv:2107.02447","type":"article-journal","author":[{"given":"Canze","family":"Zhu"},{"given":"Qunying","family":"Liao"}],"title":"Complete weight enumerators for several classes of two-weight and three-weight linear codes","issued":{"date-parts":[[2021,7,6]]},"note":"arxivid:2107.02447\narxiv_version_number:1"},{"id":"arxiv:2210.07496","type":"journal-article","publisher":"California Digital Library (CDL)","issue":"1","DOI":"10.5070/c64163844","source":"Crossref","title":"On the size of maximal binary codes with 2, 3, and 4 distances","volume":"4","author":[{"given":"Alexander","family":"Barg","sequence":"first","affiliation":[]},{"given":"Alexey","family":"Glazyrin","sequence":"additional","affiliation":[]},{"given":"Wei-Jiun","family":"Kao","sequence":"additional","affiliation":[]},{"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[]},{"given":"Pin-Chieh","family":"Tseng","sequence":"additional","affiliation":[]},{"given":"Wei-Hsuan","family":"Yu","sequence":"additional","affiliation":[]}],"container-title":"Combinatorial Theory","original-title":[],"issued":{"date-parts":[[2024,7,1]]},"URL":"http://dx.doi.org/10.5070/C64163844","ISSN":["2766-1334"],"note":"arxivid:2210.07496"},{"id":"doi:10.1109/TIT.2015.2440266","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2015.2440266","page":"4253-4258","source":"Crossref","title":"Classification of Binary Self-Dual Codes of Length 40","volume":"61","author":[{"given":"Iliya","family":"Bouyukliev","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Veliko Tarnovo, Bulgaria"}]},{"given":"Maria","family":"Dzhumalieva-Stoeva","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Veliko Tarnovo, Bulgaria"}]},{"given":"Venelin","family":"Monev","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Informatics, Veliko Tarnovo University, Veliko Tarnovo, Bulgaria"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2015,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2015.2440266","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/s10623-010-9394-8","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2-3","DOI":"10.1007/s10623-010-9394-8","page":"105-113","source":"Crossref","title":"Some two-character sets","volume":"56","author":[{"given":"Antonio","family":"Cossidente","sequence":"first","affiliation":[]},{"given":"Oliver H.","family":"King","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2010,4,11]]},"URL":"http://dx.doi.org/10.1007/s10623-010-9394-8","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9394"},{"id":"doi:10.1007/s10623-006-9024-7","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-006-9024-7","page":"103-107","source":"Crossref","title":"Veronese embedding and two–character sets","volume":"42","author":[{"given":"Antonio","family":"Cossidente","sequence":"first","affiliation":[]},{"given":"Giuseppe","family":"Marino","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2006,11,1]]},"URL":"http://dx.doi.org/10.1007/s10623-006-9024-7","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des Codes Crypt","note":"alternative-id:9024"},{"id":"arxiv:1506.06830","type":"article-journal","author":[{"given":"Zhengchun","family":"Zhou"},{"given":"Nian","family":"Li"},{"given":"Cuiling","family":"Fan"},{"given":"Tor","family":"Helleseth"}],"title":"Linear Codes with Two or Three Weights From Quadratic Bent Functions","issued":{"date-parts":[[2015,10,16]]},"note":"arxivid:1506.06830\narxiv_version_number:2"},{"id":"doi:10.1007/978-3-319-32595-8","type":"book","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-32595-8","source":"Crossref","title":"Bent Functions","author":[{"given":"Sihem","family":"Mesnager","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2016]]},"ISBN":["9783319325934","9783319325958"],"URL":"http://dx.doi.org/10.1007/978-3-319-32595-8","note":"publisher-location:Cham"},{"id":"manual:-J.-L.-Massey-Threshold-Decod","type":"book","author":[{"family":"Massey","given":"J.L."}],"title":"Threshold Decoding","publisher":"M.I.T. Press","issued":"1963","publisher-place":"Cambridge, MA"},{"id":"arxiv:2305.02484","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2024.3510698","page":"930-935","source":"Crossref","title":"A Deterministic Construction of a Large Distance Code From the Wozencraft Ensemble","volume":"71","author":[{"ORCID":"https://orcid.org/0000-0001-7926-3396","authenticated-orcid":false,"given":"Venkatesan","family":"Guruswami","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences (EECS), the Department of Mathematics, and the Simons Institute for the Theory of Computing, University of California at Berkeley, Berkeley, CA, USA"}]},{"ORCID":"https://orcid.org/0000-0001-5765-0432","authenticated-orcid":false,"given":"Shilun","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California at Berkeley, Berkeley, CA, USA"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2025,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3510698","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2305.02484"},{"id":"arxiv:0708.4164","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2008.928288","page":"3865-3872","source":"Crossref","title":"Asymptotic Improvement of the Gilbert–Varshamov Bound for Linear Codes","volume":"54","author":[{"given":"Philippe","family":"Gaborit","sequence":"first","affiliation":[]},{"given":"Gilles","family":"Zemor","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.928288","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0708.4164"},{"id":"manual:-E.-R.-Berlekamp-Negacyclic-c","type":"paper-conference","author":[{"family":"Berlekamp","given":"E.R."}],"title":"Negacyclic codes for the Lee metric","container-title":"Proceedings of the Conference on Combinatorial Mathematics and its Applications","publisher":"University of North Carolina Press","issued":"1968","publisher-place":"Chapel Hill"},{"id":"doi:10.14492/hokmj/1351001213","type":"journal-article","publisher":"Department of Mathematics, Hokkaido University","issue":"2","DOI":"10.14492/hokmj/1351001213","source":"Crossref","title":"Self-dual codes over rings and the Chinese remainder theorem","volume":"28","author":[{"given":"Steven T.","family":"DOUGHERTY","sequence":"first","affiliation":[]},{"given":"Masaaki","family":"HARADA","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"SOLÉ","sequence":"additional","affiliation":[]}],"container-title":"Hokkaido Mathematical Journal","original-title":[],"issued":{"date-parts":[[1999,2,1]]},"URL":"http://dx.doi.org/10.14492/hokmj/1351001213","ISSN":["0385-4035"],"container-title-short":"Hokkaido Math. J."},{"id":"doi:10.1007/BF01198572","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/bf01198572","page":"201-207","source":"Crossref","title":"Selbstduale Codes �ber dem Ring der ganzen Zahlen modulo 4","volume":"53","author":[{"given":"Michael","family":"Klemm","sequence":"first","affiliation":[]}],"container-title":"Archiv der Mathematik","original-title":[],"language":"de","issued":{"date-parts":[[1989,8]]},"URL":"http://dx.doi.org/10.1007/BF01198572","ISSN":["0003-889X","1420-8938"],"container-title-short":"Arch. Math","note":"alternative-id:BF01198572"},{"id":"arxiv:2208.06832","type":"article-journal","author":[{"given":"Nuh","family":"Aydin"},{"given":"Yiang","family":"Lu"},{"given":"Vishad R.","family":"Onta"}],"title":"An Updated Database of $\\mathbb{Z}_4$ Codes","issued":{"date-parts":[[2022,8,14]]},"note":"arxivid:2208.06832\narxiv_version_number:1"},{"id":"doi:10.1016/0097-3165(93)90070-O","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0097-3165(93)90070-o","page":"30-45","source":"Crossref","title":"Self-dual codes over the integers modulo 4","volume":"62","author":[{"given":"J.H","family":"Conway","sequence":"first","affiliation":[]},{"given":"N.J.A","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1993,1]]},"URL":"http://dx.doi.org/10.1016/0097-3165(93)90070-O","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:009731659390070O"},{"id":"doi:10.1109/18.661525","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.661525","page":"800-809","source":"Crossref","title":"Decompositions and extremal type II codes over Z/sub 4/","volume":"44","author":[{"given":"W.C.","family":"Huffman","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998,3]]},"URL":"http://dx.doi.org/10.1109/18.661525","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1023/A:1008335912135","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1023/a:1008335912135","page":"257-269","source":"Crossref","title":"Construction of Extremal Type II Codes over ℤ4","volume":"16","author":[{"given":"Philippe","family":"Gaborit","sequence":"first","affiliation":[]},{"given":"Masaaki","family":"Harada","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[1999,5]]},"URL":"http://dx.doi.org/10.1023/A:1008335912135","ISSN":["0925-1022","1573-7586"],"container-title-short":"Designs, Codes and Cryptography","note":"alternative-id:199820"},{"id":"doi:10.1109/18.651058","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.651058","page":"311-322","source":"Crossref","title":"All self-dual Z/sub 4/ codes of length 15 or less are known","volume":"44","author":[{"given":"J.","family":"Fields","sequence":"first","affiliation":[]},{"given":"P.","family":"Gaborit","sequence":"additional","affiliation":[]},{"given":"J.S.","family":"Leon","sequence":"additional","affiliation":[]},{"given":"V.","family":"Pless","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.651058","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1006/jcta.1996.2750","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1006/jcta.1996.2750","page":"32-50","source":"Crossref","title":"All Z4Codes of Type II and Length 16 Are Known","volume":"78","author":[{"given":"Vera","family":"Pless","sequence":"first","affiliation":[]},{"given":"Jeffrey S.","family":"Leon","sequence":"additional","affiliation":[]},{"given":"Joe","family":"Fields","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1997,4]]},"URL":"http://dx.doi.org/10.1006/jcta.1996.2750","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S0097316596927508"},{"id":"doi:10.1007/s10623-008-9243-1","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s10623-008-9243-1","page":"55-68","source":"Crossref","title":"Independence of vectors in codes over rings","volume":"51","author":[{"given":"Steven T.","family":"Dougherty","sequence":"first","affiliation":[]},{"given":"Hongwei","family":"Liu","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2008,11,6]]},"URL":"http://dx.doi.org/10.1007/s10623-008-9243-1","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9243"},{"id":"doi:10.1007/s10623-008-9220-8","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s10623-008-9220-8","page":"147-162","source":"Crossref","title":"Modular independence and generator matrices for codes over $${\\mathbb {Z}_m}$$","volume":"50","author":[{"given":"Young Ho","family":"Park","sequence":"first","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2008,6,27]]},"URL":"http://dx.doi.org/10.1007/s10623-008-9220-8","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9220"},{"id":"doi:10.1007/978-3-319-59806-2","type":"book","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-59806-2","source":"Crossref","title":"Algebraic Coding Theory Over Finite Commutative Rings","author":[{"given":"Steven T.","family":"Dougherty","sequence":"first","affiliation":[]}],"container-title":"SpringerBriefs in Mathematics","original-title":[],"issued":{"date-parts":[[2017]]},"ISBN":["9783319598055","9783319598062"],"URL":"http://dx.doi.org/10.1007/978-3-319-59806-2","ISSN":["2191-8198","2191-8201"],"note":"publisher-location:Cham"},{"id":"doi:10.1109/TIT.1958.1057446","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1958.1057446","page":"77-82","source":"Crossref","title":"Some properties of nonbinary error-correcting codes","volume":"4","author":[{"given":"C.","family":"Lee","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[1958,6]]},"URL":"http://dx.doi.org/10.1109/TIT.1958.1057446","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-Uniform-Code-Council-(UCC)-U","type":"document","note":"Uniform Code Council (UCC), Uniform Product Code (U.P.C."},{"id":"manual:-P.-Sole-ed.-2009.-Codes-over","type":"book","editor":[{"family":"Sole","given":"P."}],"title":"Codes over rings","volume":"6","publisher":"World Scientific","issued":"2009"},{"id":"doi:10.1109/18.57208","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.57208","page":"1098-1100","source":"Crossref","title":"Optimized permutation modulation","volume":"36","author":[{"given":"I.","family":"Ingemarsson","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1990]]},"URL":"http://dx.doi.org/10.1109/18.57208","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1971.1054601","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1971.1054601","page":"157-161","source":"Crossref","title":"How does a porcupine separate its quills?","volume":"17","author":[{"given":"H.","family":"Landau","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1971,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1971.1054601","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1023/A:1018723416627","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1023/a:1018723416627","page":"279-288","source":"Crossref","title":"Spherical 7-Designs in 2\n                n\n              -Dimensional Euclidean Space","volume":"10","author":[{"given":"V.M.","family":"Sidelnikov","sequence":"first","affiliation":[]}],"container-title":"Journal of Algebraic Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[1999,11]]},"URL":"http://dx.doi.org/10.1023/A:1018723416627","ISSN":["0925-9899","1572-9192"],"container-title-short":"Journal of Algebraic Combinatorics","note":"alternative-id:234956"},{"id":"manual:-V.-M.-Sidelnikov-Orbital-sph","type":"article-journal","author":[{"family":"Sidelnikov","given":"V.M."}],"title":"Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial","volume":"11","container-title":"Algebra i Analiz","issue":"4","issued":"1999","page":"183–203"},{"id":"doi:10.1002/j.1538-7305.1968.tb02486","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1002/j.1538-7305.1968.tb02486.x","page":"575-602","source":"Crossref","title":"Group Codes for the Gaussian Channel","volume":"47","author":[{"given":"David","family":"Slepian","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1968,4]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1968.tb02486.x","ISSN":["0005-8580"]},{"id":"doi:10.1109/18.485721","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/18.485721","page":"519-528","source":"Crossref","title":"Group codes generated by finite reflection groups","volume":"42","author":[{"given":"T.","family":"Mittelholzer","sequence":"first","affiliation":[]},{"given":"J.","family":"Lahtonen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1996,3]]},"URL":"http://dx.doi.org/10.1109/18.485721","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-N.-Biggs.-Finite-groups-of-a","type":"article-journal","author":[{"family":"Biggs","given":"N."}],"title":"Finite groups of automorphisms: course given at the University of Southampton","volume":"6","publisher":"CUP Archive","container-title":"October-December","issued":"1969"},{"id":"doi:10.1109/TIT.1977.1055744","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.1977.1055744","page":"500-503","source":"Crossref","title":"On the existence of[M, n]group codes for the Gaussian channel with[M, n]odd","volume":"23","author":[{"given":"C.","family":"Downey","sequence":"first","affiliation":[]},{"given":"J.","family":"Karlof","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1977,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1977.1055744","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1137/0134060","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","DOI":"10.1137/0134060","page":"715-716","source":"Crossref","title":"Odd Group Codes for The Gaussian Channel","volume":"34","author":[{"given":"John","family":"Karlof","sequence":"first","affiliation":[]},{"given":"Charles","family":"Downey","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1978,6]]},"URL":"http://dx.doi.org/10.1137/0134060","ISSN":["0036-1399","1095-712X"],"container-title-short":"SIAM J. Appl. Math.","note":"alternative-id:10.1137/0134060"},{"id":"doi:10.1007/s10623-008-9183-9","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1-3","DOI":"10.1007/s10623-008-9183-9","page":"307-321","source":"Crossref","title":"Flat tori, lattices and bounds for commutative group codes","volume":"49","author":[{"given":"Rogério M.","family":"Siqueira","sequence":"first","affiliation":[]},{"given":"Sueli I. R.","family":"Costa","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2008,3,12]]},"URL":"http://dx.doi.org/10.1007/s10623-008-9183-9","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9183"},{"id":"doi:10.1007/BFb0042068","type":"book-chapter","publisher":"Springer-Verlag","DOI":"10.1007/bfb0042068","page":"73-108","source":"Crossref","title":"Group Codes for the Gaussian Channel","author":[{"given":"Ingemar","family":"Ingemarsson","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Control and Information Sciences","original-title":[],"language":"en","issued":{"date-parts":[[null]]},"ISBN":["3540514058"],"URL":"http://dx.doi.org/10.1007/BFb0042068","note":"publisher-location:Berlin/Heidelberg"},{"id":"doi:10.2969/jmsj/03620341","type":"journal-article","publisher":"Mathematical Society of Japan (Project Euclid)","issue":"2","DOI":"10.2969/jmsj/03620341","source":"Crossref","title":"Spherical t-designs which are orbits of finite groups","volume":"36","author":[{"given":"Eiichi","family":"BANNAI","sequence":"first","affiliation":[]}],"container-title":"Journal of the Mathematical Society of Japan","original-title":[],"issued":{"date-parts":[[1984,4,1]]},"URL":"http://dx.doi.org/10.2969/jmsj/03620341","ISSN":["0025-5645"],"container-title-short":"J. Math. Soc. Japan"},{"id":"doi:10.1016/S0195-6698(03)00101-X","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0195-6698(03)00101-x","page":"213-227","source":"Crossref","title":"Spherical designs and finite group representations (some results of E. Bannai)","volume":"25","author":[{"given":"Pierre de la","family":"Harpe","sequence":"first","affiliation":[]},{"given":"Claude","family":"Pache","sequence":"additional","affiliation":[]}],"container-title":"European Journal of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2004,2]]},"URL":"http://dx.doi.org/10.1016/S0195-6698(03)00101-X","ISSN":["0195-6698"],"container-title-short":"European Journal of Combinatorics","note":"alternative-id:S019566980300101X"},{"id":"arxiv:2308.02499","type":"article-journal","author":[{"given":"Mozhgan","family":"Mohammadpour"},{"given":"Shayne","family":"Waldron"}],"title":"Complex spherical designs from group orbits","issued":{"date-parts":[[2024,4,1]]},"note":"arxivid:2308.02499\narxiv_version_number:3"},{"id":"doi:10.1007/978-0-387-34149-1_21","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-0-387-34149-1_21","page":"461-466","source":"Crossref","title":"Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations","author":[{"given":"S. L.","family":"Sobolev","sequence":"first","affiliation":[]}],"container-title":"Selected Works of S.L. Sobolev","original-title":[],"language":"en","issued":{"date-parts":[[null]]},"ISBN":["9780387341484"],"URL":"http://dx.doi.org/10.1007/978-0-387-34149-1_21","note":"publisher-location:Boston, MA"},{"id":"arxiv:2303.09000","type":"article-journal","author":[{"given":"Masatake","family":"Hirao"},{"given":"Hiroshi","family":"Nozaki"},{"given":"Koji","family":"Tasaka"}],"title":"Spherical designs and modular forms of the $D_4$ lattice","issued":{"date-parts":[[2023,9,28]]},"note":"arxivid:2303.09000\narxiv_version_number:3"},{"id":"doi:10.1007/BF01442795","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf01442795","page":"366-389","source":"Crossref","title":"Sur les formes quadratiques","volume":"6","author":[{"given":"A.","family":"Korkine","sequence":"first","affiliation":[]},{"given":"G.","family":"Zolotareff","sequence":"additional","affiliation":[]}],"container-title":"Mathematische Annalen","original-title":[],"language":"fr","issued":{"date-parts":[[1873,9]]},"URL":"http://dx.doi.org/10.1007/BF01442795","ISSN":["0025-5831","1432-1807"],"container-title-short":"Math. Ann.","note":"alternative-id:BF01442795"},{"id":"doi:10.4153/CJM-1981-038-7","type":"journal-article","publisher":"Canadian Mathematical Society","issue":"2","abstract":"<jats:p>In this paper we show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in <jats:bold>R</jats:bold><jats:sup>8</jats:sup> (resp. <jats:bold>R</jats:bold><jats:sup>24</jats:sup>) so that they all touch another unit sphere, and only one way of arranging 56 (resp. 4600) spheres in <jats:bold>R</jats:bold><jats:sup>8</jats:sup> (resp. <jats:bold>R</jats:bold><jats:sup>24</jats:sup>) so that they all touch two further, touching spheres. The following tight spherical <jats:italic>t</jats:italic>-designs are unique: the 5-design in Ω<jats:sub>7</jats:sub>, the 7-designs in Ω<jats:sub>8</jats:sub> and Ω<jats:sub>23</jats:sub>, and the 11-design in Ω<jats:sub>24</jats:sub>. It was shown in <jats:bold>[20]</jats:bold> that the maximum number of nonoverlapping unit spheres in <jats:bold>R</jats:bold><jats:sup>8</jats:sup> (resp. <jats:bold>R</jats:bold><jats:sup>24</jats:sup>) that can touch another unit sphere is 240 (resp. 196560). Arrangements of spheres meeting these bounds can be obtained from the E<jats:sub>8</jats:sub> and Leech lattices, respectively. The present paper shows that these are the only arrangements meeting these bounds.</jats:p>","DOI":"10.4153/cjm-1981-038-7","page":"437-449","source":"Crossref","title":"Uniqueness of Certain Spherical Codes","volume":"33","author":[{"given":"Eiichi","family":"Bannai","sequence":"first","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1981,4,1]]},"URL":"http://dx.doi.org/10.4153/CJM-1981-038-7","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j. math.","note":"alternative-id:S0008414X00032521"},{"id":"doi:10.1007/s00010-024-01036-6","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","abstract":"<jats:title>Abstract</jats:title><jats:p>We characterize the cases of existence of spherical designs of an odd strength attaining the Fazekas–Levenshtein bound for covering and prove some of their properties. We also find all universal minima of the potential of regular spherical configurations in two new cases: the demihypercube on <jats:inline-formula><jats:alternatives><jats:tex-math>$$S^d$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msup>\n                    <mml:mi>S</mml:mi>\n                    <mml:mi>d</mml:mi>\n                  </mml:msup>\n                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$d\\ge 4$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d</mml:mi>\n                    <mml:mo>≥</mml:mo>\n                    <mml:mn>4</mml:mn>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula>, and the <jats:inline-formula><jats:alternatives><jats:tex-math>$$2_{41}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mn>2</mml:mn>\n                    <mml:mn>41</mml:mn>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula> polytope on <jats:inline-formula><jats:alternatives><jats:tex-math>$$S^7$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msup>\n                    <mml:mi>S</mml:mi>\n                    <mml:mn>7</mml:mn>\n                  </mml:msup>\n                </mml:math></jats:alternatives></jats:inline-formula> (which is dual to the <jats:inline-formula><jats:alternatives><jats:tex-math>$$E_8$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mi>E</mml:mi>\n                    <mml:mn>8</mml:mn>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula> lattice).\n</jats:p>","DOI":"10.1007/s00010-024-01036-6","page":"509-533","source":"Crossref","title":"Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials","volume":"98","author":[{"given":"Sergiy","family":"Borodachov","sequence":"first","affiliation":[]}],"container-title":"Aequationes mathematicae","original-title":[],"language":"en","issued":{"date-parts":[[2024,3,6]]},"URL":"http://dx.doi.org/10.1007/s00010-024-01036-6","ISSN":["0001-9054","1420-8903"],"container-title-short":"Aequat. Math.","note":"alternative-id:1036"},{"id":"doi:10.1007/978-3-642-76709-8_5","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-76709-8_5","page":"71-107","source":"Crossref","title":"The Cell Structures of Certain Lattices","author":[{"given":"John H.","family":"Conway","sequence":"first","affiliation":[]},{"given":"Neil J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Miscellanea Mathematica","original-title":[],"language":"en","issued":{"date-parts":[[1991]]},"ISBN":["9783642767111","9783642767098"],"URL":"http://dx.doi.org/10.1007/978-3-642-76709-8_5","note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/TIT.1981.1056351","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1981.1056351","page":"327-338","source":"Crossref","title":"Tables of sphere packings and spherical codes","volume":"27","author":[{"given":"N.","family":"Sloane","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1981,5]]},"URL":"http://dx.doi.org/10.1109/TIT.1981.1056351","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/18.265491","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.265491","page":"1799-1808","source":"Crossref","title":"Shaping multidimensional signal spaces. I. Optimum shaping, shell mapping","volume":"39","author":[{"given":"A.K.","family":"Khandani","sequence":"first","affiliation":[]},{"given":"P.","family":"Kabal","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1993]]},"URL":"http://dx.doi.org/10.1109/18.265491","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-B.-B.-Venkov-Even-unimodular","type":"article-journal","author":[{"family":"Venkov","given":"B.B."}],"title":"Even unimodular extremal lattices, algebraic geometry and its applications","volume":"165","container-title":"Trudy Matematicheskogo Instituta imeni V. A. Steklova","issued":"1984","page":"43–48"},{"id":"preset:Venkov01","type":"chapter","author":[{"family":"Venkov","given":"B."}],"title":"Réseaux et designs sphériques","container-title":"Réseaux euclidiens, designs sphériques et formes modulaires","volume":"37","collection-title":"of Monogr. Enseign. Math","publisher":"Enseignement Math","issued":"2001","page":"10–86","publisher-place":"Geneva"},{"id":"doi:10.1016/j.ejc.2008.11.007","type":"journal-article","publisher":"Elsevier BV","issue":"6","DOI":"10.1016/j.ejc.2008.11.007","page":"1392-1425","source":"Crossref","title":"A survey on spherical designs and algebraic combinatorics on spheres","volume":"30","author":[{"given":"Eiichi","family":"Bannai","sequence":"first","affiliation":[]},{"given":"Etsuko","family":"Bannai","sequence":"additional","affiliation":[]}],"container-title":"European Journal of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2009,8]]},"URL":"http://dx.doi.org/10.1016/j.ejc.2008.11.007","ISSN":["0195-6698"],"container-title-short":"European Journal of Combinatorics","note":"alternative-id:S0195669808002400"},{"id":"doi:10.1016/0021-8693(89)90260-3","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0021-8693(89)90260-3","page":"387-408","source":"Crossref","title":"Vector stabilizers and subgroups of Leech lattice groups","volume":"127","author":[{"given":"Robert A","family":"Wilson","sequence":"first","affiliation":[]}],"container-title":"Journal of Algebra","original-title":[],"language":"en","issued":{"date-parts":[[1989,12]]},"URL":"http://dx.doi.org/10.1016/0021-8693(89)90260-3","ISSN":["0021-8693"],"container-title-short":"Journal of Algebra","note":"alternative-id:0021869389902603"},{"id":"arxiv:math/0607448","type":"article-journal","author":[{"given":"Henry","family":"Cohn"},{"given":"Abhinav","family":"Kumar"}],"title":"Uniqueness of the (22,891,1/4) spherical code","issued":{"date-parts":[[2007,6,14]]},"note":"arxivid:math/0607448\narxiv_version_number:2"},{"id":"doi:10.1016/0097-3165(79)90074-8","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0097-3165(79)90074-8","page":"210-214","source":"Crossref","title":"New bounds on the number of unit spheres that can touch a unit sphere in n dimensions","volume":"26","author":[{"given":"A.M","family":"Odlyzko","sequence":"first","affiliation":[]},{"given":"N.J.A","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[1979,3]]},"URL":"http://dx.doi.org/10.1016/0097-3165(79)90074-8","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:0097316579900748"},{"id":"manual:-N.-N.-Andreev-Location-of-po","type":"article-journal","author":[{"family":"Andreev","given":"N.N."}],"title":"Location of points on a sphere with minimal energy","volume":"219","container-title":"Priblizh. Garmon. Anal., 27–31; translation in Proceedings of the Steklov Institute of Mathematics 1997","issue":"219","issued":"1997","page":"20–24"},{"id":"arxiv:0906.1427","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/s00454-010-9266-z","page":"904-911","source":"Crossref","title":"The Contact Polytope of the Leech Lattice","volume":"44","author":[{"given":"Mathieu","family":"Dutour Sikirić","sequence":"first","affiliation":[]},{"given":"Achill","family":"Schürmann","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Vallentin","sequence":"additional","affiliation":[]}],"container-title":"Discrete &amp; Computational Geometry","original-title":[],"language":"en","issued":{"date-parts":[[2010,6,3]]},"URL":"http://dx.doi.org/10.1007/s00454-010-9266-z","ISSN":["0179-5376","1432-0444"],"container-title-short":"Discrete Comput Geom","note":"alternative-id:9266\narxivid:0906.1427"},{"id":"doi:10.1002/j.1538-7305.1948.tb00917","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1002/j.1538-7305.1948.tb00917.x","page":"623-656","source":"Crossref","title":"A Mathematical Theory of Communication","volume":"27","author":[{"given":"C. E.","family":"Shannon","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1948,10]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1948.tb00917.x","ISSN":["0005-8580"]},{"id":"arxiv:1202.0533","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2012.6284250","page":"546-550","source":"Crossref","title":"Polar coding to achieve the Holevo capacity of a pure-loss optical channel","author":[{"given":"Saikat","family":"Guha","sequence":"first","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"event":"2012 IEEE International Symposium on Information Theory - ISIT","container-title":"2012 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2012,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2012.6284250","note":"arxivid:1202.0533"},{"id":"manual:-International-Telecommunicat","type":"document","author":[{"family":"Union-T","given":"International Telecommunication"}],"title":"Recommendation V.21: 300 bits per second duplex modem standardized for use in the general switched telephone network","issued":"1984"},{"id":"manual:-International-Telecommunicat","type":"document","title":"International Telecommunication Union-T, Recommendation V.23: 600/1200-baud modem standardized for use in the general switched telephone network","issued":"1988"},{"id":"manual:-M.-Lahanas.-Ancient-Greek-Co","type":"document","author":[{"family":"Lahanas","given":"M."}],"title":"Ancient Greek Communication Methods","note":"Available at \\url{https://web.archive.org/web/20141102224501/https://www.mlahanas.de/Greeks/Communication.htm}.","URL":"https://web.archive.org/web/20141102224501/https://www.mlahanas.de/Greeks/Communication.htm"},{"id":"doi:10.1049/iet-com.2019.0051","type":"journal-article","publisher":"Institution of Engineering and Technology (IET)","issue":"16","abstract":"<jats:p>This survey article discusses the revolutions of wireless communication technologies right from the prehistoric man's fire signals to today's space optical communications. A brief description of evaluations of communication technologies from the ancient Greece to today's mature telecommunication fields is discussed in this review paper. Principle, significance, demonstration and development of free space optical (FSO) communication technology over different decades are discussed. Major FSO channel limitations, temporal and spatial challenges of FSO communication system and their state‐of‐art mitigation techniques are presented. Classical quantitative analysis of reliability of FSO communication, developments on radio over FSO (RoFSO) and hybrid FSO/RF systems are also explained. Advanced developments of FSO communication techniques such as optical free space wavelength division multiplexing (WDM), sub‐carrier multiplexing (SCM), worldwide interoperability for microwave access (WiMAX), visible light communications (VLC) and vehicular visible light communications (VVLC) are reported. Deep space optical communication systems and next generation FSO wireless terrestrial/global network architecture are summarised. Research challenges of FSO systems for internet of things/everything (IoT/IoE), 5G communication, mobile‐network, terahertz spectrum, quantum communication and underwater optical applications are presented. Based on the review, we outlined the challenges that need to be addressed in near‐future researches to realize full potential FSO communication systems.</jats:p>","DOI":"10.1049/iet-com.2019.0051","page":"2405-2419","source":"Crossref","title":"Historical perspective of free space optical communications: from the early dates to today's developments","volume":"13","author":[{"given":"Arockia Bazil","family":"Raj","sequence":"first","affiliation":[{"name":"Defence Institute of Advanced Technology Pune India"}]},{"given":"Arun K.","family":"Majumder","sequence":"additional","affiliation":[{"name":"Colorado State University‐Pueblo California USA"}]}],"container-title":"IET Communications","original-title":[],"language":"en","issued":{"date-parts":[[2019,10]]},"URL":"http://dx.doi.org/10.1049/iet-com.2019.0051","ISSN":["1751-8628","1751-8636"],"container-title-short":"IET Communications","note":"alternative-id:10.1049/iet-com.2019.0051"},{"id":"manual:-J.-L.-Massey-Convolutional-c","type":"chapter","author":[{"family":"Massey","given":"J.L."}],"title":"Convolutional codes over rings","container-title":"Fourth Joint Swedish-Soviet International Workshop on Information Theory","issued":"1989"},{"id":"manual:-J.-L.-Massey-Ring-convolutio","type":"chapter","author":[{"family":"Massey","given":"J.L."}],"title":"Ring convolutional codes for phase modulation","container-title":"IEEE Int. Symp. on Information Theory","issued":"Jan","page":"14–19","publisher-place":"San Diego, CA"},{"id":"manual:-International-Telecommunicat","type":"document","title":"International Telecommunication Union-T, Recommendation V.27ter: 4800/2400 Bits Per Second Modem Standardized For Use in the General Switched Telephone Network","issued":"1984"},{"id":"doi:10.1109/TCE.1962.6373228","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tce.1962.6373228","page":"166-171","source":"Crossref","title":"Phase-modulation data sets for serial transmission at 2,000 and 2,400 bits per second","volume":"81","author":[{"given":"P. A.","family":"Baker","sequence":"first","affiliation":[{"name":"Bell Telephone Laboratories Inc., Murray Hill, N. J."}]}],"container-title":"Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics","original-title":[],"issued":{"date-parts":[[1962,7]]},"URL":"http://dx.doi.org/10.1109/TCE.1962.6373228","ISSN":["0097-2452","2379-674X"],"container-title-short":"Trans. AIEE, Part I: Comm. Electron."},{"id":"doi:10.1109/25.289416","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/25.289416","page":"355-365","source":"Crossref","title":"Modems for emerging digital cellular-mobile radio system","volume":"40","author":[{"given":"K.","family":"Feher","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Vehicular Technology","original-title":[],"issued":{"date-parts":[[1991,5]]},"URL":"http://dx.doi.org/10.1109/25.289416","ISSN":["0018-9545"],"container-title-short":"IEEE Trans. Veh. Technol."},{"id":"manual:-International-Telecommunicat","type":"document","author":[{"family":"Union-T","given":"International Telecommunication"}],"title":"Recommendation V.24: List of definitions for interchange circuits between data terminal equipment (DTE) and data circuit-terminating equipment (DCE","issued":"1988"},{"id":"doi:10.1139/v77-246","type":"journal-article","publisher":"Canadian Science Publishing","issue":"10","abstract":"<jats:p> We review the formulation and solutions of a number of extremal problems associated with points and unit charges on the surface of a sphere in E<jats:sup>3</jats:sup>. For one of these problems, namely[Formula: see text]where d<jats:sub>pq</jats:sub> is the Euclidean distance between points P and Q and m is the number of points, we discuss the results for m ≤ 16 and 1 ≤ n ≤ ∞. For the cases m = 5, 11, 13–16 we find hitherto undiscovered solutions. Our solutions for m = 5 and 11 correct earlier results in the literature. We also sharpen the existing literature results for m = 7 and 10. </jats:p>","DOI":"10.1139/v77-246","page":"1745-1761","source":"Crossref","title":"Extremal arrangements of points and unit charges on a sphere: equilibrium configurations revisited","volume":"55","author":[{"given":"Theodor William","family":"Melnyk","sequence":"first","affiliation":[]},{"given":"Osvald","family":"Knop","sequence":"additional","affiliation":[]},{"given":"William Robert","family":"Smith","sequence":"additional","affiliation":[]}],"container-title":"Canadian Journal of Chemistry","original-title":[],"language":"en","issued":{"date-parts":[[1977,5,15]]},"URL":"http://dx.doi.org/10.1139/v77-246","ISSN":["0008-4042","1480-3291"],"container-title-short":"Can. J. Chem.","note":"alternative-id:10.1139/v77-246"},{"id":"doi:10.1109/TIT.1987.1057277","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.1987.1057277","page":"116-123","source":"Crossref","title":"Using simulated annealing to design good codes","volume":"33","author":[{"given":"A.E.","family":"Gamal","sequence":"first","affiliation":[]},{"given":"L.","family":"Hemachandra","sequence":"additional","affiliation":[]},{"given":"I.","family":"Shperling","sequence":"additional","affiliation":[]},{"given":"V.","family":"Wei","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1987,1]]},"URL":"http://dx.doi.org/10.1109/TIT.1987.1057277","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1107/S0108767390011370","type":"journal-article","publisher":"International Union of Crystallography (IUCr)","issue":"3","DOI":"10.1107/s0108767390011370","page":"158-165","source":"Crossref","title":"The densest packing of equal circles on a sphere","volume":"47","author":[{"given":"D. A.","family":"Kottwitz","sequence":"first","affiliation":[]}],"container-title":"Acta Crystallographica Section A Foundations of Crystallography","original-title":[],"issued":{"date-parts":[[1991,5,1]]},"URL":"http://dx.doi.org/10.1107/S0108767390011370","ISSN":["0108-7673"],"container-title-short":"Acta Crystallogr A Found Crystallogr","note":"alternative-id:S0108767390011370"},{"id":"doi:10.1109/18.641545","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.641545","page":"1786-1798","source":"Crossref","title":"Asymptotically dense spherical codes .II. laminated spherical codes","volume":"43","author":[{"given":"J.","family":"Hamkins","sequence":"first","affiliation":[]},{"given":"K.","family":"Zeger","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1109/18.641545","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/ISIT.2019.8849464","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2019.8849464","page":"1612-1616","source":"Crossref","title":"Constructive spherical codes in 2<sup>k</sup> dimensions","author":[{"given":"Henrique K.","family":"Miyamoto","sequence":"first","affiliation":[]},{"given":"Henrique N.","family":"Sa Earp","sequence":"additional","affiliation":[]},{"given":"Sueli I. R.","family":"Costa","sequence":"additional","affiliation":[]}],"event":"2019 IEEE International Symposium on Information Theory (ISIT)","container-title":"2019 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2019,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2019.8849464"},{"id":"doi:10.1109/ISIT.2009.5205529","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2009.5205529","page":"2033-2037","source":"Crossref","title":"Spherical codes on torus layers","author":[{"given":"Cristiano","family":"Torezzan","sequence":"first","affiliation":[]},{"given":"Sueli I. R.","family":"Costa","sequence":"additional","affiliation":[]},{"given":"Vinay A.","family":"Vaishampayan","sequence":"additional","affiliation":[]}],"event":"2009 IEEE International Symposium on Information Theory - ISIT","container-title":"2009 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2009,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2009.5205529"},{"id":"doi:10.1007/978-3-319-67882-5","type":"book","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-67882-5","source":"Crossref","title":"Lattices Applied to Coding for Reliable and Secure Communications","author":[{"given":"Sueli I.R.","family":"Costa","sequence":"first","affiliation":[]},{"given":"Frédérique","family":"Oggier","sequence":"additional","affiliation":[]},{"given":"Antonio","family":"Campello","sequence":"additional","affiliation":[]},{"given":"Jean-Claude","family":"Belfiore","sequence":"additional","affiliation":[]},{"given":"Emanuele","family":"Viterbo","sequence":"additional","affiliation":[]}],"container-title":"SpringerBriefs in Mathematics","original-title":[],"issued":{"date-parts":[[2017]]},"ISBN":["9783319678818","9783319678825"],"URL":"http://dx.doi.org/10.1007/978-3-319-67882-5","ISSN":["2191-8198","2191-8201"],"note":"publisher-location:Cham"},{"id":"doi:10.1109/18.641544","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.641544","page":"1774-1785","source":"Crossref","title":"Asymptotically dense spherical codes. I. Wrapped spherical codes","volume":"43","author":[{"given":"J.","family":"Hamkins","sequence":"first","affiliation":[]},{"given":"K.","family":"Zeger","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1997]]},"URL":"http://dx.doi.org/10.1109/18.641544","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"preset:albertPC25","type":"document","author":[{"family":"Albert","given":"V.V."}],"genre":"private communication,","issued":"2025"},{"id":"manual:-R.-Klitzing.-N-ap.-Polytopes","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"N-ap","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/n-ap.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/n-ap.htm}"},{"id":"manual:-R.-Klitzing.-Squap.-Polytope","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Squap","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/squap.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/squap.htm}"},{"id":"doi:10.1007/BF01343127","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01343127","page":"325-334","source":"Crossref","title":"Das Problem der dreizehn Kugeln","volume":"125","author":[{"given":"K.","family":"Schütte","sequence":"first","affiliation":[]},{"given":"B. L.","family":"van der Waerden","sequence":"additional","affiliation":[]}],"container-title":"Mathematische Annalen","original-title":[],"language":"de","issued":{"date-parts":[[1952,12]]},"URL":"http://dx.doi.org/10.1007/BF01343127","ISSN":["0025-5831","1432-1807"],"container-title-short":"Math. Ann.","note":"alternative-id:BF01343127"},{"id":"manual:-R.-Klitzing.-Co.-Polytopes-t","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Co","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/co.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/co.htm}"},{"id":"doi:10.7312/hold92776","type":"edited-book","publisher":"Columbia University Press","DOI":"10.7312/hold92776","source":"Crossref","title":"Shapes, Space, and Symmetry","author":[{"given":"Alan","family":"Holden","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1971,12,31]]},"ISBN":["9780231890403"],"URL":"http://dx.doi.org/10.7312/hold92776","note":"alternative-id:10.7312/hold92776"},{"id":"manual:-R.-Klitzing.-Doe.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Doe","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/doe.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/doe.htm}"},{"id":"arxiv:2302.11593","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","DOI":"10.1038/s41567-024-02496-y","page":"1300-1305","source":"Crossref","title":"Quantum spherical codes","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0003-3409-1126","authenticated-orcid":false,"given":"Shubham P.","family":"Jain","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3383-1946","authenticated-orcid":false,"given":"Joseph T.","family":"Iosue","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8972-4413","authenticated-orcid":false,"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":false,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,15]]},"URL":"http://dx.doi.org/10.1038/s41567-024-02496-y","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2496\narxivid:2302.11593"},{"id":"manual:-R.-Klitzing.-Ike.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Ike","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/ike.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/ike.htm}"},{"id":"manual:-N.-N.-Andreev-An-extremal-pr","type":"article-journal","author":[{"family":"Andreev","given":"N.N."}],"title":"An extremal property of the icosahedron","volume":"2","container-title":"East J. Approx","issue":"4","issued":"1996","page":"459–462"},{"id":"arxiv:2305.07023","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.131.240601","source":"Crossref","title":"Family of Quantum Codes with Exotic Transversal Gates","volume":"131","author":[{"ORCID":"https://orcid.org/0000-0001-5451-0290","authenticated-orcid":true,"given":"Eric","family":"Kubischta","sequence":"first","affiliation":[]},{"given":"Ian","family":"Teixeira","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.131.240601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240601","note":"arxivid:2305.07023"},{"id":"manual:-R.-Klitzing.-M3m5o.-Polytope","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"M3m5o","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/m3m5o.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/m3m5o.htm}"},{"id":"doi:10.1007/978-1-4612-5648-9_13","type":"book-chapter","publisher":"Springer New York","DOI":"10.1007/978-1-4612-5648-9_13","page":"203-218","source":"Crossref","title":"Cubature Formulae, Polytopes, and Spherical Designs","author":[{"given":"J. M.","family":"Goethals","sequence":"first","affiliation":[]},{"given":"J. J.","family":"Seidel","sequence":"additional","affiliation":[]}],"container-title":"The Geometric Vein","original-title":[],"language":"en","issued":{"date-parts":[[1981]]},"ISBN":["9781461256502","9781461256489"],"URL":"http://dx.doi.org/10.1007/978-1-4612-5648-9_13","note":"publisher-location:New York, NY"},{"id":"doi:10.1016/j.laa.2020.08.010","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.laa.2020.08.010","page":"84-106","source":"Crossref","title":"Spherical (t,t)-designs with a small number of vectors","volume":"608","author":[{"ORCID":"https://orcid.org/0000-0002-6506-2582","authenticated-orcid":false,"given":"Daniel","family":"Hughes","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5612-2371","authenticated-orcid":false,"given":"Shayne","family":"Waldron","sequence":"additional","affiliation":[]}],"container-title":"Linear Algebra and its Applications","original-title":[],"language":"en","issued":{"date-parts":[[2021,1]]},"URL":"http://dx.doi.org/10.1016/j.laa.2020.08.010","ISSN":["0024-3795"],"container-title-short":"Linear Algebra and its Applications","note":"special_numbering:C\nalternative-id:S0024379520303839"},{"id":"arxiv:2403.07457","type":"article-journal","author":[{"given":"Sergiy","family":"Borodachov"},{"given":"Peter","family":"Boyvalenkov"},{"given":"Peter","family":"Dragnev"},{"given":"Douglas","family":"Hardin"},{"given":"Edward","family":"Saff"},{"given":"Maya","family":"Stoyanova"}],"title":"Energy bounds for weighted spherical codes and designs via linear programming","issued":{"date-parts":[[2024,12,19]]},"note":"arxivid:2403.07457\narxiv_version_number:2"},{"id":"arxiv:1712.07666","type":"journal-article","publisher":"The Royal Society","issue":"2123","abstract":"<jats:p>We explore a distance-3 homological CSS quantum code, namely the small stellated dodecahedron code, for dense storage of quantum information and we compare its performance with the distance-3 surface code. The data and ancilla qubits of the small stellated dodecahedron code can be located on the edges respectively vertices of a small stellated dodecahedron, making this code suitable for three-dimensional connectivity. This code encodes eight logical qubits into 30 physical qubits (plus 22 ancilla qubits for parity check measurements) in contrast with one logical qubit into nine physical qubits (plus eight ancilla qubits) for the surface code. We develop fault-tolerant parity check circuits and a decoder for this code, allowing us to numerically assess the circuit-based pseudo-threshold.</jats:p>\n          <jats:p>This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.</jats:p>","DOI":"10.1098/rsta.2017.0323","page":"20170323","source":"Crossref","title":"The small stellated dodecahedron code and friends","volume":"376","author":[{"given":"J.","family":"Conrad","sequence":"first","affiliation":[{"name":"JARA Institute for Quantum Information, RWTH Aachen University, Aachen 52056, Germany"}]},{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":false,"given":"C.","family":"Chamberland","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"}]},{"given":"N. P.","family":"Breuckmann","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London WC1E 6BT, UK"}]},{"given":"B. M.","family":"Terhal","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, The Netherlands"},{"name":"Institute for Theoretical Nanoelectronics, Forschungszentrum Juelich, 52425 Juelich, Germany"}]}],"container-title":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,28]]},"URL":"http://dx.doi.org/10.1098/rsta.2017.0323","ISSN":["1364-503X","1471-2962"],"container-title-short":"Phil. Trans. R. Soc. A.","note":"alternative-id:10.1098/rsta.2017.0323\narxivid:1712.07666"},{"id":"manual:-R.-Klitzing.-Rhode.-Polytope","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Rhode","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/rhode.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/rhode.htm}"},{"id":"manual:-R.-Klitzing.-Sirco.-Polytope","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Sirco","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/sirco.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/sirco.htm}"},{"id":"arxiv:2403.16874","type":"article-journal","author":[{"given":"Henry","family":"Cohn"},{"given":"David","family":"de Laat"},{"given":"Nando","family":"Leijenhorst"}],"title":"Optimality of spherical codes via exact semidefinite programming bounds","issued":{"date-parts":[[2024,3,25]]},"note":"arxivid:2403.16874\narxiv_version_number:1"},{"id":"manual:-R.-Klitzing.-Snic.-Polytopes","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Snic","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/snic.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/snic.htm}"},{"id":"arxiv:1309.7530","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","DOI":"10.1007/s10701-014-9830-0","page":"1085-1095","source":"Crossref","title":"Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell","volume":"44","author":[{"given":"Mordecai","family":"Waegell","sequence":"first","affiliation":[]},{"given":"P. K.","family":"Aravind","sequence":"additional","affiliation":[]}],"container-title":"Foundations of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,7]]},"URL":"http://dx.doi.org/10.1007/s10701-014-9830-0","ISSN":["0015-9018","1572-9516"],"container-title-short":"Found Phys","note":"alternative-id:9830\narxivid:1309.7530"},{"id":"doi:10.3390/sym2031423","type":"journal-article","publisher":"MDPI AG","issue":"3","abstract":"<jats:p>The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may be used in the numerical computation of solid-state nuclear magnetic resonance spectra, and in spherical tensor analysis procedures.</jats:p>","DOI":"10.3390/sym2031423","page":"1423-1449","source":"Crossref","title":"Orientational Sampling Schemes Based on Four Dimensional Polytopes","volume":"2","author":[{"given":"Salvatore","family":"Mamone","sequence":"first","affiliation":[{"name":"School of Chemistry, University of Southampton, University Road, Southampton SO17 1BJ, UK"}]},{"given":"Giuseppe","family":"Pileio","sequence":"additional","affiliation":[{"name":"School of Chemistry, University of Southampton, University Road, Southampton SO17 1BJ, UK"}]},{"given":"Malcolm H.","family":"Levitt","sequence":"additional","affiliation":[{"name":"School of Chemistry, University of Southampton, University Road, Southampton SO17 1BJ, UK"}]}],"container-title":"Symmetry","original-title":[],"language":"en","issued":{"date-parts":[[2010,7,7]]},"URL":"http://dx.doi.org/10.3390/sym2031423","ISSN":["2073-8994"],"container-title-short":"Symmetry","note":"alternative-id:sym2031423"},{"id":"doi:10.1007/978-3-0348-4117-7_8","type":"book-chapter","publisher":"Springer Basel","DOI":"10.1007/978-3-0348-4117-7_8","page":"198-218","source":"Crossref","title":"An Attempt to determine the twenty-seven Lines upon a Surface of the third Order, and to divide such Surfaces into Species in Reference to the Reality of the Lines upon the Surface","author":[{"given":"L.","family":"Kollros","sequence":"first","affiliation":[]}],"container-title":"Gesammelte Mathematische Abhandlungen","original-title":[],"language":"de","issued":{"date-parts":[[1953]]},"ISBN":["9783034840453","9783034841177"],"URL":"http://dx.doi.org/10.1007/978-3-0348-4117-7_8","note":"publisher-location:Basel"},{"id":"manual:-R.-Klitzing.-Hi.-Polytopes-t","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Hi","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/hi.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/hi.htm}"},{"id":"manual:-P.-H.-Schoute-(1903).-Mehrdi","type":"article-journal","author":[{"family":"Schoute","given":"P.H."}],"volume":"2","publisher":"Die Polytope","container-title":"Mehrdimensionale Geometrie","issued":"1903"},{"id":"arxiv:0911.2289","type":"journal-article","publisher":"IOP Publishing","issue":"10","DOI":"10.1088/1751-8113/43/10/105304","page":"105304","source":"Crossref","title":"Critical noncolorings of the 600-cell proving the Bell–Kochen–Specker theorem","volume":"43","author":[{"given":"Mordecai","family":"Waegell","sequence":"first","affiliation":[]},{"given":"P K","family":"Aravind","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2010,2,18]]},"URL":"http://dx.doi.org/10.1088/1751-8113/43/10/105304","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(10)37660-8\narxivid:0911.2289"},{"id":"preset:Schlafli1901","type":"document","author":[{"family":"Schläfli","given":"L."}],"title":"\\emph{Theorie der vielfachen Kontinuität}","volume":"38","issued":"1901"},{"id":"arxiv:math/0309430","type":"article-journal","author":[{"given":"Oleg R.","family":"Musin"}],"title":"The kissing number in four dimensions","issued":{"date-parts":[[2006,10,4]]},"note":"arxivid:math/0309430\narxiv_version_number:3"},{"id":"manual:-R.-Klitzing.-Ico.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Ico","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/ico.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/ico.htm}"},{"id":"arxiv:math/0607447","type":"journal-article","publisher":"Informa UK Limited","issue":"3","DOI":"10.1080/10586458.2007.10129008","page":"313-320","source":"Crossref","title":"The<i>D</i><sub>4</sub>Root System Is Not Universally Optimal","volume":"16","author":[{"given":"Henry","family":"Cohn","sequence":"first","affiliation":[]},{"given":"John H.","family":"Conway","sequence":"additional","affiliation":[]},{"given":"Noam D.","family":"Elkies","sequence":"additional","affiliation":[]},{"given":"Abhinav","family":"Kumar","sequence":"additional","affiliation":[]}],"container-title":"Experimental Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2007,1]]},"URL":"http://dx.doi.org/10.1080/10586458.2007.10129008","ISSN":["1058-6458","1944-950X"],"container-title-short":"Experimental Mathematics","note":"alternative-id:10.1080/10586458.2007.10129008\narxivid:math/0607447"},{"id":"preset:Boroczky78","type":"article-journal","author":[{"family":"Boroczky","given":"K."}],"title":"Packing of spheres in spaces of constant curvature","volume":"32","container-title":"Acta Math. Acad. Sci. Hung","issued":"1978","page":"243–261"},{"id":"doi:10.1007/PL00000504","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/pl00000504","page":"360-368","source":"Crossref","title":"Uniqueness of the 120-point spherical 11-design in four dimensions","volume":"77","author":[{"given":"P.","family":"Boyvalenkov","sequence":"first","affiliation":[]},{"given":"D.","family":"Danev","sequence":"additional","affiliation":[]}],"container-title":"Archiv der Mathematik","original-title":[],"language":"en","issued":{"date-parts":[[2001,10]]},"URL":"http://dx.doi.org/10.1007/PL00000504","ISSN":["0003-889X"],"container-title-short":"Arch. Math.","note":"alternative-id:E698"},{"id":"manual:-R.-Klitzing.-Ex.-Polytopes-t","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Ex","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/ex.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/ex.htm}"},{"id":"doi:10.1109/ITW.2003.1216742","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.2003.1216742","page":"253-258","source":"Crossref","title":"Spherical designs in four dimensions","author":[{"given":"N.J.A.","family":"Sloane","sequence":"first","affiliation":[]},{"given":"R.H.","family":"Hardin","sequence":"additional","affiliation":[]},{"given":"P.","family":"Cara","sequence":"additional","affiliation":[]}],"event":"2003 IEEE Information Theory Workshop","container-title":"Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674)","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ITW.2003.1216742"},{"id":"doi:10.2307/2371466","type":"journal-article","publisher":"JSTOR","issue":"1/4","DOI":"10.2307/2371466","page":"457","source":"Crossref","title":"The Polytope 2 21 Whose Twenty-Seven Vertices Correspond to the Lines to the General Cubic Surface","volume":"62","author":[{"given":"H. S. M.","family":"Coxeter","sequence":"first","affiliation":[]}],"container-title":"American Journal of Mathematics","original-title":[],"issued":{"date-parts":[[1940]]},"URL":"http://dx.doi.org/10.2307/2371466","ISSN":["0002-9327"],"container-title-short":"American Journal of Mathematics"},{"id":"doi:10.1007/BF02417955","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf02417955","page":"83-119","source":"Crossref","title":"The classes and representations of the groups of 27 lines and 28 bitangents","volume":"32","author":[{"given":"J. S.","family":"Frame","sequence":"first","affiliation":[]}],"container-title":"Annali di Matematica Pura ed Applicata","original-title":[],"language":"en","issued":{"date-parts":[[1951,12]]},"URL":"http://dx.doi.org/10.1007/BF02417955","ISSN":["0373-3114","1618-1891"],"container-title-short":"Annali di Matematica","note":"alternative-id:BF02417955"},{"id":"preset:coxeterComplex","type":"book","author":[{"family":"Coxeter","given":"H.S.M."}],"title":"\\emph{Regular Complex Polytopes}","publisher":"Cambridge University Press","issued":"1991"},{"id":"manual:-P.-H.-Schoute-On-the-relatio","type":"article-journal","author":[{"family":"Schoute","given":"P.H."}],"title":"On the relation between the vertices of a definite six-dimensional polytope and the lines of a cubic surface","volume":"13","container-title":"Proc. Roy. Acad. Amsterdam","issued":"1910"},{"id":"arxiv:1911.05809","type":"article-journal","author":[{"given":"Blake C.","family":"Stacey"}],"title":"Sporadic SICs and Exceptional Lie Algebras","issued":{"date-parts":[[2019,11,13]]},"note":"arxivid:1911.05809\narxiv_version_number:1"},{"id":"manual:-R.-Klitzing.-3-3-3-3-3.-Comp","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"3-3-3-3-3","container-title":"Complex Polytopes","URL":"https://bendwavy.org/klitzing/complex/3-3-3-3-3.htm","note":"Available at \\url{https://bendwavy.org/klitzing/complex/3-3-3-3-3.htm}"},{"id":"manual:-R.-Klitzing.-Jak.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Jak","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/jak.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/jak.htm}"},{"id":"arxiv:1104.4692","type":"article-journal","author":[{"given":"Aidan","family":"Roy"},{"given":"Sho","family":"Suda"}],"title":"Complex spherical designs and codes","issued":{"date-parts":[[2011,4,25]]},"note":"arxivid:1104.4692\narxiv_version_number:1"},{"id":"manual:-R.-Klitzing.-3-3-3-4-2.-Comp","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"3-3-3-4-2","container-title":"Complex Polytopes","URL":"https://bendwavy.org/klitzing/complex/3-3-3-4-2.htm","note":"Available at \\url{https://bendwavy.org/klitzing/complex/3-3-3-4-2.htm}"},{"id":"manual:-R.-Klitzing.-Mo.-Polytopes-t","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Mo","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/mo.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/mo.htm}"},{"id":"manual:-R.-Klitzing.-Laq.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Laq","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/laq.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/laq.htm}"},{"id":"manual:-R.-Klitzing.-Naq.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Naq","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/naq.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/naq.htm}"},{"id":"doi:10.1515/dma.1994.4.2.143","type":"journal-article","publisher":"Walter de Gruyter GmbH","issue":"2","DOI":"10.1515/dma.1994.4.2.143","source":"Crossref","title":"On Korkin-Zolotarev’s construction","volume":"4","author":[{"given":"A. V.","family":"KOLUSHOV","sequence":"first","affiliation":[]},{"given":"V. A.","family":"YUDIN","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics and Applications","original-title":[],"issued":{"date-parts":[[1994]]},"URL":"http://dx.doi.org/10.1515/dma.1994.4.2.143","ISSN":["0924-9265","1569-3929"]},{"id":"manual:-R.-Klitzing.-Bay.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Bay","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/bay.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/bay.htm}"},{"id":"manual:-R.-Penrose-On-Bell-non-local","type":"article-journal","author":[{"family":"Penrose","given":"R."}],"title":"On Bell non-locality without probabilities: some curious geometry","container-title":"Quantum Reflections","issued":"2000","page":"1–27"},{"id":"doi:10.1016/0039-3681(93)90061-N","type":"journal-article","publisher":"Elsevier BV","issue":"5","DOI":"10.1016/0039-3681(93)90061-n","page":"697-720","source":"Crossref","title":"On bell non-locality without probabilities: More curious geometry","volume":"24","author":[{"given":"Jason","family":"Zimba","sequence":"first","affiliation":[]},{"given":"Roger","family":"Penrose","sequence":"additional","affiliation":[]}],"container-title":"Studies in History and Philosophy of Science Part A","original-title":[],"language":"en","issued":{"date-parts":[[1993,12]]},"URL":"http://dx.doi.org/10.1016/0039-3681(93)90061-N","ISSN":["0039-3681"],"container-title-short":"Studies in History and Philosophy of Science Part A","note":"alternative-id:003936819390061N"},{"id":"doi:10.1119/1.19336","type":"journal-article","publisher":"American Association of Physics Teachers (AAPT)","issue":"7","abstract":"<jats:p>This paper gives an elementary account of the “Penrose dodecahedron,” a set of 40 states of a spin-32 particle used by Zimba and Penrose [Stud. Hist. Phil. Sci. 24, 697–720 (1993)] to give a proof of Bell’s nonlocality theorem. The Penrose rays are constructed here from the rotation operator of a spin-32 particle and the geometry of a dodecahedron, and their orthogonality properties are derived and illustrated from a couple of different viewpoints. After recalling how the proof of Bell’s theorem can be reduced to a coloring problem on the Penrose rays, a “proof-tree” argument is used to establish the noncolorability of the Penrose rays and hence prove Bell’s theorem.</jats:p>","DOI":"10.1119/1.19336","page":"631-638","source":"Crossref","title":"The Penrose dodecahedron revisited","volume":"67","author":[{"given":"Jordan E.","family":"Massad","sequence":"first","affiliation":[{"name":"Physics Department, Worcester Polytechnic Institute, Worcester, Massachusetts 01609"}]},{"given":"P. K.","family":"Aravind","sequence":"additional","affiliation":[{"name":"Physics Department, Worcester Polytechnic Institute, Worcester, Massachusetts 01609"}]}],"container-title":"American Journal of Physics","original-title":[],"language":"en","issued":{"date-parts":[[1999,7,1]]},"URL":"http://dx.doi.org/10.1119/1.19336","ISSN":["0002-9505","1943-2909"]},{"id":"arxiv:1701.06512","type":"journal-article","publisher":"Elsevier BV","issue":"22","DOI":"10.1016/j.physleta.2017.03.039","page":"1853-1857","source":"Crossref","title":"The Penrose dodecahedron and the Witting polytope are identical in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" altimg=\"si1.gif\" overflow=\"scroll\"><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">CP</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>","volume":"381","author":[{"given":"Mordecai","family":"Waegell","sequence":"first","affiliation":[]},{"given":"P.K.","family":"Aravind","sequence":"additional","affiliation":[]}],"container-title":"Physics Letters A","original-title":[],"language":"en","issued":{"date-parts":[[2017,6]]},"URL":"http://dx.doi.org/10.1016/j.physleta.2017.03.039","ISSN":["0375-9601"],"container-title-short":"Physics Letters A","note":"updated-by:{\"DOI\":\"10.1016/j.physleta.2020.126331\",\"type\":\"erratum\",\"label\":\"Erratum\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2020,4,20]],\"date-time\":\"2020-04-20T00:00:00Z\",\"timestamp\":1587340800000}}\nalternative-id:S037596011730292X\narxivid:1701.06512"},{"id":"manual:-R.-Klitzing.-3-3-3-3-3-3-3.-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"3-3-3-3-3-3-3","container-title":"Complex Polytopes","URL":"https://bendwavy.org/klitzing/complex/3-3-3-3-3-3-3.htm","note":"Available at \\url{https://bendwavy.org/klitzing/complex/3-3-3-3-3-3-3.htm}"},{"id":"manual:-R.-Klitzing.-Fy.-Polytopes-t","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Fy","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/fy.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/fy.htm}"},{"id":"manual:-R.-Klitzing.-Hex.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Hex","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/hex.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/hex.htm}"},{"id":"preset:Stroud71","type":"book","author":[{"family":"Stroud","given":"A.H."}],"title":"\\emph{Approximate Calculation of Multiple Integrals}","publisher":"Prentice Hall","issued":"1971"},{"id":"manual:-R.-Klitzing.-Tes.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Tes","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/tes.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/tes.htm}"},{"id":"arxiv:2008.05051","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Ben W.","family":"Reichardt"}],"title":"Short Shor-style syndrome sequences","issued":{"date-parts":[[2020,8,12]]},"note":"arxivid:2008.05051\narxiv_version_number:1"},{"id":"arxiv:2112.03785","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.110.012419","source":"Crossref","title":"Distance-four quantum codes with combined postselection and error correction","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0001-8324-3996","authenticated-orcid":true,"given":"Prithviraj","family":"Prabhu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03taz7m60","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Southern California"}]},{"ORCID":"https://orcid.org/0000-0002-4934-8732","authenticated-orcid":true,"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03taz7m60","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Southern California"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.012419","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012419","note":"arxivid:2112.03785"},{"id":"manual:-R.-Klitzing.-Pen.-Polytopes-","type":"chapter","author":[{"family":"Klitzing","given":"R."}],"title":"Pen","container-title":"Polytopes \\& their Incidence Matrices","URL":"https://bendwavy.org/klitzing/incmats/pen.htm","note":"Available at \\url{https://bendwavy.org/klitzing/incmats/pen.htm}"},{"id":"arxiv:0708.3947","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.jcta.2008.05.001","page":"195-204","source":"Crossref","title":"Optimality and uniqueness of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" altimg=\"si1.gif\" overflow=\"scroll\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>10</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math> spherical code","volume":"116","author":[{"given":"Christine","family":"Bachoc","sequence":"first","affiliation":[]},{"given":"Frank","family":"Vallentin","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Theory, Series A","original-title":[],"language":"en","issued":{"date-parts":[[2009,1]]},"URL":"http://dx.doi.org/10.1016/j.jcta.2008.05.001","ISSN":["0097-3165"],"container-title-short":"Journal of Combinatorial Theory, Series A","note":"alternative-id:S0097316508000733\narxivid:0708.3947"},{"id":"arxiv:1503.08217","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The constituent parts of a quantum computer are inherently vulnerable to errors. To this end, we have developed quantum error-correcting codes to protect quantum information from noise. However, discovering codes that are capable of a universal set of computational operations with the minimal cost in quantum resources remains an important and ongoing challenge. One proposal of significant recent interest is the gauge color code. Notably, this code may offer a reduced resource cost over other well-studied fault-tolerant architectures by using a new method, known as gauge fixing, for performing the non-Clifford operations that are essential for universal quantum computation. Here we examine the gauge color code when it is subject to noise. Specifically, we make use of single-shot error correction to develop a simple decoding algorithm for the gauge color code, and we numerically analyse its performance. Remarkably, we find threshold error rates comparable to those of other leading proposals. Our results thus provide the first steps of a comparative study between the gauge color code and other promising computational architectures.</jats:p>","DOI":"10.1038/ncomms12302","source":"Crossref","title":"Fault-tolerant error correction with the gauge color code","volume":"7","author":[{"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[]},{"given":"Naomi H.","family":"Nickerson","sequence":"additional","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2016,7,29]]},"URL":"http://dx.doi.org/10.1038/ncomms12302","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"12302","note":"alternative-id:BFncomms12302\narxivid:1503.08217"},{"id":"arxiv:2309.07522","type":"article-journal","author":[{"given":"Ronan","family":"Egan"}],"title":"A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes","issued":{"date-parts":[[2023,11,7]]},"note":"arxivid:2309.07522\narxiv_version_number:2"},{"id":"arxiv:2311.00354","type":"article-journal","author":[{"given":"Minjia","family":"Shi"},{"given":"Danni","family":"Lu"},{"given":"Andrés","family":"Armario"},{"given":"Ronan","family":"Egan"},{"given":"Ferruh","family":"Ozbudak"},{"given":"Patrick","family":"Solé"}],"title":"Butson Hadamard matrices, bent sequences, and spherical codes","issued":{"date-parts":[[2023,11,1]]},"note":"arxivid:2311.00354\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0512154","type":"article-journal","author":[{"given":"Wojciech","family":"Tadej"},{"given":"Karol","family":"Zyczkowski"}],"title":"A concise guide to complex Hadamard matrices","issued":{"date-parts":[[2006,5,25]]},"note":"arxivid:quant-ph/0512154\narxiv_version_number:2"},{"id":"arxiv:2407.13527","type":"article-journal","author":[{"given":"Bulent","family":"Sarac"},{"given":"Damla","family":"Acar"}],"title":"A Construction of Quantum Stabilizer Codes from Classical Codes and Butson Hadamard Matrices","issued":{"date-parts":[[2024,7,18]]},"note":"arxivid:2407.13527\narxiv_version_number:1"},{"id":"manual:-H.-K-nig-Isometric-imbedding","type":"article-journal","author":[{"family":"König","given":"H."}],"title":"Isometric imbeddings of Euclidean spaces into finite dimensional lp-spaces","volume":"34","container-title":"Banach Center Publications","issue":"1","issued":"1995","page":"79–87","URL":"http://matwbn.icm.edu.pl/ksiazki/bcp/bcp34/bcp3418.pdf","note":"Available at \\url{http://matwbn.icm.edu.pl/ksiazki/bcp/bcp34/bcp3418.pdf}"},{"id":"doi:10.1016/B978-0-12-189420-7.50028","type":"book-chapter","publisher":"Elsevier","DOI":"10.1016/b978-0-12-189420-7.50028-1","page":"290-297","source":"Crossref","title":"QUADRATIC FORMS OVER GF(2)","author":[{"given":"P.J.","family":"CAMERON","sequence":"first","affiliation":[]},{"given":"J.J.","family":"SEIDEL","sequence":"additional","affiliation":[]}],"container-title":"Geometry and Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[1991]]},"ISBN":["9780121894207"],"URL":"http://dx.doi.org/10.1016/B978-0-12-189420-7.50028-1"},{"id":"arxiv:1509.07837","type":"article-journal","author":[{"given":"P. G.","family":"Boyvalenkov"},{"given":"P. D.","family":"Dragnev"},{"given":"D. P.","family":"Hardin"},{"given":"E. B.","family":"Saff"},{"given":"M. M.","family":"Stoyanova"}],"title":"Universal upper and lower bounds on energy of spherical designs","issued":{"date-parts":[[2015,9,25]]},"note":"arxivid:1509.07837\narxiv_version_number:1"},{"id":"doi:10.1109/TIT.1965.1053807","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.1965.1053807","page":"385-395","source":"Crossref","title":"A class of codes for polyphase signals on a bandlimited Gaussian channel","volume":"11","author":[{"given":"L.-H.","family":"Zetterberg","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1965,7]]},"URL":"http://dx.doi.org/10.1109/TIT.1965.1053807","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1109/TIT.1966.1053876","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1966.1053876","page":"153-161","source":"Crossref","title":"Detection of a class of coded and phase-modulated signals","volume":"12","author":[{"given":"L.-H.","family":"Zetterberg","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1966,4]]},"URL":"http://dx.doi.org/10.1109/TIT.1966.1053876","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-G.-Einarsson-Polyphase-codin","type":"article-journal","author":[{"family":"Einarsson","given":"G."}],"title":"Polyphase coding for a Gaussian channel(Polyphase coding for Gaussian channel, investigating PM signal transmission over channel disturbed by additive white Gaussian noise","volume":"24","container-title":"Ericsson Technics","issue":"2","issued":"1968","page":"75–130"},{"id":"manual:-G.-Einarsson.-Performance-of","type":"document","author":[{"family":"Einarsson","given":"G."}],"title":"Performance of polyphase signals on a Gaussian channel","issued":"1966"},{"id":"manual:-R.-Ottoson-Performance-of-ph","type":"article-journal","author":[{"family":"Ottoson","given":"R."}],"title":"Performance of phase- and amplitude-modulated signals on a Gaussian channel(Phase and amplitude modulated signals transmission over band limited channel disturbed by additive white Gaussian noise","volume":"25","container-title":"Ericsson Technics","issue":"3","issued":"1969","page":"153–198"},{"id":"manual:-M.-Nilsson-Linear-block-code","type":"thesis","author":[{"family":"Nilsson","given":"M."}],"title":"Linear block codes over rings for phase shift keying","genre":"PhD thesis no. 331,","publisher":"Linkoping University","issued":"1993"},{"id":"doi:10.1109/TIT.1986.1057230","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.1986.1057230","page":"760-767","source":"Crossref","title":"Bounds for codes over the unit circle","volume":"32","author":[{"given":"P.","family":"Piret","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1986,11]]},"URL":"http://dx.doi.org/10.1109/TIT.1986.1057230","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"manual:-V.-V.-Ginzburg-Multidimensio","type":"article-journal","author":[{"family":"Ginzburg","given":"V.V."}],"title":"Multidimensional Signals for a Continuous Channel","volume":"20","container-title":"Problemy Peredachi Informatsii","issue":"1","issued":"1984","page":"28–46"},{"id":"manual:-S.-L.-Portnoi-Characterizati","type":"article-journal","author":[{"family":"Portnoi","given":"S.L."}],"title":"Characterizations of modulation and encoding systems as concatenated codes","volume":"21","container-title":"Problems of Information Transmission","issue":"3","issued":"1985","page":"14–27"},{"id":"manual:-V.-V.-Zyablov-and-S.-L.-Port","type":"article-journal","author":[{"family":"Zyablov","given":"V.V."},{"family":"Portnoi","given":"S.L."}],"title":"Modulation/Coding System for a Gaussian Channel","volume":"23","container-title":"Problemy Peredachi Informatsii","issue":"3","issued":"1987","page":"18–26"},{"id":"manual:-V.-A.-Zinoviev-S.-N.-Litsyn-","type":"article-journal","author":[{"family":"Zinoviev","given":"V.A."},{"family":"Litsyn","given":"S.N."},{"family":"Portnoi","given":"S.L."}],"title":"Cascade codes in Euclidean space","volume":"25","container-title":"Problems of Information Transmission","issue":"3","issued":"1989","page":"62–75,"},{"id":"doi:10.1016/0021-8693(78)90220-X","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0021-8693(78)90220-x","page":"257-280","source":"Crossref","title":"Strongly regular graphs having strongly regular subconstituents","volume":"55","author":[{"given":"P.J","family":"Cameron","sequence":"first","affiliation":[]},{"given":"J.M","family":"Goethals","sequence":"additional","affiliation":[]},{"given":"J.J","family":"Seidel","sequence":"additional","affiliation":[]}],"container-title":"Journal of Algebra","original-title":[],"language":"en","issued":{"date-parts":[[1978,12]]},"URL":"http://dx.doi.org/10.1016/0021-8693(78)90220-X","ISSN":["0021-8693"],"container-title-short":"Journal of Algebra","note":"alternative-id:002186937890220X"},{"id":"manual:-J.-McLaughlin.-A-simple-grou","type":"document","author":[{"family":"McLaughlin","given":"J."}],"title":"A simple group of order 898,128,000. Theory of finite groups","issued":"1969","page":"109–111","publisher-place":"Benjamin New York"},{"id":"doi:10.1016/B978-0-12-189420-7.50019","type":"book-chapter","publisher":"Elsevier","DOI":"10.1016/b978-0-12-189420-7.50019-0","page":"177-192","source":"Crossref","title":"THE REGULAR TWO-GRAPH ON 276 VERTICES","author":[{"given":"J.M.","family":"GOETHALS","sequence":"first","affiliation":[]},{"given":"J.J.","family":"SEIDEL","sequence":"additional","affiliation":[]}],"container-title":"Geometry and Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[1991]]},"ISBN":["9780121894207"],"URL":"http://dx.doi.org/10.1016/B978-0-12-189420-7.50019-0"},{"id":"arxiv:math/0611451","type":"journal-article","publisher":"Informa UK Limited","issue":"3","DOI":"10.1080/10586458.2009.10129052","page":"257-283","source":"Crossref","title":"Experimental Study of Energy-Minimizing Point Configurations on Spheres","volume":"18","author":[{"given":"Brandon","family":"Ballinger","sequence":"first","affiliation":[]},{"given":"Grigoriy","family":"Blekherman","sequence":"additional","affiliation":[]},{"given":"Henry","family":"Cohn","sequence":"additional","affiliation":[]},{"given":"Noah","family":"Giansiracusa","sequence":"additional","affiliation":[]},{"given":"Elizabeth","family":"Kelly","sequence":"additional","affiliation":[]},{"given":"Achill","family":"Schürmann","sequence":"additional","affiliation":[]}],"container-title":"Experimental Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2009,1]]},"URL":"http://dx.doi.org/10.1080/10586458.2009.10129052","ISSN":["1058-6458","1944-950X"],"container-title-short":"Experimental Mathematics","note":"alternative-id:10.1080/10586458.2009.10129052\narxivid:math/0611451"},{"id":"preset:Sloane04","type":"document","author":[{"family":"Sloane","given":"N.J.A."},{"family":"Hardin","given":"R.H."},{"family":"Smith","given":"W.D."}],"title":"Tables of spherical codes","note":"Published electronically at","issued":"2004","URL":"https://neilsloane.com/packings/"},{"id":"arxiv:1407.8282","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/j.jco.2015.02.003","page":"293-326","source":"Crossref","title":"Distributing many points on spheres: Minimal energy and designs","volume":"31","author":[{"given":"Johann S.","family":"Brauchart","sequence":"first","affiliation":[]},{"given":"Peter J.","family":"Grabner","sequence":"additional","affiliation":[]}],"container-title":"Journal of Complexity","original-title":[],"language":"en","issued":{"date-parts":[[2015,6]]},"URL":"http://dx.doi.org/10.1016/j.jco.2015.02.003","ISSN":["0885-064X"],"container-title-short":"Journal of Complexity","note":"alternative-id:S0885064X15000205\narxivid:1407.8282"},{"id":"manual:-V.-A.-Yudin-Minimum-potentia","type":"article-journal","author":[{"family":"Yudin","given":"V.A."}],"title":"Minimum potential energy of a point system of charges","volume":"4","container-title":"Diskretnaya Matematika","issue":"2","issued":"1992","page":"115–121"},{"id":"doi:10.1007/BF02789828","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf02789828","page":"25-34","source":"Crossref","title":"Extremal dispositions of points on the sphere","volume":"23","author":[{"given":"A. V.","family":"Kolushov","sequence":"first","affiliation":[]},{"given":"V. A.","family":"Yudin","sequence":"additional","affiliation":[]},{"given":"А. В.","family":"Колущов","sequence":"additional","affiliation":[]},{"given":"В. А.","family":"Удин","sequence":"additional","affiliation":[]}],"container-title":"Analysis Mathematica","original-title":[],"language":"en","issued":{"date-parts":[[1997,1]]},"URL":"http://dx.doi.org/10.1007/BF02789828","ISSN":["0133-3852","1588-273X"],"container-title-short":"Anal Math","note":"alternative-id:BF02789828"},{"id":"doi:10.1007/BF03024331","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf03024331","page":"5-11","source":"Crossref","title":"Distributing many points on a sphere","volume":"19","author":[{"given":"E. B.","family":"Saff","sequence":"first","affiliation":[]},{"given":"A. B. J.","family":"Kuijlaars","sequence":"additional","affiliation":[]}],"container-title":"The Mathematical Intelligencer","original-title":[],"language":"en","issued":{"date-parts":[[1997,12]]},"URL":"http://dx.doi.org/10.1007/BF03024331","ISSN":["0343-6993","1866-7414"],"container-title-short":"The Mathematical Intelligencer","note":"alternative-id:BF03024331"},{"id":"doi:10.2307/1969393","type":"journal-article","publisher":"JSTOR","issue":"4","DOI":"10.2307/1969393","page":"1062","source":"Crossref","title":"On the Closest Packing of Spheres in n Dimensions","volume":"48","author":[{"given":"R. A.","family":"Rankin","sequence":"first","affiliation":[]}],"container-title":"The Annals of Mathematics","original-title":[],"issued":{"date-parts":[[1947,10]]},"URL":"http://dx.doi.org/10.2307/1969393","ISSN":["0003-486X"],"container-title-short":"The Annals of Mathematics"},{"id":"manual:-H.-Davenport-and-G.-Hajos-(1","type":"article-journal","author":[{"family":"Davenport","given":"H."},{"family":"Hajos","given":"G."}],"title":"Aufgabe 35","volume":"58","container-title":"Math. Lapok","issued":"1951"},{"id":"doi:10.1017/S2040618500033219","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"3","abstract":"<jats:p>Let <jats:italic>S</jats:italic><jats:sub>n</jats:sub> denote the “surface” of an <jats:italic>n</jats:italic>-dimensional unit sphere in Euclidean space of <jats:italic>n</jats:italic> dimensions. We may suppose that the sphere is centred at the origin of coordinates <jats:italic>O</jats:italic>, so that the points <jats:italic>P</jats:italic>(<jats:italic>x</jats:italic>1, <jats:italic>x</jats:italic>2, …, <jats:italic>x</jats:italic><jats:sub>n</jats:sub>) of <jats:italic>S<jats:sub>n</jats:sub></jats:italic> satisfy</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S2040618500033219_eqn1\" /></jats:disp-formula></jats:p><jats:p>We suppose that <jats:italic>n</jats:italic>≥2.</jats:p>","DOI":"10.1017/s2040618500033219","page":"139-144","source":"Crossref","title":"The Closest Packing of Spherical Caps in <i>n</i> Dimensions","volume":"2","author":[{"given":"R. A.","family":"Rankin","sequence":"first","affiliation":[]}],"container-title":"Proceedings of the Glasgow Mathematical Association","original-title":[],"language":"en","issued":{"date-parts":[[1955,7]]},"URL":"http://dx.doi.org/10.1017/S2040618500033219","ISSN":["2040-6185","2051-2104"],"container-title-short":"Proceedings of the Glasgow Mathematical Association","note":"alternative-id:S2040618500033219"},{"id":"manual:-L.-Fejes-Toth-Uber-die-Absch","type":"article-journal","author":[{"family":"Toth","given":"L.Fejes"}],"title":"Uber die Abschatzung des kiirzesten Abstandes zweier Punkte eines auf einer Kugelflache liegenden Punktsystemes","volume":"53","container-title":"Jber. Deut. Math. Verein","issued":"1943","page":"66–68,"},{"id":"doi:10.1002/j.1538-7305.1965.tb04170","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1002/j.1538-7305.1965.tb04170.x","page":"1061-1122","source":"Crossref","title":"Capabilities of Bounded Discrepancy Decoding","volume":"44","author":[{"given":"A. D.","family":"Wyner","sequence":"first","affiliation":[]}],"container-title":"Bell System Technical Journal","original-title":[],"language":"en","issued":{"date-parts":[[1965,7,8]]},"URL":"http://dx.doi.org/10.1002/j.1538-7305.1965.tb04170.x","ISSN":["0005-8580"]},{"id":"arxiv:2410.23126","type":"article-journal","author":[{"given":"Jerry Yao-Chieh","family":"Hu"},{"given":"Dennis","family":"Wu"},{"given":"Han","family":"Liu"}],"title":"Provably Optimal Memory Capacity for Modern Hopfield Models: Transformer-Compatible Dense Associative Memories as Spherical Codes","issued":{"date-parts":[[2024,10,31]]},"note":"arxivid:2410.23126\narxiv_version_number:2"},{"id":"arxiv:2402.13725","type":"article-journal","author":[{"given":"Saul","family":"Santos"},{"given":"Vlad","family":"Niculae"},{"given":"Daniel","family":"McNamee"},{"given":"Andre F. T.","family":"Martins"}],"title":"Sparse and Structured Hopfield Networks","issued":{"date-parts":[[2024,6,4]]},"note":"arxivid:2402.13725\narxiv_version_number:2"},{"id":"arxiv:1009.4407","type":"article-journal","author":[{"given":"Andriy","family":"Bondarenko"},{"given":"Danylo","family":"Radchenko"},{"given":"Maryna","family":"Viazovska"}],"title":"Optimal asymptotic bounds for spherical designs","issued":{"date-parts":[[2011,3,7]]},"note":"arxivid:1009.4407\narxiv_version_number:3"},{"id":"doi:10.1080/10652469308819013","type":"journal-article","publisher":"Informa UK Limited","issue":"2","DOI":"10.1080/10652469308819013","page":"105-117","source":"Crossref","title":"Spherical faraday cage for the case of equal point charges and chebyshev-type quadrature on the sphere","volume":"1","author":[{"given":"J.","family":"Korevaar","sequence":"first","affiliation":[]},{"given":"J.L.H.","family":"Meyers","sequence":"additional","affiliation":[]}],"container-title":"Integral Transforms and Special Functions","original-title":[],"language":"en","issued":{"date-parts":[[1993,10]]},"URL":"http://dx.doi.org/10.1080/10652469308819013","ISSN":["1065-2469","1476-8291"],"container-title-short":"Integral Transforms and Special Functions","note":"alternative-id:10.1080/10652469308819013"},{"id":"doi:10.5802/alco.213","type":"journal-article","publisher":"Cellule MathDoc/CEDRAM","issue":"2","DOI":"10.5802/alco.213","page":"347-369","source":"Crossref","title":"Explicit spherical designs","volume":"5","author":[{"given":"Ziqing","family":"Xiang","sequence":"first","affiliation":[]}],"container-title":"Algebraic Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,5]]},"URL":"http://dx.doi.org/10.5802/alco.213","ISSN":["2589-5486"],"note":"alternative-id:10.5802/alco.213"},{"id":"arxiv:1110.6298","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tsp.2011.2166394","page":"5876-5887","source":"Crossref","title":"A Novel Sampling Theorem on the Sphere","volume":"59","author":[{"given":"Jason D.","family":"McEwen","sequence":"first","affiliation":[]},{"given":"Yves","family":"Wiaux","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Signal Processing","original-title":[],"issued":{"date-parts":[[2011,12]]},"URL":"http://dx.doi.org/10.1109/TSP.2011.2166394","ISSN":["1053-587X","1941-0476"],"container-title-short":"IEEE Trans. Signal Process.","note":"arxivid:1110.6298"},{"id":"doi:10.1017/S0962492900002701","type":"journal-article","publisher":"Cambridge University Press (CUP)","abstract":"<jats:p>In this paper we present a general, theoretical foundation for the construction of cubature formulae to approximate multivariate integrals. The focus is on cubature formulae that are exact for certain vector spaces of polynomials. Our main quality criteria are the algebraic and trigonometric degrees. The constructions using ideal theory and invariant theory are outlined. The known lower bounds for the number of points are surveyed and characterizations of minimal cubature formulae are given. We include references to all known minimal cubature formulae. Finally, some methods to construct cubature formulae illustrate the previously introduced concepts and theorems.</jats:p>","DOI":"10.1017/s0962492900002701","page":"1-54","source":"Crossref","title":"Constructing cubature formulae: the science behind the art","volume":"6","author":[{"given":"Ronald","family":"Cools","sequence":"first","affiliation":[]}],"container-title":"Acta Numerica","original-title":[],"language":"en","issued":{"date-parts":[[1997,1]]},"URL":"http://dx.doi.org/10.1017/S0962492900002701","ISSN":["0962-4929","1474-0508"],"container-title-short":"Acta Numerica","note":"alternative-id:S0962492900002701"},{"id":"doi:10.1016/S0885-064X(03)00011-6","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s0885-064x(03)00011-6","page":"445-453","source":"Crossref","title":"An encyclopaedia of cubature formulas","volume":"19","author":[{"given":"Ronald","family":"Cools","sequence":"first","affiliation":[]}],"container-title":"Journal of Complexity","original-title":[],"language":"en","issued":{"date-parts":[[2003,6]]},"URL":"http://dx.doi.org/10.1016/S0885-064X(03)00011-6","ISSN":["0885-064X"],"container-title-short":"Journal of Complexity","note":"alternative-id:S0885064X03000116"},{"id":"doi:10.18434/M3167","type":"dataset","language":"en","author":[{"family":"Olver","given":"Frank W. J."},{"family":"Olde Daalhuis","given":"Adri B."},{"family":"Lozier","given":"Daniel W."},{"family":"Schneider","given":"Barry"},{"family":"Boisvert","given":"Ronald F."},{"family":"Clark","given":"Charles W."},{"family":"Miller","given":"Bruce R."},{"family":"Saunders","given":"Bonita  V."},{"family":"Cohl","given":"Howard S."},{"family":"McClain","given":"Marjorie A."}],"contributor":[{"literal":"Barry I. Schneider"}],"issued":{"date-parts":[[2019,5,16]]},"abstract":"The Digital Library of Mathematical Functions (DLMF) is an online reference on the properties of the special functions of applied mathematics. It contains three methodological chapters and 33 chapters focused on individual classes of functions. Information on functions include definitions, notations, series expansions, integrals, connection formulae, asymptotics, special values, methods of computation, etc. It includes hundreds of visualizations of the functions, including interactive  three-dimensional representations of complex-valued functions. A math-aware search engine is also available. Extensive links to available software and external references are also provided.","DOI":"10.18434/M3167","publisher":"National Institute of Standards and Technology","title":"Digital Library of Mathematical Functions","URL":"https://data.nist.gov/od/id/FDB59097469F5200E043065706813E54167","version":"1.0.2","note":"copyright:NIST Licensing"},{"id":"doi:10.1016/j.ejc.2009.03.035","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/j.ejc.2009.03.035","page":"423-441","source":"Crossref","title":"Cubature formulas in numerical analysis and Euclidean tight designs","volume":"31","author":[{"given":"Eiichi","family":"Bannai","sequence":"first","affiliation":[]},{"given":"Etsuko","family":"Bannai","sequence":"additional","affiliation":[]},{"given":"Masatake","family":"Hirao","sequence":"additional","affiliation":[]},{"given":"Masanori","family":"Sawa","sequence":"additional","affiliation":[]}],"container-title":"European Journal of Combinatorics","original-title":[],"language":"en","issued":{"date-parts":[[2010,2]]},"URL":"http://dx.doi.org/10.1016/j.ejc.2009.03.035","ISSN":["0195-6698"],"container-title-short":"European Journal of Combinatorics","note":"alternative-id:S0195669809000857"},{"id":"arxiv:2211.05127","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>t</a:mi></a:math>-designs from [Blume-Kohout , ], no state designs exist for <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>t</c:mi><c:mo>≥</c:mo><c:mn>2</c:mn></c:math>. Similarly, we prove that no CV unitary <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>t</e:mi></e:math>-designs exist for <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:mi>t</g:mi><g:mo>≥</g:mo><g:mn>2</g:mn></g:math>. We propose an alternative definition for CV state designs, which we call rigged <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:mi>t</i:mi></i:math>-designs, and provide explicit constructions for <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:mi>t</k:mi><k:mo>=</k:mo><k:mn>2</k:mn></k:math>. As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevx.14.011013","source":"Crossref","title":"Continuous-Variable Quantum State Designs: Theory and Applications","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0003-3383-1946","authenticated-orcid":true,"given":"Joseph T.","family":"Iosue","sequence":"first","affiliation":[{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3132-1088","authenticated-orcid":true,"given":"Kunal","family":"Sharma","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2024,2,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.14.011013","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011013","note":"arxivid:2211.05127"},{"id":"doi:10.1017/S0963548300001917","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"2","abstract":"<jats:p>We describe a very simple method of randomly permuting the cube {0, 1}<jats:sup><jats:italic>n</jats:italic></jats:sup> such that the sample space is very small, but, given any <jats:italic>m</jats:italic> distinct points in {0, 1}<jats:sup><jats:italic>n</jats:italic></jats:sup>, the images of those points under the random permutation are approximately uniformly distributed over all sequences of <jats:italic>m</jats:italic> distinct points.</jats:p>","DOI":"10.1017/s0963548300001917","page":"119-130","source":"Crossref","title":"An Almost <i>m</i>-wise Independent Random Permutation of the Cube","volume":"5","author":[{"given":"W. T.","family":"Gowers","sequence":"first","affiliation":[]}],"container-title":"Combinatorics, Probability and Computing","original-title":[],"language":"en","issued":{"date-parts":[[1996,6]]},"URL":"http://dx.doi.org/10.1017/S0963548300001917","ISSN":["0963-5483","1469-2163"],"container-title-short":"Combinator. Probab. Comp.","note":"alternative-id:S0963548300001917"},{"id":"arxiv:math/0405366","type":"article-journal","author":[{"given":"Greg","family":"Kuperberg"}],"title":"Numerical cubature from Archimedes' hat-box theorem","issued":{"date-parts":[[2004,9,22]]},"note":"arxivid:math/0405366\narxiv_version_number:2"},{"id":"arxiv:2311.13479","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Toric <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>-designs, or equivalently <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>-designs on the diagonal subgroup of the unitary group, are sets of points on the torus over which sums reproduce integrals of degree <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math> monomials over the full torus. Motivated by the projective structure of quantum mechanics, we develop the notion of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>-designs on the projective torus, which have a much more restricted structure than their counterparts on full tori. We provide various new constructions of toric and projective toric designs and prove bounds on their size. We draw connections between projective toric designs and a diverse set of mathematical objects, including difference and Sidon sets from the field of additive combinatorics, symmetric, informationally complete positive operator valued measures and complete sets of mutually unbiased bases (MUBs) from quantum information theory, and crystal ball sequences of certain root lattices. Using these connections, we prove bounds on the maximal size of dense <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>B</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo lspace=\"thickmathspace\" rspace=\"thickmathspace\">mod</mml:mo><mml:mi>m</mml:mi></mml:math> sets. We also use projective toric designs to construct families of quantum state designs. In particular, we construct families of (uniformly-weighted) quantum state <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn></mml:math>-designs in dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> of size exactly <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math> that do not form complete sets of MUBs, thereby disproving a conjecture concerning the relationship between designs and MUBs (Zhu 2015). We then propose a modification of Zhu's conjecture and discuss potential paths towards proving this conjecture. We prove a fundamental distinction between complete sets of MUBs in prime-power dimensions versus in dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>6</mml:mn></mml:math> (and, we conjecture, in all non-prime-power dimensions), the distinction relating to group structure of the corresponding projective toric design. Finally, we discuss many open questions about the properties of these projective toric designs and how they relate to other questions in number theory, geometry, and quantum information.</jats:p>","DOI":"10.22331/q-2024-12-03-1546","page":"1546","source":"Crossref","title":"Projective toric designs, quantum state designs, and mutually unbiased bases","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0003-3383-1946","authenticated-orcid":false,"given":"Joseph T.","family":"Iosue","sequence":"first","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0001-9727-6967","authenticated-orcid":false,"given":"T. C.","family":"Mooney","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0002-3167-6519","authenticated-orcid":false,"given":"Adam","family":"Ehrenberg","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0003-0509-3421","authenticated-orcid":false,"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,3]]},"URL":"http://dx.doi.org/10.22331/q-2024-12-03-1546","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2311.13479"},{"id":"doi:10.2307/2002483","type":"journal-article","publisher":"JSTOR","issue":"55","DOI":"10.2307/2002483","page":"130","source":"Crossref","title":"Numerical Integration Over Simplexes and Cones","volume":"10","author":[{"given":"P. C.","family":"Hammer","sequence":"first","affiliation":[]},{"given":"O. J.","family":"Marlowe","sequence":"additional","affiliation":[]},{"given":"A. H.","family":"Stroud","sequence":"additional","affiliation":[]}],"container-title":"Mathematical Tables and Other Aids to Computation","original-title":[],"issued":{"date-parts":[[1956,7]]},"URL":"http://dx.doi.org/10.2307/2002483","ISSN":["0891-6837"],"container-title-short":"Mathematical Tables and Other Aids to Computation"},{"id":"doi:10.2307/2002484","type":"journal-article","publisher":"JSTOR","issue":"55","DOI":"10.2307/2002484","page":"137","source":"Crossref","title":"Numerical Integration Over Simplexes","volume":"10","author":[{"given":"Preston C.","family":"Hammer","sequence":"first","affiliation":[]},{"given":"Arthur H.","family":"Stroud","sequence":"additional","affiliation":[]}],"container-title":"Mathematical Tables and Other Aids to Computation","original-title":[],"issued":{"date-parts":[[1956,7]]},"URL":"http://dx.doi.org/10.2307/2002484","ISSN":["0891-6837"],"container-title-short":"Mathematical Tables and Other Aids to Computation"},{"id":"doi:10.4036/iis.2018.S.02","type":"journal-article","publisher":"Graduate School of Information Sciences, Tohoku University","issue":"2","DOI":"10.4036/iis.2018.s.02","page":"181-184","source":"Crossref","title":"On Explicit Construction of Simplex &lt;i&gt;t&lt;/i&gt;-designs","volume":"24","author":[{"given":"Mohammad Samy","family":"BALADRAM","sequence":"first","affiliation":[{"name":"Graduate School of Information Sciences, Tohoku University"}]}],"container-title":"Interdisciplinary Information Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2018]]},"URL":"http://dx.doi.org/10.4036/iis.2018.S.02","ISSN":["1340-9050","1347-6157"],"container-title-short":"IIS"},{"id":"doi:10.18434/M32189","type":"dataset","categories":["Statistics","FOS: Mathematics","experimental design","time series","analysis of variance","quality control","reliability","graphics","exploratory data analysis","process characterization","regression","statistical modeling","model validation","uncertaintyanalysis"],"language":"en","author":[{"family":"Guthrie","given":"William F."}],"issued":{"date-parts":[[2020]]},"abstract":"The NIST/SEMATECH e-Handbook of Statistical Methods is a Web-based book written to help scientists and engineers incorporate statistical methods into their work as efficiently as possible. Ideally, it will serve as a reference which will help scientists and engineers design their own experiments and carry out the appropriate analyses when a statistician is not available to help. It is also hoped that it will serve as a useful educational tool that will help users of statistical methods and consumers of statistical information better understand statistical procedures and their underlying assumptions, and more clearly interpret scientific and engineering results stated in statistical terms.","DOI":"10.18434/M32189","publisher":"National Institute of Standards and Technology","title":"NIST/SEMATECH e-Handbook of Statistical Methods (NIST Handbook 151)","URL":"https://www.itl.nist.gov/div898/handbook/","note":"copyright:License Information\n      for NIST data"},{"id":"arxiv:2412.09672","type":"article-journal","author":[{"given":"Jakub","family":"Czartowski"},{"given":"Karol","family":"Życzkowski"}],"title":"Quantum Pushforward Designs","issued":{"date-parts":[[2025,2,24]]},"note":"arxivid:2412.09672\narxiv_version_number:3"},{"id":"arxiv:math/0701626","type":"article-journal","author":[{"given":"Gerald","family":"Hoehn"}],"title":"Conformal Designs based on Vertex Operator Algebras","issued":{"date-parts":[[2007,1,23]]},"note":"arxivid:math/0701626\narxiv_version_number:1"},{"id":"arxiv:2603.02659","type":"article-journal","author":[{"given":"Namit","family":"Anand"},{"given":"Jeffrey","family":"Marshall"},{"given":"Jason","family":"Saied"},{"given":"Eleanor","family":"Rieffel"},{"given":"Andrea","family":"Morello"}],"title":"Qudit Designs and Where to Find Them","issued":{"date-parts":[[2026,3,3]]},"note":"arxivid:2603.02659\narxiv_version_number:1"},{"id":"doi:10.1016/0001-8708(84)90022-7","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0001-8708(84)90022-7","page":"213-240","source":"Crossref","title":"Averaging sets: A generalization of mean values and spherical designs","volume":"52","author":[{"given":"P.D","family":"Seymour","sequence":"first","affiliation":[]},{"given":"Thomas","family":"Zaslavsky","sequence":"additional","affiliation":[]}],"container-title":"Advances in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1984,6]]},"URL":"http://dx.doi.org/10.1016/0001-8708(84)90022-7","ISSN":["0001-8708"],"container-title-short":"Advances in Mathematics","note":"alternative-id:0001870884900227"},{"id":"arxiv:1111.5900","type":"article-journal","author":[{"given":"Isaac Z.","family":"Pesenson"},{"given":"Daryl","family":"Geller"}],"title":"Cubature formulas and discrete fourier transform on compact manifolds","issued":{"date-parts":[[2011,11,25]]},"note":"arxivid:1111.5900\narxiv_version_number:1"},{"id":"arxiv:1112.4900","type":"article-journal","author":[{"given":"Daniel M.","family":"Kane"}],"title":"Small Designs for Path Connected Spaces and Path Connected Homogeneous Spaces","issued":{"date-parts":[[2012,6,25]]},"note":"arxivid:1112.4900\narxiv_version_number:3"},{"id":"arxiv:1110.1042","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-014-1894-3","page":"755-771","source":"Crossref","title":"The Curious Nonexistence of Gaussian 2-Designs","volume":"326","author":[{"given":"Robin","family":"Blume-Kohout","sequence":"first","affiliation":[]},{"given":"Peter S.","family":"Turner","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,20]]},"URL":"http://dx.doi.org/10.1007/s00220-014-1894-3","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1894\narxivid:1110.1042"},{"id":"arxiv:2412.17909","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/dy4m-gq5c","source":"Crossref","title":"Continuous-Variable Designs and Design-Based Shadow Tomography from Random Lattices","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0001-6120-9930","authenticated-orcid":true,"given":"Jonathan","family":"Conrad","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02s376052","id-type":"ROR","asserted-by":"publisher"}],"name":"École Polytechnique Fédérale de Lausanne (EPFL)"}]},{"ORCID":"https://orcid.org/0000-0003-3383-1946","authenticated-orcid":true,"given":"Joseph T.","family":"Iosue","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"},{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"}]},{"ORCID":"https://orcid.org/0000-0002-6172-7962","authenticated-orcid":true,"given":"Ansgar G.","family":"Burchards","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"}]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,5]]},"URL":"http://dx.doi.org/10.1103/dy4m-gq5c","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"060802","note":"arxivid:2412.17909"},{"id":"manual:-A.-S.-Holevo-Bounds-for-the-","type":"article-journal","author":[{"family":"Holevo","given":"A.S."}],"title":"Bounds for the Quantity of Information Transmitted by a Quantum Communication Channel","volume":"9","container-title":"Problemy Peredachi Informatsii","issue":"3","issued":"1973","page":"3–11"},{"id":"doi:10.1103/PhysRevA.56.131","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.56.131","page":"131-138","source":"Crossref","title":"Sending classical information via noisy quantum channels","volume":"56","author":[{"given":"Benjamin","family":"Schumacher","sequence":"first","affiliation":[]},{"given":"Michael D.","family":"Westmoreland","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,7,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.56.131","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"doi:10.1109/18.651037","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.651037","page":"269-273","source":"Crossref","title":"The capacity of the quantum channel with general signal states","volume":"44","author":[{"given":"A.S.","family":"Holevo","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1998]]},"URL":"http://dx.doi.org/10.1109/18.651037","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:0809.3972","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1038/nphys1224","page":"255-257","source":"Crossref","title":"Superadditivity of communication capacity using entangled inputs","volume":"5","author":[{"given":"M. B.","family":"Hastings","sequence":"first","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,15]]},"URL":"http://dx.doi.org/10.1038/nphys1224","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nature Phys","note":"alternative-id:BFnphys1224\narxivid:0809.3972"},{"id":"arxiv:1308.6503","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s00220-015-2382-0","page":"103-137","source":"Crossref","title":"Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels","volume":"338","author":[{"given":"Marco","family":"Tomamichel","sequence":"first","affiliation":[]},{"given":"Vincent Y. F.","family":"Tan","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2015,5,7]]},"URL":"http://dx.doi.org/10.1007/s00220-015-2382-0","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2382\narxivid:1308.6503"},{"id":"arxiv:1709.05258","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2019.2898656","page":"4609-4619","source":"Crossref","title":"On Converse Bounds for Classical Communication Over Quantum Channels","volume":"65","author":[{"ORCID":"https://orcid.org/0000-0002-0641-3186","authenticated-orcid":false,"given":"Xin","family":"Wang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9232-6846","authenticated-orcid":false,"given":"Kun","family":"Fang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5410-3329","authenticated-orcid":false,"given":"Marco","family":"Tomamichel","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2019,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2019.2898656","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1709.05258"},{"id":"arxiv:1409.3562","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s00220-017-2928-4","page":"373-426","source":"Crossref","title":"Strong Converse Exponent for Classical-Quantum Channel Coding","volume":"355","author":[{"ORCID":"https://orcid.org/0000-0002-5973-5533","authenticated-orcid":false,"given":"Milán","family":"Mosonyi","sequence":"first","affiliation":[]},{"given":"Tomohiro","family":"Ogawa","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,6,29]]},"URL":"http://dx.doi.org/10.1007/s00220-017-2928-4","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2928\narxivid:1409.3562"},{"id":"arxiv:quant-ph/9703013","type":"article-journal","author":[{"given":"M. V.","family":"Burnashev"},{"given":"A. S.","family":"Holevo"}],"title":"On Reliability Function of Quantum Communication Channel","issued":{"date-parts":[[1998,8,12]]},"note":"arxivid:quant-ph/9703013\narxiv_version_number:4"},{"id":"arxiv:quant-ph/9907087","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.868501","page":"2256-2261","source":"Crossref","title":"Reliability function of general classical-quantum channel","volume":"46","author":[{"given":"A.S.","family":"Holevo","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2000]]},"URL":"http://dx.doi.org/10.1109/18.868501","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/9907087"},{"id":"arxiv:quant-ph/0206186","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2002.1023343","page":"71","source":"Crossref","title":"A general formula for the classical capacity of a general quantum channel","author":[{"given":"M.","family":"Hayashi","sequence":"first","affiliation":[]},{"given":"H.","family":"Nagaoka","sequence":"additional","affiliation":[]}],"event":"IEEE International Symposium on Information Theory","container-title":"Proceedings IEEE International Symposium on Information Theory,","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2002.1023343","note":"arxivid:quant-ph/0206186"},{"id":"arxiv:quant-ph/0611013","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.76.062301","source":"Crossref","title":"Error exponent in asymmetric quantum hypothesis testing and its application to classical-quantum channel coding","volume":"76","author":[{"given":"Masahito","family":"Hayashi","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,12,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.062301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062301","note":"arxivid:quant-ph/0611013"},{"id":"arxiv:0805.4092","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-009-0825-1","page":"1087-1098","source":"Crossref","title":"Universal Coding for Classical-Quantum Channel","volume":"289","author":[{"given":"Masahito","family":"Hayashi","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2009,5,13]]},"URL":"http://dx.doi.org/10.1007/s00220-009-0825-1","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:825\narxivid:0805.4092"},{"id":"arxiv:1007.5456","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.108.200501","source":"Crossref","title":"One-Shot Classical-Quantum Capacity and Hypothesis Testing","volume":"108","author":[{"given":"Ligong","family":"Wang","sequence":"first","affiliation":[]},{"given":"Renato","family":"Renner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,5,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.108.200501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200501","note":"arxivid:1007.5456"},{"id":"arxiv:1312.3822","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2014.2361632","page":"7980-7986","source":"Crossref","title":"Quantum Achievability Proof via Collision Relative Entropy","volume":"60","author":[{"given":"Salman","family":"Beigi","sequence":"first","affiliation":[]},{"given":"Amin","family":"Gohari","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2361632","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1312.3822"},{"id":"arxiv:2208.02132","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/prxquantum.4.040330","source":"Crossref","title":"Simple and Tighter Derivation of Achievability for Classical Communication Over Quantum Channels","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0003-4499-4679","authenticated-orcid":true,"given":"Hao-Chung","family":"Cheng","sequence":"first","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,22]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.040330","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040330","note":"arxivid:2208.02132"},{"id":"arxiv:2303.04138","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.109.062416","source":"Crossref","title":"One-shot and asymptotic classical capacity in general physical theories","volume":"109","author":[{"ORCID":"https://orcid.org/0000-0002-8637-629X","authenticated-orcid":true,"given":"Shintaro","family":"Minagawa","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04chrp450","id-type":"ROR","asserted-by":"publisher"}],"name":"Nagoya University"}]},{"ORCID":"https://orcid.org/0000-0002-6780-1246","authenticated-orcid":true,"given":"Hayato","family":"Arai","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN, Center for Quantum Computing"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.109.062416","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062416","note":"arxivid:2303.04138"},{"id":"arxiv:2310.09014","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2024.3446614","page":"7882-7891","source":"Crossref","title":"Lower Bounds on Error Exponents via a New Quantum Decoder","volume":"70","author":[{"ORCID":"https://orcid.org/0000-0003-3588-4662","authenticated-orcid":false,"given":"Salman","family":"Beigi","sequence":"first","affiliation":[{"name":"School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran"}]},{"ORCID":"https://orcid.org/0000-0001-5410-3329","authenticated-orcid":false,"given":"Marco","family":"Tomamichel","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering and the Centre for Quantum Technologies, National University of Singapore, Queenstown, Singapore"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2024,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3446614","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2310.09014"},{"id":"arxiv:2207.08899","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw55543.2023.10161621","page":"170-174","source":"Crossref","title":"Achievable error exponents of data compression with quantum side information and communication over symmetric classical-quantum channels","author":[{"given":"Joseph M.","family":"Renes","sequence":"first","affiliation":[{"name":"ETH Z&#x00FC;rich,Institute for Theoretical Physics,Z&#x00FC;rich,Switzerland,8093"}]}],"event":"2023 IEEE Information Theory Workshop (ITW)","container-title":"2023 IEEE Information Theory Workshop (ITW)","original-title":[],"issued":{"date-parts":[[2023,4,23]]},"URL":"http://dx.doi.org/10.1109/ITW55543.2023.10161621","note":"arxivid:2207.08899"},{"id":"arxiv:1201.5411","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2013.2283794","page":"8027-8056","source":"Crossref","title":"Lower Bounds on the Probability of Error for Classical and Classical-Quantum Channels","volume":"59","author":[{"given":"Marco","family":"Dalai","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2283794","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1201.5411"},{"id":"arxiv:2407.12403","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.134.010802","source":"Crossref","title":"Reliability Function of Classical-Quantum Channels","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0002-3944-8449","authenticated-orcid":true,"given":"Ke","family":"Li","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01yqg2h08","id-type":"ROR","asserted-by":"publisher"}],"name":"Harbin Institute of Technology"}]},{"given":"Dong","family":"Yang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.010802","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"010802","note":"arxivid:2407.12403"},{"id":"arxiv:2507.06232","type":"article-journal","author":[{"given":"Hao-Chung","family":"Cheng"},{"given":"Po-Chieh","family":"Liu"}],"title":"Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem","issued":{"date-parts":[[2025,9,24]]},"note":"arxivid:2507.06232\narxiv_version_number:3"},{"id":"doi:10.1109/ISIT.2013.6620209","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2013.6620209","page":"166-170","source":"Crossref","title":"Achieving the Holevo capacity of a pure state classical-quantum channel via unambiguous state discrimination","author":[{"given":"Masahiro","family":"Takeoka","sequence":"first","affiliation":[]},{"given":"Hari","family":"Krovi","sequence":"additional","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]}],"event":"2013 IEEE International Symposium on Information Theory (ISIT)","container-title":"2013 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2013,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2013.6620209"},{"id":"arxiv:1102.1963","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2011.6034073","page":"274-278","source":"Crossref","title":"On quantum limit of optical communications: Concatenated codes and joint-detection receivers","author":[{"given":"Saikat","family":"Guha","sequence":"first","affiliation":[]},{"given":"Zachary","family":"Dutton","sequence":"additional","affiliation":[]},{"given":"Jeffrey H.","family":"Shapiro","sequence":"additional","affiliation":[]}],"event":"2011 IEEE International Symposium on Information Theory - ISIT","container-title":"2011 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2011,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2011.6034073","note":"arxivid:1102.1963"},{"id":"doi:10.1109/JSTQE.2009.2024959","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/jstqe.2009.2024959","page":"1547-1569","source":"Crossref","title":"The Quantum Theory of Optical Communications","volume":"15","author":[{"given":"J.H.","family":"Shapiro","sequence":"first","affiliation":[]}],"container-title":"IEEE Journal of Selected Topics in Quantum Electronics","original-title":[],"issued":{"date-parts":[[2009]]},"URL":"http://dx.doi.org/10.1109/JSTQE.2009.2024959","ISSN":["1077-260X"],"container-title-short":"IEEE J. Select. Topics Quantum Electron."},{"id":"arxiv:2002.05766","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/jlt.2020.2973890","page":"2741-2754","source":"Crossref","title":"Quantum Limits in Optical Communications","volume":"38","author":[{"ORCID":"https://orcid.org/0000-0002-5389-6897","authenticated-orcid":false,"given":"Konrad","family":"Banaszek","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0613-7070","authenticated-orcid":false,"given":"Ludwig","family":"Kunz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2163-5139","authenticated-orcid":false,"given":"Michal","family":"Jachura","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9989-1648","authenticated-orcid":false,"given":"Marcin","family":"Jarzyna","sequence":"additional","affiliation":[]}],"container-title":"Journal of Lightwave Technology","original-title":[],"issued":{"date-parts":[[2020,5,15]]},"URL":"http://dx.doi.org/10.1109/JLT.2020.2973890","ISSN":["0733-8724","1558-2213"],"container-title-short":"J. Lightwave Technol.","note":"arxivid:2002.05766"},{"id":"doi:10.1515/9783110642490","type":"edited-book","publisher":"De Gruyter","DOI":"10.1515/9783110642490","source":"Crossref","title":"Quantum Systems, Channels, Information","author":[{"given":"Alexander S.","family":"Holevo","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2019,7,9]]},"ISBN":["9783110642490"],"URL":"http://dx.doi.org/10.1515/9783110642490","note":"alternative-id:10.1515/9783110642490"},{"id":"arxiv:quant-ph/0308012","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.92.027902","source":"Crossref","title":"Classical Capacity of the Lossy Bosonic Channel: The Exact Solution","volume":"92","author":[{"given":"V.","family":"Giovannetti","sequence":"first","affiliation":[]},{"given":"S.","family":"Guha","sequence":"additional","affiliation":[]},{"given":"S.","family":"Lloyd","sequence":"additional","affiliation":[]},{"given":"L.","family":"Maccone","sequence":"additional","affiliation":[]},{"given":"J. H.","family":"Shapiro","sequence":"additional","affiliation":[]},{"given":"H. P.","family":"Yuen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2004,1,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.92.027902","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"027902","note":"arxivid:quant-ph/0308012"},{"id":"arxiv:1312.6225","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","DOI":"10.1038/nphoton.2014.216","page":"796-800","source":"Crossref","title":"Ultimate classical communication rates of quantum optical channels","volume":"8","author":[{"given":"V.","family":"Giovannetti","sequence":"first","affiliation":[]},{"given":"R.","family":"García-Patrón","sequence":"additional","affiliation":[]},{"given":"N. J.","family":"Cerf","sequence":"additional","affiliation":[]},{"given":"A. S.","family":"Holevo","sequence":"additional","affiliation":[]}],"container-title":"Nature Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,21]]},"URL":"http://dx.doi.org/10.1038/nphoton.2014.216","ISSN":["1749-4885","1749-4893"],"container-title-short":"Nature Photon","note":"alternative-id:BFnphoton2014216\narxivid:1312.6225"},{"id":"doi:10.1103/PhysRevLett.70.363","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.70.363","page":"363-366","source":"Crossref","title":"Ultimate information carrying limit of quantum systems","volume":"70","author":[{"given":"Horace P.","family":"Yuen","sequence":"first","affiliation":[]},{"given":"Masanao","family":"Ozawa","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1993,1,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.70.363","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"doi:10.1103/RevModPhys.66.481","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/revmodphys.66.481","page":"481-537","source":"Crossref","title":"Quantum limits on bosonic communication rates","volume":"66","author":[{"given":"Carlton M.","family":"Caves","sequence":"first","affiliation":[]},{"given":"P. D.","family":"Drummond","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[1994,4,1]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.66.481","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys."},{"id":"manual:-J.-P.-Gordon-in-Advances-in-","type":"chapter","author":[{"family":"P","given":"J."}],"title":"Gordon","container-title":"Advances in Quantum Electronics","editor":[{"family":"Singer","given":"J.R."}],"publisher":"Columbia University","issued":"1961","page":"509","publisher-place":"New York"},{"id":"doi:10.1109/JRPROC.1962.288169","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/jrproc.1962.288169","page":"1898-1908","source":"Crossref","title":"Quantum Effects in Communications Systems","volume":"50","author":[{"given":"J.","family":"Gordon","sequence":"first","affiliation":[]}],"container-title":"Proceedings of the IRE","original-title":[],"issued":{"date-parts":[[1962,9]]},"URL":"http://dx.doi.org/10.1109/JRPROC.1962.288169","ISSN":["0096-8390"],"container-title-short":"Proc. IRE"},{"id":"manual:-J.-P.-Gordon-in-Quantum-Elec","type":"paper-conference","author":[{"family":"P","given":"J."}],"title":"Gordon","container-title":"Quantum Electronics and Coherent Light, Proceedings of the International School of Physics \"Enrico Fermi,\" Course XXXI","editor":[{"given":"P.A."}],"publisher":"Miles (Academic","issued":"1964","page":"156","publisher-place":"New York"},{"id":"manual:-G.-D.-Forney-Jr.-S.-M.-thesi","type":"thesis","author":[{"family":"G. D. Forney","given":"S.M.","suffix":"Jr."}],"genre":"thesis,","publisher":"Massachusetts Institute of Technology","note":"unpublished","issued":"1963"},{"id":"arxiv:cond-mat/9907500","type":"journal-article","publisher":"American Association of Physics Teachers (AAPT)","issue":"10","abstract":"<jats:p>It has been well known since the pioneering work of Claude Shannon in the 1940s that a message transmitted with optimal efficiency over a channel of limited bandwidth is indistinguishable from random noise to a receiver who is unfamiliar with the language in which the message is written. We derive some similar results about electromagnetic transmissions. In particular, we show that if electromagnetic radiation is used as a transmission medium, the most information-efficient format for a given message is indistinguishable from blackbody radiation. The characteristic temperature of the radiation is set by the amount of energy used to make the transmission. If information is not encoded in the direction of the radiation, but only in its timing, energy, and polarization, then the most efficient format has the form of a one-dimensional blackbody spectrum.</jats:p>","DOI":"10.1119/1.1773578","page":"1290-1293","source":"Crossref","title":"The physical limits of communication <i>or</i> Why any sufficiently advanced technology is indistinguishable from noise","volume":"72","author":[{"given":"Michael","family":"Lachmann","sequence":"first","affiliation":[{"name":"Max Planck Institute for Evolutionary Anthropology, Deutscher Platz 6, 04103 Leipzig, Germany"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501"}]},{"given":"M. E. J.","family":"Newman","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501"}]},{"given":"Cristopher","family":"Moore","sequence":"additional","affiliation":[{"name":"Computer Science Department and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501"}]}],"container-title":"American Journal of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2004,9,13]]},"URL":"http://dx.doi.org/10.1119/1.1773578","ISSN":["0002-9505","1943-2909"],"note":"arxivid:cond-mat/9907500"},{"id":"doi:10.1109/TIT.1975.1055351","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.1975.1055351","page":"125-134","source":"Crossref","title":"Optimum testing of multiple hypotheses in quantum detection theory","volume":"21","author":[{"given":"H.","family":"Yuen","sequence":"first","affiliation":[]},{"given":"R.","family":"Kennedy","sequence":"additional","affiliation":[]},{"given":"M.","family":"Lax","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1975,3]]},"URL":"http://dx.doi.org/10.1109/TIT.1975.1055351","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1116/5.0036959","type":"journal-article","publisher":"American Vacuum Society","issue":"2","abstract":"<jats:p>Communication is an integral part of human life. Today, optical pulses are the preferred information carriers for long-distance communication. The exponential growth in data leads to a “capacity crunch” in the underlying physical systems. One of the possible methods to deter the exponential growth of physical resources for communication is to use quantum, rather than classical measurement at the receiver. Quantum measurement improves the energy efficiency of optical communication protocols by enabling discrimination of optical coherent states with the discrimination error rate below the shot-noise limit. In this review article, the authors focus on quantum receivers that can be practically implemented at the current state of technology, first and foremost displacement-based receivers. The authors present the experimentalist view on the progress in quantum-enhanced receivers and discuss their potential.</jats:p>","DOI":"10.1116/5.0036959","source":"Crossref","title":"Practical quantum-enhanced receivers for classical communication","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-5695-8339","authenticated-orcid":false,"given":"I. A.","family":"Burenkov","sequence":"first","affiliation":[{"name":"National Institute of Standards and Technology 1 , Gaithersburg, Maryland 20899, USA"},{"name":"Joint Quantum Institute and University of Maryland 2 , College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0001-6022-6085","authenticated-orcid":false,"given":"M. V.","family":"Jabir","sequence":"additional","affiliation":[{"name":"National Institute of Standards and Technology 1 , Gaithersburg, Maryland 20899, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4601-0250","authenticated-orcid":false,"given":"S. V.","family":"Polyakov","sequence":"additional","affiliation":[{"name":"National Institute of Standards and Technology 1 , Gaithersburg, Maryland 20899, USA"},{"name":"Department of Physics, University of Maryland 3 , College Park, Maryland 20742, USA"}]}],"container-title":"AVS Quantum Science","original-title":[],"language":"en","issued":{"date-parts":[[2021,4,20]]},"URL":"http://dx.doi.org/10.1116/5.0036959","ISSN":["2639-0213"],"page":"025301"},{"id":"arxiv:1101.1550","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.106.240502","source":"Crossref","title":"Structured Optical Receivers to Attain Superadditive Capacity and the Holevo Limit","volume":"106","author":[{"given":"Saikat","family":"Guha","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,6,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.106.240502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240502","note":"arxivid:1101.1550"},{"id":"arxiv:2109.00008","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are prime candidates as information carriers for optical communications. A large body of literature exists on their quantum-limited estimation and discrimination. However, very little is known about the practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and outline a theory of USD with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, auxiliary vacuum modes, and on-off photon detection. Our results indicate that, in some regimes, these currently-available optical components are typically sufficient to achieve near-optimal unambiguous discrimination of multiple, multimode coherent states.</jats:p>","DOI":"10.22331/q-2023-05-31-1025","page":"1025","source":"Crossref","title":"Linear optics and photodetection achieve near-optimal unambiguous coherent state discrimination","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-6167-8224","authenticated-orcid":false,"given":"Jasminder S.","family":"Sidhu","sequence":"first","affiliation":[{"name":"SUPA Department of Physics, The University of Strathclyde, Glasgow, G4 0NG, UK"}]},{"ORCID":"https://orcid.org/0000-0002-3528-7473","authenticated-orcid":false,"given":"Michael S.","family":"Bullock","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"https://orcid.org/0000-0002-2581-4380","authenticated-orcid":false,"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, Arizona 85721, USA"},{"name":"College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"https://orcid.org/0000-0002-5227-4009","authenticated-orcid":false,"given":"Cosmo","family":"Lupo","sequence":"additional","affiliation":[{"name":"Dipartimento Interateneo di Fisica, Politecnico & Università di Bari, 70126 Bari, Italy"},{"name":"INFN, Sezione di Bari, 70126 Bari, Italy"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,5,31]]},"URL":"http://dx.doi.org/10.22331/q-2023-05-31-1025","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2109.00008"},{"id":"doi:10.1080/17442507508833114","type":"journal-article","publisher":"Informa UK Limited","issue":"1-4","DOI":"10.1080/17442507508833114","page":"315-345","source":"Crossref","title":"Optimal multiple quantum statistical hypothesis testing","volume":"1","author":[{"given":"V. P.","family":"Belavkin","sequence":"first","affiliation":[]}],"container-title":"Stochastics","original-title":[],"language":"en","issued":{"date-parts":[[1975,1]]},"URL":"http://dx.doi.org/10.1080/17442507508833114","ISSN":["0090-9491"],"container-title-short":"Stochastics","note":"alternative-id:10.1080/17442507508833114"},{"id":"doi:10.1137/1123048","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","DOI":"10.1137/1123048","page":"411-415","source":"Crossref","title":"On Asymptotically Optimal Hypothesis Testing in Quantum Statistics","volume":"23","author":[{"given":"A. S.","family":"Kholevo","sequence":"first","affiliation":[]}],"container-title":"Theory of Probability &amp; Its Applications","original-title":[],"language":"en","issued":{"date-parts":[[1979,3]]},"URL":"http://dx.doi.org/10.1137/1123048","ISSN":["0040-585X","1095-7219"],"container-title-short":"Theory Probab. Appl.","note":"alternative-id:10.1137/1123048"},{"id":"doi:10.1080/09500349414552221","type":"journal-article","publisher":"Informa UK Limited","issue":"12","DOI":"10.1080/09500349414552221","page":"2385-2390","source":"Crossref","title":"A ‘Pretty Good’ Measurement for Distinguishing Quantum States","volume":"41","author":[{"given":"Paul","family":"Hausladen","sequence":"first","affiliation":[]},{"given":"William K.","family":"Wootters","sequence":"additional","affiliation":[]}],"container-title":"Journal of Modern Optics","original-title":[],"language":"en","issued":{"date-parts":[[1994,12]]},"URL":"http://dx.doi.org/10.1080/09500349414552221","ISSN":["0950-0340","1362-3044"],"container-title-short":"Journal of Modern Optics","note":"alternative-id:10.1080/09500349414552221"},{"id":"arxiv:quant-ph/0005132","type":"article-journal","author":[{"given":"Yonina C.","family":"Eldar"},{"given":"G. David","family":"Forney"}],"title":"On Quantum Detection and the Square-Root Measurement","issued":{"date-parts":[[2000,8,29]]},"note":"arxivid:quant-ph/0005132\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0211111","type":"article-journal","author":[{"given":"Yonina C.","family":"Eldar"},{"given":"Alexandre","family":"Megretski"},{"given":"George C.","family":"Verghese"}],"title":"Optimal Detection of Symmetric Mixed Quantum States","issued":{"date-parts":[[2002,11,19]]},"note":"arxivid:quant-ph/0211111\narxiv_version_number:1"},{"id":"arxiv:2203.09822","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit45174.2021.9517959","page":"902-905","source":"Crossref","title":"Performance of Coherent Frequency-Shift Keying for Classical Communication on Quantum Channels","author":[{"given":"Matteo","family":"Rosati","sequence":"first","affiliation":[]}],"event":"2021 IEEE International Symposium on Information Theory (ISIT)","container-title":"2021 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2021,7,12]]},"URL":"http://dx.doi.org/10.1109/ISIT45174.2021.9517959","note":"arxivid:2203.09822"},{"id":"arxiv:2501.12376","type":"journal-article","publisher":"IOP Publishing","issue":"41","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>A geometrically uniform (GU) ensemble is a uniformly weighted quantum state ensemble generated from a fixed state by a unitary representation of a finite group <jats:italic>G</jats:italic>. In this work we analyze the problem of discriminating GU ensembles from various angles. Assuming that the representation of <jats:italic>G</jats:italic> is irreducible, we first show that a particular optimal measurement can be understood as the limit of weighted ‘pretty good measurements’ (PGMs). This naturally provides examples of state discrimination for which the unweighted PGM is provably sub-optimal. We extend this analysis to certain reducible representations, and use Schur–Weyl duality to discuss two particular examples of GU ensembles in terms of Werner-type and permutation-invariant generator states. For the case of pure-state GU ensembles we give a streamlined proof of optimality of the PGM first proved in Eldar <jats:italic>et al</jats:italic> (2004 <jats:italic>IEEE Trans. Inf. Theory</jats:italic>\n                  <jats:bold>50</jats:bold> 1198–207). We use this result to give simplified proofs of the optimality of the PGM, along with expressions for the corresponding success probabilities, for two tasks: the hidden subgroup problem (HSP) over semidirect product groups (first proved in Bacon <jats:italic>et al</jats:italic> (2005 <jats:italic>46th Annual IEEE Symp. Foundations of Computer Science (FOCS’05)</jats:italic> pp 469–78), and port-based teleportation (first proved in Mozrzymas <jats:italic>et al</jats:italic> (2019 <jats:italic>New J. Phys.</jats:italic>\n                  <jats:bold>20</jats:bold> 053006) and Leditzky (2022 <jats:italic>Lett. Math. Phys.</jats:italic>\n                  <jats:bold>112</jats:bold> 98). Finally, we consider the <jats:italic>n</jats:italic>-copy setting and adapt a result of Montanaro (2007 <jats:italic>Commun. Math. Phys.</jats:italic>\n                  <jats:bold>273</jats:bold> 619–36) to derive a compact and easily evaluated lower bound on the success probability of the PGM for this task. This result can be applied to the HSP to obtain a new proof for an upper bound on the sample complexity by Hayashi <jats:italic>et al</jats:italic> (2006 <jats:italic>Quantum Inf. Comput.</jats:italic>\n                  <jats:bold>8</jats:bold> 345–58).</jats:p>","DOI":"10.1088/1751-8121/ae0a95","page":"415303","source":"Crossref","title":"On the distinguishability of geometrically uniform quantum states","volume":"58","author":[{"ORCID":"https://orcid.org/0009-0000-0994-8082","authenticated-orcid":true,"given":"Juntai","family":"Zhou","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2771-8330","authenticated-orcid":true,"given":"Stefano","family":"Chessa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6990-7821","authenticated-orcid":true,"given":"Eric","family":"Chitambar","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1073-9795","authenticated-orcid":true,"given":"Felix","family":"Leditzky","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2025,10,8]]},"URL":"http://dx.doi.org/10.1088/1751-8121/ae0a95","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:2501.12376"},{"id":"arxiv:1504.04908","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.91.042334","source":"Crossref","title":"Optimality of square-root measurements in quantum state discrimination","volume":"91","author":[{"given":"Nicola","family":"Dalla Pozza","sequence":"first","affiliation":[]},{"given":"Gianfranco","family":"Pierobon","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.042334","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042334","note":"arxivid:1504.04908"},{"id":"arxiv:1507.04737","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.92.062333","source":"Crossref","title":"Optimal measurements for symmetric quantum states with applications to optical communication","volume":"92","author":[{"given":"Hari","family":"Krovi","sequence":"first","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]},{"given":"Zachary","family":"Dutton","sequence":"additional","affiliation":[]},{"given":"Marcus P.","family":"da Silva","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,12,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.92.062333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062333","note":"arxivid:1507.04737"},{"id":"arxiv:quant-ph/9907073","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.61.010303","source":"Crossref","title":"Continuous variable quantum cryptography","volume":"61","author":[{"given":"T. C.","family":"Ralph","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1999,12,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.61.010303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"010303","note":"arxivid:quant-ph/9907073"},{"id":"arxiv:quant-ph/0109084","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.88.057902","source":"Crossref","title":"Continuous Variable Quantum Cryptography Using Coherent States","volume":"88","author":[{"given":"Frédéric","family":"Grosshans","sequence":"first","affiliation":[]},{"given":"Philippe","family":"Grangier","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2002,1,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.88.057902","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"057902","note":"arxivid:quant-ph/0109084"},{"id":"arxiv:quant-ph/0312016","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6920","DOI":"10.1038/nature01289","page":"238-241","source":"Crossref","title":"Quantum key distribution using gaussian-modulated coherent states","volume":"421","author":[{"given":"Frédéric","family":"Grosshans","sequence":"first","affiliation":[]},{"given":"Gilles","family":"Van Assche","sequence":"additional","affiliation":[]},{"given":"Jérôme","family":"Wenger","sequence":"additional","affiliation":[]},{"given":"Rosa","family":"Brouri","sequence":"additional","affiliation":[]},{"given":"Nicolas J.","family":"Cerf","sequence":"additional","affiliation":[]},{"given":"Philippe","family":"Grangier","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2003,1]]},"URL":"http://dx.doi.org/10.1038/nature01289","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature01289\narxivid:quant-ph/0312016"},{"id":"doi:10.1002/9783527628285","type":"monograph","publisher":"Wiley","DOI":"10.1002/9783527628285","source":"Crossref","title":"Coherent States in Quantum Physics","author":[{"given":"Jean‐Pierre","family":"Gazeau","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2009,9,23]]},"ISBN":["9783527407095","9783527628285"],"URL":"http://dx.doi.org/10.1002/9783527628285","note":"edition-number:1\nalternative-id:10.1002/9783527628285"},{"id":"arxiv:0710.4858","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.101.130503","source":"Crossref","title":"Experimentally Feasible Quantum Erasure-Correcting Code for Continuous Variables","volume":"101","author":[{"given":"J.","family":"Niset","sequence":"first","affiliation":[]},{"given":"U. L.","family":"Andersen","sequence":"additional","affiliation":[]},{"given":"N. J.","family":"Cerf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,9,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.101.130503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130503","note":"arxivid:0710.4858"},{"id":"arxiv:1006.3941","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","DOI":"10.1038/nphoton.2010.168","page":"700-705","source":"Crossref","title":"Quantum optical coherence can survive photon losses using a continuous-variable quantum erasure-correcting code","volume":"4","author":[{"given":"Mikael","family":"Lassen","sequence":"first","affiliation":[]},{"given":"Metin","family":"Sabuncu","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Huck","sequence":"additional","affiliation":[]},{"given":"Julien","family":"Niset","sequence":"additional","affiliation":[]},{"given":"Gerd","family":"Leuchs","sequence":"additional","affiliation":[]},{"given":"Nicolas J.","family":"Cerf","sequence":"additional","affiliation":[]},{"given":"Ulrik L.","family":"Andersen","sequence":"additional","affiliation":[]}],"container-title":"Nature Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2010,7,25]]},"URL":"http://dx.doi.org/10.1038/nphoton.2010.168","ISSN":["1749-4885","1749-4893"],"container-title-short":"Nature Photon","note":"alternative-id:BFnphoton2010168\narxivid:1006.3941"},{"id":"doi:10.1103/PhysRevA.61.032309","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.61.032309","source":"Crossref","title":"Experimental proposal for achieving superadditive communication capacities with a binary quantum alphabet","volume":"61","author":[{"given":"J. R.","family":"Buck","sequence":"first","affiliation":[]},{"given":"S. J.","family":"van Enk","sequence":"additional","affiliation":[]},{"given":"Christopher A.","family":"Fuchs","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,2,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.61.032309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032309"},{"id":"arxiv:1103.5592","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/physrevlett.106.250503","source":"Crossref","title":"Quantum Receiver beyond the Standard Quantum Limit of Coherent Optical Communication","volume":"106","author":[{"given":"Kenji","family":"Tsujino","sequence":"first","affiliation":[]},{"given":"Daiji","family":"Fukuda","sequence":"additional","affiliation":[]},{"given":"Go","family":"Fujii","sequence":"additional","affiliation":[]},{"given":"Shuichiro","family":"Inoue","sequence":"additional","affiliation":[]},{"given":"Mikio","family":"Fujiwara","sequence":"additional","affiliation":[]},{"given":"Masahiro","family":"Takeoka","sequence":"additional","affiliation":[]},{"given":"Masahide","family":"Sasaki","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,6,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.106.250503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"250503","note":"arxivid:1103.5592"},{"id":"manual:-R.-S.-Kennedy-A-near-optimum","type":"article-journal","author":[{"family":"Kennedy","given":"R.S."}],"title":"A near-optimum receiver for the binary coherent state quantum channel","volume":"108","container-title":"Quarterly Progress Report","issued":"1973","page":"219–225"},{"id":"arxiv:0706.1038","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.78.022320","source":"Crossref","title":"Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers","volume":"78","author":[{"given":"Masahiro","family":"Takeoka","sequence":"first","affiliation":[]},{"given":"Masahide","family":"Sasaki","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,8,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.022320","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022320","note":"arxivid:0706.1038"},{"id":"arxiv:1807.05199","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.121.023603","source":"Crossref","title":"Robust Measurement for the Discrimination of Binary Coherent States","volume":"121","author":[{"given":"M. T.","family":"DiMario","sequence":"first","affiliation":[]},{"given":"F. E.","family":"Becerra","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2018,7,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.121.023603","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"023603","note":"arxivid:1807.05199"},{"id":"arxiv:1404.5033","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.117.200501","source":"Crossref","title":"Practical Receiver for Optimal Discrimination of Binary Coherent Signals","volume":"117","author":[{"given":"Denis","family":"Sych","sequence":"first","affiliation":[]},{"given":"Gerd","family":"Leuchs","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,11,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.117.200501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200501","note":"arxivid:1404.5033"},{"id":"arxiv:2410.21800","type":"article-journal","author":[{"given":"Aakash","family":"Warke"},{"given":"Janis","family":"Nötzel"},{"given":"Kan","family":"Takase"},{"given":"Warit","family":"Asavanant"},{"given":"Hironari","family":"Nagayoshi"},{"given":"Kosuke","family":"Fukui"},{"given":"Shuntaro","family":"Takeda"},{"given":"Akira","family":"Furusawa"},{"given":"Peter","family":"van Loock"}],"title":"Photonic Quantum Receiver Attaining the Helstrom Bound","issued":{"date-parts":[[2024,10,29]]},"note":"arxivid:2410.21800\narxiv_version_number:1"},{"id":"arxiv:2310.05889","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41467-025-59107-4","source":"Crossref","title":"Superadditive communication with the green machine as a practical demonstration of nonlocality without entanglement","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0003-4517-8980","authenticated-orcid":false,"given":"Chaohan","family":"Cui","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0009-0000-2149-3279","authenticated-orcid":false,"given":"Jack","family":"Postlewaite","sequence":"additional","affiliation":[]},{"given":"Babak N.","family":"Saif","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5498-7352","authenticated-orcid":false,"given":"Linran","family":"Fan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2581-4380","authenticated-orcid":false,"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2025,4,22]]},"URL":"http://dx.doi.org/10.1038/s41467-025-59107-4","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"3760","note":"alternative-id:59107\narxivid:2310.05889"},{"id":"arxiv:0809.4953","type":"journal-article","publisher":"American Physical Society (APS)","issue":"21","DOI":"10.1103/physrevlett.101.210501","source":"Crossref","title":"Demonstration of Near-Optimal Discrimination of Optical Coherent States","volume":"101","author":[{"given":"Christoffer","family":"Wittmann","sequence":"first","affiliation":[]},{"given":"Masahiro","family":"Takeoka","sequence":"additional","affiliation":[]},{"given":"Katiúscia N.","family":"Cassemiro","sequence":"additional","affiliation":[]},{"given":"Masahide","family":"Sasaki","sequence":"additional","affiliation":[]},{"given":"Gerd","family":"Leuchs","sequence":"additional","affiliation":[]},{"given":"Ulrik L.","family":"Andersen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,11,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.101.210501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"210501","note":"arxivid:0809.4953"},{"id":"arxiv:1911.08932","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.101.032306","source":"Crossref","title":"Sub-shot-noise-limited fiber-optic quantum receiver","volume":"101","author":[{"given":"M. L.","family":"Shcherbatenko","sequence":"first","affiliation":[]},{"given":"M. S.","family":"Elezov","sequence":"additional","affiliation":[]},{"given":"G. N.","family":"Goltsman","sequence":"additional","affiliation":[]},{"given":"D. V.","family":"Sych","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.101.032306","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032306","note":"arxivid:1911.08932"},{"id":"arxiv:1907.12515","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The achievable rate of information transfer in optical communications is determined by the physical properties of the communication channel, such as the intrinsic channel noise. Bosonic phase noise channels, a class of non-Gaussian channels, have emerged as a relevant noise model in quantum information and optical communication. However, while the fundamental limits for communication over Gaussian channels have been extensively studied, the properties of communication over Bosonic phase noise channels are not well understood. Here we propose and demonstrate experimentally the concept of optimized communication strategies for communication over phase noise channels to enhance information transfer beyond what is possible with conventional methods of modulation and detection. Two key ingredients are generalized constellations of coherent states that interpolate between standard on-off keying and binary phase-shift keying formats, and non-Gaussian measurements based on photon number resolving detection of the coherently displaced signal. For a given power constraint and channel noise strength, these novel strategies rely on joint optimization of the input alphabet and the measurement to provide enhanced communication capability over a non-Gaussian channel characterized in terms of the error rate as well as mutual information.</jats:p>","DOI":"10.1038/s41534-019-0177-4","source":"Crossref","title":"Optimized communication strategies with binary coherent states over phase noise channels","volume":"5","author":[{"given":"M. T.","family":"DiMario","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0613-7070","authenticated-orcid":false,"given":"L.","family":"Kunz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5389-6897","authenticated-orcid":false,"given":"K.","family":"Banaszek","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2928-310X","authenticated-orcid":false,"given":"F. E.","family":"Becerra","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,31]]},"URL":"http://dx.doi.org/10.1038/s41534-019-0177-4","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"65","note":"alternative-id:177\narxivid:1907.12515"},{"id":"doi:10.1103/PhysRevResearch.2.023384","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevresearch.2.023384","source":"Crossref","title":"Phase tracking for sub-shot-noise-limited receivers","volume":"2","author":[{"given":"M. T.","family":"DiMario","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2928-310X","authenticated-orcid":true,"given":"F. E.","family":"Becerra","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.023384","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023384"},{"id":"arxiv:2007.11109","type":"article-journal","author":[{"given":"Renzhi","family":"Yuan"},{"given":"Mufei","family":"Zhao"},{"given":"Shuai","family":"Han"},{"given":"Julian","family":"Cheng"}],"title":"Optimally Displaced Threshold Detection for Discriminating Binary Coherent States Using Imperfect Devices","issued":{"date-parts":[[2020,7,21]]},"note":"arxivid:2007.11109\narxiv_version_number:1"},{"id":"arxiv:2207.12234","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum state discrimination is a central problem in quantum measurement theory, with applications spanning from quantum communication to computation. Typical measurement paradigms for state discrimination involve a minimum probability of error or unambiguous discrimination with a minimum probability of inconclusive results. Alternatively, an optimal inconclusive measurement, a non-projective measurement, achieves minimal error for a given inconclusive probability. This more general measurement encompasses the standard measurement paradigms for state discrimination and provides a much more powerful tool for quantum information and communication. Here, we experimentally demonstrate the optimal inconclusive measurement for the discrimination of binary coherent states using linear optics and single-photon detection. Our demonstration uses coherent displacement operations based on interference, single-photon detection, and fast feedback to prepare the optimal feedback policy for the optimal non-projective quantum measurement with high fidelity. This generalized measurement allows us to transition among standard measurement paradigms in an optimal way from minimum error to unambiguous measurements for binary coherent states. As a particular case, we use this general measurement to implement the optimal minimum error measurement for phase-coherent states, which is the optimal modulation for communications under the average power constraint. Moreover, we propose a hybrid measurement that leverages the binary optimal inconclusive measurement in conjunction with sequential, unambiguous state elimination to realize higher dimensional inconclusive measurements of coherent states.</jats:p>","DOI":"10.1038/s41534-022-00595-3","source":"Crossref","title":"Demonstration of optimal non-projective measurement of binary coherent states with photon counting","volume":"8","author":[{"given":"M. T.","family":"DiMario","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2928-310X","authenticated-orcid":false,"given":"F. E.","family":"Becerra","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,18]]},"URL":"http://dx.doi.org/10.1038/s41534-022-00595-3","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"84","note":"alternative-id:595\narxivid:2207.12234"},{"id":"arxiv:2102.13052","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.106.032613","source":"Crossref","title":"Demonstration of a quantum advantage by a joint detection receiver for optical communication using quantum belief propagation on a trapped-ion device","volume":"106","author":[{"given":"Conor","family":"Delaney","sequence":"first","affiliation":[{"name":"Aliro Technologies, Inc., Boston, Massachusetts 02135, USA"}]},{"given":"Kaushik P.","family":"Seshadreesan","sequence":"additional","affiliation":[{"name":"College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA"},{"name":"School of Computing and Information, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA"}]},{"given":"Ian","family":"MacCormack","sequence":"additional","affiliation":[{"name":"Aliro Technologies, Inc., Boston, Massachusetts 02135, USA"},{"name":"Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA"},{"name":"Department of Physics, Princeton University, Princeton, New Jersey 08544, USA"}]},{"given":"Alexey","family":"Galda","sequence":"additional","affiliation":[{"name":"James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[{"name":"College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3956-4594","authenticated-orcid":true,"given":"Prineha","family":"Narang","sequence":"additional","affiliation":[{"name":"School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.106.032613","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032613","note":"arxivid:2102.13052"},{"id":"doi:10.1109/18.915636","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/18.915636","page":"858-872","source":"Crossref","title":"On quantum detection and the square-root measurement","volume":"47","author":[{"given":"Y.C.","family":"Eldar","sequence":"first","affiliation":[]},{"given":"G.D.","family":"Forney","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2001,3]]},"URL":"http://dx.doi.org/10.1109/18.915636","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1038/nphoton.2014.280","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/nphoton.2014.280","page":"48-53","source":"Crossref","title":"Photon number resolution enables quantum receiver for realistic coherent optical communications","volume":"9","author":[{"given":"F. E.","family":"Becerra","sequence":"first","affiliation":[]},{"given":"J.","family":"Fan","sequence":"additional","affiliation":[]},{"given":"A.","family":"Migdall","sequence":"additional","affiliation":[]}],"container-title":"Nature Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2014,11,17]]},"URL":"http://dx.doi.org/10.1038/nphoton.2014.280","ISSN":["1749-4885","1749-4893"],"container-title-short":"Nature Photon","note":"alternative-id:BFnphoton2014280"},{"id":"preset:Helstrom","type":"book","author":[{"family":"Helstrom","given":"C.W."}],"title":"\\emph{Quantum Detection and Estimation Theory}","publisher":"Elsevier","issued":"1976"},{"id":"doi:10.1109/26.752130","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/26.752130","page":"248-254","source":"Crossref","title":"Quantum detection and mutual information for QAM and PSK signals","volume":"47","author":[{"given":"K.","family":"Kato","sequence":"first","affiliation":[]},{"given":"M.","family":"Osaki","sequence":"additional","affiliation":[]},{"given":"M.","family":"Sasaki","sequence":"additional","affiliation":[]},{"given":"O.","family":"Hirota","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[1999]]},"URL":"http://dx.doi.org/10.1109/26.752130","ISSN":["0090-6778"],"container-title-short":"IEEE Trans. Commun."},{"id":"arxiv:1410.5282","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.90.042309","source":"Crossref","title":"Gaussian states and geometrically uniform symmetry","volume":"90","author":[{"given":"Gianfranco","family":"Cariolaro","sequence":"first","affiliation":[]},{"given":"Roberto","family":"Corvaja","sequence":"additional","affiliation":[]},{"given":"Gianfranco","family":"Pierobon","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,10,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.042309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042309","note":"arxivid:1410.5282"},{"id":"arxiv:quant-ph/0410133","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.71.022318","source":"Crossref","title":"Implementation of projective measurements with linear optics and continuous photon counting","volume":"71","author":[{"given":"Masahiro","family":"Takeoka","sequence":"first","affiliation":[]},{"given":"Masahide","family":"Sasaki","sequence":"additional","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]},{"given":"Norbert","family":"Lütkenhaus","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,2,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.022318","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022318","note":"arxivid:quant-ph/0410133"},{"id":"doi:10.1103/PhysRevA.84.062324","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.84.062324","source":"Crossref","title":"<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>M</mml:mi></mml:math>-ary-state phase-shift-keying discrimination below the homodyne limit","volume":"84","author":[{"given":"F. E.","family":"Becerra","sequence":"first","affiliation":[]},{"given":"J.","family":"Fan","sequence":"additional","affiliation":[]},{"given":"G.","family":"Baumgartner","sequence":"additional","affiliation":[]},{"given":"S. V.","family":"Polyakov","sequence":"additional","affiliation":[]},{"given":"J.","family":"Goldhar","sequence":"additional","affiliation":[]},{"given":"J. T.","family":"Kosloski","sequence":"additional","affiliation":[]},{"given":"A.","family":"Migdall","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,12,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.062324","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062324"},{"id":"arxiv:0905.2496","type":"journal-article","publisher":"Informa UK Limited","issue":"3","DOI":"10.1080/09500340903145031","page":"213-217","source":"Crossref","title":"Discrimination of optical coherent states using a photon number resolving detector","volume":"57","author":[{"given":"Christoffer","family":"Wittmann","sequence":"first","affiliation":[]},{"given":"Ulrik L.","family":"Andersen","sequence":"additional","affiliation":[]},{"given":"Gerd","family":"Leuchs","sequence":"additional","affiliation":[]}],"container-title":"Journal of Modern Optics","original-title":[],"language":"en","issued":{"date-parts":[[2010,2,10]]},"URL":"http://dx.doi.org/10.1080/09500340903145031","ISSN":["0950-0340","1362-3044"],"container-title-short":"Journal of Modern Optics","note":"alternative-id:10.1080/09500340903145031\narxivid:0905.2496"},{"id":"arxiv:1208.1815","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.86.042328","source":"Crossref","title":"Displacement receiver for phase-shift-keyed coherent states","volume":"86","author":[{"given":"Shuro","family":"Izumi","sequence":"first","affiliation":[]},{"given":"Masahiro","family":"Takeoka","sequence":"additional","affiliation":[]},{"given":"Mikio","family":"Fujiwara","sequence":"additional","affiliation":[]},{"given":"Nicola Dalla","family":"Pozza","sequence":"additional","affiliation":[]},{"given":"Antonio","family":"Assalini","sequence":"additional","affiliation":[]},{"given":"Kazuhiro","family":"Ema","sequence":"additional","affiliation":[]},{"given":"Masahide","family":"Sasaki","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,10,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.042328","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042328","note":"arxivid:1208.1815"},{"id":"arxiv:1302.2691","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.87.042328","source":"Crossref","title":"Quantum receivers with squeezing and photon-number-resolving detectors for<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>M</mml:mi></mml:math>-ary coherent state discrimination","volume":"87","author":[{"given":"Shuro","family":"Izumi","sequence":"first","affiliation":[]},{"given":"Masahiro","family":"Takeoka","sequence":"additional","affiliation":[]},{"given":"Kazuhiro","family":"Ema","sequence":"additional","affiliation":[]},{"given":"Masahide","family":"Sasaki","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,4,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.042328","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042328","note":"arxivid:1302.2691"},{"id":"arxiv:1304.7316","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"22","DOI":"10.1109/lpt.2013.2282155","page":"2182-2184","source":"Crossref","title":"Suppressing the Errors Due to Mode Mismatch for $M$-Ary PSK Quantum Receivers Using Photon-Number-Resolving Detector","volume":"25","author":[{"given":"Ke","family":"Li","sequence":"first","affiliation":[]},{"given":"Yuan","family":"Zuo","sequence":"additional","affiliation":[]},{"given":"Bing","family":"Zhu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Photonics Technology Letters","original-title":[],"issued":{"date-parts":[[2013,11]]},"URL":"http://dx.doi.org/10.1109/LPT.2013.2282155","ISSN":["1041-1135","1941-0174"],"container-title-short":"IEEE Photon. Technol. Lett.","note":"arxivid:1304.7316"},{"id":"arxiv:1802.08287","type":"journal-article","publisher":"Optica Publishing Group","issue":"3","DOI":"10.1364/optica.5.000227","page":"227","source":"Crossref","title":"Quantum receiver for large alphabet communication","volume":"5","author":[{"given":"I. A.","family":"Burenkov","sequence":"first","affiliation":[]},{"given":"O. V.","family":"Tikhonova","sequence":"additional","affiliation":[]},{"given":"S. V.","family":"Polyakov","sequence":"additional","affiliation":[]}],"container-title":"Optica","original-title":[],"language":"en","issued":{"date-parts":[[2018,2,21]]},"URL":"http://dx.doi.org/10.1364/OPTICA.5.000227","ISSN":["2334-2536"],"container-title-short":"Optica","note":"arxivid:1802.08287"},{"id":"doi:10.1038/ncomms3028","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/ncomms3028","source":"Crossref","title":"Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states","volume":"4","author":[{"given":"F. E.","family":"Becerra","sequence":"first","affiliation":[]},{"given":"J.","family":"Fan","sequence":"additional","affiliation":[]},{"given":"A.","family":"Migdall","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2013,6,18]]},"URL":"http://dx.doi.org/10.1038/ncomms3028","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"2028","note":"alternative-id:BFncomms3028"},{"id":"doi:10.1038/nphoton.2012.316","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1038/nphoton.2012.316","page":"147-152","source":"Crossref","title":"Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination","volume":"7","author":[{"given":"F. E.","family":"Becerra","sequence":"first","affiliation":[]},{"given":"J.","family":"Fan","sequence":"additional","affiliation":[]},{"given":"G.","family":"Baumgartner","sequence":"additional","affiliation":[]},{"given":"J.","family":"Goldhar","sequence":"additional","affiliation":[]},{"given":"J. T.","family":"Kosloski","sequence":"additional","affiliation":[]},{"given":"A.","family":"Migdall","sequence":"additional","affiliation":[]}],"container-title":"Nature Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2013,1,6]]},"URL":"http://dx.doi.org/10.1038/nphoton.2012.316","ISSN":["1749-4885","1749-4893"],"container-title-short":"Nature Photon","note":"alternative-id:BFnphoton2012316"},{"id":"arxiv:2001.05902","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevapplied.13.054015","source":"Crossref","title":"Experimental Demonstration of a Quantum Receiver Beating the Standard Quantum Limit at Telecom Wavelength","volume":"13","author":[{"ORCID":"https://orcid.org/0000-0002-2567-5069","authenticated-orcid":true,"given":"Shuro","family":"Izumi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8585-0068","authenticated-orcid":true,"given":"Jonas S.","family":"Neergaard-Nielsen","sequence":"additional","affiliation":[]},{"given":"Shigehito","family":"Miki","sequence":"additional","affiliation":[]},{"given":"Hirotaka","family":"Terai","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1990-7687","authenticated-orcid":true,"given":"Ulrik L.","family":"Andersen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2020,5,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.13.054015","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"054015","note":"arxivid:2001.05902"},{"id":"arxiv:1711.00074","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Measurements approaching the ultimate quantum limits of sensitivity are central in quantum information processing, quantum metrology, and communication. Quantum measurements to discriminate multiple states at the single-photon level are essential for optimizing information transfer in low-power optical communications and quantum communications, and can enhance the capabilities of many quantum information protocols. Here, we theoretically investigate and experimentally demonstrate the discrimination of multiple coherent states of light with sensitivities surpassing the quantum noise limit (QNL) at the single-photon level under realistic conditions of loss and noise based on strategies implementing globally-optimized adaptive measurements with single photon counting and displacement operations. These discrimination strategies can provide realistic advantages to enhance information transfer at low powers, and are compatible with photon number resolving detection, which provides robustness at high powers, thus allowing for surpassing the QNL at arbitrary input power levels under realistic conditions.</jats:p>","DOI":"10.1038/s41534-017-0042-2","source":"Crossref","title":"Multi-state discrimination below the quantum noise limit at the single-photon level","volume":"3","author":[{"given":"A. R.","family":"Ferdinand","sequence":"first","affiliation":[]},{"given":"M. T.","family":"DiMario","sequence":"additional","affiliation":[]},{"given":"F. E.","family":"Becerra","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,16]]},"URL":"http://dx.doi.org/10.1038/s41534-017-0042-2","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"43","note":"alternative-id:42\narxivid:1711.00074"},{"id":"doi:10.1364/CLEO_QELS.2020.FF1D.1","type":"proceedings-article","publisher":"Optica Publishing Group","abstract":"<jats:p>We developed Coherent Frequency Shift Keying and Hybrid Frequency-Phase Shift Keying protocols optimized for quantum measurement. Paired with our new quantum receiver, these protocols experimentally demonstrate the record energy efficiency and improve communication channel capacity.</jats:p>","DOI":"10.1364/cleo_qels.2020.ff1d.1","page":"FF1D.1","source":"Crossref","title":"Experimental demonstration of time resolving quantum receiver for bandwidth and power efficient communications","author":[{"given":"Ivan A.","family":"Burenkov","sequence":"first","affiliation":[]},{"given":"M. V.","family":"Jabir","sequence":"additional","affiliation":[]},{"given":"N. Fajar R.","family":"Annafianto","sequence":"additional","affiliation":[]},{"given":"Abdella","family":"Battou","sequence":"additional","affiliation":[]},{"given":"Sergey V.","family":"Polyakov","sequence":"additional","affiliation":[]}],"event":"CLEO: QELS_Fundamental Science","container-title":"Conference on Lasers and Electro-Optics","original-title":[],"issued":{"date-parts":[[2020]]},"URL":"http://dx.doi.org/10.1364/CLEO_QELS.2020.FF1D.1","note":"publisher-location:Washington, D.C."},{"id":"doi:10.1116/5.0123880","type":"journal-article","publisher":"American Vacuum Society","issue":"1","abstract":"<jats:p>We experimentally demonstrate a quantum-measurement-based receiver for a range of modulation schemes and alphabet lengths in a telecom C-band. We attain symbol error rates below the shot noise limit for all the studied modulation schemes and the alphabet lengths 4≤M≤16. In doing so, we achieve the record energy sensitivity for telecom receivers. We investigate the trade-off between energy and bandwidth use and its dependence on the alphabet length. We identify the combined (energy and bandwidth) resource efficiency as a figure of merit and experimentally confirm that the quantum-inspired hybrid frequency/phase encoding has the highest combined resource efficiency.</jats:p>","DOI":"10.1116/5.0123880","source":"Crossref","title":"Versatile quantum-enabled telecom receiver","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0001-6022-6085","authenticated-orcid":false,"given":"M. V.","family":"Jabir","sequence":"first","affiliation":[{"name":"National Institute of Standards and Technology 1 , Gaithersburg, Maryland 20899, USA"}]},{"given":"N. Fajar R.","family":"Annafianto","sequence":"additional","affiliation":[{"name":"National Institute of Standards and Technology 1 , Gaithersburg, Maryland 20899, USA"}]},{"ORCID":"https://orcid.org/0000-0001-5695-8339","authenticated-orcid":false,"given":"I. A.","family":"Burenkov","sequence":"additional","affiliation":[{"name":"National Institute of Standards and Technology 1 , Gaithersburg, Maryland 20899, USA"},{"name":"Joint Quantum Institute, University of Maryland 2 , College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0002-6758-0586","authenticated-orcid":false,"given":"M.","family":"Dagenais","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Maryland 3 , College Park, Maryland 20742, USA"}]},{"given":"A.","family":"Battou","sequence":"additional","affiliation":[{"name":"National Institute of Standards and Technology 1 , Gaithersburg, Maryland 20899, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4601-0250","authenticated-orcid":false,"given":"S. V.","family":"Polyakov","sequence":"additional","affiliation":[{"name":"National Institute of Standards and Technology 1 , Gaithersburg, Maryland 20899, USA"},{"name":"Department of Physics, University of Maryland 4 , College Park, Maryland 20742, USA"}]}],"container-title":"AVS Quantum Science","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,1]]},"URL":"http://dx.doi.org/10.1116/5.0123880","ISSN":["2639-0213"],"page":"015001"},{"id":"arxiv:2009.02558","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.2.020305","source":"Crossref","title":"Adaptive Generalized Measurement for Unambiguous State Discrimination of Quaternary Phase-Shift-Keying Coherent States","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-2567-5069","authenticated-orcid":true,"given":"Shuro","family":"Izumi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8585-0068","authenticated-orcid":true,"given":"Jonas S.","family":"Neergaard-Nielsen","sequence":"additional","affiliation":[]},{"given":"Ulrik L.","family":"Andersen","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,4,13]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.020305","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020305","note":"arxivid:2009.02558"},{"id":"doi:10.1103/PRXQuantum.1.010308","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.1.010308","source":"Crossref","title":"Time-Resolving Quantum Measurement Enables Energy-Efficient, Large-Alphabet Communication","volume":"1","author":[{"ORCID":"https://orcid.org/0000-0001-5695-8339","authenticated-orcid":true,"given":"I.A.","family":"Burenkov","sequence":"first","affiliation":[]},{"given":"M.V.","family":"Jabir","sequence":"additional","affiliation":[]},{"given":"A.","family":"Battou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4601-0250","authenticated-orcid":true,"given":"S.V.","family":"Polyakov","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,21]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.1.010308","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010308"},{"id":"doi:10.1364/OL.18.001896","type":"journal-article","publisher":"Optica Publishing Group","issue":"22","DOI":"10.1364/ol.18.001896","page":"1896","source":"Crossref","title":"Near-quantum optimum receivers for the phase-quadrature coherent-state channel","volume":"18","author":[{"given":"Roy S.","family":"Bondurant","sequence":"first","affiliation":[]}],"container-title":"Optics Letters","original-title":[],"language":"en","issued":{"date-parts":[[1993,11,15]]},"URL":"http://dx.doi.org/10.1364/OL.18.001896","ISSN":["0146-9592","1539-4794"],"container-title-short":"Opt. Lett."},{"id":"doi:10.1364/OSAC.409200","type":"journal-article","publisher":"Optica Publishing Group","issue":"12","abstract":"<jats:p>We implement the cyclic quantum receiver based on the theoretical proposal of Roy Bondurant and demonstrate experimentally below the shot-noise limit (SNL) discrimination of quadrature phase-shift keying signals (PSK). We also experimentally test the receiver generalized for longer communication alphabet lengths and coherent frequency shift keying (CFSK) encoding. Using off-the-shelf components, we obtain state discrimination error rates that are 3 dB and 4.6 dB below the SNLs of ideal classical receivers for quadrature PSK and CFSK encodings, respectively. The receiver unconditionally surpasses the SNL for M=8 PSK and CFSK. This receiver can be used for the simple and robust practical implementation of quantum-enhanced optical communication.</jats:p>","DOI":"10.1364/osac.409200","page":"3324","source":"Crossref","title":"Experimental demonstration of the near-quantum optimal receiver","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-6022-6085","authenticated-orcid":true,"given":"M. V.","family":"Jabir","sequence":"first","affiliation":[]},{"given":"I. A.","family":"Burenkov","sequence":"additional","affiliation":[]},{"given":"N. Fajar R.","family":"Annafianto","sequence":"additional","affiliation":[]},{"given":"A.","family":"Battou","sequence":"additional","affiliation":[]},{"given":"S. V.","family":"Polyakov","sequence":"additional","affiliation":[]}],"container-title":"OSA Continuum","original-title":[],"language":"en","issued":{"date-parts":[[2020,11,19]]},"URL":"http://dx.doi.org/10.1364/OSAC.409200","ISSN":["2578-7519"],"container-title-short":"OSA Continuum"},{"id":"doi:10.1038/s41534-022-00573-9","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>We present a systematic study of quantum receivers and modulation methods enabling resource efficient quantum-enhanced optical communication. We introduce quantum-inspired modulation schemes that theoretically yield a better resource efficiency than legacy protocols. Experimentally, we demonstrate below the shot-noise limit symbol error rates for <jats:italic>M</jats:italic> ≤ 16 legacy and quantum-inspired communication alphabets using software-configurable optical communication time-resolving quantum receiver testbed. Further, we experimentally verify that our quantum-inspired modulation schemes boost the accuracy of practical quantum measurements and significantly optimize the combined use of energy and bandwidth for communication alphabets that are longer than <jats:italic>M</jats:italic> = 4 symbols.</jats:p>","DOI":"10.1038/s41534-022-00573-9","source":"Crossref","title":"Energy and bandwidth efficiency optimization of quantum-enabled optical communication channels","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0001-6022-6085","authenticated-orcid":false,"given":"M. V.","family":"Jabir","sequence":"first","affiliation":[]},{"given":"N. Fajar R.","family":"Annafianto","sequence":"additional","affiliation":[]},{"given":"I. A.","family":"Burenkov","sequence":"additional","affiliation":[]},{"given":"A.","family":"Battou","sequence":"additional","affiliation":[]},{"given":"S. V.","family":"Polyakov","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,27]]},"URL":"http://dx.doi.org/10.1038/s41534-022-00573-9","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"63","note":"alternative-id:573"},{"id":"doi:10.1038/nature05655","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7137","DOI":"10.1038/nature05655","page":"774-777","source":"Crossref","title":"Optical coherent state discrimination using a closed-loop quantum measurement","volume":"446","author":[{"given":"Robert L.","family":"Cook","sequence":"first","affiliation":[]},{"given":"Paul J.","family":"Martin","sequence":"additional","affiliation":[]},{"given":"J. M.","family":"Geremia","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2007,4]]},"URL":"http://dx.doi.org/10.1038/nature05655","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature05655"},{"id":"manual:-S.-J.-Dolinar-An-optimum-rec","type":"article-journal","author":[{"family":"Dolinar","given":"S.J."}],"title":"An optimum receiver for the binary coherent state quantum channel","volume":"11","container-title":"Research Laboratory of Electronics, MIT, Quarterly Progress Report","issued":"1973","page":"115–120"},{"id":"arxiv:1002.2819","type":"journal-article","publisher":"Optica Publishing Group","issue":"8","DOI":"10.1364/oe.18.008107","page":"8107","source":"Crossref","title":"Sub-shot-noise-limit discrimination of on-off keyed coherent signals via a quantum receiver with a superconducting transition edge sensor","volume":"18","author":[{"given":"Kenji","family":"Tsujino","sequence":"first","affiliation":[]},{"given":"Daiji","family":"Fukuda","sequence":"additional","affiliation":[]},{"given":"Go","family":"Fujii","sequence":"additional","affiliation":[]},{"given":"Shuichiro","family":"Inoue","sequence":"additional","affiliation":[]},{"given":"Mikio","family":"Fujiwara","sequence":"additional","affiliation":[]},{"given":"Masahiro","family":"Takeoka","sequence":"additional","affiliation":[]},{"given":"Masahide","family":"Sasaki","sequence":"additional","affiliation":[]}],"container-title":"Optics Express","original-title":[],"language":"en","issued":{"date-parts":[[2010,4,1]]},"URL":"http://dx.doi.org/10.1364/OE.18.008107","ISSN":["1094-4087"],"container-title-short":"Opt. Express","note":"arxivid:1002.2819"},{"id":"arxiv:1111.4017","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1038/nphoton.2012.113","page":"374-379","source":"Crossref","title":"Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver","volume":"6","author":[{"given":"Jian","family":"Chen","sequence":"first","affiliation":[]},{"given":"Jonathan L.","family":"Habif","sequence":"additional","affiliation":[]},{"given":"Zachary","family":"Dutton","sequence":"additional","affiliation":[]},{"given":"Richard","family":"Lazarus","sequence":"additional","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]}],"container-title":"Nature Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2012,5,20]]},"URL":"http://dx.doi.org/10.1038/nphoton.2012.113","ISSN":["1749-4885","1749-4893"],"container-title-short":"Nature Photon","note":"alternative-id:BFnphoton2012113\narxivid:1111.4017"},{"id":"manual:-S.-J.-Dolinar-Jr.-A-near-opt","type":"article-journal","author":[{"family":"Dolinar","given":"S.J.","suffix":"Jr"}],"title":"A near-optimum receiver structure for the detection of M-ary optical PPM signals","volume":"42 72","publisher":"NASA","container-title":"The Telecommunications and Data Acquisition Progress Report","issued":"1982-12","publisher-place":"Pasadena, CA"},{"id":"arxiv:2105.06233","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We introduce parity quantum optimization with the aim of solving optimization problems consisting of arbitrary <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>-body interactions and side conditions using planar quantum chip architectures. The method introduces a decomposition of the problem graph with arbitrary <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>-body terms using generalized closed cycles of a hypergraph. Side conditions of the optimization problem in form of hard constraints can be included as open cycles containing the terms involved in the side conditions. The generalized parity mapping thus circumvents the need to translate optimization problems to a quadratic unconstrained binary optimization problem (QUBO) and allows for the direct encoding of higher-order constrained binary optimization problems (HCBO) on a square lattice and full parallelizability of gates.</jats:p>","DOI":"10.22331/q-2023-03-17-950","page":"950","source":"Crossref","title":"Parity Quantum Optimization: Compiler","volume":"7","author":[{"given":"Kilian","family":"Ender","sequence":"first","affiliation":[{"name":"Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria"},{"name":"Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria"}]},{"given":"Roeland","family":"ter Hoeven","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria"},{"name":"Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria"}]},{"given":"Benjamin E.","family":"Niehoff","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria"}]},{"given":"Maike","family":"Drieb-Schön","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria"},{"name":"Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria"}]},{"given":"Wolfgang","family":"Lechner","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria"},{"name":"Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,17]]},"URL":"http://dx.doi.org/10.22331/q-2023-03-17-950","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2105.06233"},{"id":"doi:10.1126/sciadv.1500838","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"9","abstract":"<jats:p>An architecture for adiabatic quantum computing devices, which is programmed by local fields and features all-to-all connectivity.</jats:p>","DOI":"10.1126/sciadv.1500838","source":"Crossref","title":"A quantum annealing architecture with all-to-all connectivity from local interactions","volume":"1","author":[{"given":"Wolfgang","family":"Lechner","sequence":"first","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, 6020 Innsbruck, Austria."},{"name":"Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria."}]},{"given":"Philipp","family":"Hauke","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, 6020 Innsbruck, Austria."},{"name":"Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria."}]},{"ORCID":"https://orcid.org/0000-0003-4014-1505","authenticated-orcid":false,"given":"Peter","family":"Zoller","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, 6020 Innsbruck, Austria."},{"name":"Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria."}]}],"container-title":"Science Advances","original-title":[],"language":"en","issued":{"date-parts":[[2015,10,2]]},"URL":"http://dx.doi.org/10.1126/sciadv.1500838","ISSN":["2375-2548"],"container-title-short":"Sci. Adv.","note":"alternative-id:10.1126/sciadv.1500838"},{"id":"arxiv:1511.00004","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.93.052325","source":"Crossref","title":"Error correction for encoded quantum annealing","volume":"93","author":[{"given":"Fernando","family":"Pastawski","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,5,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.052325","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052325","note":"arxivid:1511.00004"},{"id":"arxiv:2303.08602","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce57702.2023.00022","page":"120-130","source":"Crossref","title":"Constant Depth Code Deformations in the Parity Architecture","author":[{"given":"Anette","family":"Messinger","sequence":"first","affiliation":[{"name":"Parity Quantum Computing GmbH,Innsbruck,Austria,A-6020"}]},{"given":"Michael","family":"Fellner","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH,Innsbruck,Austria,A-6020"}]},{"given":"Wolfgang","family":"Lechner","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH,Innsbruck,Austria,A-6020"}]}],"event":"2023 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2023 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2023,9,17]]},"URL":"http://dx.doi.org/10.1109/QCE57702.2023.00022","note":"arxivid:2303.08602"},{"id":"arxiv:2205.09505","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.129.180503","source":"Crossref","title":"Universal Parity Quantum Computing","volume":"129","author":[{"ORCID":"https://orcid.org/0000-0002-6980-5806","authenticated-orcid":true,"given":"Michael","family":"Fellner","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3208-3974","authenticated-orcid":true,"given":"Anette","family":"Messinger","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3773-9530","authenticated-orcid":true,"given":"Kilian","family":"Ender","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3662-1020","authenticated-orcid":true,"given":"Wolfgang","family":"Lechner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,10,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.129.180503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180503","note":"arxivid:2205.09505"},{"id":"arxiv:2404.11332","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>We present a fault-tolerant universal quantum computing architecture based on a code concatenation of biased-noise qubits and the parity architecture. The parity architecture can be understood as a low-density parity-check code tailored specifically to obtain any desired logical connectivity from nearest-neighbor physical interactions. The code layout can be dynamically adjusted to algorithmic requirements on the fly. This allows for implementations with any desired code distance with a universal set of fault-tolerant gates. In addition to the previously explored toolsets for concatenated cat codes, our approach features parallelizable interactions between arbitrary sets of qubits by directly addressing the parity qubits in the code. The proposed scheme supports codes with lower physical qubit overhead than the repetition code at the same code distances, while requiring only weight-3 and weight-4 stabilizers and nearest-neighbor connectivity on a two-dimensional square lattice.</jats:p>","DOI":"10.1103/physrevapplied.23.044032","source":"Crossref","title":"Fault-tolerant quantum computing with the parity code and biased-noise qubits","volume":"23","author":[{"ORCID":"https://orcid.org/0000-0003-3208-3974","authenticated-orcid":true,"given":"Anette","family":"Messinger","sequence":"first","affiliation":[{"name":"Parity Quantum Computing GmbH"}]},{"ORCID":"https://orcid.org/0009-0000-7788-1115","authenticated-orcid":true,"given":"Valentin","family":"Torggler","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing Germany GmbH"}]},{"ORCID":"https://orcid.org/0009-0004-0169-9340","authenticated-orcid":true,"given":"Berend","family":"Klaver","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH"},{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"}]},{"ORCID":"https://orcid.org/0000-0002-6980-5806","authenticated-orcid":true,"given":"Michael","family":"Fellner","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH"},{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"}]},{"given":"Wolfgang","family":"Lechner","sequence":"additional","affiliation":[{"name":"Parity Quantum Computing GmbH"},{"name":"Parity Quantum Computing Germany GmbH"},{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"}]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2025,4,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.23.044032","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"044032","note":"arxivid:2404.11332"},{"id":"arxiv:1109.2591","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2012.2218792","page":"1175-1187","source":"Crossref","title":"Polar Codes for Classical-Quantum Channels","volume":"59","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2012.2218792","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1109.2591"},{"id":"arxiv:1701.03397","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2018.2869460","page":"7424-7442","source":"Crossref","title":"Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs","volume":"64","author":[{"ORCID":"https://orcid.org/0000-0003-0057-1201","authenticated-orcid":false,"given":"Rajai","family":"Nasser","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2302-8025","authenticated-orcid":false,"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2018.2869460","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1701.03397"},{"id":"arxiv:1701.05583","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2017.2754921","page":"577-592","source":"Crossref","title":"Duality of Channels and Codes","volume":"64","author":[{"ORCID":"https://orcid.org/0000-0003-2302-8025","authenticated-orcid":false,"given":"Joseph M.","family":"Renes","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2018,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2017.2754921","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1701.05583"},{"id":"arxiv:1607.04833","type":"journal-article","publisher":"IOP Publishing","issue":"7","DOI":"10.1088/1367-2630/aa7c78","page":"072001","source":"Crossref","title":"Belief propagation decoding of quantum channels by passing quantum messages","volume":"19","author":[{"ORCID":"https://orcid.org/0000-0003-2302-8025","authenticated-orcid":false,"given":"Joseph M","family":"Renes","sequence":"first","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2017,7,28]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aa7c78","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1607.04833"},{"id":"arxiv:2109.08170","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Recently, Renes proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel \\cite{renes_2017}, i.e., a channel with classical input and pure-state quantum output. The algorithm presents a genuine quantum counterpart to decoding based on the classical belief propagation algorithm, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>e</mml:mi><mml:mi>t</mml:mi></mml:math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>a</mml:mi><mml:mi>l</mml:mi><mml:mo>.</mml:mo></mml:math> \\cite{rengaswamy_2020} observed that BPQM implements the optimal decoder on a small example code, in that it implements the optimal measurement that distinguishes the quantum output states for the set of input codewords with highest achievable probability. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following  contributions. First, we prove analytically that BPQM realizes optimal decoding for any binary linear code with tree Tanner graph. We also provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. In so doing, we identify a key flaw overlooked in the original algorithm and subsequent works which implies quantum circuit realizations will be exponentially large in the code dimension. Although BPQM passes quantum messages, other information required by the algorithm is processed globally. We remedy this problem by formulating a truly message-passing algorithm which approximates BPQM and has quantum circuit complexity <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>poly&amp;#xA0;</mml:mtext><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mtext>polylog&amp;#xA0;</mml:mtext><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&amp;#x03F5;</mml:mi></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> is the code length and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> is the approximation error. Finally, we also propose a novel method for extending BPQM to factor graphs containing cycles by making use of approximate cloning. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder.</jats:p>","DOI":"10.22331/q-2022-08-23-784","page":"784","source":"Crossref","title":"Quantum message-passing algorithm for optimal and efficient decoding","volume":"6","author":[{"given":"Christophe","family":"Piveteau","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, ETH Zürich, Switzerland"}]},{"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics, ETH Zürich, Switzerland"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,23]]},"URL":"http://dx.doi.org/10.22331/q-2022-08-23-784","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2109.08170"},{"id":"arxiv:2207.04984","type":"article-journal","author":[{"given":"S.","family":"Brandsen"},{"given":"Avijit","family":"Mandal"},{"given":"Henry D.","family":"Pfister"}],"title":"Belief Propagation with Quantum Messages for Symmetric Classical-Quantum Channels","issued":{"date-parts":[[2022,7,11]]},"note":"arxivid:2207.04984\narxiv_version_number:1"},{"id":"arxiv:2003.04356","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>For space-based laser communications, when the mean photon number per received optical pulse is much smaller than one, there is a large gap between communications capacity achievable with a receiver that performs individual pulse-by-pulse detection, and the quantum-optimal “joint-detection receiver” that acts collectively on long codeword-blocks of modulated pulses; an effect often termed “superadditive capacity”. In this paper, we consider the simplest scenario where a large superadditive capacity is known: a pure-loss channel with a coherent-state binary phase-shift keyed (BPSK) modulation. The two BPSK states can be mapped conceptually to two non-orthogonal states of a qubit, described by an inner product that is a function of the mean photon number per pulse. Using this map, we derive an explicit construction of the quantum circuit of a joint-detection receiver based on a recent idea of “belief-propagation with quantum messages” (BPQM). We quantify its performance improvement over the Dolinar receiver that performs optimal pulse-by-pulse detection, which represents the best “classical” approach. We analyze the scheme rigorously and show that it achieves the quantum limit of minimum average error probability in discriminating 8 (BPSK) codewords of a length-5 binary linear code with a tree factor graph. Our result suggests that a BPQM receiver might attain the Holevo capacity of this BPSK-modulated pure-loss channel. Moreover, our receiver circuit provides an alternative proposal for a quantum supremacy experiment, targeted at a specific application that can potentially be implemented on a small, special-purpose, photonic quantum computer capable of performing cat-basis universal qubit logic.</jats:p>","DOI":"10.1038/s41534-021-00422-1","source":"Crossref","title":"Belief propagation with quantum messages for quantum-enhanced classical communications","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[]},{"given":"Kaushik P.","family":"Seshadreesan","sequence":"additional","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]},{"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2021,6,15]]},"URL":"http://dx.doi.org/10.1038/s41534-021-00422-1","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"97","note":"alternative-id:422\narxivid:2003.04356"},{"id":"doi:10.1109/MBITS.2023.3285848","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/mbits.2023.3285848","page":"20-32","source":"Crossref","title":"Belief Propagation for Classical and Quantum Systems: Overview and Recent Results","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0001-5521-4397","authenticated-orcid":false,"given":"Henry D.","family":"Pfister","sequence":"first","affiliation":[{"name":"Duke University, Durham, NC, USA"}]},{"ORCID":"https://orcid.org/0000-0002-0591-9740","authenticated-orcid":false,"given":"Christophe","family":"Piveteau","sequence":"additional","affiliation":[{"name":"ETH Z&#x00FC;rich, Z&#x00FC;rich, Switzerland"}]},{"ORCID":"https://orcid.org/0000-0003-2302-8025","authenticated-orcid":false,"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[{"name":"ETH Z&#x00FC;rich, Z&#x00FC;rich, Switzerland"}]},{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"University of Arizona, Tucson, AZ, USA"}]}],"container-title":"IEEE BITS the Information Theory Magazine","original-title":[],"issued":{"date-parts":[[2022,12,1]]},"URL":"http://dx.doi.org/10.1109/MBITS.2023.3285848","ISSN":["2692-4110","2692-4080"],"container-title-short":"IEEE BITS Inform. Theory Mag."},{"id":"arxiv:1704.07036","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.97.012315","source":"Crossref","title":"Beating the classical limits of information transmission using a quantum decoder","volume":"97","author":[{"given":"Robert J.","family":"Chapman","sequence":"first","affiliation":[]},{"given":"Akib","family":"Karim","sequence":"additional","affiliation":[]},{"given":"Zixin","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"Marco","family":"Tomamichel","sequence":"additional","affiliation":[]},{"given":"Alberto","family":"Peruzzo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,1,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.012315","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012315","note":"arxivid:1704.07036"},{"id":"arxiv:0802.2414","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.78.012341","source":"Crossref","title":"Classical enhancement of quantum-error-correcting codes","volume":"78","author":[{"given":"Isaac","family":"Kremsky","sequence":"first","affiliation":[]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,7,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.012341","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012341","note":"arxivid:0802.2414"},{"id":"arxiv:0807.2080","type":"article-journal","author":[{"given":"Min-Hsiu","family":"Hsieh"}],"title":"Entanglement-assisted Coding Theory","issued":{"date-parts":[[2008,7,15]]},"note":"arxivid:0807.2080\narxiv_version_number:2"},{"id":"arxiv:1911.12260","type":"article-journal","author":[{"given":"Andrew","family":"Nemec"},{"given":"Andreas","family":"Klappenecker"}],"title":"Infinite Families of Quantum-Classical Hybrid Codes","issued":{"date-parts":[[2020,9,9]]},"note":"arxivid:1911.12260\narxiv_version_number:2"},{"id":"arxiv:2202.02184","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2023.3277808","page":"5857-5868","source":"Crossref","title":"Singleton Bounds for Entanglement-Assisted Classical and Quantum Error Correcting Codes","volume":"69","author":[{"ORCID":"https://orcid.org/0000-0002-8157-3972","authenticated-orcid":false,"given":"Manideep","family":"Mamindlapally","sequence":"first","affiliation":[{"name":"Department of Electronics and Electronic Communication Engineering, IIT Kharagpur, Kharagpur, India"}]},{"ORCID":"https://orcid.org/0000-0001-6344-4870","authenticated-orcid":false,"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[{"name":"Departament de F&#x00ED;sica, Grup d&#x2019;Informaci&#x00F3; Qu&#x00E0;ntica, Universitat Aut&#x00F2;noma de Barcelona, Bellaterra, Barcelona, Spain"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2023,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2023.3277808","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2202.02184"},{"id":"arxiv:0811.4227","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2010.2053903","page":"4682-4704","source":"Crossref","title":"Entanglement-Assisted Communication of Classical and Quantum Information","volume":"56","author":[{"given":"Min-Hsiu","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2053903","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0811.4227"},{"id":"arxiv:0901.3038","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2010.2054532","page":"4705-4730","source":"Crossref","title":"Trading classical communication, quantum communication, and entanglement in quantum Shannon theory","volume":"56","author":[{"family":"Min-Hsiu Hsieh","sequence":"first","affiliation":[]},{"given":"Mark M","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2054532","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0901.3038"},{"id":"arxiv:0903.3920","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.80.022306","source":"Crossref","title":"Public and private communication with a quantum channel and a secret key","volume":"80","author":[{"given":"Min-Hsiu","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.022306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022306","note":"arxivid:0903.3920"},{"id":"arxiv:quant-ph/0501045","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2008.924665","page":"3091-3113","source":"Crossref","title":"Capacity theorems for quantum multiple-access channels: classical-quantum and quantum-quantum capacity regions","volume":"54","author":[{"given":"J.","family":"Yard","sequence":"first","affiliation":[]},{"given":"P.","family":"Hayden","sequence":"additional","affiliation":[]},{"given":"I.","family":"Devetak","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.924665","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0501045"},{"id":"arxiv:quant-ph/9908066","type":"article-journal","author":[{"given":"Emanuel","family":"Knill"},{"given":"Raymond","family":"Laflamme"},{"given":"Lorenza","family":"Viola"}],"title":"Theory of Quantum Error Correction for General Noise","issued":{"date-parts":[[1999,8,19]]},"note":"arxivid:quant-ph/9908066\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0402056","type":"article-journal","author":[{"given":"J. A.","family":"Holbrook"},{"given":"D. W.","family":"Kribs"},{"given":"R.","family":"Laflamme"}],"title":"Noiseless subsystems and the structure of the commutant in quantum error correction","issued":{"date-parts":[[2004,2,6]]},"note":"arxivid:quant-ph/0402056\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0507213","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.96.050501","source":"Crossref","title":"Method to Find Quantum Noiseless Subsystems","volume":"96","author":[{"given":"Man-Duen","family":"Choi","sequence":"first","affiliation":[]},{"given":"David W.","family":"Kribs","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,2,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.96.050501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050501","note":"arxivid:quant-ph/0507213"},{"id":"arxiv:quant-ph/0603252","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.74.042301","source":"Crossref","title":"Protected realizations of quantum information","volume":"74","author":[{"given":"E.","family":"Knill","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,10,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.74.042301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042301","note":"arxivid:quant-ph/0603252"},{"id":"arxiv:0705.4282","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.100.030501","source":"Crossref","title":"Characterizing the Structure of Preserved Information in Quantum Processes","volume":"100","author":[{"given":"Robin","family":"Blume-Kohout","sequence":"first","affiliation":[]},{"given":"Hui Khoon","family":"Ng","sequence":"additional","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]},{"given":"Lorenza","family":"Viola","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,1,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.100.030501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030501","note":"arxivid:0705.4282"},{"id":"arxiv:1006.1358","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.82.062306","source":"Crossref","title":"Information-preserving structures: A general framework for quantum zero-error information","volume":"82","author":[{"given":"Robin","family":"Blume-Kohout","sequence":"first","affiliation":[]},{"given":"Hui Khoon","family":"Ng","sequence":"additional","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]},{"given":"Lorenza","family":"Viola","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,12,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.82.062306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062306","note":"arxivid:1006.1358"},{"id":"arxiv:math-ph/9807030","type":"article-journal","author":[{"given":"N. P.","family":"Landsman"}],"title":"Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics","issued":{"date-parts":[[1998,7,24]]},"note":"arxivid:math-ph/9807030\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9910016","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.63.012301","source":"Crossref","title":"Stabilizing quantum information","volume":"63","author":[{"given":"Paolo","family":"Zanardi","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,12,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.63.012301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012301","note":"arxivid:quant-ph/9910016"},{"id":"arxiv:quant-ph/0101090","type":"journal-article","publisher":"IOP Publishing","issue":"35","DOI":"10.1088/0305-4470/34/35/331","page":"7067-7079","source":"Crossref","title":"Constructing qubits in physical systems","volume":"34","author":[{"given":"Lorenza","family":"Viola","sequence":"first","affiliation":[]},{"given":"Emanuel","family":"Knill","sequence":"additional","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[2001,8,24]]},"URL":"http://dx.doi.org/10.1088/0305-4470/34/35/331","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen.","note":"alternative-id:S0305-4470(01)22120-9\narxivid:quant-ph/0101090"},{"id":"arxiv:quant-ph/0608071","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevlett.98.100502","source":"Crossref","title":"Generalization of Quantum Error Correction via the Heisenberg Picture","volume":"98","author":[{"given":"Cédric","family":"Bény","sequence":"first","affiliation":[]},{"given":"Achim","family":"Kempf","sequence":"additional","affiliation":[]},{"given":"David W.","family":"Kribs","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,3,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.98.100502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"100502","note":"arxivid:quant-ph/0608071"},{"id":"arxiv:0705.1574","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.76.042303","source":"Crossref","title":"Quantum error correction of observables","volume":"76","author":[{"given":"Cédric","family":"Bény","sequence":"first","affiliation":[]},{"given":"Achim","family":"Kempf","sequence":"additional","affiliation":[]},{"given":"David W.","family":"Kribs","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,10,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.042303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042303","note":"arxivid:0705.1574"},{"id":"arxiv:0901.3629","type":"article-journal","author":[{"given":"Cédric","family":"Bény"}],"title":"Information flow at the quantum-classical boundary","issued":{"date-parts":[[2009,1,23]]},"note":"arxivid:0901.3629\narxiv_version_number:1"},{"id":"doi:10.1142/S0219749908003839","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"supp01","abstract":"<jats:p> We give a brief introduction to the algebraic formulation of error correction in quantum computing called operator algebra quantum error correction (OAQEC). Then we extend one of the basic results for subsystem codes in operator quantum error correction (OQEC) to the OAQEC setting: Every hybrid classical-quantum code is shown to be unitarily recoverable in an appropriate sense. The algebraic approach of the proof yields a new, less technical proof for the OQEC case. </jats:p>","DOI":"10.1142/s0219749908003839","page":"597-603","source":"Crossref","title":"ALGEBRAIC FORMULATION OF QUANTUM ERROR CORRECTION","volume":"06","author":[{"given":"CÉDRIC","family":"BÉNY","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada"}]},{"given":"DAVID W.","family":"KRIBS","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada"}]},{"given":"ARON","family":"PASIEKA","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1142/S0219749908003839","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749908003839"},{"id":"arxiv:quant-ph/0403161","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.70.032307","source":"Crossref","title":"Decoherence-full subsystems and the cryptographic power of a private shared reference frame","volume":"70","author":[{"given":"Stephen D.","family":"Bartlett","sequence":"first","affiliation":[]},{"given":"Terry","family":"Rudolph","sequence":"additional","affiliation":[]},{"given":"Robert W.","family":"Spekkens","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,9,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.70.032307","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032307","note":"arxivid:quant-ph/0403161"},{"id":"arxiv:1811.10425","type":"article-journal","author":[{"given":"D. W.","family":"Kribs"},{"given":"J.","family":"Levick"},{"given":"M. I.","family":"Nelson"},{"given":"R.","family":"Pereira"},{"given":"M.","family":"Rahaman"}],"title":"Quantum Complementarity and Operator Structures","issued":{"date-parts":[[2019,2,5]]},"note":"arxivid:1811.10425\narxiv_version_number:2"},{"id":"arxiv:1806.04687","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.9.021010","source":"Crossref","title":"Cage-Net Fracton Models","volume":"9","author":[{"given":"Abhinav","family":"Prem","sequence":"first","affiliation":[]},{"given":"Sheng-Jie","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Hao","family":"Song","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Hermele","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2019,4,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.021010","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021010","note":"arxivid:1806.04687"},{"id":"arxiv:1812.01613","type":"journal-article","publisher":"Stichting SciPost","issue":"4","abstract":"<jats:p>Foliated fracton order is a qualitatively new kind of phase of\nmatter. It is similar to topological order, but with the fundamental\ndifference that a layered structure, referred to as a foliation, plays\nan essential role and determines the mobility restrictions of the\ntopological excitations. In this work, we introduce a new kind of field\ntheory to describe these phases: a foliated field theory. We also\nintroduce a new lattice model and string-membrane-net condensation\npicture of these phases, which is analogous to the string-net\ncondensation picture of topological order.</jats:p>","DOI":"10.21468/scipostphys.6.4.043","source":"Crossref","title":"Foliated field theory and string-membrane-net condensation picture of fracton order","volume":"6","author":[{"given":"Kevin","family":"Slagle","sequence":"first","affiliation":[{"name":"California Institute of Technology"}]},{"given":"David","family":"Aasen","sequence":"additional","affiliation":[{"name":"California Institute of Technology"},{"name":"Kavli Institute for Theoretical Physics"},{"name":"Microsoft"}]},{"given":"Dominic","family":"Williamson","sequence":"additional","affiliation":[{"name":"Yale University"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2019,4,12]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.6.4.043","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"043","note":"arxivid:1812.01613"},{"id":"manual:-K.-Fredenhagen-Superselectio","type":"article-journal","author":[{"family":"Fredenhagen","given":"K."}],"title":"Superselection sectors with infinite statistical dimension","volume":"1993","container-title":"Subfactors (Kyuzeso","issued":"1994","page":"242–258"},{"id":"arxiv:2106.12577","type":"article-journal","author":[{"given":"Ryan","family":"Thorngren"},{"given":"Yifan","family":"Wang"}],"title":"Fusion Category Symmetry II: Categoriosities at c = 1 and Beyond","issued":{"date-parts":[[2021,6,23]]},"note":"arxivid:2106.12577\narxiv_version_number:1"},{"id":"manual:-K.-Walker.-href-https://www.","type":"document","author":[{"family":"Walker","given":"K."}],"title":"\\href{https://www.birs.ca/events/2023/5-day-workshops/23w5091/videos/watch/202312061131-Walker.html}{Lie group symmetries and non-unital higher categories","genre":"Subfactors and Fusion (2-)Categories},","publisher":"Banff International Research Station","issued":"2023"},{"id":"arxiv:2503.14596","type":"article-journal","author":[{"given":"Adrià","family":"Marín-Salvador"}],"title":"Continuous Tambara-Yamagami tensor categories","issued":{"date-parts":[[2025,3,18]]},"note":"arxivid:2503.14596\narxiv_version_number:1"},{"id":"arxiv:gr-qc/0303060","type":"article-journal","author":[{"given":"C. J.","family":"Isham"}],"title":"A New Approach to Quantising Space-Time: I. Quantising on a General Category","issued":{"date-parts":[[2003,12,24]]},"note":"arxivid:gr-qc/0303060\narxiv_version_number:2"},{"id":"arxiv:1804.03199","type":"article-journal","author":[{"given":"Johannes","family":"Berger"},{"given":"Tobias J.","family":"Osborne"}],"title":"Perfect tangles","issued":{"date-parts":[[2018,4,9]]},"note":"arxivid:1804.03199\narxiv_version_number:1"},{"id":"arxiv:math/0307200","type":"article-journal","author":[{"given":"John C.","family":"Baez"},{"given":"Aaron D.","family":"Lauda"}],"title":"Higher-Dimensional Algebra V: 2-Groups","issued":{"date-parts":[[2004,10,27]]},"note":"arxivid:math/0307200\narxiv_version_number:3"},{"id":"arxiv:hep-th/0304074","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/s0003-4916(03)00147-7","page":"447-477","source":"Crossref","title":"Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics","volume":"308","author":[{"given":"Hendryk","family":"Pfeiffer","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2003,12]]},"URL":"http://dx.doi.org/10.1016/S0003-4916(03)00147-7","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491603001477\narxivid:hep-th/0304074"},{"id":"arxiv:hep-th/0412325","type":"article-journal","author":[{"given":"John","family":"Baez"},{"given":"Urs","family":"Schreiber"}],"title":"Higher Gauge Theory: 2-Connections on 2-Bundles","issued":{"date-parts":[[2004,12,30]]},"note":"arxivid:hep-th/0412325\narxiv_version_number:1"},{"id":"arxiv:math/0511710","type":"article-journal","author":[{"given":"John C.","family":"Baez"},{"given":"Urs","family":"Schreiber"}],"title":"Higher Gauge Theory","issued":{"date-parts":[[2006,6,28]]},"note":"arxivid:math/0511710\narxiv_version_number:2"},{"id":"arxiv:1002.4636","type":"article-journal","author":[{"given":"Soo-Jong","family":"Rey"},{"given":"Fumihiko","family":"Sugino"}],"title":"A Nonperturbative Proposal for Nonabelian Tensor Gauge Theory and Dynamical Quantum Yang-Baxter Maps","issued":{"date-parts":[[2010,2,25]]},"note":"arxivid:1002.4636\narxiv_version_number:1"},{"id":"arxiv:1003.4485","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"9","DOI":"10.1007/s10714-010-1070-9","page":"2335-2392","source":"Crossref","title":"An invitation to higher gauge theory","volume":"43","author":[{"given":"John C.","family":"Baez","sequence":"first","affiliation":[]},{"given":"John","family":"Huerta","sequence":"additional","affiliation":[]}],"container-title":"General Relativity and Gravitation","original-title":[],"language":"en","issued":{"date-parts":[[2010,8,19]]},"URL":"http://dx.doi.org/10.1007/s10714-010-1070-9","ISSN":["0001-7701","1572-9532"],"container-title-short":"Gen Relativ Gravit","note":"alternative-id:1070\narxivid:1003.4485"},{"id":"arxiv:1307.4793","type":"article-journal","author":[{"given":"Sergei","family":"Gukov"},{"given":"Anton","family":"Kapustin"}],"title":"Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories","issued":{"date-parts":[[2013,8,14]]},"note":"arxivid:1307.4793\narxiv_version_number:2"},{"id":"arxiv:1404.2634","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"9","DOI":"10.1007/jhep09(2014)034","source":"Crossref","title":"Lattice gerbe theory","volume":"2014","author":[{"given":"Arthur E.","family":"Lipstein","sequence":"first","affiliation":[]},{"given":"Ronald A.","family":"Reid-Edwards","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,9]]},"URL":"http://dx.doi.org/10.1007/JHEP09(2014)034","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"34","note":"alternative-id:8816\narxivid:1404.2634"},{"id":"arxiv:1308.2926","type":"article-journal","author":[{"given":"Anton","family":"Kapustin"},{"given":"Ryan","family":"Thorngren"}],"title":"Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement","issued":{"date-parts":[[2013,9,20]]},"note":"arxivid:1308.2926\narxiv_version_number:2"},{"id":"arxiv:1309.4721","type":"article-journal","author":[{"given":"Anton","family":"Kapustin"},{"given":"Ryan","family":"Thorngren"}],"title":"Higher symmetry and gapped phases of gauge theories","issued":{"date-parts":[[2015,2,12]]},"note":"arxivid:1309.4721\narxiv_version_number:2"},{"id":"arxiv:1702.00868","type":"journal-article","publisher":"World Scientific Pub Co Pte Ltd","issue":"04","abstract":"<jats:p> Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. </jats:p><jats:p> The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented. </jats:p>","DOI":"10.1142/s0129055x20500117","page":"2050011","source":"Crossref","title":"Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry","volume":"32","author":[{"given":"Alex","family":"Bullivant","sequence":"first","affiliation":[{"name":"School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK"},{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"}]},{"given":"Marcos","family":"Calçada","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"},{"name":"Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, Brazil"}]},{"given":"Zoltán","family":"Kádár","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"}]},{"given":"João Faria","family":"Martins","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"}]},{"given":"Paul","family":"Martin","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK"}]}],"container-title":"Reviews in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,11,4]]},"URL":"http://dx.doi.org/10.1142/S0129055X20500117","ISSN":["0129-055X","1793-6659"],"container-title-short":"Rev. Math. Phys.","note":"alternative-id:10.1142/S0129055X20500117\narxivid:1702.00868"},{"id":"arxiv:1606.06639","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.95.155118","source":"Crossref","title":"Topological phases from higher gauge symmetry in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>dimensions","volume":"95","author":[{"given":"Alex","family":"Bullivant","sequence":"first","affiliation":[]},{"given":"Marcos","family":"Calçada","sequence":"additional","affiliation":[]},{"given":"Zoltán","family":"Kádár","sequence":"additional","affiliation":[]},{"given":"Paul","family":"Martin","sequence":"additional","affiliation":[]},{"given":"João Faria","family":"Martins","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,4,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.95.155118","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155118","note":"arxivid:1606.06639"},{"id":"arxiv:1802.10104","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>We consider exactly solvable models in (3+1)d whose ground states are described by topological lattice gauge theories. Using simplicial arguments, we emphasize how the consistency condition of the unitary map performing a local change of triangulation is equivalent to the coherence relation of the pentagonator 2-morphism of a monoidal 2-category. By weakening some axioms of such 2-category, we obtain a cohomological model whose underlying 1-category is a 2-group. Topological models from 2-groups together with their lattice realization are then studied from a higher gauge theory point of view. Symmetry protected topological phases protected by higher symmetry structures are explicitly constructed, and the gauging procedure which yields the corresponding topological gauge theories is discussed in detail. We finally study the correspondence between symmetry protected topological phases and ’t Hooft anomalies in the context of these higher group symmetries.</jats:p>","DOI":"10.1007/jhep10(2018)049","source":"Crossref","title":"From gauge to higher gauge models of topological phases","volume":"2018","author":[{"given":"Clement","family":"Delcamp","sequence":"first","affiliation":[]},{"given":"Apoorv","family":"Tiwari","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2018,10]]},"URL":"http://dx.doi.org/10.1007/JHEP10(2018)049","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"49","note":"alternative-id:9147\narxivid:1802.10104"},{"id":"arxiv:1901.02249","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>We explore 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space <jats:italic>B</jats:italic>\n            <jats:sup>2</jats:sup>\n            <jats:italic>G</jats:italic> of the symmetry group <jats:italic>G</jats:italic>, and they are classified by cohomology classes of <jats:italic>B</jats:italic>\n            <jats:sup>2</jats:sup>\n            <jats:italic>G</jats:italic>. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of <jats:italic>B</jats:italic>\n            <jats:sup>2</jats:sup>\n            <jats:italic>G</jats:italic> as provided by the so-called <jats:italic>W</jats:italic> -construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of <jats:italic>G</jats:italic>-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model.</jats:p>","DOI":"10.1007/jhep05(2019)064","source":"Crossref","title":"On 2-form gauge models of topological phases","volume":"2019","author":[{"ORCID":"https://orcid.org/0000-0003-2337-5772","authenticated-orcid":false,"given":"Clement","family":"Delcamp","sequence":"first","affiliation":[]},{"given":"Apoorv","family":"Tiwari","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,5]]},"URL":"http://dx.doi.org/10.1007/JHEP05(2019)064","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"64","note":"alternative-id:10525\narxivid:1901.02249"},{"id":"arxiv:1904.00994","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevd.100.085012","source":"Crossref","title":"Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory","volume":"100","author":[{"given":"Zheyan","family":"Wan","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9396-9010","authenticated-orcid":true,"given":"Juven","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Yunqin","family":"Zheng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2019,10,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.100.085012","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"085012","note":"update-to:{\"updated\":{\"date-parts\":[[2019,10,31]],\"date-time\":\"2019-10-31T00:00:00Z\",\"timestamp\":1572480000000},\"DOI\":\"10.1103/physrevd.100.085012\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2019,10,31]],\"date-time\":\"2019-10-31T00:00:00Z\",\"timestamp\":1572480000000},\"DOI\":\"10.1103/physrevd.100.085012\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:1904.00994"},{"id":"doi:10.1142/S0218216593000076","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"01","abstract":"<jats:p> Using techniques from [14] we construct topological quantum field theories using an algebraic model of a homotopy 2-type as initial data. </jats:p>","DOI":"10.1142/s0218216593000076","page":"113-123","source":"Crossref","title":"TQFT’S FROM HOMOTOPY 2-TYPES","volume":"02","author":[{"given":"DAVID N.","family":"YETTER","sequence":"first","affiliation":[{"name":"Department of Mathematics, Kansas State University, Manhattan, KS 66502–2602, U.S.A."}]}],"container-title":"Journal of Knot Theory and Its Ramifications","original-title":[],"language":"en","issued":{"date-parts":[[1993,3]]},"URL":"http://dx.doi.org/10.1142/S0218216593000076","ISSN":["0218-2165","1793-6527"],"container-title-short":"J. Knot Theory Ramifications","note":"alternative-id:10.1142/S0218216593000076"},{"id":"doi:10.1112/S0024610798006838","type":"journal-article","publisher":"Wiley","issue":"3","DOI":"10.1112/s0024610798006838","page":"723-732","source":"Crossref","title":"Topological Quantum Field Theories from Homotopy <i>n</i>\n-Types","volume":"58","author":[{"given":"Tim","family":"Porter","sequence":"first","affiliation":[]}],"container-title":"Journal of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1998,12]]},"URL":"http://dx.doi.org/10.1112/S0024610798006838","ISSN":["0024-6107"],"container-title-short":"Journal of the London Mathematical Society"},{"id":"doi:10.1142/S0218216596000400","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"05","DOI":"10.1142/s0218216596000400","page":"687-720","source":"Crossref","title":"INTERPRETATIONS OF YETTER’S NOTION OF G-COLORING: SIMPLICIAL FIBRE BUNDLES AND NON-ABELIAN COHOMOLOGY","volume":"05","author":[{"given":"TIM","family":"PORTER","sequence":"first","affiliation":[{"name":"School of Mathematics, University of Wales, Bangor, Gwynedd, LL57 1UT, UK"}]}],"container-title":"Journal of Knot Theory and Its Ramifications","original-title":[],"language":"en","issued":{"date-parts":[[1996,10]]},"URL":"http://dx.doi.org/10.1142/S0218216596000400","ISSN":["0218-2165","1793-6527"],"container-title-short":"J. Knot Theory Ramifications","note":"alternative-id:10.1142/S0218216596000400"},{"id":"arxiv:math/9903003","type":"article-journal","author":[{"given":"Marco","family":"Mackaay"}],"title":"Finite groups, spherical 2-categories, and 4-manifold invariants","issued":{"date-parts":[[1999,11,9]]},"note":"arxivid:math/9903003\narxiv_version_number:3"},{"id":"arxiv:1909.07937","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>We consider an exactly solvable model for topological phases in (3+1) d whose input data is a strict 2-group. This model, which has a higher gauge theory interpretation, provides a lattice Hamiltonian realisation of the Yetter homotopy 2-type topological quantum field theory. The Hamiltonian yields bulk flux and charge composite excitations that are either point-like or loop-like. Applying a generalised tube algebra approach, we reveal the algebraic structure underlying these excitations and derive the irreducible modules of this algebra, which in turn classify the elementary excitations of the model. As a further application of the tube algebra approach, we demonstrate that the ground state subspace of the three-torus is described by the central subalgebra of the tube algebra for torus boundary, demonstrating the ground state degeneracy is given by the number of elementary loop-like excitations.</jats:p>","DOI":"10.1007/jhep01(2020)107","source":"Crossref","title":"Excitations in strict 2-group higher gauge models of topological phases","volume":"2020","author":[{"given":"Alex","family":"Bullivant","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2337-5772","authenticated-orcid":false,"given":"Clement","family":"Delcamp","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,1,17]]},"URL":"http://dx.doi.org/10.1007/JHEP01(2020)107","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"107","note":"alternative-id:12180\narxivid:1909.07937"},{"id":"arxiv:2206.09941","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>In this, the third paper in our series describing the excitations of the higher-lattice gauge theory model for topological phases, we will examine the (3+1)-dimensional case in detail. We will explicitly construct the ribbon and membrane operators which create the topological excitations, and use these creation operators to find the pattern of condensation and confinement. We also use these operators to find the braiding relations of the excitations, and to construct charge measurement operators which project to states of definite topological charge.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevb.109.035152","source":"Crossref","title":"Excitations in the higher-lattice gauge theory model for topological phases. III. The (3+1)-dimensional case","volume":"109","author":[{"ORCID":"https://orcid.org/0000-0002-4857-0091","authenticated-orcid":true,"given":"Joe","family":"Huxford","sequence":"first","affiliation":[{"name":"Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Oxford OX1 3PU, United Kingdom"},{"name":"Department of Physics, University of Toronto, Ontario, Canada M5S 1A7"}]},{"ORCID":"https://orcid.org/0000-0001-7757-5978","authenticated-orcid":true,"given":"Steven H.","family":"Simon","sequence":"additional","affiliation":[{"name":"Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Oxford OX1 3PU, United Kingdom"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.109.035152","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"035152","note":"arxivid:2206.09941"},{"id":"arxiv:2202.08294","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.108.245132","source":"Crossref","title":"Excitations in the higher-lattice gauge theory model for topological phases. I. Overview","volume":"108","author":[{"ORCID":"https://orcid.org/0000-0002-4857-0091","authenticated-orcid":true,"given":"Joe","family":"Huxford","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7757-5978","authenticated-orcid":true,"given":"Steven H.","family":"Simon","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.108.245132","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"245132","note":"arxivid:2202.08294"},{"id":"preset:Ocneanu93","type":"article-journal","author":[{"family":"Ocneanu","given":"A."}],"title":"Chirality for operator algebras","volume":"39","container-title":"Subfactors (Kyuzeso","issued":"1993"},{"id":"doi:10.2969/aspm/03110235","type":"proceedings-article","publisher":"Mathematical Society of Japan","DOI":"10.2969/aspm/03110235","page":"235-263","source":"Crossref","title":"Operator Algebras, Topology and Subgroups of Quantum Symmetry – Construction of Subgroups of Quantum Groups –","author":[{"given":"Adrian","family":"Ocneanu","sequence":"first","affiliation":[]}],"event":"Taniguchi Conference on Mathematics","container-title":"Advanced Studies in Pure Mathematics","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.2969/aspm/03110235","ISSN":["0920-1971"],"note":"publisher-location:Tokyo, Japan"},{"id":"arxiv:2305.17165","type":"article-journal","author":[{"given":"Thomas","family":"Bartsch"},{"given":"Mathew","family":"Bullimore"},{"given":"Andrea","family":"Grigoletto"}],"title":"Representation theory for categorical symmetries","issued":{"date-parts":[[2023,5,26]]},"note":"arxivid:2305.17165\narxiv_version_number:1"},{"id":"arxiv:1606.07144","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2016.12.018","page":"311-344","source":"Crossref","title":"Hamiltonian models for topological phases of matter in three spatial dimensions","volume":"377","author":[{"given":"Dominic J.","family":"Williamson","sequence":"first","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,2]]},"URL":"http://dx.doi.org/10.1016/j.aop.2016.12.018","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491616302871\narxivid:1606.07144"},{"id":"manual:-P.-S.-Hsin-private-communica","type":"document","author":[{"family":"Hsin","given":"P.S."}],"genre":"private communication,","issued":"2024"},{"id":"arxiv:2110.14644","type":"journal-article","publisher":"Stichting SciPost","issue":"5","abstract":"<jats:p>We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has an anomalous boundary <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> topological order with fermionic particle and fermionic loop excitations that have mutual <jats:inline-formula><jats:alternatives><jats:tex-math>\\pi</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>π</mml:mi></mml:math></jats:alternatives></jats:inline-formula> statistics. We argue that this construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.</jats:p>","DOI":"10.21468/scipostphys.14.5.089","source":"Crossref","title":"Exactly solvable lattice Hamiltonians and gravitational anomalies","volume":"14","author":[{"given":"Yu-An","family":"Chen","sequence":"first","affiliation":[{"name":"University of Maryland, College Park"},{"name":"Walter Burke Institute for Theoretical Physics"}]},{"given":"Po-Shen","family":"Hsin","sequence":"additional","affiliation":[{"name":"Mani L. Bhaumik Institute for Theoretical Physics"},{"name":"Walter Burke Institute for Theoretical Physics"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2023,5,2]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.14.5.089","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"089","note":"arxivid:2110.14644"},{"id":"arxiv:2511.02900","type":"article-journal","author":[{"given":"Ryohei","family":"Kobayashi"},{"given":"Guanyu","family":"Zhu"},{"given":"Po-Shen","family":"Hsin"}],"title":"Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic","issued":{"date-parts":[[2025,12,14]]},"note":"arxivid:2511.02900\narxiv_version_number:2"},{"id":"arxiv:2412.15336","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","abstract":"<jats:title>\n                    A\n                    <jats:sc>bstract</jats:sc>\n                  </jats:title>\n                  <jats:p>\n                    We propose weak Hopf symmetry as a general framework to explore (1+1)D topological phases that exhibit non-invertible symmetries. Inspired by the Symmetry Topological Field Theory (SymTFT) description of quantum phases with non-invertible symmetry, we construct a lattice model by introducing two distinct topological boundary conditions for a weak Hopf lattice gauge theory. One boundary encodes the topological symmetry information, while the other incorporates the non-topological dynamics. The resulting model is termed the cluster ladder model. We demonstrate that the cluster state model is a special case of this broader class of lattice models exhibiting weak Hopf symmetry\n                    <jats:italic>H</jats:italic>\n                    ×\n                    <jats:italic>Ĥ</jats:italic>\n                    , where\n                    <jats:italic>H</jats:italic>\n                    is a weak Hopf algebra and\n                    <jats:italic>Ĥ</jats:italic>\n                    is its dual weak Hopf algebra. On a closed manifold, the symmetry reduces to Cocom(\n                    <jats:italic>H</jats:italic>\n                    ) × Cocom(\n                    <jats:italic>Ĥ</jats:italic>\n                    ), corresponding to the cocommutative subalgebras of\n                    <jats:italic>H</jats:italic>\n                    ×\n                    <jats:italic>Ĥ</jats:italic>\n                    . An essential weak Hopf sub-symmetry is Cocom(\n                    <jats:italic>H</jats:italic>\n                    ) × Rep(\n                    <jats:italic>H</jats:italic>\n                    ), which, in the finite group case, reduces to the familiar symmetry\n                    <jats:italic>G</jats:italic>\n                    × Rep(\n                    <jats:italic>G</jats:italic>\n                    ). To exactly solve the lattice model, we introduce a weak Hopf tensor network. Furthermore, we demonstrate how to construct the lattice realization of an arbitrary fusion category symmetry\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$ \\mathcal{S} $$</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                          <mml:mi>S</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    via combining Tannaka-Krein reconstruction or weak Hopf tube algebra and the cluster ladder model.\n                  </jats:p>","DOI":"10.1007/jhep03(2026)154","source":"Crossref","title":"Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory","volume":"2026","author":[{"ORCID":"https://orcid.org/0000-0001-8588-173X","authenticated-orcid":false,"given":"Zhian","family":"Jia","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00f1zfq44","id-type":"ROR","asserted-by":"publisher"}],"name":"Central South University"},{"id":[{"id":"https://ror.org/01tgyzw49","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Singapore"},{"id":[{"id":"https://ror.org/02j1m6098","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Singapore"}]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,16]]},"URL":"http://dx.doi.org/10.1007/JHEP03(2026)154","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"154","note":"alternative-id:28460\narxivid:2412.15336"},{"id":"arxiv:2405.09277","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"9","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.</jats:p>","DOI":"10.1007/jhep09(2024)147","source":"Crossref","title":"Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation","volume":"2024","author":[{"ORCID":"https://orcid.org/0000-0001-8588-173X","authenticated-orcid":false,"given":"Zhian","family":"Jia","sequence":"first","affiliation":[{"name":"National University of Singapore"},{"id":[{"id":"https://ror.org/01tgyzw49","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Singapore"}]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,20]]},"URL":"http://dx.doi.org/10.1007/JHEP09(2024)147","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"147","note":"alternative-id:24524\narxivid:2405.09277"},{"id":"arxiv:2602.10183","type":"article-journal","author":[{"given":"Da-Chuan","family":"Lu"},{"given":"Arkya","family":"Chatterjee"},{"given":"Nathanan","family":"Tantivasadakarn"}],"title":"Generalized Kramers-Wannier Self-Duality in Hopf-Ising Models","issued":{"date-parts":[[2026,2,10]]},"note":"arxivid:2602.10183\narxiv_version_number:1"},{"id":"arxiv:1007.5283","type":"journal-article","publisher":"AIP Publishing","issue":"1","abstract":"<jats:p>We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [Ann. Phys. 303, 2 (2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.</jats:p>","DOI":"10.1063/1.4773316","source":"Crossref","title":"A hierarchy of topological tensor network states","volume":"54","author":[{"given":"Oliver","family":"Buerschaper","sequence":"first","affiliation":[{"name":"Perimeter Institute for Theoretical Physics 1 , 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5"},{"name":"Max-Planck-Institut für Quantenoptik 2 , Hans-Kopfermann-Straße 1, 85748 Garching, Germany"}]},{"given":"Juan Martín","family":"Mombelli","sequence":"additional","affiliation":[{"name":"Ciudad Universitaria 3 Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Medina Allende s/n, , 5000 Córdoba, Argentina"}]},{"given":"Matthias","family":"Christandl","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics 4 , ETH Zurich, 8093 Zurich, Switzerland"}]},{"given":"Miguel","family":"Aguado","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut für Quantenoptik 2 , Hans-Kopfermann-Straße 1, 85748 Garching, Germany"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2013,1,1]]},"URL":"http://dx.doi.org/10.1063/1.4773316","ISSN":["0022-2488","1089-7658"],"note":"arxivid:1007.5283"},{"id":"arxiv:1206.2308","type":"article-journal","author":[{"given":"Benjamin","family":"Balsam"},{"given":"Alexander","family":"Kirillov"}],"title":"Kitaev's Lattice Model and Turaev-Viro TQFTs","issued":{"date-parts":[[2012,6,11]]},"note":"arxivid:1206.2308\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9707021","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0003-4916(02)00018-0","page":"2-30","source":"Crossref","title":"Fault-tolerant quantum computation by anyons","volume":"303","author":[{"given":"A.Yu.","family":"Kitaev","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2003,1]]},"URL":"http://dx.doi.org/10.1016/S0003-4916(02)00018-0","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491602000180\narxivid:quant-ph/9707021"},{"id":"arxiv:2207.03970","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>The generalized quantum double lattice realization of 2<jats:italic>d</jats:italic> topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2<jats:italic>d</jats:italic> surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.</jats:p>","DOI":"10.1007/jhep07(2023)160","source":"Crossref","title":"Boundary and domain wall theories of 2d generalized quantum double model","volume":"2023","author":[{"ORCID":"https://orcid.org/0000-0001-8588-173X","authenticated-orcid":false,"given":"Zhian","family":"Jia","sequence":"first","affiliation":[]},{"given":"Dagomir","family":"Kaszlikowski","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-3318-9942","authenticated-orcid":false,"given":"Sheng","family":"Tan","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,21]]},"URL":"http://dx.doi.org/10.1007/JHEP07(2023)160","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"160","note":"alternative-id:21372\narxivid:2207.03970"},{"id":"arxiv:2208.06317","type":"article-journal","author":[{"given":"Alexander","family":"Cowtan"},{"given":"Shahn","family":"Majid"}],"title":"Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model","issued":{"date-parts":[[2022,8,12]]},"note":"arxivid:2208.06317\narxiv_version_number:1"},{"id":"arxiv:1006.5823","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/j.nuclphysb.2013.08.014","page":"619-636","source":"Crossref","title":"Electric–magnetic duality of lattice systems with topological order","volume":"876","author":[{"given":"Oliver","family":"Buerschaper","sequence":"first","affiliation":[]},{"given":"Matthias","family":"Christandl","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Kong","sequence":"additional","affiliation":[]},{"given":"Miguel","family":"Aguado","sequence":"additional","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[2013,11]]},"URL":"http://dx.doi.org/10.1016/j.nuclphysb.2013.08.014","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"alternative-id:S0550321313004367\narxivid:1006.5823"},{"id":"arxiv:1309.4181","type":"journal-article","publisher":"AIP Publishing","issue":"4","abstract":"<jats:p>We establish a generalization of Kitaev models based on unitary quantum groupoids. In particular, when inputting a Kitaev-Kong quantum groupoid \\documentclass[12pt]{minimal}\\begin{document}$H_\\mathcal {C}$\\end{document}HC, we show that the ground state manifold of the generalized model is canonically isomorphic to that of the Levin-Wen model based on a unitary fusion category \\documentclass[12pt]{minimal}\\begin{document}$\\mathcal {C}$\\end{document}C. Therefore, the generalized Kitaev models provide realizations of the target space of the Turaev-Viro topological quantum field theory based on \\documentclass[12pt]{minimal}\\begin{document}$\\mathcal {C}$\\end{document}C.</jats:p>","DOI":"10.1063/1.4869326","source":"Crossref","title":"Kitaev models based on unitary quantum groupoids","volume":"55","author":[{"given":"Liang","family":"Chang","sequence":"first","affiliation":[{"name":"Texas A&M University Department of Mathematics, , College Station, Texas 77843-3368, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,4,1]]},"URL":"http://dx.doi.org/10.1063/1.4869326","ISSN":["0022-2488","1089-7658"],"page":"041703","note":"arxivid:1309.4181"},{"id":"arxiv:math/0203060","type":"article-journal","author":[{"given":"Pavel","family":"Etingof"},{"given":"Dmitri","family":"Nikshych"},{"given":"Viktor","family":"Ostrik"}],"title":"On fusion categories","issued":{"date-parts":[[2017,4,28]]},"note":"arxivid:math/0203060\narxiv_version_number:11"},{"id":"arxiv:2204.05940","type":"article-journal","author":[{"given":"Andras","family":"Molnar"},{"given":"Alberto Ruiz","family":"de Alarcón"},{"given":"José","family":"Garre-Rubio"},{"given":"Norbert","family":"Schuch"},{"given":"J. Ignacio","family":"Cirac"},{"given":"David","family":"Pérez-García"}],"title":"Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states","issued":{"date-parts":[[2022,6,29]]},"note":"arxivid:2204.05940\narxiv_version_number:2"},{"id":"arxiv:2302.08131","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-023-04792-9","page":"3045-3107","source":"Crossref","title":"On Weak Hopf Symmetry and Weak Hopf Quantum Double Model","volume":"402","author":[{"ORCID":"https://orcid.org/0000-0001-8588-173X","authenticated-orcid":false,"given":"Zhian","family":"Jia","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-3318-9942","authenticated-orcid":false,"given":"Sheng","family":"Tan","sequence":"additional","affiliation":[]},{"given":"Dagomir","family":"Kaszlikowski","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Chang","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,19]]},"URL":"http://dx.doi.org/10.1007/s00220-023-04792-9","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:4792\narxivid:2302.08131"},{"id":"arxiv:2210.07909","type":"journal-article","publisher":"IOP Publishing","issue":"29","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Kitaev’s toric code is constructed using a finite gauge group from gauge theory. Such gauge theories can be extended with the gauge group generalized to any finite-dimensional semisimple Hopf algebra. This also leads to extensions of the toric code. Here we consider the simple case where the gauge group is unchanged but furnished with a non-trivial quasitriangular structure (R-matrix), which modifies the construction of the gauge theory. This leads to some interesting phenomena; for example, the space of functions on the group becomes a non-commutative algebra. We also obtain simple Hamiltonian models generalizing the toric code, which are of the same overall topological type as the toric code, except that the various species of particles created by string operators in the model are permuted in a way that depends on the R-matrix. In the case of <jats:inline-formula>\n                     <jats:tex-math/>\n                     <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                        <mml:msub>\n                           <mml:mrow>\n                              <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n                           </mml:mrow>\n                           <mml:mrow>\n                              <mml:mi>N</mml:mi>\n                           </mml:mrow>\n                        </mml:msub>\n                     </mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"aacdf9aieqn1.gif\" xlink:type=\"simple\"/>\n                  </jats:inline-formula> gauge theory, we find that the introduction of a non-trivial R-matrix amounts to flux attachment.</jats:p>","DOI":"10.1088/1751-8121/acdf9a","page":"295302","source":"Crossref","title":"Modified toric code models with flux attachment from Hopf algebra gauge theory","volume":"56","author":[{"ORCID":"https://orcid.org/0000-0002-5958-6152","authenticated-orcid":true,"given":"A","family":"Conlon","sequence":"first","affiliation":[]},{"given":"D","family":"Pellegrino","sequence":"additional","affiliation":[]},{"given":"J K","family":"Slingerland","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2023,6,30]]},"URL":"http://dx.doi.org/10.1088/1751-8121/acdf9a","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:2210.07909"},{"id":"arxiv:0907.2670","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.80.155136","source":"Crossref","title":"Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models","volume":"80","author":[{"given":"Oliver","family":"Buerschaper","sequence":"first","affiliation":[]},{"given":"Miguel","family":"Aguado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,10,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.80.155136","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"155136","note":"arxivid:0907.2670"},{"id":"arxiv:1412.6589","type":"journal-article","publisher":"IOP Publishing","issue":"12","DOI":"10.1088/1751-8113/48/12/12ft01","page":"12FT01","source":"Crossref","title":"On enriching the Levin–Wen model with symmetry","volume":"48","author":[{"given":"Liang","family":"Chang","sequence":"first","affiliation":[]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[]},{"given":"Shawn X","family":"Cui","sequence":"additional","affiliation":[]},{"given":"Yuting","family":"Hu","sequence":"additional","affiliation":[]},{"given":"Wei","family":"Jin","sequence":"additional","affiliation":[]},{"given":"Ramis","family":"Movassagh","sequence":"additional","affiliation":[]},{"given":"Pieter","family":"Naaijkens","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Amanda","family":"Young","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2015,3,2]]},"URL":"http://dx.doi.org/10.1088/1751-8113/48/12/12FT01","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1412.6589"},{"id":"arxiv:1002.2816","type":"journal-article","publisher":"Elsevier BV","issue":"12","DOI":"10.1016/j.aop.2010.08.001","page":"2707-2749","source":"Crossref","title":"Quantum computation with Turaev–Viro codes","volume":"325","author":[{"given":"Robert","family":"Koenig","sequence":"first","affiliation":[]},{"given":"Greg","family":"Kuperberg","sequence":"additional","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2010,12]]},"URL":"http://dx.doi.org/10.1016/j.aop.2010.08.001","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491610001375\narxivid:1002.2816"},{"id":"arxiv:cond-mat/0404617","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.71.045110","source":"Crossref","title":"String-net condensation: A physical mechanism for topological phases","volume":"71","author":[{"given":"Michael A.","family":"Levin","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2005,1,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.71.045110","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"045110","note":"arxivid:cond-mat/0404617"},{"id":"arxiv:2209.06202","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.4.020339","source":"Crossref","title":"Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"first","affiliation":[]},{"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":true,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,9]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.020339","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020339","note":"arxivid:2209.06202"},{"id":"arxiv:2407.21761","type":"article-journal","author":[{"given":"Sary","family":"Bseiso"},{"given":"Joel","family":"Pommerening"},{"given":"Richard R.","family":"Allen"},{"given":"Steven H.","family":"Simon"},{"given":"Layla","family":"Hormozi"}],"title":"Minimal Quantum Circuits for Simulating Fibonacci Anyons","issued":{"date-parts":[[2024,8,2]]},"note":"arxivid:2407.21761\narxiv_version_number:2"},{"id":"arxiv:1806.02358","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.125.050502","source":"Crossref","title":"Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits","volume":"125","author":[{"ORCID":"https://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"first","affiliation":[]},{"given":"Ali","family":"Lavasani","sequence":"additional","affiliation":[]},{"given":"Maissam","family":"Barkeshli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.050502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050502","note":"arxivid:1806.02358"},{"id":"arxiv:1806.06078","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.102.075105","source":"Crossref","title":"Instantaneous braids and Dehn twists in topologically ordered states","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"first","affiliation":[]},{"given":"Ali","family":"Lavasani","sequence":"additional","affiliation":[]},{"given":"Maissam","family":"Barkeshli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.102.075105","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"075105","note":"arxivid:1806.06078"},{"id":"arxiv:quant-ph/0001108","type":"article-journal","author":[{"given":"Michael","family":"Freedman"},{"given":"Michael","family":"Larsen"},{"given":"Zhenghan","family":"Wang"}],"title":"A modular functor which is universal for quantum computation","issued":{"date-parts":[[2000,2,1]]},"note":"arxivid:quant-ph/0001108\narxiv_version_number:2"},{"id":"arxiv:1310.4150","type":"journal-article","publisher":"American Physical Society (APS)","issue":"14","DOI":"10.1103/physrevlett.112.140504","source":"Crossref","title":"Asymptotically Optimal Topological Quantum Compiling","volume":"112","author":[{"given":"Vadym","family":"Kliuchnikov","sequence":"first","affiliation":[]},{"given":"Alex","family":"Bocharov","sequence":"additional","affiliation":[]},{"given":"Krysta M.","family":"Svore","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,4,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.112.140504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"140504","note":"arxivid:1310.4150"},{"id":"arxiv:2008.10790","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.2.010334","source":"Crossref","title":"Fibonacci Anyons Versus Majorana Fermions: A Monte Carlo Approach to the Compilation of Braid Circuits in \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><mml:mi>SU</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>k</mml:mi></mml:msub></mml:math>\n Anyon Models","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0003-4064-3177","authenticated-orcid":true,"given":"Emil","family":"Génetay Johansen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0730-9126","authenticated-orcid":true,"given":"Tapio","family":"Simula","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,3,1]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.010334","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010334","note":"update-to:{\"updated\":{\"date-parts\":[[2021,8,6]],\"date-time\":\"2021-08-06T00:00:00Z\",\"timestamp\":1628208000000},\"DOI\":\"10.1103/prxquantum.2.010334\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2021,8,6]],\"date-time\":\"2021-08-06T00:00:00Z\",\"timestamp\":1628208000000},\"DOI\":\"10.1103/prxquantum.2.010334\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2008.10790"},{"id":"arxiv:1506.03815","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.95.022309","source":"Crossref","title":"Classical simulation of quantum error correction in a Fibonacci anyon code","volume":"95","author":[{"given":"Simon","family":"Burton","sequence":"first","affiliation":[]},{"given":"Courtney G.","family":"Brell","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,2,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.95.022309","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022309","note":"arxivid:1506.03815"},{"id":"arxiv:1607.02159","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-017-2923-9","page":"519-560","source":"Crossref","title":"Fault-Tolerant Quantum Error Correction for non-Abelian Anyons","volume":"355","author":[{"ORCID":"https://orcid.org/0000-0002-6909-5891","authenticated-orcid":false,"given":"Guillaume","family":"Dauphinais","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,7,10]]},"URL":"http://dx.doi.org/10.1007/s00220-017-2923-9","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2923\narxivid:1607.02159"},{"id":"arxiv:2012.04610","type":"article-journal","author":[{"given":"Alexis","family":"Schotte"},{"given":"Guanyu","family":"Zhu"},{"given":"Lander","family":"Burgelman"},{"given":"Frank","family":"Verstraete"}],"title":"Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code","issued":{"date-parts":[[2021,4,9]]},"note":"arxivid:2012.04610\narxiv_version_number:2"},{"id":"arxiv:2301.00054","type":"article-journal","author":[{"given":"Alexis","family":"Schotte"},{"given":"Lander","family":"Burgelman"},{"given":"Guanyu","family":"Zhu"}],"title":"Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature","issued":{"date-parts":[[2022,12,30]]},"note":"arxivid:2301.00054\narxiv_version_number:1"},{"id":"arxiv:1711.05752","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevresearch.2.013285","source":"Crossref","title":"Quantum origami: Transversal gates for quantum computation and measurement of topological order","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1679-4880","authenticated-orcid":true,"given":"Mohammad","family":"Hafezi","sequence":"additional","affiliation":[]},{"given":"Maissam","family":"Barkeshli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.013285","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013285","note":"arxivid:1711.05752"},{"id":"arxiv:2210.12145","type":"article-journal","author":[{"given":"Yu-ang","family":"Fan"},{"given":"Yingcheng","family":"Li"},{"given":"Yuting","family":"Hu"},{"given":"Yishan","family":"Li"},{"given":"Xinyue","family":"Long"},{"given":"Hongfeng","family":"Liu"},{"given":"Xiaodong","family":"Yang"},{"given":"Xinfang","family":"Nie"},{"given":"Jun","family":"Li"},{"given":"Tao","family":"Xin"},{"given":"Dawei","family":"Lu"},{"given":"Yidun","family":"Wan"}],"title":"Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons","issued":{"date-parts":[[2022,10,21]]},"note":"arxivid:2210.12145\narxiv_version_number:1"},{"id":"arxiv:2404.00091","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"9","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.</jats:p>","DOI":"10.1038/s41567-024-02529-6","page":"1469-1475","source":"Crossref","title":"Non-Abelian braiding of Fibonacci anyons with a superconducting processor","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0002-8546-4255","authenticated-orcid":false,"given":"Shibo","family":"Xu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7370-4525","authenticated-orcid":false,"given":"Zheng-Zhi","family":"Sun","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0307-5457","authenticated-orcid":false,"given":"Ke","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3914-4979","authenticated-orcid":false,"given":"Hekang","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8219-2311","authenticated-orcid":false,"given":"Zitian","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1852-4094","authenticated-orcid":false,"given":"Hang","family":"Dong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4076-1608","authenticated-orcid":false,"given":"Jinfeng","family":"Deng","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6102-0035","authenticated-orcid":false,"given":"Xu","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0006-6444-2416","authenticated-orcid":false,"given":"Jiachen","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1137-4660","authenticated-orcid":false,"given":"Yaozu","family":"Wu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4710-1752","authenticated-orcid":false,"given":"Chuanyu","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0789-8207","authenticated-orcid":false,"given":"Feitong","family":"Jin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1318-8367","authenticated-orcid":false,"given":"Xuhao","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4138-8983","authenticated-orcid":false,"given":"Yu","family":"Gao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0002-0599-5361","authenticated-orcid":false,"given":"Aosai","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-3884-9997","authenticated-orcid":false,"given":"Ning","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-2838-9552","authenticated-orcid":false,"given":"Yiren","family":"Zou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0001-8858-2193","authenticated-orcid":false,"given":"Ziqi","family":"Tan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0003-5289-419X","authenticated-orcid":false,"given":"Fanhao","family":"Shen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0001-3631-6774","authenticated-orcid":false,"given":"Jiarun","family":"Zhong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0007-9164-714X","authenticated-orcid":false,"given":"Zehang","family":"Bao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7137-5390","authenticated-orcid":false,"given":"Weikang","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6537-8712","authenticated-orcid":false,"given":"Wenjie","family":"Jiang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5849-4029","authenticated-orcid":false,"given":"Li-Wei","family":"Yu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1089-9251","authenticated-orcid":false,"given":"Zixuan","family":"Song","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8908-4228","authenticated-orcid":false,"given":"Pengfei","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8539-9510","authenticated-orcid":false,"given":"Liang","family":"Xiang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1093-3405","authenticated-orcid":false,"given":"Qiujiang","family":"Guo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6369-8010","authenticated-orcid":false,"given":"Zhen","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1853-9392","authenticated-orcid":false,"given":"Chao","family":"Song","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5380-4297","authenticated-orcid":false,"given":"H.","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1042-4646","authenticated-orcid":false,"given":"Dong-Ling","family":"Deng","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,1]]},"URL":"http://dx.doi.org/10.1038/s41567-024-02529-6","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2529\narxivid:2404.00091"},{"id":"arxiv:2406.12820","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n          <jats:p>The remarkable complexity of a topologically ordered many-body quantum system is encoded in the characteristics of its anyons. Quintessential predictions emanating from this complexity employ the Fibonacci string net condensate (Fib SNC) and its anyons: sampling Fib-SNC would estimate chromatic polynomials while exchanging its anyons would implement universal quantum computation. However, physical realizations remained elusive. We introduce a scalable dynamical string net preparation (DSNP) that constructs Fib SNC and its anyons on reconfigurable graphs suitable for near-term superconducting processors. Coupling the DSNP approach with composite error-mitigation on deep circuits, we create, measure, and braids Fibonacci anyons; charge measurements show 94% accuracy, and exchanging the anyons yields the expected golden ratio <jats:italic>ϕ</jats:italic> with 98% average accuracy. We then sample the Fib SNC to estimate chromatic polynomial at <jats:italic>ϕ</jats:italic> + 2 for several graphs. Our results establish the proof of principle for using Fib-SNC and its anyons for fault-tolerant universal quantum computation and aim at a classically hard problem.</jats:p>","DOI":"10.1038/s41467-025-61493-8","source":"Crossref","title":"Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0003-3759-1847","authenticated-orcid":false,"given":"Zlatko K.","family":"Minev","sequence":"first","affiliation":[]},{"given":"Khadijeh","family":"Najafi","sequence":"additional","affiliation":[]},{"given":"Swarnadeep","family":"Majumder","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9396-9010","authenticated-orcid":false,"given":"Juven","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9493-268X","authenticated-orcid":false,"given":"Ady","family":"Stern","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9554-4443","authenticated-orcid":false,"given":"Eun-Ah","family":"Kim","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4876-8417","authenticated-orcid":false,"given":"Chao-Ming","family":"Jian","sequence":"additional","affiliation":[]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,6]]},"URL":"http://dx.doi.org/10.1038/s41467-025-61493-8","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"6225","note":"alternative-id:61493\narxivid:2406.12820"},{"id":"doi:10.1112/blms/19.2.113","type":"journal-article","publisher":"Wiley","issue":"2","DOI":"10.1112/blms/19.2.113","page":"113-134","source":"Crossref","title":"From Groups to Groupoids: a Brief Survey","volume":"19","author":[{"given":"Ronald","family":"Brown","sequence":"first","affiliation":[]}],"container-title":"Bulletin of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1987,3]]},"URL":"http://dx.doi.org/10.1112/blms/19.2.113","ISSN":["0024-6093"],"container-title-short":"Bulletin of the London Mathematical Society"},{"id":"arxiv:2212.01021","type":"article-journal","author":[{"given":"Pramod","family":"Padmanabhan"},{"given":"Indrajit","family":"Jana"}],"title":"Groupoid Toric Codes","issued":{"date-parts":[[2022,12,2]]},"note":"arxivid:2212.01021\narxiv_version_number:1"},{"id":"arxiv:2307.12552","type":"article-journal","author":[{"given":"Corey","family":"Jones"},{"given":"Pieter","family":"Naaijkens"},{"given":"David","family":"Penneys"},{"given":"Daniel","family":"Wallick"}],"title":"Local topological order and boundary algebras","issued":{"date-parts":[[2025,2,12]]},"note":"arxivid:2307.12552\narxiv_version_number:2"},{"id":"arxiv:1106.6033","type":"article-journal","author":[{"given":"Alexander","family":"Kirillov"}],"title":"String-net model of Turaev-Viro invariants","issued":{"date-parts":[[2011,6,29]]},"note":"arxivid:1106.6033\narxiv_version_number:1"},{"id":"arxiv:2008.11187","type":"journal-article","publisher":"Stichting SciPost","issue":"3","abstract":"<jats:p>We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, we use this generalisation to build explicit PEPS realisations of domain walls between different topological phases as constructed by Kitaev and Kong [Commun. Math. Phys. 313 (2012) 351-373]. While the prevailing abstract categorical approach is sufficient to describe the structure of topological phases, explicit tensor network representations are required to simulate these systems on a computer, such as needed for calculating thresholds of quantum error-correcting codes based on string-nets with boundaries. Finally, we show that all these string-net PEPS representations can be understood as specific instances of Turaev-Viro state-sum models of topological field theory on three-manifolds with a physical boundary, thereby putting these tensor network constructions on a mathematically rigorous footing.</jats:p>","DOI":"10.21468/scipostphys.10.3.053","source":"Crossref","title":"Matrix product operator symmetries and intertwiners in string-nets with  domain walls","volume":"10","author":[{"given":"Laurens","family":"Lootens","sequence":"first","affiliation":[{"name":"Ghent University"}]},{"given":"Jürgen","family":"Fuchs","sequence":"additional","affiliation":[{"name":"Karlstad University"}]},{"given":"Jutho","family":"Haegeman","sequence":"additional","affiliation":[{"name":"Ghent University"}]},{"given":"Christoph","family":"Schweigert","sequence":"additional","affiliation":[{"name":"University of Hamburg"}]},{"given":"Frank","family":"Verstraete","sequence":"additional","affiliation":[{"name":"Ghent University"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2021,3,1]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.10.3.053","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"053","note":"arxivid:2008.11187"},{"id":"arxiv:2112.12757","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevb.105.085130","source":"Crossref","title":"Mapping between Morita-equivalent string-net states with a constant depth quantum circuit","volume":"105","author":[{"ORCID":"https://orcid.org/0000-0002-1364-4863","authenticated-orcid":true,"given":"Laurens","family":"Lootens","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0080-2030","authenticated-orcid":true,"given":"Bram","family":"Vancraeynest-De Cuiper","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6494-8616","authenticated-orcid":true,"given":"Norbert","family":"Schuch","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Verstraete","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.105.085130","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"085130","note":"arxivid:2112.12757"},{"id":"arxiv:2303.04711","type":"article-journal","author":[{"given":"Jessica","family":"Christian"},{"given":"David","family":"Green"},{"given":"Peter","family":"Huston"},{"given":"David","family":"Penneys"}],"title":"A lattice model for condensation in Levin-Wen systems","issued":{"date-parts":[[2023,3,14]]},"note":"arxivid:2303.04711\narxiv_version_number:2"},{"id":"arxiv:0907.2204","type":"article-journal","author":[{"given":"Seung-Moon","family":"Hong"}],"title":"On symmetrization of 6j-symbols and Levin-Wen Hamiltonian","issued":{"date-parts":[[2009,7,13]]},"note":"arxivid:0907.2204\narxiv_version_number:1"},{"id":"arxiv:2004.07045","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevb.102.115154","source":"Crossref","title":"Generalized string-net model for unitary fusion categories without tetrahedral symmetry","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0002-4152-9854","authenticated-orcid":true,"given":"Alexander","family":"Hahn","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9404-5781","authenticated-orcid":true,"given":"Ramona","family":"Wolf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.102.115154","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"115154","note":"arxivid:2004.07045"},{"id":"arxiv:2004.11982","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2020.168318","page":"168318","source":"Crossref","title":"Ground subspaces of topological phases of matter as error correcting codes","volume":"422","author":[{"given":"Yang","family":"Qiu","sequence":"first","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,11]]},"URL":"http://dx.doi.org/10.1016/j.aop.2020.168318","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491620302529\narxivid:2004.11982"},{"id":"arxiv:2110.02020","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/prxquantum.3.040315","source":"Crossref","title":"Methods for Simulating String-Net States and Anyons on a Digital Quantum Computer","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-7657-9464","authenticated-orcid":true,"given":"Yu-Jie","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3857-3141","authenticated-orcid":true,"given":"Kirill","family":"Shtengel","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6744-4592","authenticated-orcid":true,"given":"Adam","family":"Smith","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Pollmann","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,11,7]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.040315","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040315","note":"arxivid:2110.02020"},{"id":"arxiv:0712.0348","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.100.070404","source":"Crossref","title":"Entanglement Renormalization and Topological Order","volume":"100","author":[{"given":"Miguel","family":"Aguado","sequence":"first","affiliation":[]},{"given":"Guifré","family":"Vidal","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,2,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.100.070404","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070404","note":"arxivid:0712.0348"},{"id":"arxiv:0806.4583","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevb.79.195123","source":"Crossref","title":"Exact entanglement renormalization for string-net models","volume":"79","author":[{"given":"Robert","family":"König","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]},{"given":"Guifré","family":"Vidal","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,5,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.79.195123","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"195123","note":"arxivid:0806.4583"},{"id":"arxiv:0809.3031","type":"article-journal","author":[{"given":"Pavel","family":"Etingof"},{"given":"Dmitri","family":"Nikshych"},{"given":"Victor","family":"Ostrik"}],"title":"Weakly group-theoretical and solvable fusion categories","issued":{"date-parts":[[2009,7,22]]},"note":"arxivid:0809.3031\narxiv_version_number:2"},{"id":"arxiv:2411.04985","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>The classification of topological phases of matter is a fundamental challenge in quantum many-body physics, with applications to quantum technology. Recently, this classification has been extended to the setting of adaptive finite-depth local unitary (AFDLU) circuits, which allow global classical communication. In this setting, the trivial phase is the collection of all topological states that can be prepared via AFDLU. Here, we propose a complete classification of the trivial phase by showing how to prepare all solvable anyon theories that admit a gapped boundary via AFDLU, extending recent results on solvable groups. Our construction includes non-Abelian anyons with irrational quantum dimensions, such as Ising anyons, and more general acyclic anyons. Specifically, we introduce a sequential gauging procedure, with an AFDLU implementation, to produce a string-net ground state in any topological phase described by a solvable anyon theory with gapped boundary. In addition, we introduce a sequential ungauging and regauging procedure, with an AFDLU implementation, to apply string operators of arbitrary length for anyons and symmetry twist defects in solvable anyon theories. We apply our procedure to the quantum double of the group <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:msub><a:mi>S</a:mi><a:mn>3</a:mn></a:msub></a:math> and to several examples that are beyond solvable groups, including the doubled Ising theory, the <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:msub><c:mi mathvariant=\"double-struck\">Z</c:mi><c:mn>3</c:mn></c:msub></c:math> Tambara-Yamagami string net, and doubled <f:math xmlns:f=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><f:mi>SU</f:mi><f:mo stretchy=\"false\">(</f:mo><f:mn>2</f:mn><f:msub><f:mo stretchy=\"false\">)</f:mo><f:mn>4</f:mn></f:msub></f:math> anyons.</jats:p>","DOI":"10.1103/b9hf-gx4f","source":"Crossref","title":"Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits","volume":"15","author":[{"given":"Yuanjie","family":"Ren","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":true,"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,29]]},"URL":"http://dx.doi.org/10.1103/b9hf-gx4f","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031060","note":"arxivid:2411.04985"},{"id":"arxiv:1901.11029","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>A basic question in the theory of fault-tolerant quantum computation is to understand the fundamental resource costs for performing a universal logical set of gates on encoded qubits to arbitrary accuracy. Here we consider qubits encoded with constant space overhead (i.e. finite encoding rate) in the limit of arbitrarily large code distance<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>through the use of topological codes associated to triangulations of hyperbolic surfaces. We introduce explicit protocols to demonstrate how Dehn twists of the hyperbolic surface can be implemented on the code through constant depth unitary circuits, without increasing the space overhead. The circuit for a given Dehn twist consists of a permutation of physical qubits, followed by a constant depth local unitary circuit, where locality here is defined with respect to a hyperbolic metric that defines the code. Applying our results to the hyperbolic Fibonacci Turaev-Viro code implies the possibility of applying universal logical gate sets on encoded qubits through constant depth unitary circuits and with constant space overhead. Our circuits are inherently protected from errors as they map local operators to local operators while changing the size of their support by at most a constant factor; in the presence of noisy syndrome measurements, our results suggest the possibility of universal fault tolerant quantum computation with constant space overhead and time overhead of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. For quantum circuits that allow parallel gate operations, this yields the optimal scaling of space-time overhead known to date.</jats:p>","DOI":"10.22331/q-2019-08-26-180","page":"180","source":"Crossref","title":"Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes","volume":"3","author":[{"given":"Ali","family":"Lavasani","sequence":"first","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Maissam","family":"Barkeshli","sequence":"additional","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,26]]},"URL":"http://dx.doi.org/10.22331/q-2019-08-26-180","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1901.11029"},{"id":"arxiv:math/0103200","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s002200200636","page":"177-199","source":"Crossref","title":"The Two-Eigenvalue Problem and Density¶of Jones Representation of Braid Groups","volume":"228","author":[{"given":"Michael H.","family":"Freedman","sequence":"first","affiliation":[]},{"given":"Michael J.","family":"Larsen","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"issued":{"date-parts":[[2002,6,1]]},"URL":"http://dx.doi.org/10.1007/s002200200636","ISSN":["0010-3616","1432-0916"],"container-title-short":"Communications in Mathematical Physics","note":"alternative-id:64HBE7H8UBXHQNTQ\narxivid:math/0103200"},{"id":"arxiv:1206.6048","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevb.86.165113","source":"Crossref","title":"Quantum circuits for measuring Levin-Wen operators","volume":"86","author":[{"given":"N. E.","family":"Bonesteel","sequence":"first","affiliation":[]},{"given":"D. P.","family":"DiVincenzo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2012,10,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.86.165113","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"165113","note":"arxivid:1206.6048"},{"id":"arxiv:2405.17379","type":"article-journal","author":[{"given":"Isaac H.","family":"Kim"},{"given":"Daniel","family":"Ranard"}],"title":"Classifying 2D topological phases: mapping ground states to string-nets","issued":{"date-parts":[[2024,5,27]]},"note":"arxivid:2405.17379\narxiv_version_number:1"},{"id":"arxiv:2408.02664","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"11","abstract":"<jats:title>\n                    A\n                    <jats:sc>bstract</jats:sc>\n                  </jats:title>\n                  <jats:p>We develop a systematic framework for understanding symmetries in topological phases in 2 + 1 dimensions using the string-net model, encompassing both gauge invariances that preserve anyon types and global symmetries permuting anyon types, including both invertible symmetries describable by groups and noninvertible symmetries described by categories. As an archetypal example, we reveal the first noninvertible categorical gauge invariance of topological orders in 2 + 1 dimensions: the Fibonacci gauge invariance of the doubled Fibonacci topological order, described by the Fibonacci fusion 2-category. Our approach involves two steps: first, classifying and establishing dualities between different string-net models describing the same topological order; and second, constructing symmetry transformations within the same string-net model when the dual models have isomorphic input data, achieved by composing duality maps with isomorphisms of degrees of freedom between the dual models.</jats:p>","DOI":"10.1007/jhep11(2025)138","source":"Crossref","title":"Noninvertible gauge invariance in (2+1)d topological orders: a string-net model realization","volume":"2025","author":[{"ORCID":"https://orcid.org/0009-0005-4673-2383","authenticated-orcid":false,"given":"Yu","family":"Zhao","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/013q1eq08","id-type":"ROR","asserted-by":"publisher"}],"name":"Fudan University"}]},{"ORCID":"https://orcid.org/0000-0003-4318-8464","authenticated-orcid":false,"given":"Yidun","family":"Wan","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013q1eq08","id-type":"ROR","asserted-by":"publisher"}],"name":"Fudan University"},{"id":[{"id":"https://ror.org/034t30j35","id-type":"ROR","asserted-by":"publisher"}],"name":"Shanghai Research Center for Quantum Sciences"},{"id":[{"id":"https://ror.org/04c4dkn09","id-type":"ROR","asserted-by":"publisher"}],"name":"Hefei National Laboratory"}]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,24]]},"URL":"http://dx.doi.org/10.1007/JHEP11(2025)138","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"138","note":"alternative-id:27668\narxivid:2408.02664"},{"id":"arxiv:1211.3695","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevb.87.125114","source":"Crossref","title":"Twisted quantum double model of topological phases in two dimensions","volume":"87","author":[{"given":"Yuting","family":"Hu","sequence":"first","affiliation":[]},{"given":"Yidun","family":"Wan","sequence":"additional","affiliation":[]},{"given":"Yong-Shi","family":"Wu","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2013,3,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.87.125114","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"125114","note":"arxivid:1211.3695"},{"id":"arxiv:1701.00747","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.95.245126","source":"Crossref","title":"Fracton topological order via coupled layers","volume":"95","author":[{"given":"Han","family":"Ma","sequence":"first","affiliation":[]},{"given":"Ethan","family":"Lake","sequence":"additional","affiliation":[]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Hermele","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,6,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.95.245126","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"245126","note":"arxivid:1701.00747"},{"id":"arxiv:1806.08625","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2019.167922","page":"167922","source":"Crossref","title":"Fractional excitations in foliated fracton phases","volume":"410","author":[{"given":"Wilbur","family":"Shirley","sequence":"first","affiliation":[]},{"given":"Kevin","family":"Slagle","sequence":"additional","affiliation":[]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,11]]},"URL":"http://dx.doi.org/10.1016/j.aop.2019.167922","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491619301770\narxivid:1806.08625"},{"id":"arxiv:1904.01111","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevb.100.085127","source":"Crossref","title":"Foliated fracton order in the Majorana checkerboard model","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0001-7227-915X","authenticated-orcid":true,"given":"Taige","family":"Wang","sequence":"first","affiliation":[]},{"given":"Wilbur","family":"Shirley","sequence":"additional","affiliation":[]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.100.085127","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"085127","note":"arxivid:1904.01111"},{"id":"arxiv:1903.11625","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevb.100.195136","source":"Crossref","title":"Fracton fusion and statistics","volume":"100","author":[{"given":"Shriya","family":"Pai","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2594-1176","authenticated-orcid":true,"given":"Michael","family":"Hermele","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,11,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.100.195136","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"195136","note":"arxivid:1903.11625"},{"id":"arxiv:1410.4540","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevb.100.115147","source":"Crossref","title":"Symmetry fractionalization, defects, and gauging of topological phases","volume":"100","author":[{"given":"Maissam","family":"Barkeshli","sequence":"first","affiliation":[]},{"given":"Parsa","family":"Bonderson","sequence":"additional","affiliation":[]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,9,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.100.115147","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"115147","note":"arxivid:1410.4540"},{"id":"arxiv:1610.07628","type":"journal-article","publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"4","abstract":"<jats:p>\n            We construct a state-sum type invariant of smooth closed oriented 4-manifolds out of a \n            <jats:inline-formula>\n              <jats:tex-math>G</jats:tex-math>\n            </jats:inline-formula>\n            -crossed braided spherical fusion category (\n            <jats:inline-formula>\n              <jats:tex-math>G</jats:tex-math>\n            </jats:inline-formula>\n            -BSFC) for \n            <jats:inline-formula>\n              <jats:tex-math>G</jats:tex-math>\n            </jats:inline-formula>\n             a finite group. The construction can be extended to obtain a \n            <jats:inline-formula>\n              <jats:tex-math>(3+1)</jats:tex-math>\n            </jats:inline-formula>\n            -dimensional topological quantum field theory (TQFT). The invariant of 4-manifolds generalizes several known invariants in literature such as the Crane–Yetter invariant from a ribbon fusion category and Yetter's invariant from homotopy 2-types. If the \n            <jats:inline-formula>\n              <jats:tex-math>G</jats:tex-math>\n            </jats:inline-formula>\n            -BSFC is concentrated only at the sector indexed by the trivial group element, a cohomology class in \n            <jats:inline-formula>\n              <jats:tex-math>H^4(G,U(1))</jats:tex-math>\n            </jats:inline-formula>\n             can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf–Witten theory in a special case. Although not proven, it is believed that our invariants are strictly different from other known invariants. It remains to be seen if the invariants are sensitive to smooth structures. It is expected that the most general input to the state-sum type construction of \n            <jats:inline-formula>\n              <jats:tex-math>(3+1)</jats:tex-math>\n            </jats:inline-formula>\n            -TQFTs is a spherical fusion 2-category. We show that a \n            <jats:inline-formula>\n              <jats:tex-math>G</jats:tex-math>\n            </jats:inline-formula>\n            -BSFC corresponds to a monoidal 2-category with certain extra structure, but that structure does not satisfy all the axioms of a spherical fusion 2-category given by M. Mackaay. Thus the question of what axioms properly define a spherical fusion 2-category is open.\n          </jats:p>","DOI":"10.4171/qt/128","page":"593-676","source":"Crossref","title":"Four dimensional topological quantum field theories from $G$-crossed braided categories","volume":"10","author":[{"given":"Shawn X.","family":"Cui","sequence":"first","affiliation":[{"name":"Stanford University, USA"}]}],"container-title":"Quantum Topology","original-title":[],"issued":{"date-parts":[[2019,9,6]]},"URL":"http://dx.doi.org/10.4171/QT/128","ISSN":["1663-487X","1664-073X"],"container-title-short":"Quantum Topol.","note":"arxivid:1610.07628"},{"id":"arxiv:1405.5763","type":"article-journal","author":[{"given":"Rinat","family":"Kashaev"}],"title":"A simple model of 4d-TQFT","issued":{"date-parts":[[2014,5,22]]},"note":"arxivid:1405.5763\narxiv_version_number:1"},{"id":"arxiv:1504.01979","type":"article-journal","author":[{"given":"Rinat","family":"Kashaev"}],"title":"On realizations of Pachner moves in 4D","issued":{"date-parts":[[2015,4,8]]},"note":"arxivid:1504.01979\narxiv_version_number:1"},{"id":"arxiv:2003.11553","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevresearch.2.043033","source":"Crossref","title":"Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0001-8978-4531","authenticated-orcid":true,"given":"Daniel","family":"Bulmash","sequence":"first","affiliation":[]},{"given":"Maissam","family":"Barkeshli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,10,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.043033","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"043033","note":"arxivid:2003.11553"},{"id":"arxiv:1104.2632","type":"article-journal","author":[{"given":"Kevin","family":"Walker"},{"given":"Zhenghan","family":"Wang"}],"title":"(3+1)-TQFTs and Topological Insulators","issued":{"date-parts":[[2011,4,28]]},"note":"arxivid:1104.2632\narxiv_version_number:2"},{"id":"arxiv:hep-th/9301062","type":"article-journal","author":[{"given":"Louis","family":"Crane"},{"given":"David N.","family":"Yetter"}],"title":"A categorical construction of 4D TQFTs","issued":{"date-parts":[[1993,1,15]]},"note":"arxivid:hep-th/9301062\narxiv_version_number:1"},{"id":"arxiv:hep-th/9309063","type":"article-journal","author":[{"given":"Louis","family":"Crane"},{"given":"Louis H.","family":"Kauffman"},{"given":"David N.","family":"Yetter"}],"title":"Evaluating the Crane-Yetter Invariant","issued":{"date-parts":[[1993,9,10]]},"note":"arxivid:hep-th/9309063\narxiv_version_number:1"},{"id":"arxiv:hep-th/9409167","type":"article-journal","author":[{"given":"Louis","family":"Crane"},{"given":"Louis H.","family":"Kauffman"},{"given":"David N.","family":"Yetter"}],"title":"State-Sum Invariants of 4-Manifolds I","issued":{"date-parts":[[1994,9,27]]},"note":"arxivid:hep-th/9409167\narxiv_version_number:1"},{"id":"arxiv:2011.04693","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>We present a scheme for universal topological quantum computation based on Clifford-complete braiding and fusion of symmetry defects in the 3-fermion anyon theory, supplemented with magic state injection. We formulate a fault-tolerant measurement-based realization of this computational scheme on the lattice using ground states of the Walker-Wang model for the 3-fermion anyon theory with symmetry defects. The Walker-Wang measurement-based topological quantum computation paradigm that we introduce provides a general construction of computational resource states with thermally stable symmetry-protected topological order. We also demonstrate how symmetry defects of the 3-fermion anyon theory can be realized in a two-dimensional subsystem code due to Bombín—making contact with an alternative implementation of the 3-fermion defect-computation scheme via code deformations.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.010315","source":"Crossref","title":"3-Fermion Topological Quantum Computation","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-4652-389X","authenticated-orcid":true,"given":"Sam","family":"Roberts","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia"}]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,2,2]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.010315","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010315","note":"arxivid:2011.04693"},{"id":"arxiv:1208.5128","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.87.045107","source":"Crossref","title":"Three-dimensional topological lattice models with surface anyons","volume":"87","author":[{"given":"C. W.","family":"von Keyserlingk","sequence":"first","affiliation":[]},{"given":"F. J.","family":"Burnell","sequence":"additional","affiliation":[]},{"given":"S. H.","family":"Simon","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2013,1,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.87.045107","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"045107","note":"arxivid:1208.5128"},{"id":"arxiv:1812.01625","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s00220-022-04528-1","page":"469-540","source":"Crossref","title":"Nontrivial Quantum Cellular Automata in Higher Dimensions","volume":"398","author":[{"ORCID":"https://orcid.org/0000-0002-1087-6853","authenticated-orcid":false,"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]},{"given":"Lukasz","family":"Fidkowski","sequence":"additional","affiliation":[]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,11,10]]},"URL":"http://dx.doi.org/10.1007/s00220-022-04528-1","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:4528\narxivid:1812.01625"},{"id":"arxiv:2205.09141","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/s00220-025-05239-z","source":"Crossref","title":"Topological Phases of Unitary Dynamics: Classification in Clifford Category","volume":"406","author":[{"ORCID":"https://orcid.org/0000-0002-1087-6853","authenticated-orcid":false,"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,5]]},"URL":"http://dx.doi.org/10.1007/s00220-025-05239-z","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","page":"76","note":"alternative-id:5239\narxivid:2205.09141"},{"id":"arxiv:2208.03397","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevb.107.085134","source":"Crossref","title":"Disentangling modular Walker-Wang models via fermionic invertible boundaries","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0002-4881-2003","authenticated-orcid":true,"given":"Andreas","family":"Bauer","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2023,2,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.107.085134","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"085134","note":"arxivid:2208.03397"},{"id":"arxiv:1302.7072","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.90.245122","source":"Crossref","title":"Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order","volume":"90","author":[{"given":"F. J.","family":"Burnell","sequence":"first","affiliation":[]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Lukasz","family":"Fidkowski","sequence":"additional","affiliation":[]},{"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.90.245122","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"245122","note":"arxivid:1302.7072"},{"id":"arxiv:1907.02075","type":"journal-article","publisher":"AIP Publishing","issue":"9","abstract":"<jats:p>We study locality preserving automorphisms of operator algebras on D-dimensional uniform lattices of prime p-dimensional qudits quantum cellular automata (QCAs), specializing in those that are translation invariant (TI), and map every prime p-dimensional Pauli matrix to a tensor product of Pauli matrices (Clifford). We associate antihermitian forms of the unit determinant over Laurent polynomial rings to TI Clifford QCA with lattice boundaries and prove that the form determines the QCA up to Clifford circuits and shifts (trivial). It follows that every 2D TI Clifford QCA is trivial since the antihermitian form in this case is always trivial. Furthermore, we prove that for any D, the fourth power of any TI Clifford QCA is trivial. We present explicit examples of nontrivial TI Clifford QCA for D = 3 and any odd prime p and show that the Witt group of the finite field Fp is a subgroup of the group C(D=3,p) of all TI Clifford QCA modulo trivial ones. That is, C(D=3,p≡1mod4)⊇Z2×Z2 and C(D=3,p≡3mod4)⊇Z4. The examples are found by disentangling the ground state of a commuting Pauli Hamiltonian, which is constructed by coupling layers of prime dimensional toric codes such that an exposed surface has an anomalous topological order that is not realizable by commuting Pauli Hamiltonians strictly in two dimensions. In an appendix independent of the main body of this paper, we revisit a recent theorem of Freedman and Hastings that any two-dimensional QCA, which is not necessarily Clifford or translation invariant, is a constant depth quantum circuit followed by a shift. We give a more direct proof of the theorem without using any ancillas.</jats:p>","DOI":"10.1063/5.0022185","source":"Crossref","title":"Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D","volume":"62","author":[{"ORCID":"https://orcid.org/0000-0002-1087-6853","authenticated-orcid":false,"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft Quantum and Microsoft Research , Redmond, Washington 98052, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,9,1]]},"URL":"http://dx.doi.org/10.1063/5.0022185","ISSN":["0022-2488","1089-7658"],"page":"092202","note":"arxivid:1907.02075"},{"id":"arxiv:2202.05442","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.3.030326","source":"Crossref","title":"Three-Dimensional Quantum Cellular Automata from Chiral Semion Surface Topological Order and beyond","volume":"3","author":[{"given":"Wilbur","family":"Shirley","sequence":"first","affiliation":[]},{"given":"Yu-An","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1740-6889","authenticated-orcid":true,"given":"Tyler D.","family":"Ellison","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,23]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.030326","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030326","note":"arxivid:2202.05442"},{"id":"arxiv:2211.03798","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories–this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msubsup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mn>4</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:math> anyon theory with degenerate braiding relations and the chiral semion theory–both of which cannot be captured by topological stabilizer codes. The construction proceeds by \"gauging out\" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.</jats:p>","DOI":"10.22331/q-2023-10-12-1137","page":"1137","source":"Crossref","title":"Pauli topological subsystem codes from Abelian anyon theories","volume":"7","author":[{"given":"Tyler D.","family":"Ellison","sequence":"first","affiliation":[{"name":"Department of Physics, Yale University, New Haven, CT 06511, USA"}]},{"given":"Yu-An","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Physics, Condensed Matter Theory Center, Joint Quantum Institute, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA"}]},{"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA"}]},{"given":"Wilbur","family":"Shirley","sequence":"additional","affiliation":[{"name":"School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA"}]},{"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA"},{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA"}]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,12]]},"URL":"http://dx.doi.org/10.22331/q-2023-10-12-1137","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2211.03798"},{"id":"arxiv:0708.2142","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.76.062313","source":"Crossref","title":"General entanglement-assisted quantum error-correcting codes","volume":"76","author":[{"given":"Min-Hsiu","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"Igor","family":"Devetak","sequence":"additional","affiliation":[]},{"given":"Todd","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,12,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.062313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062313","note":"arxivid:0708.2142"},{"id":"doi:10.1109/ISIT.2007.4557160","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2007.4557160","page":"2101-2105","source":"Crossref","title":"General entanglement-assisted quantum error-correcting codes","author":[{"given":"Todd A.","family":"Brun","sequence":"first","affiliation":[]},{"given":"Igor","family":"Devetak","sequence":"additional","affiliation":[]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[]}],"event":"2007 IEEE International Symposium on Information Theory","container-title":"2007 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2007,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2007.4557160"},{"id":"arxiv:quant-ph/0205117","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.66.052313","source":"Crossref","title":"Entanglement required in achieving entanglement-assisted channel capacities","volume":"66","author":[{"given":"Garry","family":"Bowen","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,11,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.66.052313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052313","note":"arxivid:quant-ph/0205117"},{"id":"arxiv:0904.1175","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2010.5513540","source":"Crossref","title":"Entanglement generation with a quantum channel and a shared state","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[]}],"event":"2010 IEEE International Symposium on Information Theory - ISIT","container-title":"2010 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2010,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2010.5513540","note":"arxivid:0904.1175"},{"id":"arxiv:quant-ph/0106052","type":"article-journal","author":[{"given":"Charles H.","family":"Bennett"},{"given":"Peter W.","family":"Shor"},{"given":"John A.","family":"Smolin"},{"given":"Ashish V.","family":"Thapliyal"}],"title":"Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem","issued":{"date-parts":[[2002,5,14]]},"note":"arxivid:quant-ph/0106052\narxiv_version_number:2"},{"id":"arxiv:2210.02939","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2024.3354319","page":"2655-2673","source":"Crossref","title":"Fault-Tolerant Coding for Entanglement-Assisted Communication","volume":"70","author":[{"ORCID":"https://orcid.org/0000-0003-0834-613X","authenticated-orcid":false,"given":"Paula","family":"Belzig","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark"}]},{"ORCID":"https://orcid.org/0000-0003-2281-3355","authenticated-orcid":false,"given":"Matthias","family":"Christandl","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark"}]},{"ORCID":"https://orcid.org/0000-0002-9252-2695","authenticated-orcid":false,"given":"Alexander","family":"Müller-Hermes","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Oslo, Blindern Campus, Oslo, Norway"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2024,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3354319","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2210.02939"},{"id":"arxiv:1610.04013","type":"article-journal","author":[{"given":"Todd","family":"Brun"},{"given":"Min-Hsiu","family":"Hsieh"}],"title":"Entanglement-Assisted Quantum Error-Correcting Codes","issued":{"date-parts":[[2016,10,13]]},"note":"arxivid:1610.04013\narxiv_version_number:1"},{"id":"doi:10.1017/CBO9781139034807.009","type":"book-chapter","publisher":"Cambridge University Press","DOI":"10.1017/cbo9781139034807.009","page":"181-200","source":"Crossref","title":"Entanglement-assisted quantum error-correcting codes","author":[{"given":"Todd A.","family":"Brun","sequence":"first","affiliation":[]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[]}],"container-title":"Quantum Error Correction","original-title":[],"issued":{"date-parts":[[2013,9,12]]},"URL":"http://dx.doi.org/10.1017/CBO9781139034807.009","note":"edition-number:1"},{"id":"arxiv:2204.08162","type":"article-journal","author":[{"given":"Cedric","family":"Beny"},{"given":"Jason","family":"Crann"},{"given":"Hun Hee","family":"Lee"},{"given":"Sang-Jun","family":"Park"},{"given":"Sang-Gyun","family":"Youn"}],"title":"Gaussian quantum information over general quantum kinematical systems I: Gaussian states","issued":{"date-parts":[[2022,4,18]]},"note":"arxivid:2204.08162\narxiv_version_number:1"},{"id":"arxiv:2004.13860","type":"article-journal","author":[{"given":"Sang Jun","family":"Park"},{"given":"Cedric","family":"Beny"},{"given":"Hun Hee","family":"Lee"}],"title":"Twisted Fourier analysis and pseudo-probability distributions","issued":{"date-parts":[[2020,5,16]]},"note":"arxivid:2004.13860\narxiv_version_number:3"},{"id":"arxiv:1408.6237","type":"journal-article","publisher":"IOP Publishing","issue":"2","DOI":"10.1088/1367-2630/17/2/023029","page":"023029","source":"Crossref","title":"Generalized cluster states based on finite groups","volume":"17","author":[{"given":"Courtney G","family":"Brell","sequence":"first","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2015,2,10]]},"URL":"http://dx.doi.org/10.1088/1367-2630/17/2/023029","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1408.6237"},{"id":"arxiv:1911.00099","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.10.031050","source":"Crossref","title":"Robust Encoding of a Qubit in a Molecule","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5104-6883","authenticated-orcid":true,"given":"Jacob P.","family":"Covey","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2421-4762","authenticated-orcid":true,"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.10.031050","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031050","note":"arxivid:1911.00099"},{"id":"arxiv:2111.12096","type":"article-journal","author":[{"given":"Victor V.","family":"Albert"},{"given":"David","family":"Aasen"},{"given":"Wenqing","family":"Xu"},{"given":"Wenjie","family":"Ji"},{"given":"Jason","family":"Alicea"},{"given":"John","family":"Preskill"}],"title":"Spin chains, defects, and quantum wires for the quantum-double edge","issued":{"date-parts":[[2021,11,23]]},"note":"arxivid:2111.12096\narxiv_version_number:1"},{"id":"arxiv:2312.09272","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Despite growing interest in beyond-group symmetries in quantum condensed matter systems, there are relatively few microscopic lattice models explicitly realizing these symmetries, and many phenomena have yet to be studied at the microscopic level. We introduce a one-dimensional stabilizer Hamiltonian composed of group-based Pauli operators whose ground state is a <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>G</a:mi><a:mo>×</a:mo><a:mi>Rep</a:mi><a:mo stretchy=\"false\">(</a:mo><a:mi>G</a:mi><a:mo stretchy=\"false\">)</a:mo></a:math>-symmetric state: the <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>G</e:mi></e:math>-cluster state introduced by Brell []. We show that this state lies in a symmetry-protected topological (SPT) phase protected by <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:mi>G</g:mi><g:mo>×</g:mo><g:mi>Rep</g:mi><g:mo stretchy=\"false\">(</g:mo><g:mi>G</g:mi><g:mo stretchy=\"false\">)</g:mo></g:math> symmetry, distinct from the symmetric product state by a duality argument. We identify several signatures of SPT order, namely, protected edge modes, string order parameters, and topological response. We discuss how <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:mi>G</k:mi></k:math>-cluster states may be used as a universal resource for measurement-based quantum computation, explicitly working out the case where <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:mi>G</m:mi></m:math> is a semidirect product of Abelian groups.</jats:p>","DOI":"10.1103/physrevx.15.011058","source":"Crossref","title":"Noninvertible Symmetry-Protected Topological Order in a Group-Based Cluster State","volume":"15","author":[{"ORCID":"https://orcid.org/0000-0002-2590-8221","authenticated-orcid":true,"given":"Christopher","family":"Fechisin","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST/University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"name":"Microsoft Quantum"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST/University of Maryland, College Park"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.15.011058","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011058","note":"arxivid:2312.09272"},{"id":"arxiv:1903.08807","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevd.100.034518","source":"Crossref","title":"General methods for digital quantum simulation of gauge theories","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0003-3033-0791","authenticated-orcid":true,"given":"Henry","family":"Lamm","sequence":"first","affiliation":[]},{"given":"Scott","family":"Lawrence","sequence":"additional","affiliation":[]},{"given":"Yukari","family":"Yamauchi","sequence":"additional","affiliation":[]},{"name":"NuQS Collaboration","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.100.034518","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"034518","note":"arxivid:1903.08807"},{"id":"arxiv:2208.12309","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevd.106.114501","source":"Crossref","title":"Primitive quantum gates for an \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>\n discrete subgroup: Binary tetrahedral","volume":"106","author":[{"ORCID":"https://orcid.org/0000-0001-7217-5692","authenticated-orcid":true,"given":"Erik J.","family":"Gustafson","sequence":"first","affiliation":[{"name":"Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3033-0791","authenticated-orcid":true,"given":"Henry","family":"Lamm","sequence":"additional","affiliation":[{"name":"Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA"}]},{"given":"Felicity","family":"Lovelace","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA"}]},{"ORCID":"https://orcid.org/0000-0001-7643-5615","authenticated-orcid":true,"given":"Damian","family":"Musk","sequence":"additional","affiliation":[{"name":"Stanford University Online High School, Redwood City, California 94063, USA"}]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.106.114501","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"114501","note":"arxivid:2208.12309"},{"id":"arxiv:2402.16780","type":"article-journal","author":[{"given":"Marcela","family":"Carena"},{"given":"Henry","family":"Lamm"},{"given":"Ying-Ying","family":"Li"},{"given":"Wanqiang","family":"Liu"}],"title":"Quantum error thresholds for gauge-redundant digitizations of lattice field theories","issued":{"date-parts":[[2024,2,26]]},"note":"arxivid:2402.16780\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9908010","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6760","DOI":"10.1038/46503","page":"390-393","source":"Crossref","title":"Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations","volume":"402","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[1999,11]]},"URL":"http://dx.doi.org/10.1038/46503","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BF46503\narxivid:quant-ph/9908010"},{"id":"arxiv:1503.08800","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.92.022312","source":"Crossref","title":"Qudit color codes and gauge color codes in all spatial dimensions","volume":"92","author":[{"given":"Fern H. E.","family":"Watson","sequence":"first","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Hussain","family":"Anwar","sequence":"additional","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,8,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.92.022312","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022312","note":"arxivid:1503.08800"},{"id":"arxiv:1509.03626","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2016.12.014","page":"387-413","source":"Crossref","title":"Gapped boundaries, group cohomology and fault-tolerant logical gates","volume":"377","author":[{"given":"Beni","family":"Yoshida","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,2]]},"URL":"http://dx.doi.org/10.1016/j.aop.2016.12.014","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491616302858\narxivid:1509.03626"},{"id":"arxiv:1601.03843","type":"article-journal","author":[{"given":"Reinhard F.","family":"Werner"}],"title":"Uncertainty relations for general phase spaces","issued":{"date-parts":[[2016,1,15]]},"note":"arxivid:1601.03843\narxiv_version_number:1"},{"id":"arxiv:1709.04460","type":"journal-article","publisher":"IOP Publishing","issue":"50","DOI":"10.1088/1751-8121/aa9314","page":"504002","source":"Crossref","title":"General phase spaces: from discrete variables to rotor and continuum limits","volume":"50","author":[{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":false,"given":"Victor V","family":"Albert","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7214-5685","authenticated-orcid":false,"given":"Saverio","family":"Pascazio","sequence":"additional","affiliation":[]},{"given":"Michel H","family":"Devoret","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2017,11,24]]},"URL":"http://dx.doi.org/10.1088/1751-8121/aa9314","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1709.04460"},{"id":"doi:10.5802/aif.1117","type":"journal-article","publisher":"Cellule MathDoc/CEDRAM","issue":"4","DOI":"10.5802/aif.1117","page":"191-205","source":"Crossref","title":"Some remarks on Koszul algebras and quantum groups","volume":"37","author":[{"given":"Yu. I.","family":"Manin","sequence":"first","affiliation":[]}],"container-title":"Annales de l’institut Fourier","original-title":[],"issued":{"date-parts":[[1987]]},"URL":"http://dx.doi.org/10.5802/aif.1117","ISSN":["0373-0956"]},{"id":"doi:10.1143/PTP.102.219","type":"journal-article","publisher":"Oxford University Press (OUP)","issue":"0","DOI":"10.1143/ptp.102.219","page":"219-228","source":"Crossref","title":"Quantized Theta-Functions","volume":"102","author":[{"given":"Y. I.","family":"Manin","sequence":"first","affiliation":[]}],"container-title":"Progress of Theoretical Physics Supplement","original-title":[],"language":"en","issued":{"date-parts":[[2013,5,16]]},"URL":"http://dx.doi.org/10.1143/PTP.102.219","ISSN":["0375-9687"],"container-title-short":"Progress of Theoretical Physics Supplement"},{"id":"arxiv:math/0307393","type":"article-journal","author":[{"given":"Yu. I.","family":"Manin"}],"title":"Functional equations for quantum theta functions","issued":{"date-parts":[[2003,7,31]]},"note":"arxivid:math/0307393\narxiv_version_number:1"},{"id":"arxiv:math/0402401","type":"journal-article","publisher":"IOP Publishing","issue":"19","DOI":"10.1088/0305-4470/38/19/014","page":"4255-4263","source":"Crossref","title":"Theta vectors and quantum theta functions","volume":"38","author":[{"given":"Ee","family":"Chang-Young","sequence":"first","affiliation":[]},{"given":"Hoil","family":"Kim","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[2005,4,25]]},"URL":"http://dx.doi.org/10.1088/0305-4470/38/19/014","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen.","note":"alternative-id:S0305-4470(05)90413-7\narxivid:math/0402401"},{"id":"arxiv:quant-ph/0501074","type":"article-journal","author":[{"given":"Annika","family":"Niehage"}],"title":"Quantum Goppa Codes over Hyperelliptic Curves","issued":{"date-parts":[[2005,1,14]]},"note":"arxivid:quant-ph/0501074\narxiv_version_number:1"},{"id":"arxiv:2501.18968","type":"article-journal","author":[{"given":"Roberto","family":"Zucchini"}],"title":"Calibrated hypergraph states: II calibrated hypergraph state construction and applications","issued":{"date-parts":[[2025,1,31]]},"note":"arxivid:2501.18968\narxiv_version_number:1"},{"id":"arxiv:0904.2560","type":"article-journal","author":[{"given":"Yong","family":"Zhang"}],"title":"Quantum Fourier Transform Over Galois Rings","issued":{"date-parts":[[2009,4,16]]},"note":"arxivid:0904.2560\narxiv_version_number:1"},{"id":"arxiv:1112.0825","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.87.022326","source":"Crossref","title":"Near-deterministic quantum teleportation and resource-efficient quantum computation using linear optics and hybrid qubits","volume":"87","author":[{"given":"Seung-Woo","family":"Lee","sequence":"first","affiliation":[]},{"given":"Hyunseok","family":"Jeong","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,2,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.022326","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022326","note":"arxivid:1112.0825"},{"id":"arxiv:1712.10206","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Light is an irreplaceable means of communication among various quantum information processing and storage devices. Due to their different physical nature, some of these devices couple more strongly to discrete, and some to continuous degrees of freedom of a quantum optical wave. It is therefore desirable to develop a technological capability to interconvert quantum information encoded in these degrees of freedom. Here we generate and characterize an entangled state between a dual-rail (polarization-encoded) single-photon qubit and a qubit encoded as a superposition of opposite-amplitude coherent states. We furthermore demonstrate the application of this state as a resource for the interfacing of quantum information between these encodings. In particular, we show teleportation of a polarization qubit onto a freely propagating continuous-variable qubit.</jats:p>","DOI":"10.1038/s41467-018-06055-x","source":"Crossref","title":"Entanglement and teleportation between polarization and wave-like encodings of an optical qubit","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0001-9105-4693","authenticated-orcid":false,"given":"Demid V.","family":"Sychev","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2211-559X","authenticated-orcid":false,"given":"Alexander E.","family":"Ulanov","sequence":"additional","affiliation":[]},{"given":"Egor S.","family":"Tiunov","sequence":"additional","affiliation":[]},{"given":"Anastasia A.","family":"Pushkina","sequence":"additional","affiliation":[]},{"given":"A.","family":"Kuzhamuratov","sequence":"additional","affiliation":[]},{"given":"Valery","family":"Novikov","sequence":"additional","affiliation":[]},{"given":"A. I.","family":"Lvovsky","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2018,9,10]]},"URL":"http://dx.doi.org/10.1038/s41467-018-06055-x","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"3672","note":"alternative-id:6055\narxivid:1712.10206"},{"id":"arxiv:2401.00450","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Hybridizing different degrees of freedom or physical platforms potentially offers various advantages in building scalable quantum architectures. Here, we introduce a fault-tolerant hybrid quantum computation by building on the advantages of both discrete-variable (DV) and continuous-variable (CV) systems. In particular, we define a CV-DV hybrid qubit with a bosonic cat code and a single photon, which is implementable in current photonic platforms. Due to the cat code encoded in the CV part, the predominant loss errors are readily correctable without multiqubit encoding, while the logical basis is inherently orthogonal due to the DV part. We design fault-tolerant architectures by concatenating hybrid qubits and an outer DV quantum error-correction code such as a topological code, exploring their potential merit in developing scalable quantum computation. We demonstrate by numerical simulations that our scheme is at least an order of magnitude more resource efficient compared to all previous proposals in photonic platforms, allowing us to achieve a record-high loss threshold among existing CV and hybrid approaches. We discuss the realization of our approach not only in all-photonic platforms but also in other hybrid platforms including superconducting and trapped-ion systems, which allows us to find various efficient routes toward fault-tolerant quantum computing.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.030322","source":"Crossref","title":"Fault-Tolerant Quantum Computation by Hybrid Qubits with Bosonic Cat Code and Single Photons","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-3011-1129","authenticated-orcid":true,"given":"Jaehak","family":"Lee","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05kzfa883","id-type":"ROR","asserted-by":"publisher"}],"name":"Korea Institute of Science and Technology (KIST)"},{"id":[{"id":"https://ror.org/04h9pn542","id-type":"ROR","asserted-by":"publisher"}],"name":"Seoul National University"}]},{"ORCID":"https://orcid.org/0009-0008-4376-236X","authenticated-orcid":true,"given":"Nuri","family":"Kang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05kzfa883","id-type":"ROR","asserted-by":"publisher"}],"name":"Korea Institute of Science and Technology (KIST)"},{"id":[{"id":"https://ror.org/047dqcg40","id-type":"ROR","asserted-by":"publisher"}],"name":"Korea University"}]},{"ORCID":"https://orcid.org/0000-0002-1207-2752","authenticated-orcid":true,"given":"Seok-Hyung","family":"Lee","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04h9pn542","id-type":"ROR","asserted-by":"publisher"}],"name":"Seoul National University"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0003-0348-3397","authenticated-orcid":true,"given":"Hyunseok","family":"Jeong","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04h9pn542","id-type":"ROR","asserted-by":"publisher"}],"name":"Seoul National University"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0003-1006-0333","authenticated-orcid":true,"given":"Seung-Woo","family":"Lee","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05kzfa883","id-type":"ROR","asserted-by":"publisher"}],"name":"Korea Institute of Science and Technology (KIST)"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,2]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.030322","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030322","note":"update-to:{\"DOI\":\"10.1103/prxquantum.5.030322\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2025,1,29]],\"date-time\":\"2025-01-29T00:00:00Z\",\"timestamp\":1738108800000}}\nupdated-by:{\"DOI\":\"10.1103/prxquantum.5.030322\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2025,1,29]],\"date-time\":\"2025-01-29T00:00:00Z\",\"timestamp\":1738108800000}}\narxivid:2401.00450"},{"id":"arxiv:1906.04864","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevlett.125.060501","source":"Crossref","title":"Resource-Efficient Topological Fault-Tolerant Quantum Computation with Hybrid Entanglement of Light","volume":"125","author":[{"ORCID":"https://orcid.org/0000-0002-3112-9101","authenticated-orcid":true,"given":"Srikrishna","family":"Omkar","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1766-6402","authenticated-orcid":true,"given":"Yong Siah","family":"Teo","sequence":"additional","affiliation":[]},{"given":"Hyunseok","family":"Jeong","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.060501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"060501","note":"arxivid:1906.04864"},{"id":"arxiv:2011.04209","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.103.032602","source":"Crossref","title":"Highly photon-loss-tolerant quantum computing using hybrid qubits","volume":"103","author":[{"given":"S.","family":"Omkar","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1766-6402","authenticated-orcid":true,"given":"Y. S.","family":"Teo","sequence":"additional","affiliation":[]},{"given":"Seung-Woo","family":"Lee","sequence":"additional","affiliation":[]},{"given":"H.","family":"Jeong","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,3,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.032602","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032602","note":"arxivid:2011.04209"},{"id":"arxiv:1409.3719","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"9","DOI":"10.1038/nphys3410","page":"713-719","source":"Crossref","title":"Hybrid discrete- and continuous-variable quantum information","volume":"11","author":[{"given":"Ulrik L.","family":"Andersen","sequence":"first","affiliation":[]},{"given":"Jonas S.","family":"Neergaard-Nielsen","sequence":"additional","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]},{"given":"Akira","family":"Furusawa","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2015,9]]},"URL":"http://dx.doi.org/10.1038/nphys3410","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nature Phys","note":"alternative-id:BFnphys3410\narxivid:1409.3719"},{"id":"arxiv:2407.10381","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>This tutorial offers a pedagogical guide to hybrid quantum processors that integrate discrete-variable (DV) qubits and continuous-variable (CV) oscillators. Aimed at computer scientists, engineers, and physicists, it provides an overview of the experimental, algorithmic, and architectural aspects of this novel and rapidly developing hardware model. Experimental realizations of this model include superconducting, trapped-ion, and neutral-atom platforms. By combining DV and CV components, hybrid oscillator-qubit processors enable a powerful new paradigm that offers complementary strengths for quantum control, error correction, computation, and simulation. Working toward the goal of a full-stack system connecting applications to CV-DV hardware, we define and formulate abstract machine models and instruction set architectures. These essential abstractions enable codesign of hardware and software, and resource estimation for exploring the potential of current and future hardware for computational and simulation tasks. Using these abstractions, we present both new and existing examples that illustrate the benefits of hybrid CV-DV processors relative to traditional DV-only hardware in computation as well as quantum simulation of physical models. Examples include algorithms for transferring states between DV and CV systems, performing the quantum Fourier transform, and simulation of lattice gauge theories. Relative to qubit-only hardware, the bosonic degrees of freedom natively available in hybrid architectures can substantially reduce the circuit complexity of simulations for physical models containing bosons. A key technique is the extension of quantum signal processing ideas to CV-DV systems. This work is intended to serve as a timely and comprehensive guide to this relatively unexplored yet promising approach to quantum computation and to provide a road map to guide future development.</jats:p>","DOI":"10.1103/4rf7-9tfx","source":"Crossref","title":"Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0003-1468-942X","authenticated-orcid":true,"given":"Yuan","family":"Liu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"},{"id":[{"id":"https://ror.org/04tj63d06","id-type":"ROR","asserted-by":"publisher"}],"name":"North Carolina State University"},{"id":[{"id":"https://ror.org/04tj63d06","id-type":"ROR","asserted-by":"publisher"}],"name":"North Carolina State University"}]},{"ORCID":"https://orcid.org/0000-0002-4921-1410","authenticated-orcid":true,"given":"Shraddha","family":"Singh","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-2397-1518","authenticated-orcid":true,"given":"Kevin C.","family":"Smith","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/02ex6cf31","id-type":"ROR","asserted-by":"publisher"}],"name":"Brookhaven National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-2752-6462","authenticated-orcid":true,"given":"Eleanor","family":"Crane","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-4065-6974","authenticated-orcid":true,"given":"John M.","family":"Martyn","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0001-5179-0215","authenticated-orcid":true,"given":"Alec","family":"Eickbusch","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-9969-7391","authenticated-orcid":true,"given":"Alexander","family":"Schuckert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute and QuICS, National Institute of Standards and Technology (NIST) and University of Maryland"}]},{"given":"Richard D.","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"given":"Jasmine","family":"Sinanan-Singh","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0001-7973-2842","authenticated-orcid":true,"given":"Micheline B.","family":"Soley","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01y2jtd41","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Wisconsin-Madison"}]},{"ORCID":"https://orcid.org/0000-0002-0869-2991","authenticated-orcid":true,"given":"Takahiro","family":"Tsunoda","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0001-7296-523X","authenticated-orcid":true,"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"},{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0001-6047-0547","authenticated-orcid":true,"given":"Nathan","family":"Wiebe","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03dbr7087","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Toronto"},{"id":[{"id":"https://ror.org/01sdtdd95","id-type":"ROR","asserted-by":"publisher"}],"name":"Canadian Institute for Advanced Research"},{"id":[{"id":"https://ror.org/05h992307","id-type":"ROR","asserted-by":"publisher"}],"name":"Pacific Northwest National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":true,"given":"Steven M.","family":"Girvin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,7]]},"URL":"http://dx.doi.org/10.1103/4rf7-9tfx","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010201","note":"arxivid:2407.10381"},{"id":"doi:10.1080/09500349708231877","type":"journal-article","publisher":"Informa UK Limited","issue":"11-12","DOI":"10.1080/09500349708231877","page":"2173-2184","source":"Crossref","title":"Hyper-entangled states","volume":"44","author":[{"given":"Paul G.","family":"Kwiat","sequence":"first","affiliation":[]}],"container-title":"Journal of Modern Optics","original-title":[],"language":"en","issued":{"date-parts":[[1997,11]]},"URL":"http://dx.doi.org/10.1080/09500349708231877","ISSN":["0950-0340","1362-3044"],"container-title-short":"Journal of Modern Optics","note":"alternative-id:10.1080/09500349708231877"},{"id":"arxiv:0806.4064","type":"article-journal","author":[{"given":"Amritanshu","family":"Prasad"},{"given":"M. K.","family":"Vemuri"}],"title":"Decomposition of phase space and classification of Heisenberg groups","issued":{"date-parts":[[2008,6,25]]},"note":"arxivid:0806.4064\narxiv_version_number:1"},{"id":"arxiv:1611.09274","type":"article-journal","author":[{"given":"Juan","family":"Bermejo-Vega"}],"title":"Normalizer Circuits and Quantum Computation","issued":{"date-parts":[[2016,11,28]]},"note":"arxivid:1611.09274\narxiv_version_number:1"},{"id":"arxiv:2509.18854","type":"article-journal","author":[{"given":"Lukas","family":"Brenner"},{"given":"Beatriz","family":"Dias"},{"given":"Robert","family":"Koenig"}],"title":"Trading modes against energy","issued":{"date-parts":[[2025,9,23]]},"note":"arxivid:2509.18854\narxiv_version_number:1"},{"id":"doi:10.1002/9783527635283","type":"monograph","publisher":"Wiley","DOI":"10.1002/9783527635283","source":"Crossref","title":"Quantum Teleportation and Entanglement","author":[{"given":"Akira","family":"Furusawa","sequence":"first","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2011,3,11]]},"ISBN":["9783527409303","9783527635283"],"URL":"http://dx.doi.org/10.1002/9783527635283","note":"edition-number:1\nalternative-id:10.1002/9783527635283"},{"id":"arxiv:0807.4906","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.79.022305","source":"Crossref","title":"Linear-optical hyperentanglement-assisted quantum error-correcting code","volume":"79","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Dmitry B.","family":"Uskov","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,2,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.022305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022305","note":"arxivid:0807.4906"},{"id":"doi:10.4153/CJM-1988-012-9","type":"journal-article","publisher":"Canadian Mathematical Society","issue":"2","abstract":"<jats:p>The non-commutative tori provide probably the most accessible interesting examples of non-commutative differentiable manifolds. We can identify an ordinary <jats:italic>n</jats:italic>-torus <jats:italic>T<jats:sup>n</jats:sup></jats:italic> with its algebra, <jats:italic>C(T<jats:sup>n</jats:sup>)</jats:italic>, of continuous complex-valued functions under pointwise multiplication. But <jats:italic>C(T<jats:sup>n</jats:sup>)</jats:italic> is the universal <jats:italic>C</jats:italic>*-algebra generated by <jats:italic>n</jats:italic> commuting unitary operators. By definition, [<jats:bold>15, 16, 50</jats:bold>], a non-commutative <jats:italic>n</jats:italic>-torus is the universal <jats:italic>C</jats:italic>*-algebra generated by <jats:italic>n</jats:italic> unitary operators which, while they need not commute, have as multiplicative commutators various fixed scalar multiples of the identity operator. As Connes has shown [<jats:bold>8, 10</jats:bold>], these algebras have a natural differentiable structure, defined by a natural ergodic action of <jats:italic>T<jats:sup>n</jats:sup></jats:italic> as a group of automorphisms. The non-commutative tori behave in inany ways like ordinary tori. For instance, it is an almost immediate consequence of the work of Pimsner and Voiculescu [<jats:bold>37</jats:bold>] that the <jats:italic>K</jats:italic>-groups of a non-commutative torus are the same as those of an ordinary torus of the same dimension. (In particular, non-commutative tori are <jats:italic>KK</jats:italic>-equivalent to ordinary tori by Corollary 7.5 of [<jats:bold>52</jats:bold>].) Furthermore, the structure constants of non-commutative tori can be continuously deformed into those for ordinary tori. (This is exploited in [<jats:bold>17</jats:bold>].)</jats:p>","DOI":"10.4153/cjm-1988-012-9","page":"257-338","source":"Crossref","title":"Projective Modules over Higher-Dimensional Non-Commutative Tori","volume":"40","author":[{"given":"Marc A.","family":"Rieffel","sequence":"first","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1988,4,1]]},"URL":"http://dx.doi.org/10.4153/CJM-1988-012-9","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j. math.","note":"alternative-id:S0008414X00000122"},{"id":"arxiv:hep-th/9805034","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/s0550-3213(98)00550-1","page":"720-738","source":"Crossref","title":"Morita equivalence and duality","volume":"534","author":[{"given":"Albert","family":"Schwarz","sequence":"first","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[1998,11]]},"URL":"http://dx.doi.org/10.1016/S0550-3213(98)00550-1","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"alternative-id:S0550321398005501\narxivid:hep-th/9805034"},{"id":"arxiv:hep-th/9711162","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"02","DOI":"10.1088/1126-6708/1998/02/003","page":"003-003","source":"Crossref","title":"Noncommutative geometry and Matrix theory","volume":"1998","author":[{"given":"Alain","family":"Connes","sequence":"first","affiliation":[]},{"given":"Michael R","family":"Douglas","sequence":"additional","affiliation":[]},{"given":"Albert","family":"Schwarz","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"issued":{"date-parts":[[1998,2,5]]},"URL":"http://dx.doi.org/10.1088/1126-6708/1998/02/003","ISSN":["1029-8479"],"container-title-short":"J. High Energy Phys.","note":"arxivid:hep-th/9711162"},{"id":"arxiv:math/9803057","type":"article-journal","author":[{"given":"Marc","family":"Rieffel"},{"given":"Albert","family":"Schwarz"}],"title":"Morita equivalence of multidimensional noncommutative tori","issued":{"date-parts":[[1998,8,20]]},"note":"arxivid:math/9803057\narxiv_version_number:4"},{"id":"arxiv:1408.6238","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.91.042333","source":"Crossref","title":"Generalized color codes supporting non-Abelian anyons","volume":"91","author":[{"given":"Courtney G.","family":"Brell","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.042333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042333","note":"arxivid:1408.6238"},{"id":"arxiv:1908.02829","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes --- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the ``log dim R'' term is included in the definition of entanglement entropy.</jats:p>","DOI":"10.22331/q-2020-09-24-331","page":"331","source":"Crossref","title":"Kitaev's quantum double model as an error correcting code","volume":"4","author":[{"given":"Shawn X.","family":"Cui","sequence":"first","affiliation":[{"name":"Departments of Mathematics, Physics and Astronomy, Purdue University, West Lafayette, IN 47907"}]},{"given":"Dawei","family":"Ding","sequence":"additional","affiliation":[{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305"}]},{"given":"Xizhi","family":"Han","sequence":"additional","affiliation":[{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305"}]},{"given":"Geoffrey","family":"Penington","sequence":"additional","affiliation":[{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305"}]},{"given":"Daniel","family":"Ranard","sequence":"additional","affiliation":[{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305"}]},{"given":"Brandon C.","family":"Rayhaun","sequence":"additional","affiliation":[{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305"}]},{"given":"Zhou","family":"Shangnan","sequence":"additional","affiliation":[{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,24]]},"URL":"http://dx.doi.org/10.22331/q-2020-09-24-331","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1908.02829"},{"id":"arxiv:2509.10876","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/vqlz-4rtz","source":"Crossref","title":"Partition function of the Kitaev quantum double model","volume":"113","author":[{"ORCID":"https://orcid.org/0000-0002-4778-2746","authenticated-orcid":true,"given":"Anna","family":"Ritz-Zwilling","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02en5vm52","id-type":"ROR","asserted-by":"publisher"}],"name":"Sorbonne Université"},{"id":[{"id":"https://ror.org/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/04zaaa143","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire de Physique Théorique de la Matière Condensée"},{"id":[{"id":"https://ror.org/04zaaa143","id-type":"ROR","asserted-by":"publisher"}],"name":"LPTMC"}]},{"given":"Benoît","family":"Douçot","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02en5vm52","id-type":"ROR","asserted-by":"publisher"}],"name":"Sorbonne Université"},{"id":[{"id":"https://ror.org/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/02mph9k76","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire de Physique Théorique des Hautes Énergies"},{"id":[{"id":"https://ror.org/02mph9k76","id-type":"ROR","asserted-by":"publisher"}],"name":"LPTHE"}]},{"ORCID":"https://orcid.org/0000-0001-7757-5978","authenticated-orcid":true,"given":"Steven H.","family":"Simon","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/052gg0110","id-type":"ROR","asserted-by":"publisher"}],"name":"Rudolf Peierls Centre for Theoretical Physics"},{"id":[{"id":"https://ror.org/052gg0110","id-type":"ROR","asserted-by":"publisher"}],"name":"Clarendon Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-1788-643X","authenticated-orcid":true,"given":"Julien","family":"Vidal","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02en5vm52","id-type":"ROR","asserted-by":"publisher"}],"name":"Sorbonne Université"},{"id":[{"id":"https://ror.org/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/04zaaa143","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire de Physique Théorique de la Matière Condensée"},{"id":[{"id":"https://ror.org/04zaaa143","id-type":"ROR","asserted-by":"publisher"}],"name":"LPTMC"}]},{"ORCID":"https://orcid.org/0000-0003-1950-8261","authenticated-orcid":true,"given":"Jean-Noël","family":"Fuchs","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02en5vm52","id-type":"ROR","asserted-by":"publisher"}],"name":"Sorbonne Université"},{"id":[{"id":"https://ror.org/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/04zaaa143","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire de Physique Théorique de la Matière Condensée"},{"id":[{"id":"https://ror.org/04zaaa143","id-type":"ROR","asserted-by":"publisher"}],"name":"LPTMC"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,6]]},"URL":"http://dx.doi.org/10.1103/vqlz-4rtz","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"165106","note":"arxivid:2509.10876"},{"id":"arxiv:0909.3305","type":"journal-article","publisher":"American Physical Society (APS)","issue":"26","DOI":"10.1103/physrevlett.103.261601","source":"Crossref","title":"Topological Entanglement Rényi Entropy and Reduced Density Matrix Structure","volume":"103","author":[{"given":"Steven T.","family":"Flammia","sequence":"first","affiliation":[]},{"given":"Alioscia","family":"Hamma","sequence":"additional","affiliation":[]},{"given":"Taylor L.","family":"Hughes","sequence":"additional","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,12,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.103.261601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"261601","note":"arxivid:0909.3305"},{"id":"arxiv:1006.5479","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-011-1294-x","page":"663-694","source":"Crossref","title":"The Quantum Double Model with Boundary: Condensations and Symmetries","volume":"306","author":[{"given":"Salman","family":"Beigi","sequence":"first","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Whalen","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,6,28]]},"URL":"http://dx.doi.org/10.1007/s00220-011-1294-x","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1294\narxivid:1006.5479"},{"id":"arxiv:2310.19661","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","abstract":"<jats:title>Abstract</jats:title>\n          <jats:p>We give a complete classification of the anyon sectors of Kitaev’s quantum double model on the infinite triangular lattice and for finite gauge group <jats:italic>G</jats:italic>, including the non-abelian case. As conjectured, the anyon sectors of the model correspond precisely to equivalence classes of irreducible representations of the quantum double algebra of <jats:italic>G</jats:italic>.</jats:p>","DOI":"10.1007/s00220-025-05363-w","source":"Crossref","title":"Classification of the Anyon Sectors of Kitaev’s Quantum Double Model","volume":"406","author":[{"ORCID":"https://orcid.org/0000-0002-0514-018X","authenticated-orcid":false,"given":"Alex","family":"Bols","sequence":"first","affiliation":[]},{"given":"Siddharth","family":"Vadnerkar","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,2]]},"URL":"http://dx.doi.org/10.1007/s00220-025-05363-w","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","page":"188","note":"alternative-id:5363\narxivid:2310.19661"},{"id":"arxiv:math/0605530","type":"article-journal","author":[{"given":"Deepak","family":"Naidu"}],"title":"Categorical Morita equivalence for group-theoretical categories","issued":{"date-parts":[[2006,9,7]]},"note":"arxivid:math/0605530\narxiv_version_number:2"},{"id":"doi:10.1007/3-540-49208-9_31","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-49208-9_31","page":"341-356","source":"Crossref","title":"Topological Quantum Computation","author":[{"given":"R.","family":"Walter Ogburn","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[1999]]},"ISBN":["9783540655145","9783540492085"],"URL":"http://dx.doi.org/10.1007/3-540-49208-9_31","ISSN":["0302-9743"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:quant-ph/0306063","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.69.032306","source":"Crossref","title":"Anyon computers with smaller groups","volume":"69","author":[{"given":"Carlos","family":"Mochon","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,3,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.69.032306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032306","note":"arxivid:quant-ph/0306063"},{"id":"doi:10.1017/CBO9780511792908","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Combining physics, mathematics and computer science, topological quantum computation is a rapidly expanding research area focused on the exploration of quantum evolutions that are immune to errors. In this book, the author presents a variety of different topics developed together for the first time, forming an excellent introduction to topological quantum computation. The makings of anyonic systems, their properties and their computational power are presented in a pedagogical way. Relevant calculations are fully explained, and numerous worked examples and exercises support and aid understanding. Special emphasis is given to the motivation and physical intuition behind every mathematical concept. Demystifying difficult topics by using accessible language, this book has broad appeal and is ideal for graduate students and researchers from various disciplines who want to get into this new and exciting research field.</jats:p>","DOI":"10.1017/cbo9780511792908","source":"Crossref","title":"Introduction to Topological Quantum Computation","author":[{"given":"Jiannis K.","family":"Pachos","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2012,4,12]]},"ISBN":["9781107005044","9780511792908"],"URL":"http://dx.doi.org/10.1017/CBO9780511792908","note":"edition-number:1"},{"id":"arxiv:0712.0190","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevb.78.115421","source":"Crossref","title":"Family of non-Abelian Kitaev models on a lattice: Topological condensation and confinement","volume":"78","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2008,9,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.78.115421","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"115421","note":"arxivid:0712.0190"},{"id":"arxiv:1706.03611","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevb.96.165138","source":"Crossref","title":"Twisted quantum double model of topological order with boundaries","volume":"96","author":[{"given":"Alex","family":"Bullivant","sequence":"first","affiliation":[]},{"given":"Yuting","family":"Hu","sequence":"additional","affiliation":[]},{"given":"Yidun","family":"Wan","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.96.165138","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"165138","note":"arxivid:1706.03611"},{"id":"arxiv:2204.05341","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.108.245133","source":"Crossref","title":"Excitations in the higher-lattice gauge theory model for topological phases. II. The \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>\n-dimensional case","volume":"108","author":[{"ORCID":"https://orcid.org/0000-0002-4857-0091","authenticated-orcid":true,"given":"Joe","family":"Huxford","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7757-5978","authenticated-orcid":true,"given":"Steven H.","family":"Simon","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.108.245133","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"245133","note":"arxivid:2204.05341"},{"id":"arxiv:quant-ph/0110143","type":"journal-article","publisher":"AIP Publishing","issue":"9","abstract":"<jats:p>We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z2 lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.</jats:p>","DOI":"10.1063/1.1499754","page":"4452-4505","source":"Crossref","title":"Topological quantum memory","volume":"43","author":[{"given":"Eric","family":"Dennis","sequence":"first","affiliation":[{"name":"Princeton University, Princeton, New Jersey 08544"}]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125"}]},{"given":"Andrew","family":"Landahl","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125"}]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2002,9,1]]},"URL":"http://dx.doi.org/10.1063/1.1499754","ISSN":["0022-2488","1089-7658"],"note":"arxivid:quant-ph/0110143"},{"id":"arxiv:2309.13440","type":"article-journal","author":[{"given":"Mario","family":"Tomba"},{"given":"Shuqi","family":"Wei"},{"given":"Brett","family":"Hungar"},{"given":"Daniel","family":"Wallick"},{"given":"Kyle","family":"Kawagoe"},{"given":"Chian Yeong","family":"Chuah"},{"given":"David","family":"Penneys"}],"title":"Boundary algebras of the Kitaev Quantum Double model","issued":{"date-parts":[[2023,9,23]]},"note":"arxivid:2309.13440\narxiv_version_number:1"},{"id":"arxiv:1101.0527","type":"journal-article","publisher":"Elsevier BV","issue":"9","DOI":"10.1016/j.aop.2011.07.007","page":"2444-2473","source":"Crossref","title":"From entanglement renormalisation to the disentanglement of quantum double models","volume":"326","author":[{"given":"Miguel","family":"Aguado","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,9]]},"URL":"http://dx.doi.org/10.1016/j.aop.2011.07.007","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491611001163\narxivid:1101.0527"},{"id":"arxiv:0901.1345","type":"journal-article","publisher":"IOP Publishing","issue":"5","DOI":"10.1088/1367-2630/11/5/053009","page":"053009","source":"Crossref","title":"Simulations of quantum double models","volume":"11","author":[{"given":"G K","family":"Brennen","sequence":"first","affiliation":[]},{"given":"M","family":"Aguado","sequence":"additional","affiliation":[]},{"given":"J I","family":"Cirac","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2009,5,22]]},"URL":"http://dx.doi.org/10.1088/1367-2630/11/5/053009","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:0901.1345"},{"id":"arxiv:2112.01519","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases—of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:msub><a:mi mathvariant=\"double-struck\">Z</a:mi><a:mn>2</a:mn></a:msub></a:math> Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><d:mi>G</d:mi><d:mo>×</d:mo><d:mi>H</d:mi></d:math> symmetry gives rise to anomalous LRE upon measuring <f:math xmlns:f=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><f:mi>G</f:mi></f:math>-charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and we uncover the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevx.14.021040","source":"Crossref","title":"Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"first","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA"}]},{"ORCID":"https://orcid.org/0000-0001-9433-3399","authenticated-orcid":true,"given":"Ryan","family":"Thorngren","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA"},{"name":"Center for Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts 02138, USA"},{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, US"}]},{"ORCID":"https://orcid.org/0000-0002-6306-2263","authenticated-orcid":true,"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA"}]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":true,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.14.021040","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021040","note":"arxivid:2112.01519"},{"id":"arxiv:2205.01933","type":"article-journal","author":[{"given":"Sergey","family":"Bravyi"},{"given":"Isaac","family":"Kim"},{"given":"Alexander","family":"Kliesch"},{"given":"Robert","family":"Koenig"}],"title":"Adaptive constant-depth circuits for manipulating non-abelian anyons","issued":{"date-parts":[[2022,9,28]]},"note":"arxivid:2205.01933\narxiv_version_number:2"},{"id":"arxiv:2112.03061","type":"article-journal","author":[{"given":"Ruben","family":"Verresen"},{"given":"Nathanan","family":"Tantivasadakarn"},{"given":"Ashvin","family":"Vishwanath"}],"title":"Efficiently preparing Schrödinger's cat, fractons and non-Abelian topological order in quantum devices","issued":{"date-parts":[[2022,1,6]]},"note":"arxivid:2112.03061\narxiv_version_number:2"},{"id":"arxiv:2409.12230","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/fy9r-hpcw","source":"Crossref","title":"Stability and Loop Models from Decohering Non-Abelian Topological Order","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0001-7512-505X","authenticated-orcid":true,"given":"Pablo","family":"Sala","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":true,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"},{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,26]]},"URL":"http://dx.doi.org/10.1103/fy9r-hpcw","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"250403","note":"arxivid:2409.12230"},{"id":"arxiv:2409.12948","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>The effect of decoherence on topological order (TO) has been most deeply understood for the toric code, the paragon of Abelian TOs. We show that certain non-Abelian TOs can be analyzed and understood to a similar degree, despite being significantly richer. We consider both wave-function deformations and quantum channels acting on <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:msub><a:mi>D</a:mi><a:mn>4</a:mn></a:msub></a:math> TO, which has recently been realized on a quantum processor. By identifying the corresponding local statistical mechanical spin or rotor model with <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:msub><c:mi>D</c:mi><c:mn>4</c:mn></c:msub></c:math> symmetry, we find a remarkable stability against proliferating non-Abelian anyons. This is shown by leveraging a reformulation in terms of the tractable O(2) loop model in the pure state case and <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>n</e:mi></e:math> coupled O(2) loop models for Rényi-<g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:mi>n</g:mi></g:math> quantities in the decoherence case—corresponding to worldlines of the proliferating anyon with quantum dimension 2. In particular, we find that the purity (<i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:mi>n</i:mi><i:mo>=</i:mo><i:mn>2</i:mn></i:math>) remains deep in the <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:msub><k:mi>D</k:mi><k:mn>4</k:mn></k:msub></k:math> TO for any decoherence strength, while the <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:mi>n</m:mi><m:mo stretchy=\"false\">→</m:mo><m:mi>∞</m:mi></m:math> limit becomes critical upon maximally decohering a particular anyon type, similar to our wave-function deformation result. The information-theoretic threshold (<p:math xmlns:p=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><p:mi>n</p:mi><p:mo stretchy=\"false\">→</p:mo><p:mn>1</p:mn></p:math>) appears to be controlled by a disordered version of these statistical mechanical models, akin to the toric code case although significantly more robust. We furthermore use Monte Carlo simulations to explore the phase diagrams when multiple anyon types proliferate at the same time, leading to a continued stability of the <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><s:msub><s:mi>D</s:mi><s:mn>4</s:mn></s:msub></s:math> TO in addition to critical phases with emergent U(1) symmetry. Instead of loop models, these are now described by net models corresponding to different anyon types coupled together according to fusion rules. This opens up the exploration of statistical mechanical models for decohered non-Abelian TO, which can inform optimal decoders and which in an ungauged formulation provides examples of non-Abelian strong-to-weak symmetry breaking.</jats:p>","DOI":"10.1103/5ywn-6d3q","source":"Crossref","title":"Decoherence and Wave-Function Deformation of \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>D</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:math>\n Non-Abelian Topological Order","volume":"15","author":[{"ORCID":"https://orcid.org/0000-0001-7512-505X","authenticated-orcid":true,"given":"Pablo","family":"Sala","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0001-9979-3423","authenticated-orcid":true,"given":"Jason","family":"Alicea","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":true,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[{"name":"University of Chicago"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"},{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,1]]},"URL":"http://dx.doi.org/10.1103/5ywn-6d3q","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031002","note":"arxivid:2409.12948"},{"id":"doi:10.1103/RevModPhys.51.659","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/revmodphys.51.659","page":"659-713","source":"Crossref","title":"An introduction to lattice gauge theory and spin systems","volume":"51","author":[{"given":"John B.","family":"Kogut","sequence":"first","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[1979,10,1]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.51.659","ISSN":["0034-6861"],"container-title-short":"Rev. Mod. Phys."},{"id":"manual:-S.-X.-Cui-Topological-quantu","type":"chapter","author":[{"family":"Cui","given":"S.X."}],"title":"Topological quantum computation","container-title":"Lecture Notes","issued":"2018"},{"id":"arxiv:1011.1942","type":"journal-article","publisher":"IOP Publishing","issue":"5","DOI":"10.1088/1367-2630/13/5/053039","page":"053039","source":"Crossref","title":"Toric codes and quantum doubles from two-body Hamiltonians","volume":"13","author":[{"given":"Courtney G","family":"Brell","sequence":"first","affiliation":[]},{"given":"Steven T","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"Stephen D","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Andrew C","family":"Doherty","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2011,5,20]]},"URL":"http://dx.doi.org/10.1088/1367-2630/13/5/053039","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1011.1942"},{"id":"doi:10.5446/35287","type":"article","categories":["Information Technology"],"language":"en","author":[{"family":"Kumar","given":"Prashant"}],"issued":{"date-parts":[[2011]]},"abstract":"We introduce a family of codes which are a generalisation of the Bacon-Shor code using the quantum double models introduced by Kitaev. In particular, we show that such codes are possible for non-Abelian quantum double models, as well as three-dimensional quantum double models. Our codes possess a structure which generalises the gauge subsystem of the Bacon-Shor code to non-Abelian models. This means that we retain the ability to infer error syndromes using two body measurements, as well as a generalised subsystem error correction procedure.","DOI":"10.5446/35287","publisher":"University of Southern California (USC)","title":"A Class of Quantum Double Subsystem Codes","URL":"https://av.tib.eu/media/35287"},{"id":"arxiv:hep-th/9511201","type":"article-journal","author":[{"given":"Mark de Wild","family":"Propitius"},{"given":"F. Alexander","family":"Bais"}],"title":"Discrete gauge theories","issued":{"date-parts":[[1996,3,20]]},"note":"arxivid:hep-th/9511201\narxiv_version_number:2"},{"id":"arxiv:2411.04181","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/q2c6-nd6r","source":"Crossref","title":"Protocols for Creating Anyons and Defects via Gauging","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0001-9579-0903","authenticated-orcid":true,"given":"Anasuya","family":"Lyons","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"given":"Chiu Fan Bowen","family":"Lo","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-6306-2263","authenticated-orcid":true,"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":true,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,13]]},"URL":"http://dx.doi.org/10.1103/q2c6-nd6r","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200405","note":"arxivid:2411.04181"},{"id":"arxiv:hep-th/9511195","type":"article-journal","author":[{"given":"Mark de Wild","family":"Propitius"}],"title":"Topological interactions in broken gauge theories","issued":{"date-parts":[[1995,11,27]]},"note":"arxivid:hep-th/9511195\narxiv_version_number:1"},{"id":"arxiv:1508.03468","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.93.155131","source":"Crossref","title":"Topological phases with generalized global symmetries","volume":"93","author":[{"given":"Beni","family":"Yoshida","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2016,4,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.93.155131","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155131","note":"arxivid:1508.03468"},{"id":"arxiv:2408.09353","type":"article-journal","author":[{"given":"Bowen","family":"Li"},{"given":"Gongxiang","family":"Liu"}],"title":"On Gauge Equivalence of Twisted Quantum Doubles","issued":{"date-parts":[[2024,8,18]]},"note":"arxivid:2408.09353\narxiv_version_number:1"},{"id":"arxiv:2408.10140","type":"article-journal","author":[{"given":"Quynh T.","family":"Nguyen"}],"title":"Good binary quantum codes with transversal CCZ gate","issued":{"date-parts":[[2024,10,14]]},"note":"arxivid:2408.10140\narxiv_version_number:2"},{"id":"preset:Heinrich21","type":"thesis","author":[{"family":"Heinrich","given":"M."}],"title":"On stabiliser techniques and their application to simulation and certification of quantum devices","genre":"PhD thesis,","publisher":"Universität zu Köln","issued":"2021"},{"id":"arxiv:0810.3557","type":"article-journal","author":[{"given":"Alastair","family":"Kay"},{"given":"Roger","family":"Colbeck"}],"title":"Quantum Self-Correcting Stabilizer Codes","issued":{"date-parts":[[2008,10,20]]},"note":"arxivid:0810.3557\narxiv_version_number:1"},{"id":"arxiv:1412.2858","type":"article-journal","author":[{"given":"Kristan","family":"Temme"}],"title":"Thermalization time bounds for Pauli stabilizer Hamiltonians","issued":{"date-parts":[[2016,1,18]]},"note":"arxivid:1412.2858\narxiv_version_number:2"},{"id":"arxiv:1601.01324","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.93.052337","source":"Crossref","title":"Necessity of an energy barrier for self-correction of Abelian quantum doubles","volume":"93","author":[{"given":"Anna","family":"Kómár","sequence":"first","affiliation":[]},{"given":"Olivier","family":"Landon-Cardinal","sequence":"additional","affiliation":[]},{"given":"Kristan","family":"Temme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,5,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.052337","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052337","note":"arxivid:1601.01324"},{"id":"arxiv:2107.01628","type":"journal-article","publisher":"Cambridge University Press (CUP)","abstract":"<jats:title>Abstract</jats:title>\n\t  <jats:p>We show that every ergodic Davies generator associated to <jats:italic>any</jats:italic> 2D Kitaev’s quantum double model has a nonvanishing spectral gap in the thermodynamic limit. This validates rigorously the extended belief that those models are useless as self-correcting quantum memories, even in the non-abelian case. The proof uses recent ideas and results regarding the characterization of the spectral gap for parent Hamiltonians associated to Projected Entangled Pair States in terms of a bulk-boundary correspondence.</jats:p>","DOI":"10.1017/fms.2023.98","source":"Crossref","title":"Thermalization in Kitaev’s quantum double models via tensor network techniques","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0003-1709-1220","authenticated-orcid":false,"given":"Angelo","family":"Lucia","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2990-791X","authenticated-orcid":false,"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8600-7083","authenticated-orcid":false,"given":"Antonio","family":"Pérez-Hernández","sequence":"additional","affiliation":[]}],"container-title":"Forum of Mathematics, Sigma","original-title":[],"language":"en","issued":{"date-parts":[[2023]]},"URL":"http://dx.doi.org/10.1017/fms.2023.98","ISSN":["2050-5094"],"container-title-short":"Forum of Mathematics, Sigma","page":"e107","note":"alternative-id:S2050509423000981\narxivid:2107.01628"},{"id":"arxiv:1707.04564","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-017-2960-4","page":"645-689","source":"Crossref","title":"Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter","volume":"355","author":[{"given":"Iris","family":"Cong","sequence":"first","affiliation":[]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,7,11]]},"URL":"http://dx.doi.org/10.1007/s00220-017-2960-4","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"updated-by:{\"DOI\":\"10.1007/s00220-018-3134-8\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2018,4,16]],\"date-time\":\"2018-04-16T00:00:00Z\",\"timestamp\":1523836800000}}\nalternative-id:2960\narxivid:1707.04564"},{"id":"arxiv:1707.05490","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.119.170504","source":"Crossref","title":"Universal Quantum Computation with Gapped Boundaries","volume":"119","author":[{"given":"Iris","family":"Cong","sequence":"first","affiliation":[]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.119.170504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"170504","note":"arxivid:1707.05490"},{"id":"arxiv:1609.02037","type":"article-journal","author":[{"given":"Iris","family":"Cong"},{"given":"Meng","family":"Cheng"},{"given":"Zhenghan","family":"Wang"}],"title":"Topological Quantum Computation with Gapped Boundaries","issued":{"date-parts":[[2016,10,17]]},"note":"arxivid:1609.02037\narxiv_version_number:2"},{"id":"arxiv:2602.19558","type":"article-journal","author":[{"given":"Ben T.","family":"McDonough"},{"given":"Jian-Hao","family":"Zhang"},{"given":"Victor V.","family":"Albert"},{"given":"Andrew","family":"Lucas"}],"title":"Calderbank-Shor-Steane codes on group-valued qudits","issued":{"date-parts":[[2026,2,23]]},"note":"arxivid:2602.19558\narxiv_version_number:1"},{"id":"arxiv:2603.05502","type":"article-journal","author":[{"given":"Naren","family":"Manjunath"},{"given":"Vieri","family":"Mattei"},{"given":"Apoorv","family":"Tiwari"},{"given":"Tyler D.","family":"Ellison"}],"title":"Universal quantum computation with group surface codes","issued":{"date-parts":[[2026,3,5]]},"note":"arxivid:2603.05502\narxiv_version_number:1"},{"id":"arxiv:1811.06738","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.100.012338","source":"Crossref","title":"Universal quantum computation in the surface code using non-Abelian islands","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0002-4772-9232","authenticated-orcid":true,"given":"Katharina","family":"Laubscher","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]},{"given":"James R.","family":"Wootton","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.100.012338","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012338","note":"arxivid:1811.06738"},{"id":"arxiv:2505.18265","type":"article-journal","author":[{"given":"Rohith","family":"Sajith"},{"given":"Zijian","family":"Song"},{"given":"Brenden","family":"Roberts"},{"given":"Varun","family":"Menon"},{"given":"Yabo","family":"Li"}],"title":"Non-Clifford gates between stabilizer codes via non-Abelian topological order","issued":{"date-parts":[[2025,5,23]]},"note":"arxivid:2505.18265\narxiv_version_number:1"},{"id":"arxiv:2510.20890","type":"article-journal","author":[{"given":"Sheng-Jie","family":"Huang"},{"given":"Alison","family":"Warman"},{"given":"Sakura","family":"Schafer-Nameki"},{"given":"Yanzhu","family":"Chen"}],"title":"Hybrid Lattice Surgery: Non-Clifford Gates via Non-Abelian Surface Codes","issued":{"date-parts":[[2026,1,5]]},"note":"arxivid:2510.20890\narxiv_version_number:2"},{"id":"arxiv:1401.7096","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","DOI":"10.1007/s11128-015-1016-y","page":"2687-2727","source":"Crossref","title":"Universal quantum computation with weakly integral anyons","volume":"14","author":[{"given":"Shawn X.","family":"Cui","sequence":"first","affiliation":[]},{"given":"Seung-Moon","family":"Hong","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2015,5,12]]},"URL":"http://dx.doi.org/10.1007/s11128-015-1016-y","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:1016\narxivid:1401.7096"},{"id":"arxiv:2411.09697","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41534-025-01063-4","source":"Crossref","title":"A universal circuit set using the S3 quantum double","volume":"11","author":[{"given":"Liyuan","family":"Chen","sequence":"first","affiliation":[]},{"given":"Yuanjie","family":"Ren","sequence":"additional","affiliation":[]},{"given":"Ruihua","family":"Fan","sequence":"additional","affiliation":[]},{"given":"Arthur","family":"Jaffe","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,3]]},"URL":"http://dx.doi.org/10.1038/s41534-025-01063-4","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"112","note":"alternative-id:1063\narxivid:2411.09697"},{"id":"arxiv:2512.13777","type":"article-journal","author":[{"given":"Alison","family":"Warman"},{"given":"Sakura","family":"Schafer-Nameki"}],"title":"Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes","issued":{"date-parts":[[2026,1,16]]},"note":"arxivid:2512.13777\narxiv_version_number:2"},{"id":"arxiv:2601.20956","type":"article-journal","author":[{"given":"Chiu Fan Bowen","family":"Lo"},{"given":"Anasuya","family":"Lyons"},{"given":"Dan","family":"Gresh"},{"given":"Michael","family":"Mills"},{"given":"Peter E.","family":"Siegfried"},{"given":"Maxwell D.","family":"Urmey"},{"given":"Nathanan","family":"Tantivasadakarn"},{"given":"Henrik","family":"Dreyer"},{"given":"Ashvin","family":"Vishwanath"},{"given":"Ruben","family":"Verresen"},{"given":"Mohsin","family":"Iqbal"}],"title":"Universal Topological Gates from Braiding and Fusing Anyons on Quantum Hardware","issued":{"date-parts":[[2026,1,28]]},"note":"arxivid:2601.20956\narxiv_version_number:1"},{"id":"arxiv:2502.14974","type":"article-journal","author":[{"given":"Chiu Fan Bowen","family":"Lo"},{"given":"Anasuya","family":"Lyons"},{"given":"Ruben","family":"Verresen"},{"given":"Ashvin","family":"Vishwanath"},{"given":"Nathanan","family":"Tantivasadakarn"}],"title":"Universal Quantum Computation with the $S_3$ Quantum Double: A Pedagogical Exposition","issued":{"date-parts":[[2025,2,20]]},"note":"arxivid:2502.14974\narxiv_version_number:1"},{"id":"arxiv:2405.11719","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    We construct a family of infinitely many candidate non-Abelian self-correcting topological quantum memories in\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>D</a:mi>\n                      <a:mo>≥</a:mo>\n                      <a:mspace width=\"0.2em\"/>\n                      <a:mn>5</a:mn>\n                      <a:mo>+</a:mo>\n                      <a:mn>1</a:mn>\n                    </a:math>\n                    spacetime dimensions without particle excitations using local commuting non-Pauli stabilizer lattice models and field theories of\n                    <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <d:msubsup>\n                        <d:mrow>\n                          <d:mi mathvariant=\"double-struck\">Z</d:mi>\n                        </d:mrow>\n                        <d:mn>2</d:mn>\n                        <d:mn>3</d:mn>\n                      </d:msubsup>\n                    </d:math>\n                    higher-form gauge fields with nontrivial topological action. We call such non-Pauli stabilizer models magic stabilizer codes. The family of topological orders have Abelian electric excitations and non-Abelian magnetic excitations that obey Ising-like fusion rules and non-Abelian braiding, including Borromean ring type braiding which is a signature of non-Abelian topological order, generalizing the dihedral group\n                    <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <g:msub>\n                        <g:mrow>\n                          <g:mi mathvariant=\"double-struck\">D</g:mi>\n                        </g:mrow>\n                        <g:mn>8</g:mn>\n                      </g:msub>\n                    </g:math>\n                    gauge theory in\n                    <j:math xmlns:j=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <j:mo stretchy=\"false\">(</j:mo>\n                      <j:mn>2</j:mn>\n                      <j:mo>+</j:mo>\n                      <j:mn>1</j:mn>\n                      <j:mo stretchy=\"false\">)</j:mo>\n                      <j:mrow>\n                        <j:mrow>\n                          <j:mi mathvariant=\"normal\">D</j:mi>\n                        </j:mrow>\n                      </j:mrow>\n                    </j:math>\n                    . The simplest example includes a non-Abelian self-correcting memory in\n                    <o:math xmlns:o=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <o:mo stretchy=\"false\">(</o:mo>\n                      <o:mn>5</o:mn>\n                      <o:mo>+</o:mo>\n                      <o:mn>1</o:mn>\n                      <o:mo stretchy=\"false\">)</o:mo>\n                      <o:mrow>\n                        <o:mrow>\n                          <o:mi mathvariant=\"normal\">D</o:mi>\n                        </o:mrow>\n                      </o:mrow>\n                    </o:math>\n                    with Abelian loop excitations and non-Abelian membrane excitations. We prove the self-correction property and the thermal stability and devise a probabilistic local cellular-automaton decoder. We also construct fault-tolerant non-Clifford CCZ logical gate using constant-depth circuits from higher-cup products in the 5D non-Abelian code. The use of higher-cup products and non-Pauli stabilizers allows us to get an\n                    <t:math xmlns:t=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <t:mi>O</t:mi>\n                      <t:mo stretchy=\"false\">(</t:mo>\n                      <t:msup>\n                        <t:mi>n</t:mi>\n                        <t:mrow>\n                          <t:mn>2</t:mn>\n                          <t:mo>/</t:mo>\n                          <t:mn>5</t:mn>\n                        </t:mrow>\n                      </t:msup>\n                      <t:mo stretchy=\"false\">)</t:mo>\n                    </t:math>\n                    distance overcoming the\n                    <x:math xmlns:x=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <x:mi>O</x:mi>\n                      <x:mo stretchy=\"false\">(</x:mo>\n                      <x:msup>\n                        <x:mi>n</x:mi>\n                        <x:mrow>\n                          <x:mn>1</x:mn>\n                          <x:mo>/</x:mo>\n                          <x:mn>3</x:mn>\n                        </x:mrow>\n                      </x:msup>\n                      <x:mo stretchy=\"false\">)</x:mo>\n                    </x:math>\n                    distance barrier in conventional topological stabilizer codes, including the 3D color code and the 6D self-correcting color code.\n                  </jats:p>","DOI":"10.1103/hkgn-hhqx","source":"Crossref","title":"Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the\n                    <i>n</i>\n                    <sup>1/3</sup>\n                    Distance Barrier","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-4764-1476","authenticated-orcid":true,"given":"Po-Shen","family":"Hsin","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/046rm7j60","id-type":"ROR","asserted-by":"publisher"}],"name":"Mani L. Bhaumik Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/0220mzb33","id-type":"ROR","asserted-by":"publisher"}],"name":"King’s College London"}]},{"given":"Ryohei","family":"Kobayashi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/00f809463","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Advanced Study"}]},{"ORCID":"https://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM T.J. Watson Research Center"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,10]]},"URL":"http://dx.doi.org/10.1103/hkgn-hhqx","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040360","note":"arxivid:2405.11719"},{"id":"arxiv:2502.00998","type":"article-journal","author":[{"given":"Sheng-Jie","family":"Huang"},{"given":"Yanzhu","family":"Chen"}],"title":"Generating logical magic states with the aid of non-Abelian topological order","issued":{"date-parts":[[2025,7,1]]},"note":"arxivid:2502.00998\narxiv_version_number:2"},{"id":"arxiv:0705.0665","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-008-0441-5","page":"845-872","source":"Crossref","title":"Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups","volume":"279","author":[{"given":"Deepak","family":"Naidu","sequence":"first","affiliation":[]},{"given":"Dmitri","family":"Nikshych","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2008,2,21]]},"URL":"http://dx.doi.org/10.1007/s00220-008-0441-5","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:441\narxivid:0705.0665"},{"id":"arxiv:2112.11394","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.3.010353","source":"Crossref","title":"Pauli Stabilizer Models of Twisted Quantum Doubles","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-1740-6889","authenticated-orcid":true,"given":"Tyler D.","family":"Ellison","sequence":"first","affiliation":[]},{"given":"Yu-An","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[]},{"given":"Wilbur","family":"Shirley","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,3,30]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.010353","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010353","note":"update-to:{\"updated\":{\"date-parts\":[[2022,5,3]],\"date-time\":\"2022-05-03T00:00:00Z\",\"timestamp\":1651536000000},\"DOI\":\"10.1103/prxquantum.3.010353\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\n{\"updated\":{\"date-parts\":[[2022,12,20]],\"date-time\":\"2022-12-20T00:00:00Z\",\"timestamp\":1671494400000},\"DOI\":\"10.1103/prxquantum.3.010353\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2022,12,20]],\"date-time\":\"2022-12-20T00:00:00Z\",\"timestamp\":1671494400000},\"DOI\":\"10.1103/prxquantum.3.010353\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2112.11394"},{"id":"doi:10.1016/0920-5632(91)90123-V","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0920-5632(91)90123-v","page":"60-72","source":"Crossref","title":"Quasi hope algebras, group cohomology and orbifold models","volume":"18","author":[{"given":"R.","family":"Dijkgraaf","sequence":"first","affiliation":[]},{"given":"V.","family":"Pasquier","sequence":"additional","affiliation":[]},{"given":"P.","family":"Roche","sequence":"additional","affiliation":[]}],"container-title":"Nuclear Physics B - Proceedings Supplements","original-title":[],"language":"en","issued":{"date-parts":[[1991,1]]},"URL":"http://dx.doi.org/10.1016/0920-5632(91)90123-V","ISSN":["0920-5632"],"container-title-short":"Nuclear Physics B - Proceedings Supplements","note":"alternative-id:092056329190123V"},{"id":"arxiv:2107.13091","type":"journal-article","publisher":"Stichting SciPost","issue":"3","abstract":"<jats:p>\n                    A 2+1-dimensional topological quantum field theory (TQFT) may or may\nnot admit topological (gapped) boundary conditions. A famous necessary,\nbut not sufficient, condition for the existence of a topological\nboundary condition is that the chiral central charge\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>c_-</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:msub>\n                            <mml:mi>c</mml:mi>\n                            <mml:mo>−</mml:mo>\n                          </mml:msub>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    has to vanish. In this paper, we consider conditions associated with\n``higher\" central charges, which have been introduced recently in\nthe math literature. In terms of these new obstructions, we identify\nnecessary and sufficient conditions for the existence of a topological\nboundary in the case of bosonic, Abelian TQFTs, providing an alternative\nto the identification of a Lagrangian subgroup. Our proof relies on\ngeneral aspects of gauging generalized global symmetries. For\nnon-Abelian TQFTs, we give a geometric way of studying topological\nboundary conditions, and explain certain necessary conditions given\nagain in terms of the higher central charges. Along the way, we find a\ncurious duality in the partition functions of Abelian TQFTs, which begs\nfor an explanation via the 3d-3d correspondence.\n                  </jats:p>","DOI":"10.21468/scipostphys.13.3.067","source":"Crossref","title":"Higher central charges and topological boundaries in 2+1-dimensional  TQFTs","volume":"13","author":[{"given":"Justin","family":"Kaidi","sequence":"first","affiliation":[{"name":"Stony Brook University"}]},{"given":"Zohar","family":"Komargodski","sequence":"additional","affiliation":[{"name":"Stony Brook University"}]},{"given":"Kantaro","family":"Ohmori","sequence":"additional","affiliation":[{"name":"University of Tokyo"},{"name":"Stony Brook University"}]},{"given":"Sahand","family":"Seifnashri","sequence":"additional","affiliation":[{"name":"Stony Brook University"}]},{"given":"Shu-Heng","family":"Shao","sequence":"additional","affiliation":[{"name":"C.N. Yang Institute for Theoretical Physics"},{"name":"Institute for Advanced Study"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2022,9,26]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.13.3.067","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"067","note":"arxivid:2107.13091"},{"id":"arxiv:2411.08675","type":"article-journal","author":[{"given":"Shawn X.","family":"Cui"},{"given":"César","family":"Galindo"},{"given":"Diego","family":"Romero"}],"title":"Twisted Kitaev quantum double model as local topological order","issued":{"date-parts":[[2025,2,25]]},"note":"arxivid:2411.08675\narxiv_version_number:2"},{"id":"arxiv:1905.08673","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","abstract":"<jats:title>A<jats:sc>bstract</jats:sc></jats:title><jats:p>We consider lattice Hamiltonian realizations of (<jats:italic>d</jats:italic>+1)-dimensional Dijkgraaf- Witten theory. In (2+1) d, it is well-known that the Hamiltonian yields point-like excita- tions classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalisation of this strategy that is valid in any dimensions. We then apply this generalisation to derive the algebraic structure of loop-like excitations in (3+1) d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1) d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and an<jats:italic>R</jats:italic>-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a man- ifold that is<jats:italic>n</jats:italic>-times compactified can be expressed in terms of another model in<jats:italic>n</jats:italic>-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues.</jats:p>","DOI":"10.1007/jhep10(2019)216","source":"Crossref","title":"Tube algebras, excitations statistics and compactification in gauge models of topological phases","volume":"2019","author":[{"given":"Alex","family":"Bullivant","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2337-5772","authenticated-orcid":false,"given":"Clement","family":"Delcamp","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,10]]},"URL":"http://dx.doi.org/10.1007/JHEP10(2019)216","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"216","note":"alternative-id:11638\narxivid:1905.08673"},{"id":"arxiv:1301.0861","type":"article-journal","author":[{"given":"Xie","family":"Chen"},{"given":"Zheng-Cheng","family":"Gu"},{"given":"Zheng-Xin","family":"Liu"},{"given":"Xiao-Gang","family":"Wen"}],"title":"Symmetry protected topological orders in interacting bosonic systems","issued":{"date-parts":[[2013,1,5]]},"note":"arxivid:1301.0861\narxiv_version_number:1"},{"id":"arxiv:1008.0654","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/j.nuclphysb.2010.12.017","page":"393-435","source":"Crossref","title":"Topological boundary conditions in abelian Chern–Simons theory","volume":"845","author":[{"given":"Anton","family":"Kapustin","sequence":"first","affiliation":[]},{"given":"Natalia","family":"Saulina","sequence":"additional","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[2011,4]]},"URL":"http://dx.doi.org/10.1016/j.nuclphysb.2010.12.017","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"alternative-id:S0550321310006723\narxivid:1008.0654"},{"id":"arxiv:2403.12119","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We propose a family of explicit geometrically local circuits on a 2-dimensional planar grid of qudits, realizing any abelian non-chiral topological phase as an actively error-corrected fault-tolerant memory. These circuits are constructed from measuring 1-form symmetries in discrete fixed-point path integrals, which we express through cellular cohomology and higher-order cup products. The specific path integral we use is the abelian Dijkgraaf-Witten state sum on a 3-dimensional cellulation, which is a spacetime representation of the twisted quantum double model. The resulting circuits are based on a syndrome extraction circuit of the (qudit) stabilizer toric code, into which we insert non-Clifford phase gates that implement the “twist&amp;apos;&amp;apos;. The overhead compared to the toric code is moderate, in contrast to known constructions for twisted abelian phases. We also show that other architectures for the (qudit) toric code phase, like measurement-based topological quantum computation or Floquet codes, can be enriched with phase gates to implement twisted quantum doubles instead of their untwisted versions. As a further result, we prove fault tolerance under arbitrary local (including non-Pauli) noise for a very general class of topological circuits that we call 1-form symmetric fixed-point circuits. This notion unifies the circuits in this paper as well as the stabilizer toric code, subsystem toric code, measurement-based topological quantum computation, or the (CSS) honeycomb Floquet code. We also demonstrate how our method can be adapted to construct fault-tolerant circuits for specific non-Abelian phases. In the appendix we present an explicit combinatorial procedure to define formulas for higher cup products on arbitrary cellulations, which might be interesting in its own right to the TQFT and topological-phases community.</jats:p>","DOI":"10.22331/q-2025-03-25-1673","page":"1673","source":"Crossref","title":"Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases","volume":"9","author":[{"given":"Andreas","family":"Bauer","sequence":"first","affiliation":[{"name":"Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"MIT Department of Mechanical Engineering, 77 Massachusetts Avenue, Cambridge, MA 02139, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,25]]},"URL":"http://dx.doi.org/10.22331/q-2025-03-25-1673","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2403.12119"},{"id":"arxiv:1810.08204","type":"journal-article","publisher":"IOP Publishing","issue":"5","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>We present a full quantum error correcting procedure with the semion code: an off-shell extension of the double-semion model. We construct open-string operators that recover the quantum memory from arbitrary errors and closed-string operators that implement the basic logical operations for information processing. Physically, the new open-string operators provide a detailed microscopic description of the creation of semions at their end-points. Remarkably, topological properties of the string operators are determined using fundamental properties of the Hamiltonian, namely, the fact that it is composed of commuting local terms squaring to the identity. In all, the semion code is a topological code that, unlike previously studied topological codes, it is of non-CSS type and fits into the stabilizer formalism. This is in sharp contrast with previous attempts yielding non-commutative codes.</jats:p>","DOI":"10.1088/1367-2630/ab1ed8","page":"053035","source":"Crossref","title":"Quantum error correction with the semion code","volume":"21","author":[{"given":"G","family":"Dauphinais","sequence":"first","affiliation":[]},{"given":"L","family":"Ortiz","sequence":"additional","affiliation":[]},{"given":"S","family":"Varona","sequence":"additional","affiliation":[]},{"given":"M A","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2019,5,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ab1ed8","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1810.08204"},{"id":"arxiv:2001.11516","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.</jats:p>","DOI":"10.22331/q-2021-02-17-398","page":"398","source":"Crossref","title":"Non-Pauli topological stabilizer codes from twisted quantum doubles","volume":"5","author":[{"given":"Julio Carlos","family":"Magdalena de la Fuente","sequence":"first","affiliation":[{"name":"JARA Institute for Quantum Information, RWTH Aachen University, Aachen, Germany"},{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Nicolas","family":"Tarantino","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,17]]},"URL":"http://dx.doi.org/10.22331/q-2021-02-17-398","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2001.11516"},{"id":"doi:10.1007/BF02096988","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/bf02096988","page":"393-429","source":"Crossref","title":"Topological gauge theories and group cohomology","volume":"129","author":[{"given":"Robbert","family":"Dijkgraaf","sequence":"first","affiliation":[]},{"given":"Edward","family":"Witten","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1990,4]]},"URL":"http://dx.doi.org/10.1007/BF02096988","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF02096988"},{"id":"arxiv:1404.5327","type":"journal-article","publisher":"AIP Publishing","issue":"5","abstract":"<jats:p>We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = eiπ/4 and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.</jats:p>","DOI":"10.1063/1.4920923","source":"Crossref","title":"A non-commuting stabilizer formalism","volume":"56","author":[{"given":"Xiaotong","family":"Ni","sequence":"first","affiliation":[{"name":"Max-Planck-Institut für Quantenoptik 1 , Hans-Kopfermann-Str. 1, 85748 Garching, Germany"}]},{"given":"Oliver","family":"Buerschaper","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics 2 , 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada"},{"name":"Freie Universität Berlin 3 Dahlem Center for Complex Quantum Systems, , Arnimallee 14, 14195 Berlin, Germany"}]},{"given":"Maarten","family":"Van den Nest","sequence":"additional","affiliation":[{"name":"Max-Planck-Institut für Quantenoptik 1 , Hans-Kopfermann-Str. 1, 85748 Garching, Germany"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2015,5,1]]},"URL":"http://dx.doi.org/10.1063/1.4920923","ISSN":["0022-2488","1089-7658"],"page":"052201","note":"arxivid:1404.5327"},{"id":"arxiv:1404.4618","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.91.075114","source":"Crossref","title":"Universal topological data for gapped quantum liquids in three dimensions and fusion algebra for non-Abelian string excitations","volume":"91","author":[{"given":"Heidar","family":"Moradi","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2015,2,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.91.075114","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"075114","note":"arxivid:1404.4618"},{"id":"arxiv:1709.04924","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","DOI":"10.1007/jhep12(2017)128","source":"Crossref","title":"Excitation basis for (3+1)d topological phases","volume":"2017","author":[{"given":"Clement","family":"Delcamp","sequence":"first","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,12]]},"URL":"http://dx.doi.org/10.1007/JHEP12(2017)128","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"128","note":"alternative-id:7338\narxivid:1709.04924"},{"id":"arxiv:1212.0835","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.87.155115","source":"Crossref","title":"Classification of symmetry enriched topological phases with exactly solvable models","volume":"87","author":[{"given":"Andrej","family":"Mesaros","sequence":"first","affiliation":[]},{"given":"Ying","family":"Ran","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2013,4,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.87.155115","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"155115","note":"arxivid:1212.0835"},{"id":"arxiv:1404.1062","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.4.031048","source":"Crossref","title":"Generalized Modular Transformations in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mi mathvariant=\"normal\">D</mml:mi></mml:mrow></mml:math>Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding","volume":"4","author":[{"given":"Shenghan","family":"Jiang","sequence":"first","affiliation":[]},{"given":"Andrej","family":"Mesaros","sequence":"additional","affiliation":[]},{"given":"Ying","family":"Ran","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.4.031048","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031048","note":"arxivid:1404.1062"},{"id":"arxiv:1807.11083","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>We extend the twisted gauge theory model of topological orders in three spatial dimensions to the case where the three spaces have two dimensional boundaries. We achieve this by systematically constructing the boundary Hamiltonians that are compatible with the bulk Hamiltonian. Given the bulk Hamiltonian defined by a gauge group <jats:italic>G</jats:italic> and a four-cocycle <jats:italic>ω</jats:italic> in the fourth cohomology group of <jats:italic>G</jats:italic> over U(1), we construct a gapped boundary Hamiltonian using {<jats:italic>K, α</jats:italic>}, with a subgroup <jats:italic>K</jats:italic> ⊆ <jats:italic>G</jats:italic> and a 3-cochain α of <jats:italic>K</jats:italic> over U(1), which satisfies the generalized Frobenius condition. The Hamiltonian is invariant under the topological renormalization group flow (via Pachner moves). Each solution {<jats:italic>K, α</jats:italic>} to the generalized Frobenius condition specifies a gapped boundary condition. We derive a closed-form formula of the ground state degeneracy of the model on a three-cylinder, which can be naturally generalized to three-spaces with more boundaries. We also derive the explicit ground-state wavefunction of the model on a three-ball. The ground state degeneracy and ground-state wavefunction are both presented solely in terms of the input data of the model, namely, {<jats:italic>G, ω, K, α</jats:italic>}.</jats:p>","DOI":"10.1007/jhep10(2018)114","source":"Crossref","title":"Gapped boundary theory of the twisted gauge theory model of three-dimensional topological orders","volume":"2018","author":[{"given":"Hongyu","family":"Wang","sequence":"first","affiliation":[]},{"given":"Yingcheng","family":"Li","sequence":"additional","affiliation":[]},{"given":"Yuting","family":"Hu","sequence":"additional","affiliation":[]},{"given":"Yidun","family":"Wan","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2018,10]]},"URL":"http://dx.doi.org/10.1007/JHEP10(2018)114","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"114","note":"alternative-id:9214\narxivid:1807.11083"},{"id":"arxiv:2006.06536","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.</jats:p>","DOI":"10.1007/jhep07(2021)025","source":"Crossref","title":"Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases","volume":"2021","author":[{"given":"Alex","family":"Bullivant","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2337-5772","authenticated-orcid":false,"given":"Clement","family":"Delcamp","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,7]]},"URL":"http://dx.doi.org/10.1007/JHEP07(2021)025","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"25","note":"alternative-id:16123\narxivid:2006.06536"},{"id":"arxiv:2401.13042","type":"article-journal","author":[{"given":"Joe","family":"Huxford"},{"given":"Dung Xuan","family":"Nguyen"},{"given":"Yong Baek","family":"Kim"}],"title":"Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge","issued":{"date-parts":[[2024,1,25]]},"note":"arxivid:2401.13042\narxiv_version_number:2"},{"id":"arxiv:1404.7854","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevb.91.035134","source":"Crossref","title":"Non-Abelian string and particle braiding in topological order: Modular<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>SL</mml:mi><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"double-struck\">Z</mml:mi><mml:mo>)</mml:mo></mml:math>representation and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-dimensional twisted gauge theory","volume":"91","author":[{"given":"Juven C.","family":"Wang","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2015,1,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.91.035134","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"035134","note":"arxivid:1404.7854"},{"id":"arxiv:1409.3216","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.92.045101","source":"Crossref","title":"Twisted gauge theory model of topological phases in three dimensions","volume":"92","author":[{"given":"Yidun","family":"Wan","sequence":"first","affiliation":[]},{"given":"Juven C.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Huan","family":"He","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2015,7,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.92.045101","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"045101","note":"arxivid:1409.3216"},{"id":"arxiv:hep-th/9111004","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf02096860","page":"435-472","source":"Crossref","title":"Chern-Simons theory with finite gauge group","volume":"156","author":[{"given":"Daniel S.","family":"Freed","sequence":"first","affiliation":[]},{"given":"Frank","family":"Quinn","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1993,10]]},"URL":"http://dx.doi.org/10.1007/BF02096860","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF02096860\narxivid:hep-th/9111004"},{"id":"arxiv:2212.03935","type":"article-journal","author":[{"given":"Eric","family":"Culf"},{"given":"Thomas","family":"Vidick"},{"given":"Victor V.","family":"Albert"}],"title":"Group coset monogamy games and an application to device-independent continuous-variable QKD","issued":{"date-parts":[[2022,12,7]]},"note":"arxivid:2212.03935\narxiv_version_number:1"},{"id":"doi:10.1103/Physics.13.s111","type":"journal-article","publisher":"American Physical Society (APS)","DOI":"10.1103/physics.13.s111","source":"Crossref","title":"Protecting Molecular Qubits from Noise","volume":"13","author":[{"given":"Erika K.","family":"Carlson","sequence":"first","affiliation":[]}],"container-title":"Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,1]]},"URL":"http://dx.doi.org/10.1103/Physics.13.s111","ISSN":["1943-2879"],"container-title-short":"Physics","page":"s111"},{"id":"arxiv:2403.04572","type":"article-journal","author":[{"given":"Victor V.","family":"Albert"},{"given":"Eric","family":"Kubischta"},{"given":"Mikhail","family":"Lemeshko"},{"given":"Lee R.","family":"Liu"}],"title":"Quantum theory of molecular orientations","issued":{"date-parts":[[2026,1,13]]},"note":"arxivid:2403.04572\narxiv_version_number:4"},{"id":"arxiv:1902.07714","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.10.041018","source":"Crossref","title":"Continuous Symmetries and Approximate Quantum Error Correction","volume":"10","author":[{"given":"Philippe","family":"Faist","sequence":"first","affiliation":[]},{"given":"Sepehr","family":"Nezami","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]},{"given":"Grant","family":"Salton","sequence":"additional","affiliation":[]},{"given":"Fernando","family":"Pastawski","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Hayden","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2020,10,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.10.041018","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041018","note":"arxivid:1902.07714"},{"id":"doi:10.1007/978-1-4614-7116-5","type":"book","publisher":"Springer New York","DOI":"10.1007/978-1-4614-7116-5","source":"Crossref","title":"Quantum Theory for Mathematicians","author":[{"given":"Brian C.","family":"Hall","sequence":"first","affiliation":[]}],"container-title":"Graduate Texts in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2013]]},"ISBN":["9781461471158","9781461471165"],"URL":"http://dx.doi.org/10.1007/978-1-4614-7116-5","ISSN":["0072-5285","2197-5612"],"note":"publisher-location:New York, NY"},{"id":"arxiv:1409.3208","type":"article-journal","author":[{"given":"Juan","family":"Bermejo-Vega"},{"given":"Cedric Yen-Yu","family":"Lin"},{"given":"Maarten Van den","family":"Nest"}],"title":"Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems","issued":{"date-parts":[[2015,1,20]]},"note":"arxivid:1409.3208\narxiv_version_number:2"},{"id":"arxiv:2311.07679","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.110.022402","source":"Crossref","title":"Multimode rotation-symmetric bosonic codes from homological rotor codes","volume":"110","author":[{"given":"Yijia","family":"Xu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"given":"Yixu","family":"Wang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"},{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.022402","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022402","note":"arxivid:2311.07679"},{"id":"doi:10.1093/acprof:oso/9780199681181","type":"book-chapter","publisher":"Oxford University PressOxford","abstract":"<jats:title>Abstract</jats:title><jats:p>This chapter introduces the basic concepts of quantum information processing with superconducting quantum circuits. It starts with a demonstration of how,in these electrodynamic systems, both the superconducting gap and the long-range Coulomb interactions contribute to the existence of collective modesthat have extremely low dissipationand that can be quantized. The chapter also gives in-depth coverage of quantum harmonic oscillators in all their glory from various perspectives. Superconducting qubits are then introduced, starting with the fundamental theory of the Josephson effect and moving on to a presentation of the variety of existing superconducting qubits. Particular attention is given to the dispersive readout of qubits through their interaction with a cavity into which flying modes can be scattered. Finally, the opportunities offered for quantum information processing by employing the cavity as a quantum bus are presented.</jats:p>","DOI":"10.1093/acprof:oso/9780199681181.003.0003","page":"113-256","source":"Crossref","title":"Circuit QED: superconducting qubits coupled to microwave photons","author":[{"given":"S. M.","family":"Girvin","sequence":"first","affiliation":[]}],"container-title":"Quantum Machines: Measurement and Control of Engineered Quantum Systems","original-title":[],"language":"en","issued":{"date-parts":[[2014,6,12]]},"ISBN":["019968118X","9780199681181","9780191761454"],"URL":"http://dx.doi.org/10.1093/acprof:oso/9780199681181.003.0003","note":"edition-number:1"},{"id":"arxiv:2303.13723","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-023-04905-4","source":"Crossref","title":"Homological Quantum Rotor Codes: Logical Qubits from Torsion","volume":"405","author":[{"ORCID":"https://orcid.org/0000-0002-3445-0179","authenticated-orcid":false,"given":"Christophe","family":"Vuillot","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8707-0532","authenticated-orcid":false,"given":"Alessandro","family":"Ciani","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0218-6614","authenticated-orcid":false,"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,2]]},"URL":"http://dx.doi.org/10.1007/s00220-023-04905-4","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","page":"53","note":"alternative-id:4905\narxivid:2303.13723"},{"id":"arxiv:cond-mat/0609441","type":"article-journal","author":[{"given":"Alexei","family":"Kitaev"}],"title":"Protected qubit based on a superconducting current mirror","issued":{"date-parts":[[2006,9,19]]},"note":"arxivid:cond-mat/0609441\narxiv_version_number:2"},{"id":"manual:-A.-Yu.-Kitaev-PROTECTED-QUBI","type":"document","author":[{"family":"Kitaev","given":"A.Yu"},{"family":"MIRROR","given":"P.R.O.T.E.C.T.E.D.Q.U.B.I.T.B.A.S.E.D.O.N.S.U.P.E.R.C.O.N.D.U.C.T.I.N.G.C.U.R.R.E.N.T."}],"title":"United States Patent","volume":"Number 7858966B2","issued":"2006"},{"id":"arxiv:1908.04615","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevb.100.224507","source":"Crossref","title":"Spectrum and coherence properties of the current-mirror qubit","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0002-1952-896X","authenticated-orcid":true,"given":"D. K.","family":"Weiss","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4542-3739","authenticated-orcid":true,"given":"Andy C. Y.","family":"Li","sequence":"additional","affiliation":[]},{"given":"D. G.","family":"Ferguson","sequence":"additional","affiliation":[]},{"given":"Jens","family":"Koch","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.100.224507","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"224507","note":"arxivid:1908.04615"},{"id":"arxiv:1302.4122","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.87.052306","source":"Crossref","title":"Protected gates for superconducting qubits","volume":"87","author":[{"given":"Peter","family":"Brooks","sequence":"first","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,5,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.052306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052306","note":"arxivid:1302.4122"},{"id":"arxiv:1710.07256","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.8.021047","source":"Crossref","title":"Disjointness of Stabilizer Codes and Limitations on Fault-Tolerant Logical Gates","volume":"8","author":[{"given":"Tomas","family":"Jochym-O’Connor","sequence":"first","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.8.021047","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021047","note":"arxivid:1710.07256"},{"id":"arxiv:2411.04993","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>\n                    We construct a family of two-dimensional topological stabilizer codes on continuous variable (CV) degrees of freedom, which generalize homological rotor codes and the toric-GKP code. Our topological codes are built using the concept of boson condensation—we start from a parent stabilizer code based on an\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi mathvariant=\"double-struck\">R</a:mi>\n                    </a:math>\n                    gauge theory and condense various bosonic excitations. This produces a large class of topological CV stabilizer codes, including ones that are characterized by the anyon theories of\n                    <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <d:mi mathvariant=\"normal\">U</d:mi>\n                      <d:mo stretchy=\"false\">(</d:mo>\n                      <d:mn>1</d:mn>\n                      <d:msub>\n                        <d:mo stretchy=\"false\">)</d:mo>\n                        <d:mrow>\n                          <d:mn>2</d:mn>\n                          <d:mi>n</d:mi>\n                        </d:mrow>\n                      </d:msub>\n                      <d:mo>×</d:mo>\n                      <d:mi mathvariant=\"normal\">U</d:mi>\n                      <d:mo stretchy=\"false\">(</d:mo>\n                      <d:mn>1</d:mn>\n                      <d:msub>\n                        <d:mo stretchy=\"false\">)</d:mo>\n                        <d:mrow>\n                          <d:mo>−</d:mo>\n                          <d:mn>2</d:mn>\n                          <d:mi>m</d:mi>\n                        </d:mrow>\n                      </d:msub>\n                    </d:math>\n                    Chern-Simons theories, for arbitrary pairs of positive integers\n                    <l:math xmlns:l=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <l:mo stretchy=\"false\">(</l:mo>\n                      <l:mi>n</l:mi>\n                      <l:mo>,</l:mo>\n                      <l:mi>m</l:mi>\n                      <l:mo stretchy=\"false\">)</l:mo>\n                    </l:math>\n                    . Most notably, this includes anyon theories that are nonchiral and nevertheless do not admit a gapped boundary. It is widely believed that such anyon theories cannot be realized by any stabilizer model on finite-dimensional systems. We conjecture that these CV codes go beyond codes obtained from concatenating a topological qudit code with a local encoding into CVs, and thus constitute the first example of topological codes that are intrinsic to CV systems. Moreover, we study the Hamiltonians associated with the topological CV stabilizer codes and show that, although they have a gapless spectrum, they can become gapped with the addition of a quadratic perturbation. We show that similar methods can be used to construct a gapped Hamiltonian whose anyon theory agrees with a\n                    <p:math xmlns:p=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <p:mi mathvariant=\"normal\">U</p:mi>\n                      <p:mo stretchy=\"false\">(</p:mo>\n                      <p:mn>1</p:mn>\n                      <p:msub>\n                        <p:mo stretchy=\"false\">)</p:mo>\n                        <p:mn>2</p:mn>\n                      </p:msub>\n                    </p:math>\n                    Chern-Simons theory. Our work initiates the study of scalable stabilizer codes that are intrinsic to CV systems and highlights how error-correcting codes can be used to design and analyze many-body systems of CVs that model lattice gauge theories.\n                  </jats:p>","DOI":"10.1103/spsy-3k8q","source":"Crossref","title":"Topological Stabilizer Models on Continuous Variables","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0002-3779-0080","authenticated-orcid":true,"given":"Julio C.","family":"Magdalena de la Fuente","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"}]},{"ORCID":"https://orcid.org/0000-0002-1740-6889","authenticated-orcid":true,"given":"Tyler D.","family":"Ellison","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/02dqehb95","id-type":"ROR","asserted-by":"publisher"}],"name":"Purdue University"},{"id":[{"id":"https://ror.org/02dqehb95","id-type":"ROR","asserted-by":"publisher"}],"name":"Purdue University"}]},{"ORCID":"https://orcid.org/0000-0001-7600-1809","authenticated-orcid":true,"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":true,"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,11]]},"URL":"http://dx.doi.org/10.1103/spsy-3k8q","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011054","note":"arxivid:2411.04993"},{"id":"manual:-J.-Haah-href-https://web.arc","type":"document","author":[{"family":"Haah","given":"J."}],"title":"\\href{https://web.archive.org/web/20210515012200/https://online.kitp.ucsb.edu/online/qinfo_c17/haah/}{KITP Conference: Frontiers of Quantum Information Physics","publisher-place":"UCSB, Santa Barbara, CA"},{"id":"arxiv:1709.04471","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.2.010326","source":"Crossref","title":"Error Correction of Quantum Reference Frame Information","volume":"2","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"given":"Sepehr","family":"Nezami","sequence":"additional","affiliation":[]},{"given":"Sandu","family":"Popescu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3191-0325","authenticated-orcid":true,"given":"Grant","family":"Salton","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,18]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.010326","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010326","note":"arxivid:1709.04471"},{"id":"arxiv:2503.15698","type":"article-journal","author":[{"given":"James I.","family":"Kwon"},{"given":"Anthony J.","family":"Brady"},{"given":"Victor V.","family":"Albert"}],"title":"Absolutely Maximal Entanglement in Continuous Variables","issued":{"date-parts":[[2025,11,17]]},"note":"arxivid:2503.15698\narxiv_version_number:2"},{"id":"arxiv:1402.7310","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevb.90.094518","source":"Crossref","title":"Understanding degenerate ground states of a protected quantum circuit in the presence of disorder","volume":"90","author":[{"given":"Joshua M.","family":"Dempster","sequence":"first","affiliation":[]},{"given":"Bo","family":"Fu","sequence":"additional","affiliation":[]},{"given":"David G.","family":"Ferguson","sequence":"additional","affiliation":[]},{"given":"D. I.","family":"Schuster","sequence":"additional","affiliation":[]},{"given":"Jens","family":"Koch","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.90.094518","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"094518","note":"arxivid:1402.7310"},{"id":"arxiv:1708.02886","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1088/1367-2630/aab7cd","page":"043053","source":"Crossref","title":"Coherence properties of the 0-<i>π</i>qubit","volume":"20","author":[{"given":"Peter","family":"Groszkowski","sequence":"first","affiliation":[]},{"given":"A Di","family":"Paolo","sequence":"additional","affiliation":[]},{"given":"A L","family":"Grimsmo","sequence":"additional","affiliation":[]},{"given":"A","family":"Blais","sequence":"additional","affiliation":[]},{"given":"D I","family":"Schuster","sequence":"additional","affiliation":[]},{"given":"A A","family":"Houck","sequence":"additional","affiliation":[]},{"given":"Jens","family":"Koch","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2018,4,25]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aab7cd","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1708.02886"},{"id":"arxiv:2503.14634","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>\n                    Protected superconducting qubits such as the 0-\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>π</a:mi>\n                    </a:math>\n                    qubit promise to substantially reduce physical error rates. However, a key challenge in the field is designing gates for these qubits that do not compromise their protection or become infeasibly slow as the protection of the qubit is improved. In this work we propose a protected phase gate that is compatible with the protected regime of the 0-\n                    <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <c:mi>π</c:mi>\n                    </c:math>\n                    qubit and does not suffer from spurious coupling to additional circuit modes. Our gate utilizes an internal mode of the circuit as an ancilla and is achieved by varying the qubit-ancilla coupling via a tunable Josephson element. Through numerical simulations, we study how the gate error scales with the circuit parameters of the 0-\n                    <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <e:mi>π</e:mi>\n                    </e:math>\n                    qubit and the tunable Josephson element that enacts the gate. Ultimately, we find that a protected gate with the 0-\n                    <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <g:mi>π</g:mi>\n                    </g:math>\n                    qubit is possible with near-term circuit parameters. Our work opens up the possibility of performing protected gates on protected superconducting qubits, which may significantly reduce hardware overheads for quantum computation.\n                  </jats:p>","DOI":"10.1103/bywc-cx98","source":"Crossref","title":"Protected Phase Gate for the 0-\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <mml:mi>π</mml:mi>\n                    </mml:math>\n                    Qubit using its Internal Modes","volume":"7","author":[{"ORCID":"https://orcid.org/0009-0005-0507-9226","authenticated-orcid":true,"given":"Xanda C.","family":"Kolesnikow","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0003-2243-0397","authenticated-orcid":true,"given":"Thomas B.","family":"Smith","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0001-7422-1517","authenticated-orcid":true,"given":"Felix","family":"Thomsen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"Abhijeet","family":"Alase","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,12]]},"URL":"http://dx.doi.org/10.1103/bywc-cx98","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010306","note":"arxivid:2503.14634"},{"id":"arxiv:1910.07542","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.2.010339","source":"Crossref","title":"Experimental Realization of a Protected Superconducting Circuit Derived from the \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><mml:mn>0</mml:mn></mml:math>\n–\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><mml:mi>π</mml:mi></mml:math>\n Qubit","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-1652-6707","authenticated-orcid":true,"given":"András","family":"Gyenis","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3316-0724","authenticated-orcid":true,"given":"Pranav S.","family":"Mundada","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5174-7310","authenticated-orcid":true,"given":"Agustin","family":"Di Paolo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1542-9286","authenticated-orcid":true,"given":"Thomas M.","family":"Hazard","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9178-9419","authenticated-orcid":true,"given":"Xinyuan","family":"You","sequence":"additional","affiliation":[]},{"given":"David I.","family":"Schuster","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5047-631X","authenticated-orcid":true,"given":"Jens","family":"Koch","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0591-2012","authenticated-orcid":true,"given":"Alexandre","family":"Blais","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9788-5874","authenticated-orcid":true,"given":"Andrew A.","family":"Houck","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,3,5]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.010339","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010339","note":"arxivid:1910.07542"},{"id":"arxiv:cond-mat/0202115","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevlett.88.227005","source":"Crossref","title":"Pairing of Cooper Pairs in a Fully Frustrated Josephson-Junction Chain","volume":"88","author":[{"given":"Benoit","family":"Douçot","sequence":"first","affiliation":[]},{"given":"Julien","family":"Vidal","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2002,5,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.88.227005","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"227005","note":"arxivid:cond-mat/0202115"},{"id":"arxiv:cond-mat/0205186","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevb.66.224503","source":"Crossref","title":"Possible realization of an ideal quantum computer in Josephson junction array","volume":"66","author":[{"given":"L. B.","family":"Ioffe","sequence":"first","affiliation":[]},{"given":"M. V.","family":"Feigel’man","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2002,12,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.66.224503","ISSN":["0163-1829","1095-3795"],"container-title-short":"Phys. Rev. B","page":"224503","note":"arxivid:cond-mat/0205186"},{"id":"arxiv:quant-ph/0008040","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.64.012310","source":"Crossref","title":"Encoding a qubit in an oscillator","volume":"64","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2001,6,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.64.012310","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012310","note":"arxivid:quant-ph/0008040"},{"id":"arxiv:1003.1201","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.81.052327","source":"Crossref","title":"Encoding many qubits in a rotor","volume":"81","author":[{"given":"Philippe","family":"Raynal","sequence":"first","affiliation":[]},{"given":"Amir","family":"Kalev","sequence":"additional","affiliation":[]},{"given":"Jun","family":"Suzuki","sequence":"additional","affiliation":[]},{"given":"Berthold-Georg","family":"Englert","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,5,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.052327","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052327","note":"arxivid:1003.1201"},{"id":"arxiv:2303.17000","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Traditional stabilizer codes operate over prime power local-dimensions. In this work we extend the stabilizer formalism using the local-dimension-invariant setting to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any traditional stabilizer code can be used for analog continuous-variable codes, and consider restrictions in phase space and discretized phase space. This puts this framework on an equivalent footing as traditional stabilizer codes. Following this, using extensions of prior ideas, we show that a stabilizer code originally designed with a finite field local-dimension can be transformed into a code with the same <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>, and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> parameters for any integral domain. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, which permits the use of traditional stabilizer codes for protecting their information as well.</jats:p>","DOI":"10.22331/q-2024-02-12-1249","page":"1249","source":"Crossref","title":"Stabilizer Codes with Exotic Local-dimensions","volume":"8","author":[{"given":"Lane G.","family":"Gunderman","sequence":"first","affiliation":[{"name":"No affiliation for this work"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,2,12]]},"URL":"http://dx.doi.org/10.22331/q-2024-02-12-1249","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2303.17000"},{"id":"arxiv:quant-ph/0012111","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.65.012308","source":"Crossref","title":"Quantum error-correcting codes associated with graphs","volume":"65","author":[{"given":"D.","family":"Schlingemann","sequence":"first","affiliation":[]},{"given":"R. F.","family":"Werner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2001,12,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.012308","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012308","note":"arxivid:quant-ph/0012111"},{"id":"arxiv:2011.00197","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2021.3130155","page":"1795-1808","source":"Crossref","title":"Mitigating Coherent Noise by Balancing Weight-2 <i>Z</i>-Stabilizers","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0002-2699-3966","authenticated-orcid":false,"given":"Jingzhen","family":"Hu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5073-9431","authenticated-orcid":false,"given":"Qingzhong","family":"Liang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2084-9717","authenticated-orcid":false,"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3130155","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2011.00197"},{"id":"arxiv:2406.02444","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.111.032438","source":"Crossref","title":"Noise-adapted qudit codes for amplitude-damping noise","volume":"111","author":[{"ORCID":"https://orcid.org/0000-0003-3442-6061","authenticated-orcid":true,"given":"Sourav","family":"Dutta","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03v0r5n49","id-type":"ROR","asserted-by":"publisher"}],"name":"Indian Institute of Technology Madras"},{"id":[{"id":"https://ror.org/03v0r5n49","id-type":"ROR","asserted-by":"publisher"}],"name":"IIT Madras"}]},{"ORCID":"https://orcid.org/0009-0001-0929-2623","authenticated-orcid":true,"given":"Debjyoti","family":"Biswas","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v0r5n49","id-type":"ROR","asserted-by":"publisher"}],"name":"Indian Institute of Technology Madras"},{"id":[{"id":"https://ror.org/03v0r5n49","id-type":"ROR","asserted-by":"publisher"}],"name":"IIT Madras"}]},{"ORCID":"https://orcid.org/0000-0002-1774-3527","authenticated-orcid":true,"given":"Prabha","family":"Mandayam","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v0r5n49","id-type":"ROR","asserted-by":"publisher"}],"name":"Indian Institute of Technology Madras"},{"id":[{"id":"https://ror.org/03v0r5n49","id-type":"ROR","asserted-by":"publisher"}],"name":"IIT Madras"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.111.032438","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032438","note":"arxivid:2406.02444"},{"id":"arxiv:2304.02732","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code’s logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.</jats:p>","DOI":"10.1038/s41467-023-42743-z","source":"Crossref","title":"Holographic codes from hyperinvariant tensor networks","volume":"14","author":[{"given":"Matthew","family":"Steinberg","sequence":"first","affiliation":[]},{"given":"Sebastian","family":"Feld","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7142-0059","authenticated-orcid":false,"given":"Alexander","family":"Jahn","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,11]]},"URL":"http://dx.doi.org/10.1038/s41467-023-42743-z","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"7314","note":"alternative-id:42743\narxivid:2304.02732"},{"id":"arxiv:quant-ph/0703112","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2002.1023317","page":"45","source":"Crossref","title":"Graphs, quadratic forms, and quantum codes","author":[{"given":"M.","family":"Grassl","sequence":"first","affiliation":[]},{"given":"A.","family":"Klappenecker","sequence":"additional","affiliation":[]},{"given":"M.","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"IEEE International Symposium on Information Theory","container-title":"Proceedings IEEE International Symposium on Information Theory,","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/ISIT.2002.1023317","note":"arxivid:quant-ph/0703112"},{"id":"arxiv:1511.05647","type":"article-journal","author":[{"given":"Yongsoo","family":"Hwang"},{"given":"Jun","family":"Heo"}],"title":"On the relation between a graph code and a graph state","issued":{"date-parts":[[2015,11,18]]},"note":"arxivid:1511.05647\narxiv_version_number:1"},{"id":"arxiv:1407.2777","type":"article-journal","author":[{"given":"Carlo","family":"Cafaro"},{"given":"Damian","family":"Markham"},{"given":"Peter","family":"van Loock"}],"title":"Scheme for constructing graphs associated with stabilizer quantum codes","issued":{"date-parts":[[2014,7,10]]},"note":"arxivid:1407.2777\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0111080","type":"article-journal","author":[{"given":"D.","family":"Schlingemann"}],"title":"Stabilizer codes can be realized as graph codes","issued":{"date-parts":[[2001,11,14]]},"note":"arxivid:quant-ph/0111080\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0308151","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.69.022316","source":"Crossref","title":"Graphical description of the action of local Clifford transformations on graph states","volume":"69","author":[{"given":"Maarten","family":"Van den Nest","sequence":"first","affiliation":[]},{"given":"Jeroen","family":"Dehaene","sequence":"additional","affiliation":[]},{"given":"Bart","family":"De Moor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,2,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.69.022316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022316","note":"arxivid:quant-ph/0308151"},{"id":"doi:10.1109/TIT.2002.800469","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2002.800469","page":"2384-2391","source":"Crossref","title":"Quantum codes [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ (p ≥ 3) exist","volume":"48","author":[{"family":"Keqin Feng","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2002,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.800469","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1607.01387","type":"journal-article","publisher":"Universidad Nacional de Colombia","issue":"2","DOI":"10.15446/recolma.v50n2.62214","page":"299","source":"Crossref","title":"Algebraic Methods for Quantum Codes on Lattices","volume":"50","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"container-title":"Revista Colombiana de Matemáticas","original-title":[],"issued":{"date-parts":[[2017,1,26]]},"URL":"http://dx.doi.org/10.15446/recolma.v50n2.62214","ISSN":["2357-4100","0034-7426"],"container-title-short":"rev.colomb.mat","note":"arxivid:1607.01387"},{"id":"arxiv:1804.05457","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevb.99.195124","source":"Crossref","title":"Locality of edge states and entanglement spectrum from strong subadditivity","volume":"99","author":[{"given":"Kohtaro","family":"Kato","sequence":"first","affiliation":[]},{"given":"Fernando G. S. L.","family":"Brandão","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,5,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.99.195124","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"195124","note":"arxivid:1804.05457"},{"id":"arxiv:2511.14699","type":"article-journal","author":[{"given":"Wojciech","family":"De Roeck"},{"given":"Martin","family":"Fraas"},{"given":"Bruno de O.","family":"Carvalho"}],"title":"Pure gapped ground states of spin chains are short-range entangled","issued":{"date-parts":[[2025,11,18]]},"note":"arxivid:2511.14699\narxiv_version_number:1"},{"id":"arxiv:2602.13386","type":"article-journal","author":[{"given":"Ainesh","family":"Bakshi"},{"given":"Soonwon","family":"Choi"},{"given":"Saúl","family":"Pilatowsky-Cameo"}],"title":"Entanglement in quantum spin chains is strictly finite at any temperature","issued":{"date-parts":[[2026,2,13]]},"note":"arxivid:2602.13386\narxiv_version_number:1"},{"id":"arxiv:2502.16132","type":"article-journal","author":[{"given":"Craig","family":"Gidney"},{"given":"Thiago","family":"Bergamaschi"}],"title":"A Constant Rate Quantum Computer on a Line","issued":{"date-parts":[[2025,2,22]]},"note":"arxivid:2502.16132\narxiv_version_number:1"},{"id":"arxiv:1305.6973","type":"article-journal","author":[{"given":"Jeongwan","family":"Haah"}],"title":"Lattice quantum codes and exotic topological phases of matter","issued":{"date-parts":[[2013,5,29]]},"note":"arxivid:1305.6973\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0401134","type":"article-journal","author":[{"given":"H.","family":"Ollivier"},{"given":"J. -P.","family":"Tillich"}],"title":"Quantum convolutional codes: fundamentals","issued":{"date-parts":[[2004,1,21]]},"note":"arxivid:quant-ph/0401134\narxiv_version_number:1"},{"id":"arxiv:0804.4468","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.101.130501","source":"Crossref","title":"One-Way Quantum Computing in the Optical Frequency Comb","volume":"101","author":[{"given":"Nicolas C.","family":"Menicucci","sequence":"first","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"Olivier","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,9,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.101.130501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130501","note":"arxivid:0804.4468"},{"id":"arxiv:0811.2799","type":"journal-article","publisher":"IOP Publishing","issue":"11","DOI":"10.1088/0953-4075/42/11/114009","page":"114009","source":"Crossref","title":"The optical frequency comb as a one-way quantum computer","volume":"42","author":[{"given":"Steven T","family":"Flammia","sequence":"first","affiliation":[]},{"given":"Nicolas C","family":"Menicucci","sequence":"additional","affiliation":[]},{"given":"Oliver","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics B: Atomic, Molecular and Optical Physics","original-title":[],"issued":{"date-parts":[[2009,5,15]]},"URL":"http://dx.doi.org/10.1088/0953-4075/42/11/114009","ISSN":["0953-4075","1361-6455"],"container-title-short":"J. Phys. B: At. Mol. Opt. Phys.","note":"alternative-id:S0953-4075(09)99404-4\narxivid:0811.2799"},{"id":"arxiv:2503.17759","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Efficient and high-performance quantum error correction is essential for achieving fault-tolerant quantum computing. Low-depth random circuits offer a promising approach to identifying effective and practical encoding strategies. In this work, we rigorously prove through information-theoretic analysis that one-dimensional logarithmic-depth random Clifford encoding circuits can achieve high quantum error correction performance. We demonstrate that these random codes typically exhibit good approximate quantum error correction capability by proving that their encoding rate achieves the hashing bound for Pauli noise and the channel capacity for erasure errors. We show that the error correction inaccuracy decays once a threshold of logarithmic depth is exceeded, resulting in negligible recovery errors. This threshold is shown to be lower than that of the simple separate block encoding, and the decay rate is higher. We further establish that these codes are optimal by proving that logarithmic depth is necessary to maintain a constant encoding rate and high error correction performance. To prove our results, we propose decoupling theorems tailored for one-dimensional low-depth circuits. These results also imply strong decoupling and rapid thermalization properties in low-depth random circuits and have potential applications in quantum information science and physics.</jats:p>","DOI":"10.1103/7rzk-2jyh","source":"Crossref","title":"Approximate Quantum Error Correction with 1D Log-Depth Circuits","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0003-3288-1394","authenticated-orcid":true,"given":"Guoding","family":"Liu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"ORCID":"https://orcid.org/0009-0006-5576-0834","authenticated-orcid":true,"given":"Zhenyu","family":"Du","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"ORCID":"https://orcid.org/0000-0002-3402-9763","authenticated-orcid":true,"given":"Zi-Wen","family":"Liu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"ORCID":"https://orcid.org/0000-0002-9441-4006","authenticated-orcid":true,"given":"Xiongfeng","family":"Ma","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,2,13]]},"URL":"http://dx.doi.org/10.1103/7rzk-2jyh","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010331","note":"arxivid:2503.17759"},{"id":"arxiv:2602.20900","type":"article-journal","author":[{"given":"Twan","family":"Kroll"},{"given":"Jonas","family":"Helsen"}],"title":"Error correction with brickwork Clifford circuits","issued":{"date-parts":[[2026,2,24]]},"note":"arxivid:2602.20900\narxiv_version_number:1"},{"id":"arxiv:2510.07512","type":"article-journal","author":[{"given":"Jon","family":"Nelson"},{"given":"Joel","family":"Rajakumar"},{"given":"Michael J.","family":"Gullans"}],"title":"Error correction phase transition in noisy random quantum circuits","issued":{"date-parts":[[2025,10,8]]},"note":"arxivid:2510.07512\narxiv_version_number:1"},{"id":"arxiv:1503.06794","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.nuclphysb.2015.04.020","page":"330-359","source":"Crossref","title":"Quantum phase transitions between a class of symmetry protected topological states","volume":"896","author":[{"given":"Lokman","family":"Tsui","sequence":"first","affiliation":[]},{"given":"Hong-Chen","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Yuan-Ming","family":"Lu","sequence":"additional","affiliation":[]},{"given":"Dung-Hai","family":"Lee","sequence":"additional","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[2015,7]]},"URL":"http://dx.doi.org/10.1016/j.nuclphysb.2015.04.020","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"special_numbering:C\nalternative-id:S0550321315001534\narxivid:1503.06794"},{"id":"arxiv:1812.11193","type":"journal-article","publisher":"AIP Publishing","issue":"1","abstract":"<jats:p>We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with the code distance being the linear system size is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code stabilizer group (Abelian discrete gauge theory). This means that under local Clifford circuits, the number of toric code copies is the complete invariant of topological Pauli stabilizer codes. Previously, the same conclusion was obtained under the assumption of nonchirality for qubit codes or the Calderbank–Shor–Steane structure for prime qudit codes; we do not assume any of these.</jats:p>","DOI":"10.1063/5.0021068","source":"Crossref","title":"Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices","volume":"62","author":[{"ORCID":"https://orcid.org/0000-0002-1087-6853","authenticated-orcid":false,"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft Quantum and Microsoft Research , Redmond, Washington 98052, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,1,1]]},"URL":"http://dx.doi.org/10.1063/5.0021068","ISSN":["0022-2488","1089-7658"],"page":"012201","note":"arxivid:1812.11193"},{"id":"arxiv:quant-ph/0308021","type":"article-journal","author":[{"given":"S.","family":"Bravyi"},{"given":"M.","family":"Vyalyi"}],"title":"Commutative version of the k-local Hamiltonian problem and common eigenspace problem","issued":{"date-parts":[[2004,12,16]]},"note":"arxivid:quant-ph/0308021\narxiv_version_number:2"},{"id":"arxiv:1102.0770","type":"article-journal","author":[{"given":"Dorit","family":"Aharonov"},{"given":"Lior","family":"Eldar"}],"title":"On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems","issued":{"date-parts":[[2011,2,14]]},"note":"arxivid:1102.0770\narxiv_version_number:2"},{"id":"arxiv:1803.02213","type":"document","categories":["local Hamiltonian complexity","commuting Hamiltonians","local Hamiltonian problem","trivial states","toric code","ground states","quantum NP","QMA","topological order","multiparticle entanglement","logical operators","ribbon"],"language":"en","author":[{"family":"Aharonov","given":"Dorit"},{"family":"Kenneth","given":"Oded"},{"family":"Vigdorovich","given":"Itamar"}],"contributor":[{"family":"Jeffery","given":"Stacey"}],"editor":[{"family":"Jeffery","given":"Stacey"}],"issued":{"date-parts":[[2018]]},"abstract":"The complexity of the commuting local Hamiltonians (CLH) problem still remains a mystery after two decades of research of quantum Hamiltonian complexity; it is only known to be contained in NP for few low parameters. Of particular interest is the tightly related question of understanding whether groundstates of CLHs can be generated by efficient quantum circuits. The two problems touch upon conceptual, physical and computational questions, including the centrality of non-commutation in quantum mechanics, quantum PCP and the area law. It is natural to try to address first the more physical case of CLHs embedded on a 2D lattice, but this problem too remained open apart from some very specific cases [Aharonov and Eldar, 2011; Hastings, 2012; Schuch, 2011]. Here we consider a wide class of two dimensional CLH instances; these are k-local CLHs, for any constant k; they are defined on qubits set on the edges of any surface complex, where we require that this surface complex is not too far from being \"Euclidean\". Each vertex and each face can be associated with an arbitrary term (as long as the terms commute). We show that this class is in NP, and moreover that the groundstates have an efficient quantum circuit that prepares them. This result subsumes that of Schuch [Schuch, 2011] which regarded the special case of 4-local Hamiltonians on a grid with qubits, and by that it removes the mysterious feature of Schuch's proof which showed containment in NP without providing a quantum circuit for the groundstate and considerably generalizes it. We believe this work and the tools we develop make a significant step towards showing that 2D CLHs are in NP.","container-title":"LIPIcs, Volume 111, TQC 2018","DOI":"10.4230/LIPICS.TQC.2018.2","volume":"111","number":"2","page":"2:1-2:21","page-first":"2:1","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"On the Complexity of Two Dimensional Commuting Local Hamiltonians","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2018.2","note":"copyright:Creative Commons Attribution 3.0 Unported license\narxivid:1803.02213"},{"id":"arxiv:2312.11170","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:msub><a:mrow><a:mi mathvariant=\"double-struck\">Z</a:mi></a:mrow><a:mi>d</a:mi></a:msub></a:math> qudits, including instances where <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><e:mi>d</e:mi></e:math> is a nonprime number. This capability allows the identification of topological orders that differ from the <h:math xmlns:h=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><h:msub><h:mrow><h:mi mathvariant=\"double-struck\">Z</h:mi></h:mrow><h:mi>d</h:mi></h:msub></h:math> toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for <l:math xmlns:l=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><l:msub><l:mrow><l:mi mathvariant=\"double-struck\">Z</l:mi></l:mrow><l:mi>p</l:mi></l:msub></l:math> qudits (with <p:math xmlns:p=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><p:mi>p</p:mi></p:math> being a prime) are equivalent to finite copies of <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><s:msub><s:mrow><s:mi mathvariant=\"double-struck\">Z</s:mi></s:mrow><s:mi>p</s:mi></s:msub></s:math> toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.030328","source":"Crossref","title":"Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions","volume":"5","author":[{"ORCID":"https://orcid.org/0009-0007-2103-1956","authenticated-orcid":true,"given":"Zijian","family":"Liang","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02v51f717","id-type":"ROR","asserted-by":"publisher"}],"name":"Peking University"},{"id":[{"id":"https://ror.org/0207yh398","id-type":"ROR","asserted-by":"publisher"}],"name":"Shandong University"}]},{"given":"Yijia","family":"Xu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, NIST/University of Maryland"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0003-3383-1946","authenticated-orcid":true,"given":"Joseph T.","family":"Iosue","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, NIST/University of Maryland"}]},{"ORCID":"https://orcid.org/0000-0002-8810-9355","authenticated-orcid":true,"given":"Yu-An","family":"Chen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02v51f717","id-type":"ROR","asserted-by":"publisher"}],"name":"Peking University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,8]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.030328","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030328","note":"arxivid:2312.11170"},{"id":"arxiv:2410.11942","type":"article-journal","author":[{"given":"Zijian","family":"Liang"},{"given":"Bowen","family":"Yang"},{"given":"Joseph T.","family":"Iosue"},{"given":"Yu-An","family":"Chen"}],"title":"Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes","issued":{"date-parts":[[2025,9,14]]},"note":"arxivid:2410.11942\narxiv_version_number:4"},{"id":"arxiv:2405.07970","type":"article-journal","author":[{"given":"Zhi","family":"Li"},{"given":"Dongjin","family":"Lee"},{"given":"Beni","family":"Yoshida"}],"title":"How Much Entanglement Is Needed for Topological Codes and Mixed States with Anomalous Symmetry?","issued":{"date-parts":[[2025,6,3]]},"note":"arxivid:2405.07970\narxiv_version_number:3"},{"id":"arxiv:1006.1362","type":"article-journal","author":[{"given":"Guillaume","family":"Duclos-Cianci"},{"given":"David","family":"Poulin"}],"title":"A renormalization group decoding algorithm for topological quantum codes","issued":{"date-parts":[[2010,6,7]]},"note":"arxivid:1006.1362\narxiv_version_number:1"},{"id":"arxiv:1809.10704","type":"journal-article","publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"2","abstract":"<jats:p>\n            We give a broad generalisation of the mapping, originally due to Dennis, Kitaev, Landahl, and Preskill, from quantum error correcting codes to statistical mechanical models. We show how the mapping can be extended to arbitrary stabiliser or subsystem codes subject to correlated Pauli noise models, including models of fault tolerance. This mapping connects the error correction threshold of the quantum code to a phase transition in the statistical mechanical model. Thus, any existing method for finding phase transitions, such as Monte Carlo simulations, can be applied to approximate the threshold of any such code, without having to perform optimal decoding. By way of example, we numerically study the threshold of the surface code under mildly correlated bit-flip noise, showing that noise with bunching correlations causes the threshold to drop to \n            <jats:inline-formula>\n              <jats:tex-math>p_{\\textup{corr}}=10.04(6)\\%</jats:tex-math>\n            </jats:inline-formula>\n            , from its known iid value of \n            <jats:inline-formula>\n              <jats:tex-math>p_{\\text{iid}}=10.917(3)\\%</jats:tex-math>\n            </jats:inline-formula>\n            . Complementing this, we show that the mapping also allows us to utilise any algorithm which can calculate/approximate partition functions of classical statistical mechanical models to perform optimal/approximately optimal decoding. Specifically, for 2D codes subject to locally correlated noise, we give a linear-time tensor network-based algorithm for approximate optimal decoding which extends the MPS decoder of Bravyi, Suchara and Vargo.\n          </jats:p>","DOI":"10.4171/aihpd/105","page":"269-321","source":"Crossref","title":"Statistical mechanical models for quantum codes with correlated noise","volume":"8","author":[{"given":"Christopher T.","family":"Chubb","sequence":"first","affiliation":[{"name":"University of Sydney, Australia"}]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[{"name":"University of Sydney, Australia"},{"name":"Yale University, New Haven, USA"}]}],"container-title":"Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions","original-title":[],"issued":{"date-parts":[[2021,5,16]]},"URL":"http://dx.doi.org/10.4171/AIHPD/105","ISSN":["2308-5827","2308-5835"],"container-title-short":"Ann. Inst. Henri Poincaré Comb. Phys. Interact.","note":"arxivid:1809.10704"},{"id":"arxiv:2211.09803","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>We explore the relationship between renormalization-group (RG) flow and error correction by constructing quantum algorithms that exactly recognize one-dimensional symmetry-protected topological (SPT) phases protected by finite internal Abelian symmetries. For each SPT phase, our algorithm runs a quantum circuit, which emulates RG flow: an arbitrary input ground-state wave function in the phase is mapped to a unique minimally entangled reference state, thereby allowing for efficient phase identification. This construction is enabled by viewing a generic input state in the phase as a collection of coherent “errors” applied to the reference state, and engineering a quantum circuit to efficiently detect and correct such errors. Importantly, the error-correction threshold is proven to coincide exactly with the phase boundary. We discuss the implications of our results in the context of condensed-matter physics, machine learning, and near-term quantum algorithms.</jats:p>","DOI":"10.1103/prxquantum.6.010350","source":"Crossref","title":"Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction","volume":"6","author":[{"given":"Ethan","family":"Lake","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"given":"Shankar","family":"Balasubramanian","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"},{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"given":"Soonwon","family":"Choi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"},{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,13]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.010350","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010350","note":"arxivid:2211.09803"},{"id":"arxiv:2401.06300","type":"article-journal","author":[{"given":"Weishun","family":"Zhong"},{"given":"Oles","family":"Shtanko"},{"given":"Ramis","family":"Movassagh"}],"title":"Advantage of Quantum Neural Networks as Quantum Information Decoders","issued":{"date-parts":[[2024,1,11]]},"note":"arxivid:2401.06300\narxiv_version_number:1"},{"id":"arxiv:2402.14906","type":"article-journal","author":[{"given":"Ali","family":"Lavasani"},{"given":"Sagar","family":"Vijay"}],"title":"The Stability of Gapped Quantum Matter and Error-Correction with Adiabatic Noise","issued":{"date-parts":[[2024,2,22]]},"note":"arxivid:2402.14906\narxiv_version_number:1"},{"id":"arxiv:0807.0287","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.102.070503","source":"Crossref","title":"Nonequilibrium Reliability of Quantum Memories","volume":"102","author":[{"given":"Alastair","family":"Kay","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,2,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.102.070503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070503","note":"arxivid:0807.0287"},{"id":"arxiv:0911.3843","type":"article-journal","author":[{"given":"Fernando","family":"Pastawski"},{"given":"Alastair","family":"Kay"},{"given":"Norbert","family":"Schuch"},{"given":"Ignacio","family":"Cirac"}],"title":"Limitations of Passive Protection of Quantum Information","issued":{"date-parts":[[2009,11,19]]},"note":"arxivid:0911.3843\narxiv_version_number:1"},{"id":"arxiv:1107.3940","type":"journal-article","publisher":"American Physical Society (APS)","issue":"27","DOI":"10.1103/physrevlett.107.270502","source":"Crossref","title":"Capabilities of a Perturbed Toric Code as a Quantum Memory","volume":"107","author":[{"given":"Alastair","family":"Kay","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,12,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.107.270502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"270502","note":"arxivid:1107.3940"},{"id":"arxiv:2510.08056","type":"article-journal","author":[{"given":"Ethan","family":"Lake"}],"title":"Local active error correction from simulated confinement","issued":{"date-parts":[[2025,10,16]]},"note":"arxivid:2510.08056\narxiv_version_number:3"},{"id":"arxiv:2301.05687","type":"article-journal","author":[{"given":"Yimu","family":"Bao"},{"given":"Ruihua","family":"Fan"},{"given":"Ashvin","family":"Vishwanath"},{"given":"Ehud","family":"Altman"}],"title":"Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions","issued":{"date-parts":[[2026,5,1]]},"note":"arxivid:2301.05687\narxiv_version_number:2"},{"id":"arxiv:1208.2317","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.87.020304","source":"Crossref","title":"Fault tolerance of quantum low-density parity check codes with sublinear distance scaling","volume":"87","author":[{"given":"Alexey A.","family":"Kovalev","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,2,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.020304","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"020304","note":"arxivid:1208.2317"},{"id":"arxiv:1311.7688","type":"article-journal","author":[{"given":"Alexey A.","family":"Kovalev"},{"given":"Leonid P.","family":"Pryadko"}],"title":"Spin glass reflection of the decoding transition for quantum error correcting codes","issued":{"date-parts":[[2014,2,10]]},"note":"arxivid:1311.7688\narxiv_version_number:2"},{"id":"arxiv:2301.05238","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.4.030317","source":"Crossref","title":"Quantum Criticality Under Decoherence or Weak Measurement","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-7387-3326","authenticated-orcid":true,"given":"Jong Yeon","family":"Lee","sequence":"first","affiliation":[]},{"given":"Chao-Ming","family":"Jian","sequence":"additional","affiliation":[]},{"given":"Cenke","family":"Xu","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,4]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.030317","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030317","note":"arxivid:2301.05238"},{"id":"arxiv:2301.05689","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Topological quantum memory can protect information against local errors up to finite error thresholds. Such thresholds are usually determined based on the success of decoding algorithms rather than the intrinsic properties of the mixed states describing corrupted memories. Here we provide an intrinsic characterization of the breakdown of topological quantum memory, which both gives a bound on the performance of decoding algorithms and provides examples of topologically distinct mixed states. We employ three information-theoretical quantities that can be regarded as generalizations of the diagnostics of ground-state topological order, and serve as a definition for topological order in error-corrupted mixed states. We consider the topological contribution to entanglement negativity and two other metrics based on quantum relative entropy and coherent information. In the concrete example of the two-dimensional (2D) Toric code with local bit-flip and phase errors, we map three quantities to observables in 2D classical spin models and analytically show they all undergo a transition at the same error threshold. This threshold is an upper bound on that achieved in any decoding algorithm and is indeed saturated by that in the optimal decoding algorithm for the Toric code.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020343","source":"Crossref","title":"Diagnostics of Mixed-State Topological Order and Breakdown of Quantum Memory","volume":"5","author":[{"given":"Ruihua","family":"Fan","sequence":"first","affiliation":[{"name":"Department of Physics, Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-5357-8821","authenticated-orcid":true,"given":"Yimu","family":"Bao","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California"}]},{"given":"Ehud","family":"Altman","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California"},{"name":"Materials Sciences Division, Lawrence Berkeley National Laboratory"}]},{"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,24]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020343","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020343","note":"update-to:{\"updated\":{\"date-parts\":[[2025,6,13]],\"date-time\":\"2025-06-13T00:00:00Z\",\"timestamp\":1749772800000},\"DOI\":\"10.1103/prxquantum.5.020343\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2025,6,13]],\"date-time\":\"2025-06-13T00:00:00Z\",\"timestamp\":1749772800000},\"DOI\":\"10.1103/prxquantum.5.020343\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2301.05689"},{"id":"arxiv:2309.11879","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.132.170602","source":"Crossref","title":"Separability Transitions in Topological States Induced by Local Decoherence","volume":"132","author":[{"ORCID":"https://orcid.org/0000-0002-6415-9710","authenticated-orcid":true,"given":"Yu-Hsueh","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Physics, University of California at San Diego, La Jolla, California 92093, USA"}]},{"given":"Tarun","family":"Grover","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California at San Diego, La Jolla, California 92093, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.132.170602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"170602","note":"arxivid:2309.11879"},{"id":"arxiv:2401.17359","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevb.110.085158","source":"Crossref","title":"Tapestry of dualities in decohered quantum error correction codes","volume":"110","author":[{"given":"Kaixiang","family":"Su","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02t274463","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California, Santa Barbara"}]},{"ORCID":"https://orcid.org/0000-0001-6248-4379","authenticated-orcid":true,"given":"Zhou","family":"Yang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University"}]},{"given":"Chao-Ming","family":"Jian","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.110.085158","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"085158","note":"arxivid:2401.17359"},{"id":"arxiv:1103.4606","type":"journal-article","publisher":"IOP Publishing","issue":"7","DOI":"10.1088/1367-2630/14/7/073048","page":"073048","source":"Crossref","title":"Universal topological phase of two-dimensional stabilizer codes","volume":"14","author":[{"given":"H","family":"Bombin","sequence":"first","affiliation":[]},{"given":"Guillaume","family":"Duclos-Cianci","sequence":"additional","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2012,7,25]]},"URL":"http://dx.doi.org/10.1088/1367-2630/14/7/073048","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1103.4606"},{"id":"arxiv:1107.2707","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-014-1893-4","page":"387-432","source":"Crossref","title":"Structure of 2D Topological Stabilizer Codes","volume":"327","author":[{"given":"Héctor","family":"Bombín","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,3,9]]},"URL":"http://dx.doi.org/10.1007/s00220-014-1893-4","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1893\narxivid:1107.2707"},{"id":"arxiv:2312.04617","type":"article-journal","author":[{"given":"Thomas","family":"Schuster"},{"given":"Nathanan","family":"Tantivasadakarn"},{"given":"Ashvin","family":"Vishwanath"},{"given":"Norman Y.","family":"Yao"}],"title":"A holographic view of topological stabilizer codes","issued":{"date-parts":[[2023,12,7]]},"note":"arxivid:2312.04617\narxiv_version_number:1"},{"id":"arxiv:2408.01769","type":"article-journal","author":[{"given":"Xingjian","family":"Li"},{"given":"Ting-Chun","family":"Lin"},{"given":"Min-Hsiu","family":"Hsieh"}],"title":"Transform Arbitrary Good Quantum LDPC Codes into Good Geometrically Local Codes in Any Dimension","issued":{"date-parts":[[2024,8,3]]},"note":"arxivid:2408.01769\narxiv_version_number:1"},{"id":"arxiv:2412.04181","type":"article-journal","author":[{"given":"Jens Niklas","family":"Eberhardt"},{"given":"Francisco Revson F.","family":"Pereira"},{"given":"Vincent","family":"Steffan"}],"title":"Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions","issued":{"date-parts":[[2024,12,5]]},"note":"arxivid:2412.04181\narxiv_version_number:1"},{"id":"arxiv:2503.04699","type":"journal-article","publisher":"American Physical Society (APS)","DOI":"10.1103/86j7-cmsw","source":"Crossref","title":"Anyon theory and topological frustration of high-efficiency quantum low-density parity-check codes","author":[{"family":"Anonymous","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,26]]},"URL":"http://dx.doi.org/10.1103/86j7-cmsw","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:2503.04699"},{"id":"arxiv:0810.1983","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1088/1367-2630/11/4/043029","page":"043029","source":"Crossref","title":"A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes","volume":"11","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Barbara","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2009,4,24]]},"URL":"http://dx.doi.org/10.1088/1367-2630/11/4/043029","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:0810.1983"},{"id":"arxiv:1209.5750","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.110.090502","source":"Crossref","title":"Local Topological Order Inhibits Thermal Stability in 2D","volume":"110","author":[{"given":"Olivier","family":"Landon-Cardinal","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2013,2,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.110.090502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090502","note":"arxivid:1209.5750"},{"id":"arxiv:1908.08049","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.100.155137","source":"Crossref","title":"Sorting topological stabilizer models in three dimensions","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"first","affiliation":[]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,10,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.100.155137","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155137","note":"arxivid:1908.08049"},{"id":"arxiv:2211.02116","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.4.030338","source":"Crossref","title":"Tailoring Three-Dimensional Topological Codes for Biased Noise","volume":"4","author":[{"given":"Eric","family":"Huang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5759-6314","authenticated-orcid":true,"given":"Arthur","family":"Pesah","sequence":"additional","affiliation":[]},{"given":"Christopher T.","family":"Chubb","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,9,20]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.030338","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030338","note":"arxivid:2211.02116"},{"id":"arxiv:1103.1885","type":"journal-article","publisher":"Elsevier BV","issue":"10","DOI":"10.1016/j.aop.2011.06.001","page":"2566-2633","source":"Crossref","title":"Feasibility of self-correcting quantum memory and thermal stability of topological order","volume":"326","author":[{"given":"Beni","family":"Yoshida","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,10]]},"URL":"http://dx.doi.org/10.1016/j.aop.2011.06.001","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491611001023\narxivid:1103.1885"},{"id":"arxiv:1101.1962","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.83.042330","source":"Crossref","title":"Local stabilizer codes in three dimensions without string logical operators","volume":"83","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,4,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.042330","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042330","note":"arxivid:1101.1962"},{"id":"arxiv:1703.00612","type":"article-journal","author":[{"given":"M. B.","family":"Hastings"}],"title":"Small Majorana Fermion Codes","issued":{"date-parts":[[2017,3,2]]},"note":"arxivid:1703.00612\narxiv_version_number:1"},{"id":"arxiv:2201.12337","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.3.010335","source":"Crossref","title":"Encoding Qubits in Multimode Grid States","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-9695-8552","authenticated-orcid":true,"given":"Baptiste","family":"Royer","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4921-1410","authenticated-orcid":true,"given":"Shraddha","family":"Singh","sequence":"additional","affiliation":[]},{"given":"S.M.","family":"Girvin","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,3,7]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.010335","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010335","note":"arxivid:2201.12337"},{"id":"arxiv:1204.1063","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-013-1810-2","page":"351-399","source":"Crossref","title":"Commuting Pauli Hamiltonians as Maps between Free Modules","volume":"324","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2013,10,10]]},"URL":"http://dx.doi.org/10.1007/s00220-013-1810-2","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1810\narxivid:1204.1063"},{"id":"doi:10.7907/GCYW-ZE58","type":"thesis","categories":["","quantum memory","Physics","stabilizer codes","topological order","quantum codes"],"language":"en","author":[{"family":"Haah","given":"Jeongwan"}],"contributor":[{"literal":"Institute for Quantum Information and Matter"},{"literal":"IQIM"}],"issued":{"date-parts":[[2013]]},"abstract":"This thesis addresses whether it is possible to build a robust memory device for quantum information. Many schemes for fault-tolerant quantum information processing have been developed so far, one of which, called topological quantum computation, makes use of degrees of freedom that are inherently insensitive to local errors. However, this scheme is not so reliable against thermal errors. Other fault-tolerant schemes achieve better reliability through active error correction, but incur a substantial overhead cost. Thus, it is of practical importance and theoretical interest to design and assess fault-tolerant schemes that work well at finite temperature without active error correction.\n\nIn this thesis, a three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square.\n\nThis spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.","DOI":"10.7907/GCYW-ZE58","publisher":"California Institute of Technology","title":"Lattice Quantum Codes and Exotic Topological Phases of Matter","URL":"https://resolver.caltech.edu/CaltechTHESIS:05292013-140541902","version":"Final","note":"copyright:No commercial reproduction, distribution, display or performance rights in this work are provided."},{"id":"arxiv:1712.02375","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevb.97.134426","source":"Crossref","title":"Recoverable information and emergent conservation laws in fracton stabilizer codes","volume":"97","author":[{"given":"A. T.","family":"Schmitz","sequence":"first","affiliation":[]},{"given":"Han","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Rahul M.","family":"Nandkishore","sequence":"additional","affiliation":[]},{"given":"S. A.","family":"Parameswaran","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2018,4,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.97.134426","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"134426","note":"arxivid:1712.02375"},{"id":"arxiv:1008.1029","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.83.012320","source":"Crossref","title":"Subsystem codes with spatially local generators","volume":"83","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,1,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.012320","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012320","note":"arxivid:1008.1029"},{"id":"arxiv:1610.06169","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubits <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>, the number of encoded qubits <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>, the code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, the accuracy parameter <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi></mml:math> that quantifies how well the erasure channel can be reversed, and the locality parameter <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x2113;</mml:mi></mml:math> that specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracy <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi></mml:math> that is exponentially small in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x2113;</mml:mi></mml:math>, which is the case for perturbations of local commuting projector codes, our bound reads <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi><mml:msup><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mn>2</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mi>n</mml:mi><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> for codes on <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>D</mml:mi></mml:math>-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceed <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>&amp;#x2113;</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> if <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B4;</mml:mi><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>&amp;#x2113;</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> and the size <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow></mml:math> of a minimal region that can support all approximate logical operators satisfies <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mo>&amp;#x2264;</mml:mo><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:msup><mml:mi>&amp;#x2113;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mi>D</mml:mi><mml:mrow><mml:mi>D</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math>, where the logical operators are accurate up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mrow class=\"MJX-TeXAtom-OPEN\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mi>&amp;#x03B4;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow class=\"MJX-TeXAtom-CLOSE\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math> in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.</jats:p>","DOI":"10.22331/q-2017-04-25-4","page":"4","source":"Crossref","title":"Limits on the storage of quantum information in a volume of space","volume":"1","author":[{"given":"Steven T.","family":"Flammia","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Australia"},{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington, USA"},{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, USA"}]},{"given":"Michael J.","family":"Kastoryano","sequence":"additional","affiliation":[{"name":"NBIA, Niels Bohr Institute, University of Copenhagen, Denmark"}]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"name":"IBM T. J. Watson Research Center, Yorktown Heights, New York, USA"},{"name":"Perimeter Institute for Theoretical Physics, Waterloo ON N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo ON N2L 3G1, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2017,4,25]]},"URL":"http://dx.doi.org/10.22331/q-2017-04-25-4","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1610.06169"},{"id":"arxiv:2009.13551","type":"journal-article","publisher":"Stichting SciPost","issue":"1","abstract":"<jats:p>\n                    We introduce a notion of homogeneous topological order, which is\nobeyed by most, if not all, known examples of topological order\nincluding fracton phases on quantum spins (qudits). The notion is a\ncondition on the ground state subspace, rather than on the Hamiltonian,\nand demands that given a collection of ball-like regions, any linear\ntransformation on the ground space be realized by an operator that\navoids the ball-like regions. We derive a bound on the ground state\ndegeneracy\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>\\mathcal D</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mstyle mathvariant=\"script\">\n                            <mml:mi>𝒟</mml:mi>\n                          </mml:mstyle>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    for systems with homogeneous topological order on an arbitrary closed\nRiemannian manifold of dimension\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>d</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mi>d</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    ,\nwhich reads [ D c (L/a)^{d-2}.] Here,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>L</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mi>L</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    is the diameter of the system,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>a</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mi>a</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    is the lattice spacing, and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>c</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mi>c</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    is a constant that only depends on the isometry class of the manifold,\nand\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>\\mu</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mi>μ</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    is a constant that only depends on the density of degrees of freedom. If\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>d=2</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mrow>\n                            <mml:mi>d</mml:mi>\n                            <mml:mo>=</mml:mo>\n                            <mml:mn>2</mml:mn>\n                          </mml:mrow>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    ,\nthe constant\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>c</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mi>c</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    is the (demi)genus of the space manifold. This bound is saturated up to\nconstants by known examples.examples.\n                  </jats:p>","DOI":"10.21468/scipostphys.10.1.011","source":"Crossref","title":"A degeneracy bound for homogeneous topological order","volume":"10","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2021,1,20]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.10.1.011","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"011","note":"arxivid:2009.13551"},{"id":"arxiv:2409.15203","type":"article-journal","author":[{"given":"Samuel","family":"Dai"},{"given":"Ray","family":"Li"}],"title":"Locality vs Quantum Codes","issued":{"date-parts":[[2024,9,23]]},"note":"arxivid:2409.15203\narxiv_version_number:1"},{"id":"arxiv:2503.17655","type":"article-journal","author":[{"given":"Nouédyn","family":"Baspin"}],"title":"Stabilizer codes of less than two dimensions have constant distance","issued":{"date-parts":[[2025,3,22]]},"note":"arxivid:2503.17655\narxiv_version_number:1"},{"id":"arxiv:1206.1609","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.110.170503","source":"Crossref","title":"Classification of Topologically Protected Gates for Local Stabilizer Codes","volume":"110","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2013,4,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.110.170503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"170503","note":"arxivid:1206.1609"},{"id":"arxiv:2211.11764","type":"journal-article","publisher":"Stichting SciPost","issue":"4","abstract":"<jats:p>A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the <jats:inline-formula><jats:alternatives><jats:tex-math>d</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>d</mml:mi></mml:math></jats:alternatives></jats:inline-formula>-group global symmetry and its ’t Hooft anomaly for topological finite group gauge theories in <jats:inline-formula><jats:alternatives><jats:tex-math>(d+1)</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mi>d</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the <jats:inline-formula><jats:alternatives><jats:tex-math>[O_5] ∈ H^5(BG, U(1))</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">[</mml:mo><mml:msub><mml:mi>O</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:mo stretchy=\"true\" form=\"postfix\">]</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mi>B</mml:mi><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow><mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> obstruction that has appeared in prior work. We also show how the <jats:inline-formula><jats:alternatives><jats:tex-math>d</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>d</mml:mi></mml:math></jats:alternatives></jats:inline-formula>-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the <jats:inline-formula><jats:alternatives><jats:tex-math>d</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>d</mml:mi></mml:math></jats:alternatives></jats:inline-formula>-group symmetry, such as a controlled Z gate in the (3+1) D <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> toric code.</jats:p>","DOI":"10.21468/scipostphys.16.4.089","source":"Crossref","title":"Higher-group symmetry in finite gauge theory and stabilizer codes","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0002-4322-9433","authenticated-orcid":false,"given":"Maissam","family":"Barkeshli","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"}]},{"ORCID":"https://orcid.org/0000-0002-8810-9355","authenticated-orcid":false,"given":"Yu-An","family":"Chen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02v51f717","id-type":"ROR","asserted-by":"publisher"}],"name":"Peking University"},{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"}]},{"ORCID":"https://orcid.org/0000-0002-4764-1476","authenticated-orcid":false,"given":"Po-Shen","family":"Hsin","sequence":"additional","affiliation":[{"name":"Mani L. Bhaumik Institute for Theoretical Physics"}]},{"given":"Ryohei","family":"Kobayashi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2024,4,3]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.16.4.089","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"089","note":"arxivid:2211.11764"},{"id":"arxiv:2503.04566","type":"article-journal","author":[{"given":"Fuchuan","family":"Wei"},{"given":"Zi-Wen","family":"Liu"}],"title":"Long-range nonstabilizerness and quantum codes, phases, and complexity","issued":{"date-parts":[[2026,2,26]]},"note":"arxivid:2503.04566\narxiv_version_number:3"},{"id":"arxiv:1408.1720","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.91.012305","source":"Crossref","title":"Fault-tolerant logical gates in quantum error-correcting codes","volume":"91","author":[{"given":"Fernando","family":"Pastawski","sequence":"first","affiliation":[]},{"given":"Beni","family":"Yoshida","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,1,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.012305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012305","note":"arxivid:1408.1720"},{"id":"doi:10.7907/AHMQ-EG82","type":"thesis","categories":["fault tolerance","Physics","error correction","fault-tolerant","computing"],"language":"en","author":[{"family":"Harrington","given":"James William"}],"issued":{"date-parts":[[2004]]},"abstract":"Quantum information theory is concerned with identifying how quantum mechanical resources, such as entangled quantum states, can be utilized for a number of information processing tasks, including data storage, computation, communication, and cryptography.  Efficient quantum algorithms and protocols have been developed for performing some tasks (e.g., factoring large numbers, securely communicating over a public channel, and simulating quantum mechanical systems) that appear to be very difficult with just classical resources.  In addition to identifying the separation between classical and quantum computational power, much of the theoretical focus in this field over the last decade has been concerned with finding novel ways of encoding quantum information that are robust against errors, which is an important step toward building practical quantum information processing devices.\n\nIn this thesis I present some results on the quantum error-correcting properties of oscillator codes (also described as symplectic lattice codes) and toric codes.  Any harmonic oscillator system, such as a mode of light, can be encoded with quantum information via symplectic lattice codes that are robust against shifts in the system's continuous quantum variables.  I show the existence of lattice codes whose achievable rates match the one-shot coherent information over the Gaussian quantum channel.  Also, I construct a family of symplectic self-dual lattices and search for optimal encodings of quantum information distributed between several oscillators.\n\nToric codes provide encodings of quantum information into two-dimensional spin lattices that are robust against local clusters of errors and which require only local quantum operations for error correction.  Numerical simulations of this system under realistic error models provide a calculation of the accuracy threshold for quantum memory using toric codes, which can be related to phase transitions in particular condensed matter models.  I also present a local classical processing scheme for correcting errors on toric codes, which demonstrates that quantum information can be maintained in two dimensions by purely local quantum and classical resources.","DOI":"10.7907/AHMQ-EG82","publisher":"California Institute of Technology","title":"Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes","URL":"https://resolver.caltech.edu/CaltechETD:etd-05122004-113132","version":"Final","note":"copyright:No commercial reproduction, distribution, display or performance rights in this work are provided."},{"id":"arxiv:1112.3252","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.111.200501","source":"Crossref","title":"Quantum Self-Correction in the 3D Cubic Code Model","volume":"111","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2013,11,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.111.200501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200501","note":"arxivid:1112.3252"},{"id":"arxiv:2411.02928","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Geometrically local quantum codes, which are error-correction codes embedded in <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\"><a:msup><a:mi mathvariant=\"double-struck\">R</a:mi><a:mi>D</a:mi></a:msup></a:math> with checks acting only on qubits within a fixed spatial distance, have garnered significant interest. Recently, it has been demonstrated how to achieve geometrically local codes that maximize both the dimension and the distance, as well as the energy barrier of the code. In this work, we focus on the constructions involving subdivision, and we show that they have an almost linear time decoder, obtained by combining the decoder of the outer good qLDPC code and a generalized version of the Union-Find decoder. This provides the first decoder for an optimal geometrically local three-dimensional code. We demonstrate the existence of a finite threshold error rate under the code capacity noise model using a minimum weight perfect matching decoder. Furthermore, we argue that this threshold is also applicable to the decoder based on the generalized Union-Find algorithm.</jats:p>","DOI":"10.1103/sxdx-qbcz","source":"Crossref","title":"Almost linear decoder for optimal geometrically local quantum codes","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-4709-3115","authenticated-orcid":true,"given":"Quinten","family":"Eggerickx","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02kcbn207","id-type":"ROR","asserted-by":"publisher"}],"name":"Imec"},{"id":[{"id":"https://ror.org/05f950310","id-type":"ROR","asserted-by":"publisher"}],"name":"KU Leuven"},{"id":[{"id":"https://ror.org/00cv9y106","id-type":"ROR","asserted-by":"publisher"}],"name":"Ghent University"}]},{"ORCID":"https://orcid.org/0009-0005-6653-8879","authenticated-orcid":true,"given":"Adam","family":"Wills","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"},{"id":[{"id":"https://ror.org/01v3aym70","id-type":"ROR","asserted-by":"publisher"}],"name":"Hon Hai Research Institute"}]},{"ORCID":"https://orcid.org/0000-0002-8994-4598","authenticated-orcid":true,"given":"Ting-Chun","family":"Lin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0168r3w48","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California San Diego"},{"id":[{"id":"https://ror.org/01v3aym70","id-type":"ROR","asserted-by":"publisher"}],"name":"Hon Hai Research Institute"}]},{"ORCID":"https://orcid.org/0000-0002-1314-9715","authenticated-orcid":true,"given":"Kristiaan","family":"De Greve","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02kcbn207","id-type":"ROR","asserted-by":"publisher"}],"name":"Imec"},{"id":[{"id":"https://ror.org/05f950310","id-type":"ROR","asserted-by":"publisher"}],"name":"KU Leuven"}]},{"ORCID":"https://orcid.org/0000-0002-3396-8427","authenticated-orcid":true,"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01v3aym70","id-type":"ROR","asserted-by":"publisher"}],"name":"Hon Hai Research Institute"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,24]]},"URL":"http://dx.doi.org/10.1103/sxdx-qbcz","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023300","note":"arxivid:2411.02928"},{"id":"arxiv:2506.03266","type":"article-journal","author":[{"given":"Ethan","family":"Lake"}],"title":"Fast offline decoding with local message-passing automata","issued":{"date-parts":[[2025,10,16]]},"note":"arxivid:2506.03266\narxiv_version_number:4"},{"id":"arxiv:2505.10403","type":"article-journal","author":[{"given":"David","family":"Aasen"},{"given":"Jeongwan","family":"Haah"},{"given":"Matthew B.","family":"Hastings"},{"given":"Zhenghan","family":"Wang"}],"title":"Geometrically Enhanced Topological Quantum Codes","issued":{"date-parts":[[2025,6,24]]},"note":"arxivid:2505.10403\narxiv_version_number:2"},{"id":"arxiv:2012.02249","type":"article-journal","author":[{"given":"Michael","family":"Freedman"},{"given":"Matthew B.","family":"Hastings"}],"title":"Building manifolds from quantum codes","issued":{"date-parts":[[2021,5,24]]},"note":"arxivid:2012.02249\narxiv_version_number:3"},{"id":"arxiv:2309.16104","type":"article-journal","author":[{"given":"Ting-Chun","family":"Lin"},{"given":"Adam","family":"Wills"},{"given":"Min-Hsiu","family":"Hsieh"}],"title":"Geometrically Local Quantum and Classical Codes from Subdivision","issued":{"date-parts":[[2024,7,3]]},"note":"arxivid:2309.16104\narxiv_version_number:2"},{"id":"arxiv:2404.16736","type":"article-journal","author":[{"given":"Virgile","family":"Guemard"}],"title":"Lifts of quantum CSS codes","issued":{"date-parts":[[2024,4,25]]},"note":"arxivid:2404.16736\narxiv_version_number:1"},{"id":"arxiv:2509.10418","type":"article-journal","author":[{"given":"Błażej","family":"Ruba"},{"given":"Bowen","family":"Yang"}],"title":"Witt Groups and Bulk-Boundary Correspondence for Stabilizer States","issued":{"date-parts":[[2025,12,6]]},"note":"arxivid:2509.10418\narxiv_version_number:2"},{"id":"arxiv:2203.17216","type":"article-journal","author":[{"given":"Renyu","family":"Wang"},{"given":"Leonid P.","family":"Pryadko"}],"title":"Distance bounds for generalized bicycle codes","issued":{"date-parts":[[2022,3,31]]},"note":"arxivid:2203.17216\narxiv_version_number:1"},{"id":"arxiv:2502.19406","type":"article-journal","author":[{"given":"Hsiang-Ku","family":"Lin"},{"given":"Xingrui","family":"Liu"},{"given":"Pak Kau","family":"Lim"},{"given":"Leonid P.","family":"Pryadko"}],"title":"Single-shot and two-shot decoding with generalized bicycle codes","issued":{"date-parts":[[2025,8,6]]},"note":"arxivid:2502.19406\narxiv_version_number:3"},{"id":"arxiv:2303.06755","type":"article-journal","author":[{"given":"Elia","family":"Portnoy"}],"title":"Local Quantum Codes from Subdivided Manifolds","issued":{"date-parts":[[2023,6,20]]},"note":"arxivid:2303.06755\narxiv_version_number:3"},{"id":"arxiv:2510.03090","type":"article-journal","author":[{"given":"Sebastian","family":"Stengele"},{"given":"Ángela","family":"Capel"},{"given":"Li","family":"Gao"},{"given":"Angelo","family":"Lucia"},{"given":"David","family":"Pérez-García"},{"given":"Antonio","family":"Pérez-Hernández"},{"given":"Cambyse","family":"Rouzé"},{"given":"Simone","family":"Warzel"}],"title":"Modified logarithmic Sobolev inequalities for CSS codes","issued":{"date-parts":[[2025,10,16]]},"note":"arxivid:2510.03090\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0301052","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.68.022312","source":"Crossref","title":"Measurement-based quantum computation on cluster states","volume":"68","author":[{"given":"Robert","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"Daniel E.","family":"Browne","sequence":"additional","affiliation":[]},{"given":"Hans J.","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,8,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.022312","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022312","note":"arxivid:quant-ph/0301052"},{"id":"arxiv:1009.3491","type":"journal-article","publisher":"Elsevier BV","issue":"7","DOI":"10.1016/j.aop.2011.03.006","page":"1656-1671","source":"Crossref","title":"Quantum computational capability of a 2D valence bond solid phase","volume":"326","author":[{"given":"Akimasa","family":"Miyake","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,7]]},"URL":"http://dx.doi.org/10.1016/j.aop.2011.03.006","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491611000455\narxivid:1009.3491"},{"id":"arxiv:1111.7173","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1007/s11128-011-0346-7","page":"1961-1968","source":"Crossref","title":"Topological order in 1D Cluster state protected by symmetry","volume":"11","author":[{"given":"W.","family":"Son","sequence":"first","affiliation":[]},{"given":"L.","family":"Amico","sequence":"additional","affiliation":[]},{"given":"V.","family":"Vedral","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2011,12,31]]},"URL":"http://dx.doi.org/10.1007/s11128-011-0346-7","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:346\narxivid:1111.7173"},{"id":"arxiv:1201.4877","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.108.240505","source":"Crossref","title":"Symmetry-Protected Phases for Measurement-Based Quantum Computation","volume":"108","author":[{"given":"Dominic V.","family":"Else","sequence":"first","affiliation":[]},{"given":"Ilai","family":"Schwarz","sequence":"additional","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,6,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.108.240505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240505","note":"arxivid:1201.4877"},{"id":"arxiv:quant-ph/0108118","type":"journal-article","publisher":"Informa UK Limited","issue":"8","DOI":"10.1080/09500340110107487","page":"1299-1306","source":"Crossref","title":"The one-way quantum computer--a non-network model of quantum computation","volume":"49","author":[{"given":"Robert","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Browne","sequence":"additional","affiliation":[]},{"given":"Hans","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Journal of Modern Optics","original-title":[],"language":"en","issued":{"date-parts":[[2002,7]]},"URL":"http://dx.doi.org/10.1080/09500340110107487","ISSN":["0950-0340","1362-3044"],"container-title-short":"Journal of Modern Optics","note":"alternative-id:10.1080/09500340110107487\narxivid:quant-ph/0108118"},{"id":"doi:10.1103/PhysRevLett.86.5188","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevlett.86.5188","page":"5188-5191","source":"Crossref","title":"A One-Way Quantum Computer","volume":"86","author":[{"given":"Robert","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"Hans J.","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2001,5,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.86.5188","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett."},{"id":"arxiv:1806.08780","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lower-dimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.</jats:p>","DOI":"10.22331/q-2019-05-20-142","page":"142","source":"Crossref","title":"Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-3150-0169","authenticated-orcid":false,"given":"David T.","family":"Stephen","sequence":"first","affiliation":[{"name":"Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany"}]},{"ORCID":"https://orcid.org/0000-0001-7815-7006","authenticated-orcid":false,"given":"Hendrik Poulsen","family":"Nautrup","sequence":"additional","affiliation":[{"name":"Institut für Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria"}]},{"ORCID":"https://orcid.org/0000-0003-3727-8092","authenticated-orcid":false,"given":"Juani","family":"Bermejo-Vega","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"ORCID":"https://orcid.org/0000-0003-3033-1292","authenticated-orcid":false,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Robert","family":"Raussendorf","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada"},{"name":"Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, V6T 1Z4, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,5,20]]},"URL":"http://dx.doi.org/10.22331/q-2019-05-20-142","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1806.08780"},{"id":"arxiv:1806.04663","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.98.022332","source":"Crossref","title":"Universal quantum computation using fractal symmetry-protected cluster phases","volume":"98","author":[{"given":"Trithep","family":"Devakul","sequence":"first","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,8,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.98.022332","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022332","note":"arxivid:1806.04663"},{"id":"arxiv:1907.13279","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>What kinds of symmetry-protected topologically ordered (SPTO) ground states can be used for universal measurement-based quantum computation in a similar fashion to the 2D cluster state? 2D SPTO states are classified not only by global on-site symmetries but also by subsystem symmetries, which are fine-grained symmetries dependent on the lattice geometry. Recently, all states within so-called SPTO cluster phases on the square and hexagonal lattices have been shown to be universal, based on the presence of subsystem symmetries and associated structures of quantum cellular automata. Motivated by this observation, we analyze the computational capability of SPTO cluster phases on all vertex-translative 2D Archimedean lattices. There are four subsystem symmetries here called ribbon, cone, fractal, and 1-form symmetries, and the former three are fundamentally in one-to-one correspondence with three classes of Clifford quantum cellular automata. We conclude that nine out of the eleven Archimedean lattices support universal cluster phases protected by one of the former three symmetries, while the remaining lattices possess 1-form symmetries and have a different capability related to error correction.</jats:p>","DOI":"10.22331/q-2020-02-10-228","page":"228","source":"Crossref","title":"Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices","volume":"4","author":[{"given":"Austin K.","family":"Daniel","sequence":"first","affiliation":[{"name":"Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA"}]},{"given":"Rafael N.","family":"Alexander","sequence":"additional","affiliation":[{"name":"Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA"}]},{"given":"Akimasa","family":"Miyake","sequence":"additional","affiliation":[{"name":"Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87131, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,2,10]]},"URL":"http://dx.doi.org/10.22331/q-2020-02-10-228","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1907.13279"},{"id":"arxiv:1803.02369","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevb.98.035112","source":"Crossref","title":"Subsystem symmetry protected topological order","volume":"98","author":[{"given":"Yizhi","family":"You","sequence":"first","affiliation":[]},{"given":"Trithep","family":"Devakul","sequence":"additional","affiliation":[]},{"given":"F. J.","family":"Burnell","sequence":"additional","affiliation":[]},{"given":"S. L.","family":"Sondhi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2018,7,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.98.035112","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"035112","note":"arxivid:1803.02369"},{"id":"arxiv:1803.00095","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.122.090501","source":"Crossref","title":"Computationally Universal Phase of Quantum Matter","volume":"122","author":[{"given":"Robert","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"Cihan","family":"Okay","sequence":"additional","affiliation":[]},{"given":"Dong-Sheng","family":"Wang","sequence":"additional","affiliation":[]},{"given":"David T.","family":"Stephen","sequence":"additional","affiliation":[]},{"given":"Hendrik Poulsen","family":"Nautrup","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,3,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.122.090501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090501","note":"arxivid:1803.00095"},{"id":"arxiv:1002.1567","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.105.020502","source":"Crossref","title":"Quantum State Reduction for Universal Measurement Based Computation","volume":"105","author":[{"given":"Xie","family":"Chen","sequence":"first","affiliation":[]},{"given":"Runyao","family":"Duan","sequence":"additional","affiliation":[]},{"given":"Zhengfeng","family":"Ji","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,7,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.105.020502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020502","note":"arxivid:1002.1567"},{"id":"arxiv:1009.2840","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.86.032328","source":"Crossref","title":"Two-dimensional Affleck-Kennedy-Lieb-Tasaki state on the honeycomb lattice is a universal resource for quantum computation","volume":"86","author":[{"given":"Tzu-Chieh","family":"Wei","sequence":"first","affiliation":[]},{"given":"Ian","family":"Affleck","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Raussendorf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,9,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.032328","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032328","note":"arxivid:1009.2840"},{"id":"arxiv:1102.5064","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.106.070501","source":"Crossref","title":"Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource","volume":"106","author":[{"given":"Tzu-Chieh","family":"Wei","sequence":"first","affiliation":[]},{"given":"Ian","family":"Affleck","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Raussendorf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,2,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.106.070501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070501","note":"arxivid:1102.5064"},{"id":"arxiv:1310.5100","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.90.042333","source":"Crossref","title":"Hybrid valence-bond states for universal quantum computation","volume":"90","author":[{"given":"Tzu-Chieh","family":"Wei","sequence":"first","affiliation":[]},{"given":"Poya","family":"Haghnegahdar","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Raussendorf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,10,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.042333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042333","note":"arxivid:1310.5100"},{"id":"arxiv:1509.02947","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.92.052309","source":"Crossref","title":"Symmetry-protected topologically ordered states for universal quantum computation","volume":"92","author":[{"given":"Hendrik Poulsen","family":"Nautrup","sequence":"first","affiliation":[]},{"given":"Tzu-Chieh","family":"Wei","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,11,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.92.052309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052309","note":"arxivid:1509.02947"},{"id":"arxiv:1611.05450","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.96.022306","source":"Crossref","title":"Symmetry-protected topological order at nonzero temperature","volume":"96","author":[{"given":"Sam","family":"Roberts","sequence":"first","affiliation":[]},{"given":"Beni","family":"Yoshida","sequence":"additional","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,8,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.96.022306","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022306","note":"arxivid:1611.05450"},{"id":"arxiv:quant-ph/0703143","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/9/6/199","page":"199-199","source":"Crossref","title":"Topological fault-tolerance in cluster state quantum computation","volume":"9","author":[{"given":"R","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"J","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"K","family":"Goyal","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2007,6,29]]},"URL":"http://dx.doi.org/10.1088/1367-2630/9/6/199","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"alternative-id:S1367-2630(07)44251-3\narxivid:quant-ph/0703143"},{"id":"arxiv:2107.04019","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.129.090501","source":"Crossref","title":"Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps","volume":"129","author":[{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6306-2263","authenticated-orcid":true,"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.129.090501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090501","note":"arxivid:2107.04019"},{"id":"arxiv:1909.11817","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Measurement-based quantum computing (MBQC) is a promising alternative to traditional circuit-based quantum computing predicated on the construction and measurement of cluster states. Recent work has demonstrated that MBQC provides a more general framework for fault-tolerance that extends beyond foliated quantum error-correcting codes. We systematically expand on that paradigm, and use combinatorial tiling theory to study and construct new examples of fault-tolerant cluster states derived from crystal structures. Included among these is a robust self-dual cluster state requiring only degree-<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>3</mml:mn></mml:math>connectivity. We benchmark several of these cluster states in the presence of circuit-level noise, and find a variety of promising candidates whose performance depends on the specifics of the noise model. By eschewing the distinction between data and ancilla, this malleable framework lays a foundation for the development of creative and competitive fault-tolerance schemes beyond conventional error-correcting codes.</jats:p>","DOI":"10.22331/q-2020-07-13-295","page":"295","source":"Crossref","title":"Generating Fault-Tolerant Cluster States from Crystal Structures","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-6640-1072","authenticated-orcid":false,"given":"Michael","family":"Newman","sequence":"first","affiliation":[{"name":"Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, NC, 27708, USA"}]},{"ORCID":"https://orcid.org/0000-0003-2491-8126","authenticated-orcid":false,"given":"Leonardo Andreta","family":"de Castro","sequence":"additional","affiliation":[{"name":"Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, NC, 27708, USA"},{"name":"Q-CTRL Pty Ltd, Sydney, NSW, Australia"}]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":false,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, NC, 27708, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,13]]},"URL":"http://dx.doi.org/10.22331/q-2020-07-13-295","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1909.11817"},{"id":"arxiv:1710.02646","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevb.96.165146","source":"Crossref","title":"Strong-weak coupling duality between two perturbed quantum many-body systems: Calderbank-Shor-Steane codes and Ising-like systems","volume":"96","author":[{"given":"Mohammad Hossein","family":"Zarei","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.96.165146","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"165146","note":"arxivid:1710.02646"},{"id":"arxiv:2103.06309","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/prxquantum.2.040101","source":"Crossref","title":"Quantum Low-Density Parity-Check Codes","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-7211-5515","authenticated-orcid":true,"given":"Nikolas P.","family":"Breuckmann","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0577-9159","authenticated-orcid":true,"given":"Jens Niklas","family":"Eberhardt","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,10,11]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.040101","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040101","note":"arxivid:2103.06309"},{"id":"doi:10.1109/ACCESS.2015.2503267","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/access.2015.2503267","page":"2492-2519","source":"Crossref","title":"Fifteen Years of Quantum LDPC Coding and Improved Decoding Strategies","volume":"3","author":[{"given":"Zunaira","family":"Babar","sequence":"first","affiliation":[]},{"given":"Panagiotis","family":"Botsinis","sequence":"additional","affiliation":[]},{"given":"Dimitrios","family":"Alanis","sequence":"additional","affiliation":[]},{"given":"Soon Xin","family":"Ng","sequence":"additional","affiliation":[]},{"given":"Lajos","family":"Hanzo","sequence":"additional","affiliation":[]}],"container-title":"IEEE Access","original-title":[],"issued":{"date-parts":[[2015]]},"URL":"http://dx.doi.org/10.1109/ACCESS.2015.2503267","ISSN":["2169-3536"],"container-title-short":"IEEE Access"},{"id":"arxiv:2507.11534","type":"article-journal","author":[{"given":"Daiki","family":"Komoto"},{"given":"Kenta","family":"Kasai"}],"title":"Sharp Error-Rate Transitions in Quantum QC-LDPC Codes under Joint BP Decoding","issued":{"date-parts":[[2025,7,16]]},"note":"arxivid:2507.11534\narxiv_version_number:2"},{"id":"manual:-K.-Kasai-GD-CSS-Decoder-(Qua","type":"document","author":[{"family":"Kasai","given":"K."}],"title":"GD-CSS Decoder (Quantum Error Correction using Non-binary LDPC over GF(q","issued":"2025","URL":"https://github.com/kasaikenta/gd-css-decoder"},{"id":"manual:-M.-A.-Perlin.-qLDPC.-url-htt","type":"document","author":[{"family":"qLDPC","given":"M.A.Perlin"}],"issued":"2023","URL":"https://github.com/Infleqtion/qLDPC","note":"Available at \\url{https://github.com/Infleqtion/qLDPC},"},{"id":"arxiv:2510.14090","type":"article-journal","author":[{"given":"Bane","family":"Vasic"},{"given":"Valentin","family":"Savin"},{"given":"Michele","family":"Pacenti"},{"given":"Shantom","family":"Borah"},{"given":"Nithin","family":"Raveendran"}],"title":"Quantum Low-Density Parity-Check Codes","issued":{"date-parts":[[2025,10,15]]},"note":"arxivid:2510.14090\narxiv_version_number:1"},{"id":"arxiv:1001.4363","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-011-1346-2","page":"609-627","source":"Crossref","title":"A Short Proof of Stability of Topological Order under Local Perturbations","volume":"307","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,9,17]]},"URL":"http://dx.doi.org/10.1007/s00220-011-1346-2","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1346\narxivid:1001.4363"},{"id":"arxiv:1001.0344","type":"journal-article","publisher":"AIP Publishing","issue":"9","abstract":"<jats:p>We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H0, we prove that there exists a constant threshold ϵ&amp;gt;0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb–Robinson bound.</jats:p>","DOI":"10.1063/1.3490195","source":"Crossref","title":"Topological quantum order: Stability under local perturbations","volume":"51","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[{"name":"IBM Watson Research Center 1 , Yorktown Heights, New York 10594, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"University of California 2 Microsoft Research Station Q, CNSI Building, , Santa Barbara, California 93106, USA"}]},{"given":"Spyridon","family":"Michalakis","sequence":"additional","affiliation":[{"name":"LANL 3 T-4 and CNLS, , Los Alamos, New Mexico 87544, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2010,9,1]]},"URL":"http://dx.doi.org/10.1063/1.3490195","ISSN":["0022-2488","1089-7658"],"page":"093512","note":"arxivid:1001.0344"},{"id":"arxiv:1810.02428","type":"journal-article","publisher":"AIP Publishing","issue":"6","abstract":"<jats:p>Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.</jats:p>","DOI":"10.1063/1.5095769","source":"Crossref","title":"Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms","volume":"60","author":[{"ORCID":"https://orcid.org/0000-0002-7835-3776","authenticated-orcid":false,"given":"Bruno","family":"Nachtergaele","sequence":"first","affiliation":[{"name":"Department of Mathematics and Center for Quantum Mathematics and Physics, University of California, Davis 1 , Davis, California 95616, USA"}]},{"ORCID":"https://orcid.org/0000-0002-2501-9651","authenticated-orcid":false,"given":"Robert","family":"Sims","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Arizona 2 , Tucson, Arizona 85721, USA"}]},{"given":"Amanda","family":"Young","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Arizona 2 , Tucson, Arizona 85721, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,6,1]]},"URL":"http://dx.doi.org/10.1063/1.5095769","ISSN":["0022-2488","1089-7658"],"page":"061101","note":"arxivid:1810.02428"},{"id":"arxiv:2010.15337","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","abstract":"<jats:title>Abstract</jats:title><jats:p>We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.</jats:p>","DOI":"10.1007/s00023-021-01086-5","page":"393-511","source":"Crossref","title":"Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States","volume":"23","author":[{"ORCID":"https://orcid.org/0000-0002-7835-3776","authenticated-orcid":false,"given":"Bruno","family":"Nachtergaele","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2501-9651","authenticated-orcid":false,"given":"Robert","family":"Sims","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1161-5663","authenticated-orcid":false,"given":"Amanda","family":"Young","sequence":"additional","affiliation":[]}],"container-title":"Annales Henri Poincaré","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,14]]},"URL":"http://dx.doi.org/10.1007/s00023-021-01086-5","ISSN":["1424-0637","1424-0661"],"container-title-short":"Ann. Henri Poincaré","note":"alternative-id:1086\narxivid:2010.15337"},{"id":"arxiv:2405.19412","type":"article-journal","author":[{"given":"Ali","family":"Lavasani"},{"given":"Michael J.","family":"Gullans"},{"given":"Victor V.","family":"Albert"},{"given":"Maissam","family":"Barkeshli"}],"title":"On stability of k-local quantum phases of matter","issued":{"date-parts":[[2024,9,7]]},"note":"arxivid:2405.19412\narxiv_version_number:2"},{"id":"arxiv:2411.01002","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Phases of matter with robust ground-state degeneracy, such as the quantum toric code, are known to be capable of robust quantum information storage. Here, we address the converse question: given a quantum error-correcting code, when does it define a stable gapped quantum phase of matter, whose ground-state degeneracy is robust against perturbations in the thermodynamic limit? We prove that a low-density parity-check (LDPC) code defines such a phase, robust against all few-body perturbations, if its code distance grows at least logarithmically in the number of degrees of freedom, and it exhibits “check soundness.” Many constant-rate quantum LDPC expander codes have such properties, and define stable phases of matter with a constant zero-temperature entropy density, violating the third law of thermodynamics. Our results also show that quantum toric-code phases are robust to spatially nonlocal few-body perturbations. Similarly, phases of matter defined by classical codes are stable against symmetric perturbations. In the classical setting, we present improved locality bounds on the quasiadiabatic evolution operator between two nearby states in the same code phase.</jats:p>","DOI":"10.1103/361k-nj4b","source":"Crossref","title":"Low-Density Parity-Check Codes as Stable Phases of Quantum Matter","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-3379-310X","authenticated-orcid":true,"given":"Chao","family":"Yin","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado"}]},{"given":"Andrew","family":"Lucas","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,19]]},"URL":"http://dx.doi.org/10.1103/361k-nj4b","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030329","note":"arxivid:2411.01002"},{"id":"arxiv:2411.02384","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:msub><a:mi>H</a:mi><a:mn>0</a:mn></a:msub></a:math> defined by <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mtext>⟦</c:mtext><c:mi>N</c:mi><c:mo>,</c:mo><c:mi>K</c:mi><c:mo>,</c:mo><c:mi>d</c:mi><c:mtext>⟧</c:mtext></c:math> LDPC codes, which obey certain topological quantum order conditions: (i) code distance <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>d</e:mi><e:mo>≥</e:mo><e:mi>c</e:mi><e:mi>log</e:mi><e:mo></e:mo><e:mo stretchy=\"false\">(</e:mo><e:mi>N</e:mi><e:mo stretchy=\"false\">)</e:mo></e:math>, implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:msub><i:mi>H</i:mi><i:mn>0</i:mn></i:msub></i:math> and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:mi>O</k:mi><k:mo stretchy=\"false\">(</k:mo><k:mn>1</k:mn><k:mo stretchy=\"false\">)</k:mo></k:math> smallest eigenvalues of <o:math xmlns:o=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><o:msub><o:mi>H</o:mi><o:mn>0</o:mn></o:msub></o:math>. The band originating from the smallest eigenvalue has <q:math xmlns:q=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><q:msup><q:mn>2</q:mn><q:mi>K</q:mi></q:msup></q:math> states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><s:mi>δ</s:mi><s:mo>=</s:mo><s:mi>C</s:mi><s:mi>N</s:mi><s:msup><s:mi>e</s:mi><s:mrow><s:mo>−</s:mo><s:mi mathvariant=\"normal\">Θ</s:mi><s:mo stretchy=\"false\">(</s:mo><s:mi>d</s:mi><s:mo stretchy=\"false\">)</s:mo></s:mrow></s:msup></s:math>, which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.</jats:p>","DOI":"10.1103/7x71-8j7k","source":"Crossref","title":"Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-8008-9484","authenticated-orcid":true,"given":"Wojciech","family":"De Roeck","sequence":"first","affiliation":[{"name":"Instituut voor Theoretische Fysica"},{"id":[{"id":"https://ror.org/05f950310","id-type":"ROR","asserted-by":"publisher"}],"name":"KU Leuven"}]},{"given":"Vedika","family":"Khemani","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f54p054","id-type":"ROR","asserted-by":"publisher"}],"name":"Stanford University"}]},{"ORCID":"https://orcid.org/0000-0003-3742-1944","authenticated-orcid":true,"given":"Yaodong","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f54p054","id-type":"ROR","asserted-by":"publisher"}],"name":"Stanford University"}]},{"ORCID":"https://orcid.org/0009-0009-5949-7326","authenticated-orcid":true,"given":"Nicholas","family":"O’Dea","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f54p054","id-type":"ROR","asserted-by":"publisher"}],"name":"Stanford University"}]},{"given":"Tibor","family":"Rakovszky","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02w42ss30","id-type":"ROR","asserted-by":"publisher"}],"name":"Budapest University of Technology and Economics"},{"id":[{"id":"https://ror.org/02w42ss30","id-type":"ROR","asserted-by":"publisher"}],"name":"Budapest University of Technology and Economics"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,19]]},"URL":"http://dx.doi.org/10.1103/7x71-8j7k","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030330","note":"arxivid:2411.02384"},{"id":"arxiv:2004.09560","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevx.11.011030","source":"Crossref","title":"Entanglement Phase Transitions in Measurement-Only Dynamics","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0003-4095-0038","authenticated-orcid":true,"given":"Matteo","family":"Ippoliti","sequence":"first","affiliation":[]},{"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[]},{"given":"Sarang","family":"Gopalakrishnan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1008-5178","authenticated-orcid":true,"given":"David A.","family":"Huse","sequence":"additional","affiliation":[]},{"given":"Vedika","family":"Khemani","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.11.011030","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011030","note":"arxivid:2004.09560"},{"id":"arxiv:2308.13277","type":"article-journal","author":[{"given":"Harriet","family":"Apel"},{"given":"Nouédyn","family":"Baspin"}],"title":"Simulating LDPC code Hamiltonians on 2D lattices","issued":{"date-parts":[[2023,8,25]]},"note":"arxivid:2308.13277\narxiv_version_number:1"},{"id":"arxiv:0802.4079","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"}],"title":"Families of LDPC Codes Derived from Nonprimitive BCH Codes and Cyclotomic Cosets","issued":{"date-parts":[[2008,2,27]]},"note":"arxivid:0802.4079\narxiv_version_number:1"},{"id":"arxiv:2401.07583","type":"article-journal","author":[{"given":"Nikolaos","family":"Koukoulekidis"},{"given":"Fedor","family":"Šimkovic"},{"given":"Martin","family":"Leib"},{"given":"Francisco Revson Fernandes","family":"Pereira"}],"title":"Small Quantum Codes from Algebraic Extensions of Generalized Bicycle Codes","issued":{"date-parts":[[2024,1,15]]},"note":"arxivid:2401.07583\narxiv_version_number:1"},{"id":"arxiv:2110.10794","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"20","abstract":"<jats:p>Vast numbers of qubits will be needed for large-scale quantum computing because of the overheads associated with error correction. We present a scheme for low-overhead fault-tolerant quantum computation based on quantum low-density parity-check (LDPC) codes, where long-range interactions enable many logical qubits to be encoded with a modest number of physical qubits. In our approach, logic gates operate via logical Pauli measurements that preserve both the protection of the LDPC codes and the low overheads in terms of the required number of additional qubits. Compared with surface codes with the same code distance, we estimate order-of-magnitude improvements in the overheads for processing around 100 logical qubits using this approach. Given the high thresholds demonstrated by LDPC codes, our estimates suggest that fault-tolerant quantum computation at this scale may be achievable with a few thousand physical qubits at comparable error rates to what is needed for current approaches.</jats:p>","DOI":"10.1126/sciadv.abn1717","source":"Crossref","title":"Low-overhead fault-tolerant quantum computing using long-range connectivity","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0003-2595-4081","authenticated-orcid":true,"given":"Lawrence Z.","family":"Cohen","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia."}]},{"ORCID":"https://orcid.org/0000-0001-7689-3157","authenticated-orcid":true,"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia."},{"name":"Department of Computer Science, UC Davis, Davis, CA 95616, USA."}]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia."}]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia."}]}],"container-title":"Science Advances","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,20]]},"URL":"http://dx.doi.org/10.1126/sciadv.abn1717","ISSN":["2375-2548"],"container-title-short":"Sci. Adv.","page":"eabn1717","note":"alternative-id:10.1126/sciadv.abn1717\narxivid:2110.10794"},{"id":"arxiv:2308.08648","type":"article-journal","author":[{"given":"Qian","family":"Xu"},{"given":"J. Pablo Bonilla","family":"Ataides"},{"given":"Christopher A.","family":"Pattison"},{"given":"Nithin","family":"Raveendran"},{"given":"Dolev","family":"Bluvstein"},{"given":"Jonathan","family":"Wurtz"},{"given":"Bane","family":"Vasic"},{"given":"Mikhail D.","family":"Lukin"},{"given":"Liang","family":"Jiang"},{"given":"Hengyun","family":"Zhou"}],"title":"Constant-Overhead Fault-Tolerant Quantum Computation with Reconfigurable Atom Arrays","issued":{"date-parts":[[2023,8,16]]},"note":"arxivid:2308.08648\narxiv_version_number:1"},{"id":"arxiv:2409.02193","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum error correction plays a prominent role in the realization of quantum computation, and quantum low-density parity-check (qLDPC) codes are believed to be practically useful stabilizer codes. While qLDPC codes are defined to have constant weight parity-checks, the weight of these parity checks could be large constants that make implementing these codes challenging. Large constants can also result in long syndrome extraction times and bad error propagation that can impact error correction performance. Hastings recently introduced weight reduction techniques for qLDPC codes that reduce the weight of the parity checks as well as the maximum number of checks that acts on any data qubit. However, the fault tolerance of these techniques remains an open question. In this paper, we analyze the effective distance of the weight-reduced code when single-ancilla syndrome extraction circuits are considered for error correction. We prove that there exists single-ancilla syndrome extraction circuits that largely preserve the effective distance of the weight-reduced qLDPC codes. In addition, we also show that the distance balancing technique introduced by Evra et al. \\cite{evra2022decodable} preserves effective distance. As a corollary, our result shows that higher-dimensional hypergraph product (HGP) codes, also known as homological product codes corresponding to the product of 1-complexes, have no troublesome hook errors when using any single-ancilla syndrome extraction circuit.</jats:p>","DOI":"10.22331/q-2025-10-28-1897","page":"1897","source":"Crossref","title":"Effective Distance of Higher Dimensional HGPs and Weight-Reduced Quantum LDPC Codes","volume":"9","author":[{"given":"Shi Jie Samuel","family":"Tan","sequence":"first","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA"},{"name":"Department of Computer Science, University of Maryland, MD 20742, USA"}]},{"given":"Lev","family":"Stambler","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA"},{"name":"Department of Computer Science, University of Maryland, MD 20742, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,28]]},"URL":"http://dx.doi.org/10.22331/q-2025-10-28-1897","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2409.02193"},{"id":"arxiv:2411.09668","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    Continuous-variable cat codes are encodings into a single photonic or phononic mode that offer a promising avenue for hardware-efficient fault-tolerant quantum computation. Protecting information in a cat code requires measuring the mode’s occupation number modulo two, but this can be relaxed to a linear occupation-number constraint using the alternative two-mode pair-cat encoding. We construct multimode codes with similar linear constraints using any two integer matrices satisfying a Calderbank-Shor-Steane-like homological condition of a quantum rotor code. Just like the pair-cat code, syndrome extraction can be performed in tandem with stabilizing dissipation using current superconducting-circuit designs. The framework includes codes with various finite- or infinite-dimensional code spaces and codes with finite or infinite Fock-state support. It encompasses two-component cat, pair-cat, dual-rail, two-mode binomial, various bosonic repetition codes, and aspects of\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>χ</a:mi>\n                    </a:math>\n                    -squared encodings, while also yielding codes from homological products, lattices, generalized coherent states, and algebraic varieties. Among our examples are analogs of repetition codes, the Shor code, and a surface-code-like construction that is not a concatenation of a known cat code with the qubit surface code. Code words are coherent states projected into a Fock-state subspace defined by an integer matrix, and their overlaps are governed by Gelfand-Kapranov-Zelevinsky hypergeometric functions.\n                  </jats:p>","DOI":"10.1103/ls5r-vj7r","source":"Crossref","title":"Letting the Tiger out of Its Cage: Bosonic Coding without Concatenation","volume":"15","author":[{"given":"Yijia","family":"Xu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"name":"University of Maryland"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0002-4104-3379","authenticated-orcid":true,"given":"Yixu","family":"Wang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"},{"id":[{"id":"https://ror.org/0489cd473","id-type":"ROR","asserted-by":"publisher"}],"name":"Shanghai Institute for Mathematics and Interdisciplinary Sciences"},{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"name":"University of Maryland"}]},{"ORCID":"https://orcid.org/0000-0002-3445-0179","authenticated-orcid":true,"given":"Christophe","family":"Vuillot","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04vfs2w97","id-type":"ROR","asserted-by":"publisher"}],"name":"Université de Lorraine"}]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"name":"University of Maryland"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,10]]},"URL":"http://dx.doi.org/10.1103/ls5r-vj7r","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041025","note":"arxivid:2411.09668"},{"id":"arxiv:quant-ph/9611001","type":"article-journal","author":[{"given":"E. M.","family":"Rains"}],"title":"Quantum shadow enumerators","issued":{"date-parts":[[1997,2,19]]},"note":"arxivid:quant-ph/9611001\narxiv_version_number:2"},{"id":"arxiv:2408.16914","type":"article-journal","author":[{"given":"Daniel","family":"Miller"},{"given":"Kyano","family":"Levi"},{"given":"Lukas","family":"Postler"},{"given":"Alex","family":"Steiner"},{"given":"Lennart","family":"Bittel"},{"given":"Gregory A. L.","family":"White"},{"given":"Yifan","family":"Tang"},{"given":"Eric J.","family":"Kuehnke"},{"given":"Antonio A.","family":"Mele"},{"given":"Sumeet","family":"Khatri"},{"given":"Lorenzo","family":"Leone"},{"given":"Jose","family":"Carrasco"},{"given":"Christian D.","family":"Marciniak"},{"given":"Ivan","family":"Pogorelov"},{"given":"Milena","family":"Guevara-Bertsch"},{"given":"Robert","family":"Freund"},{"given":"Rainer","family":"Blatt"},{"given":"Philipp","family":"Schindler"},{"given":"Thomas","family":"Monz"},{"given":"Martin","family":"Ringbauer"},{"given":"Jens","family":"Eisert"}],"title":"Experimental measurement and a physical interpretation of quantum shadow enumerators","issued":{"date-parts":[[2024,8,29]]},"note":"arxivid:2408.16914\narxiv_version_number:1"},{"id":"doi:10.1109/SFCS.1995.492581","type":"proceedings-article","publisher":"IEEE Comput. Soc. Press","DOI":"10.1109/sfcs.1995.492581","page":"512-519","source":"Crossref","title":"Linear time erasure codes with nearly optimal recovery","author":[{"given":"N.","family":"Alon","sequence":"first","affiliation":[]},{"given":"J.","family":"Edmonds","sequence":"additional","affiliation":[]},{"given":"M.","family":"Luby","sequence":"additional","affiliation":[]}],"event":"IEEE 36th Annual Foundations of Computer Science","container-title":"Proceedings of IEEE 36th Annual Foundations of Computer Science","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.1995.492581"},{"id":"doi:10.1109/18.556669","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/18.556669","page":"1732-1736","source":"Crossref","title":"A linear time erasure-resilient code with nearly optimal recovery","volume":"42","author":[{"given":"N.","family":"Alon","sequence":"first","affiliation":[]},{"given":"M.","family":"Luby","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1996]]},"URL":"http://dx.doi.org/10.1109/18.556669","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2410.10194","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum information is fragile and must be protected by a quantum error-correcting code for large-scale practical applications. Recently, highly efficient quantum codes have been discovered which require a high degree of spatial connectivity. This raises the question of how to realize these codes with minimal overhead under physical hardware connectivity constraints. Here, we introduce a general recipe to transform any quantum stabilizer code into a subsystem code that has local interactions, with weight and degree three, on a given graph. We call the subsystem codes produced by our recipe wire codes, and their code parameters depend on the input code and the given graph. Wire codes can be adapted to have a local implementation on any graph that supports a low-density embedding of the input Tanner graph, with an overhead that depends on the embedding. In particular, applying our results to a stabilizer code and a subdivision of its own Tanner graph, yields a quantum weight reduction procedure with a multiplicative qubit overhead and distance reduction that are linear in the input check degree and weight, respectively. Applying our results to hypercubic lattices leads to a construction of local subsystem codes with optimal scaling code parameters in any fixed spatial dimension. Similarly, applying our results to families of expanding graphs leads to local codes on these graphs with code parameters that depend on the degree of expansion. Our results constitute a general method to construct low-overhead subsystem codes on general graphs, which can be applied to adapt highly efficient quantum error correction procedures to hardware with restricted connectivity.</jats:p>","DOI":"10.22331/q-2026-04-24-2083","page":"2083","source":"Crossref","title":"Wire Codes","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0002-4028-3098","authenticated-orcid":false,"given":"Nouédyn","family":"Baspin","sequence":"first","affiliation":[{"name":"School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]},{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":false,"given":"Dominic","family":"Williamson","sequence":"additional","affiliation":[{"name":"School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,24]]},"URL":"http://dx.doi.org/10.22331/q-2026-04-24-2083","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2410.10194"},{"id":"arxiv:2504.15087","type":"article-journal","author":[{"given":"Jun-Ting","family":"Hsieh"},{"given":"Alexander","family":"Lubotzky"},{"given":"Sidhanth","family":"Mohanty"},{"given":"Assaf","family":"Reiner"},{"given":"Rachel Yun","family":"Zhang"}],"title":"Explicit Lossless Vertex Expanders","issued":{"date-parts":[[2025,4,21]]},"note":"arxivid:2504.15087\narxiv_version_number:1"},{"id":"arxiv:2203.03581","type":"article-journal","author":[{"given":"Ting-Chun","family":"Lin"},{"given":"Min-Hsiu","family":"Hsieh"}],"title":"Good quantum LDPC codes with linear time decoder from lossless expanders","issued":{"date-parts":[[2022,3,7]]},"note":"arxivid:2203.03581\narxiv_version_number:1"},{"id":"arxiv:1811.00277","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3313276.3316384","page":"481-490","source":"Crossref","title":"Good approximate quantum LDPC codes from spacetime circuit Hamiltonians","author":[{"given":"Thomas C.","family":"Bohdanowicz","sequence":"first","affiliation":[{"name":"California Institute of Technology, USA"}]},{"given":"Elizabeth","family":"Crosson","sequence":"additional","affiliation":[{"name":"University of New Mexico, USA"}]},{"given":"Chinmay","family":"Nirkhe","sequence":"additional","affiliation":[{"name":"University of California at Berkeley, USA"}]},{"given":"Henry","family":"Yuen","sequence":"additional","affiliation":[{"name":"University of Toronto, Canada"}]}],"event":"STOC '19: 51st Annual ACM SIGACT Symposium on the Theory of Computing","container-title":"Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[2019,6,23]]},"URL":"http://dx.doi.org/10.1145/3313276.3316384","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3313276.3316384\n10.1145/3313276\narxivid:1811.00277"},{"id":"arxiv:quant-ph/0701020","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2007.4557323","page":"806-810","source":"Crossref","title":"Quantum Quasi-Cyclic LDPC Codes","author":[{"given":"Manabu","family":"Hagiwara","sequence":"first","affiliation":[]},{"given":"Hideki","family":"Imai","sequence":"additional","affiliation":[]}],"event":"2007 IEEE International Symposium on Information Theory","container-title":"2007 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2007,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2007.4557323","note":"arxivid:quant-ph/0701020"},{"id":"arxiv:1007.1778","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2011.2167593","page":"1223-1230","source":"Crossref","title":"Quantum Error Correction Beyond the Bounded Distance Decoding Limit","volume":"58","author":[{"given":"Kenta","family":"Kasai","sequence":"first","affiliation":[]},{"given":"Manabu","family":"Hagiwara","sequence":"additional","affiliation":[]},{"given":"Hideki","family":"Imai","sequence":"additional","affiliation":[]},{"given":"Kohichi","family":"Sakaniwa","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2012,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2011.2167593","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1007.1778"},{"id":"arxiv:quant-ph/9605005","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.78.405","page":"405-408","source":"Crossref","title":"Quantum Error Correction and Orthogonal Geometry","volume":"78","author":[{"given":"A. R.","family":"Calderbank","sequence":"first","affiliation":[]},{"given":"E. M.","family":"Rains","sequence":"additional","affiliation":[]},{"given":"P. W.","family":"Shor","sequence":"additional","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1997,1,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.78.405","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9605005"},{"id":"arxiv:quant-ph/9512032","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.54.1098","page":"1098-1105","source":"Crossref","title":"Good quantum error-correcting codes exist","volume":"54","author":[{"given":"A. R.","family":"Calderbank","sequence":"first","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1996,8,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.54.1098","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9512032"},{"id":"arxiv:quant-ph/9705052","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"Stabilizer Codes and Quantum Error Correction","issued":{"date-parts":[[1997,5,28]]},"note":"arxivid:quant-ph/9705052\narxiv_version_number:1"},{"id":"preset:PreskillNotes","type":"document","author":[{"family":"Preskill","given":"J."}],"title":"\\emph{Lecture notes on Quantum Computation.}","issued":"1997","URL":"https://preskill.caltech.edu/ph219/"},{"id":"arxiv:1901.08092","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevb.100.134306","source":"Crossref","title":"Measurement-driven entanglement transition in hybrid quantum circuits","volume":"100","author":[{"given":"Yaodong","family":"Li","sequence":"first","affiliation":[]},{"given":"Xiao","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Matthew P. A.","family":"Fisher","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,10,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.100.134306","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"134306","note":"arxivid:1901.08092"},{"id":"arxiv:1601.01694","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"11","DOI":"10.1007/jhep11(2016)009","source":"Crossref","title":"Holographic duality from random tensor networks","volume":"2016","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"given":"Sepehr","family":"Nezami","sequence":"additional","affiliation":[]},{"given":"Xiao-Liang","family":"Qi","sequence":"additional","affiliation":[]},{"given":"Nathaniel","family":"Thomas","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Walter","sequence":"additional","affiliation":[]},{"given":"Zhao","family":"Yang","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2016,11]]},"URL":"http://dx.doi.org/10.1007/JHEP11(2016)009","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"9","note":"alternative-id:4945\narxivid:1601.01694"},{"id":"arxiv:1801.05289","type":"article-journal","author":[{"given":"Xiao-Liang","family":"Qi"},{"given":"Zhao","family":"Yang"}],"title":"Space-time random tensor networks and holographic duality","issued":{"date-parts":[[2018,1,16]]},"note":"arxivid:1801.05289\narxiv_version_number:1"},{"id":"arxiv:2105.12067","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>The AdS/CFT correspondence realises the holographic principle where information in the bulk of a space is encoded at its border. We are yet a long way from a full mathematical construction of AdS/CFT, but toy models in the form of holographic quantum error correcting codes (HQECC) have replicated some interesting features of the correspondence. In this work we construct new HQECCs built from random stabilizer tensors that describe a duality between models encompassing local Hamiltonians whilst exactly obeying the Ryu-Takayanagi entropy formula for all boundary regions. We also obtain complementary recovery of local bulk operators for any boundary bipartition. Existing HQECCs have been shown to exhibit these properties individually, whereas our mathematically rigorous toy models capture these features of AdS/CFT simultaneously, advancing further towards a complete construction of holographic duality.</jats:p>","DOI":"10.1007/jhep03(2022)052","source":"Crossref","title":"Holographic duality between local Hamiltonians from random tensor networks","volume":"2022","author":[{"given":"Harriet","family":"Apel","sequence":"first","affiliation":[]},{"given":"Tamara","family":"Kohler","sequence":"additional","affiliation":[]},{"given":"Toby","family":"Cubitt","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,3]]},"URL":"http://dx.doi.org/10.1007/JHEP03(2022)052","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"52","note":"alternative-id:17946\narxivid:2105.12067"},{"id":"arxiv:1709.08658","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.97.042327","source":"Crossref","title":"Towers of generalized divisible quantum codes","volume":"97","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,4,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.042327","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042327","note":"arxivid:1709.08658"},{"id":"arxiv:1301.1363","type":"article-journal","author":[{"given":"M. H.","family":"Freedman"},{"given":"M. B.","family":"Hastings"}],"title":"Quantum Systems on Non-$k$-Hyperfinite Complexes: A Generalization of Classical Statistical Mechanics on Expander Graphs","issued":{"date-parts":[[2013,7,26]]},"note":"arxivid:1301.1363\narxiv_version_number:2"},{"id":"arxiv:2207.03562","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We introduce a methodology for generating random multi-qubit stabilizer codes based on solving a constraint satisfaction problem (CSP) on random bipartite graphs. This framework allows us to enforce stabilizer commutation, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>Z</mml:mi></mml:math> balancing, finite rate, sparsity, and maximum-degree constraints simultaneously in a CSP that we can then solve numerically. Using a state-of-the-art CSP solver, we obtain convincing evidence for the existence of a satisfiability threshold. Furthermore, the extent of the satisfiable phase increases with the number of qubits. In that phase, finding sparse codes becomes an easy problem. Moreover, we observe that the sparse codes found in the satisfiable phase practically achieve the channel capacity for erasure noise. Our results show that intermediate-size finite-rate sparse quantum codes are easy to find, while also demonstrating a flexible methodology for generating good codes with custom properties. We therefore establish a complete and customizable pipeline for random quantum code discovery.</jats:p>","DOI":"10.22331/q-2023-04-20-985","page":"985","source":"Crossref","title":"Finite-rate sparse quantum codes aplenty","volume":"7","author":[{"given":"Maxime","family":"Tremblay","sequence":"first","affiliation":[{"name":"Département de physique & Institut quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1"}]},{"given":"Guillaume","family":"Duclos-Cianci","sequence":"additional","affiliation":[{"name":"Département de physique & Institut quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1"}]},{"given":"Stefanos","family":"Kourtis","sequence":"additional","affiliation":[{"name":"Département de physique & Institut quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,4,20]]},"URL":"http://dx.doi.org/10.22331/q-2023-04-20-985","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2207.03562"},{"id":"arxiv:1004.1127","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2014.2313577","page":"3117-3122","source":"Crossref","title":"Concatenated Quantum Codes Can Attain the Quantum Gilbert–Varshamov Bound","volume":"60","author":[{"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2313577","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1004.1127"},{"id":"doi:10.1017/CBO9780511976667","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>One of the most cited books in physics of all time, Quantum Computation and Quantum Information remains the best textbook in this exciting field of science. This 10th anniversary edition includes an introduction from the authors setting the work in context. This comprehensive textbook describes such remarkable effects as fast quantum algorithms, quantum teleportation, quantum cryptography and quantum error-correction. Quantum mechanics and computer science are introduced before moving on to describe what a quantum computer is, how it can be used to solve problems faster than 'classical' computers and its real-world implementation. It concludes with an in-depth treatment of quantum information. Containing a wealth of figures and exercises, this well-known textbook is ideal for courses on the subject, and will interest beginning graduate students and researchers in physics, computer science, mathematics, and electrical engineering.</jats:p>","DOI":"10.1017/cbo9780511976667","source":"Crossref","title":"Quantum Computation and Quantum Information","author":[{"given":"Michael A.","family":"Nielsen","sequence":"first","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2012,6,5]]},"ISBN":["9781107002173","9780511976667"],"URL":"http://dx.doi.org/10.1017/CBO9780511976667","note":"edition-number:1"},{"id":"arxiv:2402.00145","type":"article-journal","author":[{"given":"Dongjin","family":"Lee"},{"given":"Beni","family":"Yoshida"}],"title":"Randomly Monitored Quantum Codes","issued":{"date-parts":[[2024,1,31]]},"note":"arxivid:2402.00145\narxiv_version_number:1"},{"id":"arxiv:2304.01277","type":"article-journal","author":[{"given":"David","family":"Aasen"},{"given":"Jeongwan","family":"Haah"},{"given":"Zhi","family":"Li"},{"given":"Roger S. K.","family":"Mong"}],"title":"Measurement Quantum Cellular Automata and Anomalies in Floquet Codes","issued":{"date-parts":[[2023,8,2]]},"note":"arxivid:2304.01277\narxiv_version_number:2"},{"id":"arxiv:1707.09403","type":"journal-article","publisher":"IOP Publishing","issue":"8","DOI":"10.1088/1367-2630/aad8dd","page":"083030","source":"Crossref","title":"Rewiring stabilizer codes","volume":"20","author":[{"given":"Kristina R","family":"Colladay","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5186-5889","authenticated-orcid":false,"given":"Erich J","family":"Mueller","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2018,8,22]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aad8dd","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1707.09403"},{"id":"arxiv:1511.02596","type":"article-journal","author":[{"given":"Yongsoo","family":"Hwang"},{"given":"Byung-Soo","family":"Choi"},{"given":"Young-chai","family":"Ko"},{"given":"Jun","family":"Heo"}],"title":"Fault-tolerant conversion between stabilizer codes by Clifford operations","issued":{"date-parts":[[2015,11,9]]},"note":"arxivid:1511.02596\narxiv_version_number:1"},{"id":"arxiv:1709.09282","type":"article-journal","author":[{"given":"Cupjin","family":"Huang"},{"given":"Michael","family":"Newman"}],"title":"Transversal switching between generic stabilizer codes","issued":{"date-parts":[[2018,1,30]]},"note":"arxivid:1709.09282\narxiv_version_number:4"},{"id":"arxiv:2401.12017","type":"article-journal","author":[{"given":"Michael E.","family":"Beverland"},{"given":"Shilin","family":"Huang"},{"given":"Vadym","family":"Kliuchnikov"}],"title":"Fault tolerance of stabilizer channels","issued":{"date-parts":[[2024,2,25]]},"note":"arxivid:2401.12017\narxiv_version_number:2"},{"id":"arxiv:2409.13465","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Code switching is an established technique that facilitates a universal set of FT quantum gate operations by combining two QEC codes with complementary sets of gates, which each by themselves are easy to implement fault-tolerantly. In this work, we present a code switching scheme that respects the constraints of FT circuit design by only making use of transversal gates. These gates are intrinsically FT without additional qubit overhead. We analyze application of the scheme to low-distance color codes, which are suitable for operation in existing quantum processors, for instance based on trapped ions or neutral atoms. We briefly discuss connectivity constraints that arise for architectures based on superconducting qubits. Numerical simulations of circuit-level noise indicate that a logical <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gate, facilitated by our scheme, could outperform both flag-FT magic state injection protocols and a physical <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gate at low physical error rates. Transversal code switching naturally scales to code pairs of arbitrary code distance. We observe improved performance of a distance-5 protocol compared to both the distance-3 implementation and the physical gate for realistically attainable physical entangling gate error rates. We discuss how the scheme can be implemented with a large degree of parallelization, provided that logical auxiliary qubits can be prepared reliably enough. Our logical <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gate circumvents potentially costly magic state factories. The requirements to perform QEC and to achieve an FT universal gate set are then essentially the same: Prepare logical auxiliary qubits offline, execute transversal gates and perform fast-enough measurements. Transversal code switching thus serves to enable more practical hardware realizations of FT universal quantum computation. The scheme alleviates resource requirements for experimental demonstrations of quantum algorithms run on logical qubits.</jats:p>","DOI":"10.22331/q-2025-09-03-1846","page":"1846","source":"Crossref","title":"Efficient fault-tolerant code switching via one-way transversal CNOT gates","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0002-7581-2148","authenticated-orcid":false,"given":"Sascha","family":"Heußen","sequence":"first","affiliation":[{"name":"neQxt, 50670 Cologne, Germany"}]},{"ORCID":"https://orcid.org/0000-0001-9129-1314","authenticated-orcid":false,"given":"Janine","family":"Hilder","sequence":"additional","affiliation":[{"name":"neQxt, 63906 Erlenbach am Main, Germany"},{"name":"QUANTUM, Institut für Physik, Universität Mainz, 55128 Mainz, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,3]]},"URL":"http://dx.doi.org/10.22331/q-2025-09-03-1846","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2409.13465"},{"id":"arxiv:1210.3637","type":"article-journal","author":[{"given":"Juan","family":"Bermejo-Vega"},{"given":"Maarten Van den","family":"Nest"}],"title":"Classical simulations of Abelian-group normalizer circuits with intermediate measurements","issued":{"date-parts":[[2013,10,22]]},"note":"arxivid:1210.3637\narxiv_version_number:2"},{"id":"arxiv:1409.4800","type":"article-journal","author":[{"given":"Juan","family":"Bermejo-Vega"},{"given":"Cedric Yen-Yu","family":"Lin"},{"given":"Maarten Van den","family":"Nest"}],"title":"The computational power of normalizer circuits over black-box groups","issued":{"date-parts":[[2014,9,16]]},"note":"arxivid:1409.4800\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0109047","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.88.097904","source":"Crossref","title":"Efficient Classical Simulation of Continuous Variable Quantum Information Processes","volume":"88","author":[{"given":"Stephen D.","family":"Bartlett","sequence":"first","affiliation":[]},{"given":"Barry C.","family":"Sanders","sequence":"additional","affiliation":[]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2002,2,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.88.097904","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"097904","note":"arxivid:quant-ph/0109047"},{"id":"arxiv:1210.1783","type":"journal-article","publisher":"IOP Publishing","issue":"1","DOI":"10.1088/1367-2630/15/1/013037","page":"013037","source":"Crossref","title":"Efficient simulation scheme for a class of quantum optics experiments with non-negative Wigner representation","volume":"15","author":[{"given":"Victor","family":"Veitch","sequence":"first","affiliation":[]},{"given":"Nathan","family":"Wiebe","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Ferrie","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Emerson","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2013,1,17]]},"URL":"http://dx.doi.org/10.1088/1367-2630/15/1/013037","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1210.1783"},{"id":"arxiv:1208.3660","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.109.230503","source":"Crossref","title":"Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient","volume":"109","author":[{"given":"A.","family":"Mari","sequence":"first","affiliation":[]},{"given":"J.","family":"Eisert","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,12,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.109.230503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230503","note":"arxivid:1208.3660"},{"id":"arxiv:2601.15396","type":"article-journal","author":[{"given":"Andreas","family":"Bauer"},{"given":"Seth","family":"Lloyd"}],"title":"Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics","issued":{"date-parts":[[2026,1,21]]},"note":"arxivid:2601.15396\narxiv_version_number:1"},{"id":"arxiv:1411.3334","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2017.2663199","page":"2464-2479","source":"Crossref","title":"Sparse Quantum Codes From Quantum Circuits","volume":"63","author":[{"given":"Dave","family":"Bacon","sequence":"first","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3220-7682","authenticated-orcid":false,"given":"Aram W.","family":"Harrow","sequence":"additional","affiliation":[]},{"given":"Jonathan","family":"Shi","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2017,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2017.2663199","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1411.3334"},{"id":"arxiv:0908.4246","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.81.032301","source":"Crossref","title":"Topological subsystem codes","volume":"81","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,3,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.032301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032301","note":"arxivid:0908.4246"},{"id":"arxiv:1011.3529","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.86.032308","source":"Crossref","title":"Logical-operator tradeoff for local quantum codes","volume":"86","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,9,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.032308","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032308","note":"arxivid:1011.3529"},{"id":"arxiv:1605.03601","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevb.94.224206","source":"Crossref","title":"Symmetry constraints on many-body localization","volume":"94","author":[{"given":"Andrew C.","family":"Potter","sequence":"first","affiliation":[]},{"given":"Romain","family":"Vasseur","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2016,12,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.94.224206","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"224206","note":"arxivid:1605.03601"},{"id":"arxiv:2201.07254","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.3.030321","source":"Crossref","title":"Free-Fermion Subsystem Codes","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0003-2303-7589","authenticated-orcid":true,"given":"Adrian","family":"Chapman","sequence":"first","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"Alicia J.","family":"Kollár","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,10]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.030321","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030321","note":"arxivid:2201.07254"},{"id":"doi:10.2307/1969423","type":"journal-article","publisher":"JSTOR","issue":"1","DOI":"10.2307/1969423","page":"101","source":"Crossref","title":"Induced Representations of Locally Compact Groups I","volume":"55","author":[{"given":"George W.","family":"Mackey","sequence":"first","affiliation":[]}],"container-title":"The Annals of Mathematics","original-title":[],"issued":{"date-parts":[[1952,1]]},"URL":"http://dx.doi.org/10.2307/1969423","ISSN":["0003-486X"],"container-title-short":"The Annals of Mathematics"},{"id":"doi:10.1016/B978-1-4832-3188-4.50009","type":"book-chapter","publisher":"Elsevier","DOI":"10.1016/b978-1-4832-3188-4.50009-0","page":"57-118","source":"Crossref","title":"Induced and Subduced Representations","author":[{"given":"A.J.","family":"COLEMAN","sequence":"first","affiliation":[]}],"container-title":"Group Theory and its Applications","original-title":[],"language":"en","issued":{"date-parts":[[1968]]},"ISBN":["9781483231884"],"URL":"http://dx.doi.org/10.1016/B978-1-4832-3188-4.50009-0"},{"id":"manual:-C.-J.-Isham-(1984).-Topologi","type":"document","author":[{"family":"Isham","given":"C.J."}],"title":"Topological and global aspects of quantum theory","issued":"1984","page":"1059–1290","publisher-place":"North-Holland"},{"id":"doi:10.1016/0550-3213(91)90609-2","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0550-3213(91)90609-2","page":"121-160","source":"Crossref","title":"The geometry of inequivalent quantizations","volume":"365","author":[{"given":"N.P.","family":"Landsman","sequence":"first","affiliation":[]},{"given":"N.","family":"Linden","sequence":"additional","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[1991,11]]},"URL":"http://dx.doi.org/10.1016/0550-3213(91)90609-2","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"alternative-id:0550321391906092"},{"id":"arxiv:1304.3366","type":"article-journal","author":[{"given":"Fabio","family":"Scarabotti"},{"given":"Filippo","family":"Tolli"}],"title":"Induced representations and harmonic analysis on finite groups","issued":{"date-parts":[[2013,4,11]]},"note":"arxivid:1304.3366\narxiv_version_number:1"},{"id":"arxiv:2502.14165","type":"article-journal","author":[{"given":"Rui","family":"Okada"}],"title":"A Quantum Analog of Delsarte's Linear Programming Bounds","issued":{"date-parts":[[2025,2,20]]},"note":"arxivid:2502.14165\narxiv_version_number:1"},{"id":"arxiv:2511.14840","type":"article-journal","author":[{"given":"Eric","family":"Kubischta"},{"given":"Ian","family":"Teixeira"}],"title":"Intrinsic Quantum Codes","issued":{"date-parts":[[2026,3,26]]},"note":"arxivid:2511.14840\narxiv_version_number:2"},{"id":"arxiv:2410.18713","type":"journal-article","publisher":"American Physical Society (APS)","issue":"14","DOI":"10.1103/tljb-f7tt","source":"Crossref","title":"Tessellation Codes: Encoded Quantum Gates by Geometric Rotation","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0002-4104-3379","authenticated-orcid":true,"given":"Yixu","family":"Wang","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"given":"Yijia","family":"Xu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"}]},{"ORCID":"https://orcid.org/0000-0002-3402-9763","authenticated-orcid":true,"given":"Zi-Wen","family":"Liu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,2]]},"URL":"http://dx.doi.org/10.1103/tljb-f7tt","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"140602","note":"arxivid:2410.18713"},{"id":"arxiv:1701.06963","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2017.8006823","page":"1718-1722","source":"Crossref","title":"Codes for simultaneous transmission of quantum and classical information","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Sirui","family":"Lu","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"event":"2017 IEEE International Symposium on Information Theory (ISIT)","container-title":"2017 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2017,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2017.8006823","note":"arxivid:1701.06963"},{"id":"arxiv:quant-ph/0203105","type":"article-journal","author":[{"given":"Greg","family":"Kuperberg"}],"title":"The capacity of hybrid quantum memory","issued":{"date-parts":[[2003,1,29]]},"note":"arxivid:quant-ph/0203105\narxiv_version_number:3"},{"id":"arxiv:quant-ph/0311131","type":"article-journal","author":[{"given":"I.","family":"Devetak"},{"given":"P. W.","family":"Shor"}],"title":"The capacity of a quantum channel for simultaneous transmission of classical and quantum information","issued":{"date-parts":[[2004,10,21]]},"note":"arxivid:quant-ph/0311131\narxiv_version_number:3"},{"id":"arxiv:1911.12744","type":"article-journal","author":[{"given":"David W.","family":"Kribs"},{"given":"Ningping","family":"Cao"},{"given":"Chi-Kwong","family":"Li"},{"given":"Yiu-Tung","family":"Poon"},{"given":"Bei","family":"Zeng"},{"given":"Mike","family":"Nelson"}],"title":"Higher Rank Matricial Ranges and Hybrid Quantum Error Correction","issued":{"date-parts":[[2019,11,28]]},"note":"arxivid:1911.12744\narxiv_version_number:1"},{"id":"arxiv:2012.05896","type":"journal-article","publisher":"World Scientific Pub Co Pte Ltd","issue":"02","abstract":"<jats:p> In this paper, we show how to construct hybrid quantum-classical codes from subsystem codes by encoding the classical information into the gauge qudits using gauge fixing. Unlike previous work on hybrid codes, we allow for two separate minimum distances, one for the quantum information and one for the classical information. We give an explicit construction of hybrid codes from two classical linear codes using Bacon–Casaccino subsystem codes, as well as several new examples of good hybrid code. </jats:p>","DOI":"10.1142/s0219749921500416","source":"Crossref","title":"Encoding classical information in gauge subsystems of quantum codes","volume":"20","author":[{"given":"Andrew","family":"Nemec","sequence":"first","affiliation":[{"name":"Department of Computer Science & Engineering, Texas A&M University, College Station, TX 77843, USA"}]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[{"name":"Department of Computer Science & Engineering, Texas A&M University, College Station, TX 77843, USA"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2022,1,12]]},"URL":"http://dx.doi.org/10.1142/S0219749921500416","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","page":"2150041","note":"alternative-id:10.1142/S0219749921500416\narxivid:2012.05896"},{"id":"arxiv:quant-ph/0412076","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.94.180501","source":"Crossref","title":"Unified and Generalized Approach to Quantum Error Correction","volume":"94","author":[{"given":"David","family":"Kribs","sequence":"first","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,5,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.94.180501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180501","note":"arxivid:quant-ph/0412076"},{"id":"arxiv:quant-ph/0504189","type":"article-journal","author":[{"given":"David W.","family":"Kribs"},{"given":"Raymond","family":"Laflamme"},{"given":"David","family":"Poulin"},{"given":"Maia","family":"Lesosky"}],"title":"Operator quantum error correction","issued":{"date-parts":[[2006,1,17]]},"note":"arxivid:quant-ph/0504189\narxiv_version_number:3"},{"id":"arxiv:quant-ph/0506069","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.75.064304","source":"Crossref","title":"Algebraic and information-theoretic conditions for operator quantum error correction","volume":"75","author":[{"given":"Michael A.","family":"Nielsen","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,6,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.064304","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"064304","note":"arxivid:quant-ph/0506069"},{"id":"arxiv:0806.3145","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.78.022333","source":"Crossref","title":"Operator quantum error correction for continuous dynamics","volume":"78","author":[{"given":"Ognyan","family":"Oreshkov","sequence":"first","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,8,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.022333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022333","note":"arxivid:0806.3145"},{"id":"arxiv:quant-ph/0608045","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.74.042329","source":"Crossref","title":"Quantum error-correcting subsystems are unitarily recoverable subsystems","volume":"74","author":[{"given":"David W.","family":"Kribs","sequence":"first","affiliation":[]},{"given":"Robert W.","family":"Spekkens","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,10,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.74.042329","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042329","note":"arxivid:quant-ph/0608045"},{"id":"arxiv:0811.0947","type":"journal-article","publisher":"IOP Publishing","issue":"24","DOI":"10.1088/1751-8113/42/24/245303","page":"245303","source":"Crossref","title":"The multiplicative domain in quantum error correction","volume":"42","author":[{"given":"Man-Duen","family":"Choi","sequence":"first","affiliation":[]},{"given":"Nathaniel","family":"Johnston","sequence":"additional","affiliation":[]},{"given":"David W","family":"Kribs","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2009,5,28]]},"URL":"http://dx.doi.org/10.1088/1751-8113/42/24/245303","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(09)97767-8\narxivid:0811.0947"},{"id":"arxiv:quant-ph/9609015","type":"article-journal","author":[{"given":"Asher","family":"Peres"}],"title":"Unitary dynamics for quantum codewords","issued":{"date-parts":[[1996,9,18]]},"note":"arxivid:quant-ph/9609015\narxiv_version_number:1"},{"id":"arxiv:0709.3533","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.77.032333","source":"Crossref","title":"Robustness of operator quantum error correction with respect to initialization errors","volume":"77","author":[{"given":"Ognyan","family":"Oreshkov","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,3,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.032333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032333","note":"arxivid:0709.3533"},{"id":"arxiv:1202.5139","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.86.012335","source":"Crossref","title":"Towards a unified framework for approximate quantum error correction","volume":"86","author":[{"given":"Prabha","family":"Mandayam","sequence":"first","affiliation":[]},{"given":"Hui Khoon","family":"Ng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,7,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.012335","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012335","note":"arxivid:1202.5139"},{"id":"arxiv:0909.1584","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.80.022311","source":"Crossref","title":"Optical implementation of a unitarily correctable code","volume":"80","author":[{"given":"K. M.","family":"Schreiter","sequence":"first","affiliation":[]},{"given":"A.","family":"Pasieka","sequence":"additional","affiliation":[]},{"given":"R.","family":"Kaltenbaek","sequence":"additional","affiliation":[]},{"given":"K. J.","family":"Resch","sequence":"additional","affiliation":[]},{"given":"D. W.","family":"Kribs","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.022311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022311","note":"arxivid:0909.1584"},{"id":"arxiv:math/0404553","type":"article-journal","author":[{"given":"David W.","family":"Kribs"}],"title":"A quantum computing primer for operator theorists","issued":{"date-parts":[[2004,11,11]]},"note":"arxivid:math/0404553\narxiv_version_number:2"},{"id":"arxiv:math/0506491","type":"article-journal","author":[{"given":"David W.","family":"Kribs"}],"title":"A brief introduction to operator quantum error correction","issued":{"date-parts":[[2005,6,24]]},"note":"arxivid:math/0506491\narxiv_version_number:1"},{"id":"doi:10.1017/CBO9781139034807.008","type":"book-chapter","publisher":"Cambridge University Press","DOI":"10.1017/cbo9781139034807.008","page":"163-180","source":"Crossref","title":"Operator quantum error correction","author":[{"given":"David","family":"Kribs","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Quantum Error Correction","original-title":[],"issued":{"date-parts":[[2013,9,12]]},"URL":"http://dx.doi.org/10.1017/CBO9781139034807.008","note":"edition-number:1"},{"id":"arxiv:quant-ph/0604161","type":"article-journal","author":[{"given":"Andreas","family":"Klappenecker"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Clifford Code Constructions of Operator Quantum Error Correcting Codes","issued":{"date-parts":[[2006,10,18]]},"note":"arxivid:quant-ph/0604161\narxiv_version_number:2"},{"id":"arxiv:2310.10278","type":"article-journal","author":[{"given":"Daniel","family":"Zhang"},{"given":"Toby","family":"Cubitt"}],"title":"Quantum Error Transmutation","issued":{"date-parts":[[2023,10,16]]},"note":"arxivid:2310.10278\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9704002","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.56.2567","page":"2567-2573","source":"Crossref","title":"Approximate quantum error correction can lead to better codes","volume":"56","author":[{"given":"Debbie W.","family":"Leung","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Nielsen","sequence":"additional","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]},{"given":"Yoshihisa","family":"Yamamoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,10,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.56.2567","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9704002"},{"id":"doi:10.1103/PhysRevA.56.1114","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.56.1114","page":"1114-1125","source":"Crossref","title":"Bosonic quantum codes for amplitude damping","volume":"56","author":[{"given":"Isaac L.","family":"Chuang","sequence":"first","affiliation":[]},{"given":"Debbie W.","family":"Leung","sequence":"additional","affiliation":[]},{"given":"Yoshihisa","family":"Yamamoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,8,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.56.1114","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"doi:10.1088/0954-8998/1/2/005","type":"journal-article","publisher":"IOP Publishing","issue":"2","DOI":"10.1088/0954-8998/1/2/005","page":"131-151","source":"Crossref","title":"Probability-density-functional description of quantum photodetection processes","volume":"1","author":[{"given":"M","family":"Ueda","sequence":"first","affiliation":[]}],"container-title":"Quantum Optics: Journal of the European Optical Society Part B","original-title":[],"issued":{"date-parts":[[1989,12]]},"URL":"http://dx.doi.org/10.1088/0954-8998/1/2/005","ISSN":["0954-8998"],"container-title-short":"Quantum Opt."},{"id":"arxiv:1708.05010","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.97.032346","source":"Crossref","title":"Performance and structure of single-mode bosonic codes","volume":"97","author":[{"given":"Victor V.","family":"Albert","sequence":"first","affiliation":[]},{"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"given":"Kasper","family":"Duivenvoorden","sequence":"additional","affiliation":[]},{"given":"Dylan J.","family":"Young","sequence":"additional","affiliation":[]},{"given":"R. T.","family":"Brierley","sequence":"additional","affiliation":[]},{"given":"Philip","family":"Reinhold","sequence":"additional","affiliation":[]},{"given":"Christophe","family":"Vuillot","sequence":"additional","affiliation":[]},{"given":"Linshu","family":"Li","sequence":"additional","affiliation":[]},{"given":"Chao","family":"Shen","sequence":"additional","affiliation":[]},{"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,3,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.032346","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032346","note":"arxivid:1708.05010"},{"id":"arxiv:0909.1596","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.81.062306","source":"Crossref","title":"Stabilizing open quantum systems by Markovian reservoir engineering","volume":"81","author":[{"given":"S. G.","family":"Schirmer","sequence":"first","affiliation":[]},{"given":"Xiaoting","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,6,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.062306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062306","note":"arxivid:0909.1596"},{"id":"arxiv:2008.00477","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Evaluating capacities of quantum channels is the first purpose of quantum Shannon theory, but in most cases the task proves to be very hard. Here, we introduce the set of Multi-level Amplitude Damping quantum channels as a generalization of the standard qubit Amplitude Damping Channel to quantum systems of finite dimension <jats:italic>d</jats:italic>. In the special case of <jats:italic>d</jats:italic> = 3, by exploiting degradability, data-processing inequalities, and channel isomorphism, we compute the associated quantum and private classical capacities for a rather wide class of maps, extending the set of models whose capacity can be computed known so far. We proceed then to the evaluation of the entanglement assisted quantum and classical capacities.</jats:p>","DOI":"10.1038/s42005-021-00524-4","source":"Crossref","title":"Quantum capacity analysis of multi-level amplitude damping channels","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0003-2771-8330","authenticated-orcid":false,"given":"Stefano","family":"Chessa","sequence":"first","affiliation":[]},{"given":"Vittorio","family":"Giovannetti","sequence":"additional","affiliation":[]}],"container-title":"Communications Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,10]]},"URL":"http://dx.doi.org/10.1038/s42005-021-00524-4","ISSN":["2399-3650"],"container-title-short":"Commun Phys","page":"22","note":"alternative-id:524\narxivid:2008.00477"},{"id":"arxiv:quant-ph/0606132","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.98.130501","source":"Crossref","title":"Quantum Capacities of Bosonic Channels","volume":"98","author":[{"given":"Michael M.","family":"Wolf","sequence":"first","affiliation":[]},{"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[]},{"given":"Geza","family":"Giedke","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,3,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.98.130501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130501","note":"arxivid:quant-ph/0606132"},{"id":"arxiv:quant-ph/0405110","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.71.032314","source":"Crossref","title":"Information-capacity description of spin-chain correlations","volume":"71","author":[{"given":"Vittorio","family":"Giovannetti","sequence":"first","affiliation":[]},{"given":"Rosario","family":"Fazio","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,3,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.032314","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032314","note":"arxivid:quant-ph/0405110"},{"id":"arxiv:1309.2219","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.88.042337","source":"Crossref","title":"Classical and quantum capacities of a fully correlated amplitude damping channel","volume":"88","author":[{"given":"A.","family":"D'Arrigo","sequence":"first","affiliation":[]},{"given":"G.","family":"Benenti","sequence":"additional","affiliation":[]},{"given":"G.","family":"Falci","sequence":"additional","affiliation":[]},{"given":"C.","family":"Macchiavello","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,10,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.88.042337","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042337","note":"arxivid:1309.2219"},{"id":"arxiv:1207.5612","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s11128-014-0883-y","page":"765-782","source":"Crossref","title":"Quantum capacity of an amplitude-damping channel with memory","volume":"14","author":[{"given":"Rabia","family":"Jahangir","sequence":"first","affiliation":[]},{"given":"Nigum","family":"Arshed","sequence":"additional","affiliation":[]},{"given":"A. H.","family":"Toor","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2014,11,28]]},"URL":"http://dx.doi.org/10.1007/s11128-014-0883-y","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:883\narxivid:1207.5612"},{"id":"arxiv:1510.05313","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.92.062342","source":"Crossref","title":"Information transmission over an amplitude damping channel with an arbitrary degree of memory","volume":"92","author":[{"given":"Antonio","family":"D'Arrigo","sequence":"first","affiliation":[]},{"given":"Giuliano","family":"Benenti","sequence":"additional","affiliation":[]},{"given":"Giuseppe","family":"Falci","sequence":"additional","affiliation":[]},{"given":"Chiara","family":"Macchiavello","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,12,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.92.062342","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062342","note":"arxivid:1510.05313"},{"id":"arxiv:1001.2356","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2010.5513648","page":"2672-2676","source":"Crossref","title":"Multi-error-correcting amplitude damping codes","author":[{"given":"Runyao","family":"Duan","sequence":"first","affiliation":[]},{"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"Zhengfeng","family":"Ji","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"event":"2010 IEEE International Symposium on Information Theory - ISIT","container-title":"2010 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2010,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2010.5513648","note":"arxivid:1001.2356"},{"id":"arxiv:0712.2586","type":"article-journal","author":[{"given":"Ruitian","family":"Lang"},{"given":"Peter W.","family":"Shor"}],"title":"Nonadditive quantum error correcting codes adapted to the ampltitude damping channel","issued":{"date-parts":[[2007,12,16]]},"note":"arxivid:0712.2586\narxiv_version_number:1"},{"id":"arxiv:2208.05461","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.13.041022","source":"Crossref","title":"Erasure Qubits: Overcoming the \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>\n Limit in Superconducting Circuits","volume":"13","author":[{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":true,"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"given":"Arbel","family":"Haim","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3916-1819","authenticated-orcid":true,"given":"Yotam","family":"Vaknin","sequence":"additional","affiliation":[]},{"given":"Harry","family":"Levine","sequence":"additional","affiliation":[]},{"given":"Fernando","family":"Brandão","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Retzker","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.13.041022","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041022","note":"arxivid:2208.05461"},{"id":"arxiv:2010.00538","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Coherent errors, which arise from collective couplings, are a dominant form of noise in many realistic quantum systems, and are more damaging than oft considered stochastic errors. Here, we propose integrating stabilizer codes with constant-excitation codes by code concatenation. Namely, by concatenating an [[<jats:italic>n</jats:italic>, <jats:italic>k</jats:italic>, <jats:italic>d</jats:italic>]] stabilizer outer code with dual-rail inner codes, we obtain a [[2<jats:italic>n</jats:italic>, <jats:italic>k</jats:italic>, <jats:italic>d</jats:italic>]] constant-excitation code immune from coherent phase errors and also equivalent to a Pauli-rotated stabilizer code. When the stabilizer outer code is fault-tolerant, the constant-excitation code has a positive fault-tolerant threshold against stochastic errors. Setting the outer code as a four-qubit amplitude damping code yields an eight-qubit constant-excitation code that corrects a single amplitude damping error, and we analyze this code’s potential as a quantum memory.</jats:p>","DOI":"10.1038/s41534-021-00429-8","source":"Crossref","title":"Avoiding coherent errors with rotated concatenated stabilizer codes","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0003-1115-0074","authenticated-orcid":false,"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2021,6,2]]},"URL":"http://dx.doi.org/10.1038/s41534-021-00429-8","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"87","note":"alternative-id:429\narxivid:2010.00538"},{"id":"arxiv:quant-ph/0402067","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.70.024302","source":"Crossref","title":"Quantum error correction for continuously detected errors with any number of error channels per qubit","volume":"70","author":[{"given":"Charlene","family":"Ahn","sequence":"first","affiliation":[]},{"given":"Howard","family":"Wiseman","sequence":"additional","affiliation":[]},{"given":"Kurt","family":"Jacobs","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,8,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.70.024302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"024302","note":"arxivid:quant-ph/0402067"},{"id":"arxiv:quant-ph/0302006","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.67.052310","source":"Crossref","title":"Quantum error correction for continuously detected errors","volume":"67","author":[{"given":"Charlene","family":"Ahn","sequence":"first","affiliation":[]},{"given":"H. M.","family":"Wiseman","sequence":"additional","affiliation":[]},{"given":"G. J.","family":"Milburn","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,5,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.67.052310","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052310","note":"arxivid:quant-ph/0302006"},{"id":"arxiv:quant-ph/0103042","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevlett.86.4402","page":"4402-4405","source":"Crossref","title":"Stabilizing Distinguishable Qubits against Spontaneous Decay by Detected-Jump Correcting Quantum Codes","volume":"86","author":[{"given":"G.","family":"Alber","sequence":"first","affiliation":[]},{"given":"Th.","family":"Beth","sequence":"additional","affiliation":[]},{"given":"Ch.","family":"Charnes","sequence":"additional","affiliation":[]},{"given":"A.","family":"Delgado","sequence":"additional","affiliation":[]},{"given":"M.","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"M.","family":"Mussinger","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2001,5,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.86.4402","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/0103042"},{"id":"arxiv:quant-ph/0501184","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevlett.95.100501","source":"Crossref","title":"Loss-Tolerant Optical Qubits","volume":"95","author":[{"given":"T. C.","family":"Ralph","sequence":"first","affiliation":[]},{"given":"A. J. F.","family":"Hayes","sequence":"additional","affiliation":[]},{"given":"Alexei","family":"Gilchrist","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,8,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.95.100501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"100501","note":"arxivid:quant-ph/0501184"},{"id":"arxiv:1106.3800","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.84.042110","source":"Crossref","title":"Symmetric rotating-wave approximation for the generalized single-mode spin-boson system","volume":"84","author":[{"given":"Victor V.","family":"Albert","sequence":"first","affiliation":[]},{"given":"Gregory D.","family":"Scholes","sequence":"additional","affiliation":[]},{"given":"Paul","family":"Brumer","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,10,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.042110","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042110","note":"arxivid:1106.3800"},{"id":"arxiv:2012.06994","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.physleta.2021.127779","page":"127779","source":"Crossref","title":"Approximating the two-mode two-photon Rabi model","volume":"422","author":[{"ORCID":"https://orcid.org/0000-0003-4587-0280","authenticated-orcid":false,"given":"David H.","family":"Wu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":false,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]}],"container-title":"Physics Letters A","original-title":[],"language":"en","issued":{"date-parts":[[2022,1]]},"URL":"http://dx.doi.org/10.1016/j.physleta.2021.127779","ISSN":["0375-9601"],"container-title-short":"Physics Letters A","note":"special_numbering:C\nalternative-id:S0375960121006435\narxivid:2012.06994"},{"id":"manual:-V.-V.-Albert-and-M.-H.-Devor","type":"document","author":[{"family":"Albert","given":"V.V."},{"family":"H","given":"M."}],"genre":"Devoret, private communication,","issued":"2016"},{"id":"arxiv:1811.01450","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.122.040502","source":"Crossref","title":"Ancilla-Free Quantum Error Correction Codes for Quantum Metrology","volume":"122","author":[{"given":"David","family":"Layden","sequence":"first","affiliation":[]},{"given":"Sisi","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Paola","family":"Cappellaro","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,1,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.122.040502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"040502","note":"arxivid:1811.01450"},{"id":"arxiv:1709.05302","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.97.032323","source":"Crossref","title":"Hardware-efficient bosonic quantum error-correcting codes based on symmetry operators","volume":"97","author":[{"given":"Murphy Yuezhen","family":"Niu","sequence":"first","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]},{"given":"Jeffrey H.","family":"Shapiro","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,3,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.032323","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032323","note":"arxivid:1709.05302"},{"id":"arxiv:1809.09801","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.2019.2956142","page":"2921-2933","source":"Crossref","title":"Permutation-Invariant Constant-Excitation Quantum Codes for Amplitude Damping","volume":"66","author":[{"ORCID":"https://orcid.org/0000-0003-1115-0074","authenticated-orcid":false,"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2570-2426","authenticated-orcid":false,"given":"Rui","family":"Chao","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2020,5]]},"URL":"http://dx.doi.org/10.1109/TIT.2019.2956142","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1809.09801"},{"id":"arxiv:quant-ph/9807004","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.81.2594","page":"2594-2597","source":"Crossref","title":"Decoherence-Free Subspaces for Quantum Computation","volume":"81","author":[{"given":"D. A.","family":"Lidar","sequence":"first","affiliation":[]},{"given":"I. L.","family":"Chuang","sequence":"additional","affiliation":[]},{"given":"K. B.","family":"Whaley","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1998,9,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.81.2594","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9807004"},{"id":"arxiv:quant-ph/9902041","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.60.1944","page":"1944-1955","source":"Crossref","title":"Robustness of decoherence-free subspaces for quantum computation","volume":"60","author":[{"given":"D.","family":"Bacon","sequence":"first","affiliation":[]},{"given":"D. A.","family":"Lidar","sequence":"additional","affiliation":[]},{"given":"K. B.","family":"Whaley","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1999,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.60.1944","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9902041"},{"id":"arxiv:quant-ph/9908064","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.63.022306","source":"Crossref","title":"Decoherence-free subspaces for multiple-qubit errors. I. Characterization","volume":"63","author":[{"given":"Daniel A.","family":"Lidar","sequence":"first","affiliation":[]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[]},{"given":"Julia","family":"Kempe","sequence":"additional","affiliation":[]},{"given":"K. B.","family":"Whaley","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2001,1,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.63.022306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022306","note":"arxivid:quant-ph/9908064"},{"id":"arxiv:quant-ph/0007013","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.63.022307","source":"Crossref","title":"Decoherence-free subspaces for multiple-qubit errors. II. Universal, fault-tolerant quantum computation","volume":"63","author":[{"given":"Daniel A.","family":"Lidar","sequence":"first","affiliation":[]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[]},{"given":"Julia","family":"Kempe","sequence":"additional","affiliation":[]},{"given":"K. B.","family":"Whaley","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2001,1,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.63.022307","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022307","note":"arxivid:quant-ph/0007013"},{"id":"arxiv:quant-ph/0210072","type":"journal-article","publisher":"Informa UK Limited","issue":"8","DOI":"10.1080/09500340308235203","page":"1285-1297","source":"Crossref","title":"Combined error correction techniques for quantum computing architectures","volume":"50","author":[{"given":"Mark S.","family":"Byrd","sequence":"first","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Journal of Modern Optics","original-title":[],"language":"en","issued":{"date-parts":[[2003,5]]},"URL":"http://dx.doi.org/10.1080/09500340308235203","ISSN":["0950-0340","1362-3044"],"container-title-short":"Journal of Modern Optics","note":"alternative-id:10.1080/09500340308235203\narxivid:quant-ph/0210072"},{"id":"arxiv:quant-ph/9505011","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.52.3489","page":"3489-3496","source":"Crossref","title":"Simple quantum computer","volume":"52","author":[{"given":"Isaac L.","family":"Chuang","sequence":"first","affiliation":[]},{"given":"Yoshihisa","family":"Yamamoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1995,11,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.52.3489","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9505011"},{"id":"arxiv:quant-ph/9604031","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevlett.76.4281","page":"4281-4284","source":"Crossref","title":"Quantum Bit Regeneration","volume":"76","author":[{"given":"Isaac L.","family":"Chuang","sequence":"first","affiliation":[]},{"given":"Yoshihisa","family":"Yamamoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1996,5,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.76.4281","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9604031"},{"id":"arxiv:quant-ph/9702001","type":"journal-article","publisher":"The Royal Society","issue":"1946","DOI":"10.1098/rspa.1996.0029","page":"567-584","source":"Crossref","title":"Quantum computers and dissipation","volume":"452","container-title":"Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1996,12,31]]},"URL":"http://dx.doi.org/10.1098/rspa.1996.0029","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. Lond. A","note":"alternative-id:10.1098/rspa.1996.0029\narxivid:quant-ph/9702001"},{"id":"arxiv:2311.04423","type":"article-journal","author":[{"given":"Akshay","family":"Koottandavida"},{"given":"Ioannis","family":"Tsioutsios"},{"given":"Aikaterini","family":"Kargioti"},{"given":"Cassady R.","family":"Smith"},{"given":"Vidul R.","family":"Joshi"},{"given":"Wei","family":"Dai"},{"given":"James D.","family":"Teoh"},{"given":"Jacob C.","family":"Curtis"},{"given":"Luigi","family":"Frunzio"},{"given":"Robert J.","family":"Schoelkopf"},{"given":"Michel H.","family":"Devoret"}],"title":"Erasure detection of a dual-rail qubit encoded in a double-post superconducting cavity","issued":{"date-parts":[[2023,11,17]]},"note":"arxivid:2311.04423\narxiv_version_number:2"},{"id":"doi:10.1038/s41467-023-41104-0","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Fast, high-fidelity operations between microwave resonators are an important tool for bosonic quantum computation and simulation with superconducting circuits. An attractive approach for implementing these operations is to couple these resonators via a nonlinear converter and actuate parametric processes with RF drives. It can be challenging to make these processes simultaneously fast and high fidelity, since this requires introducing strong drives without activating parasitic processes or introducing additional decoherence channels. We show that in addition to a careful management of drive frequencies and the spectrum of environmental noise, leveraging the inbuilt symmetries of the converter Hamiltonian can suppress unwanted nonlinear interactions, preventing converter-induced decoherence. We demonstrate these principles using a differentially-driven DC-SQUID as our converter, coupled to two high-Q microwave cavities. Using this architecture, we engineer a highly-coherent beamsplitter and fast (~100 ns) swaps between the cavities, limited primarily by their intrinsic single-photon loss. We characterize this beamsplitter in the cavities’ joint single-photon subspace, and show that we can detect and post-select photon loss events to achieve a beamsplitter gate fidelity exceeding 99.98%, which to our knowledge far surpasses the current state of the art.</jats:p>","DOI":"10.1038/s41467-023-41104-0","source":"Crossref","title":"High-fidelity parametric beamsplitting with a parity-protected converter","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0002-0413-698X","authenticated-orcid":false,"given":"Yao","family":"Lu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0009-0006-8597-2985","authenticated-orcid":false,"given":"Aniket","family":"Maiti","sequence":"additional","affiliation":[]},{"given":"John W. O.","family":"Garmon","sequence":"additional","affiliation":[]},{"given":"Suhas","family":"Ganjam","sequence":"additional","affiliation":[]},{"given":"Yaxing","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6711-4585","authenticated-orcid":false,"given":"Jahan","family":"Claes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0272-5481","authenticated-orcid":false,"given":"Luigi","family":"Frunzio","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":false,"given":"Steven M.","family":"Girvin","sequence":"additional","affiliation":[]},{"given":"Robert J.","family":"Schoelkopf","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2023,9,18]]},"URL":"http://dx.doi.org/10.1038/s41467-023-41104-0","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"5767","note":"updated-by:{\"DOI\":\"10.1038/s41467-023-41822-5\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2023,9,28]],\"date-time\":\"2023-09-28T00:00:00Z\",\"timestamp\":1695859200000}}\nalternative-id:41104"},{"id":"doi:10.1038/35051009","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6816","DOI":"10.1038/35051009","page":"46-52","source":"Crossref","title":"A scheme for efficient quantum computation with linear optics","volume":"409","author":[{"given":"E.","family":"Knill","sequence":"first","affiliation":[]},{"given":"R.","family":"Laflamme","sequence":"additional","affiliation":[]},{"given":"G. J.","family":"Milburn","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2001,1]]},"URL":"http://dx.doi.org/10.1038/35051009","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BF35051009"},{"id":"arxiv:0712.1480","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1140/epjd/e2007-00303-6","page":"381-394","source":"Crossref","title":"Stabilization of quantum information by combined dynamical decoupling and detected-jump error correction","volume":"46","author":[{"given":"D.","family":"Geberth","sequence":"first","affiliation":[]},{"given":"O.","family":"Kern","sequence":"additional","affiliation":[]},{"given":"G.","family":"Alber","sequence":"additional","affiliation":[]},{"given":"I.","family":"Jex","sequence":"additional","affiliation":[]}],"container-title":"The European Physical Journal D","original-title":[],"language":"en","issued":{"date-parts":[[2007,10,31]]},"URL":"http://dx.doi.org/10.1140/epjd/e2007-00303-6","ISSN":["1434-6060","1434-6079"],"container-title-short":"Eur. Phys. J. D","note":"alternative-id:303\narxivid:0712.1480"},{"id":"arxiv:quant-ph/0307015","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.68.064303","source":"Crossref","title":"Bounds on the probability of success of postselected nonlinear sign shifts implemented with linear optics","volume":"68","author":[{"given":"E.","family":"Knill","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,12,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.064303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"064303","note":"arxivid:quant-ph/0307015"},{"id":"arxiv:2212.11196","type":"article-journal","author":[{"given":"Takahiro","family":"Tsunoda"},{"given":"James D.","family":"Teoh"},{"given":"William D.","family":"Kalfus"},{"given":"Stijn J.","family":"de Graaf"},{"given":"Benjamin J.","family":"Chapman"},{"given":"Jacob C.","family":"Curtis"},{"given":"Neel","family":"Thakur"},{"given":"Steven M.","family":"Girvin"},{"given":"Robert J.","family":"Schoelkopf"}],"title":"Error-detectable bosonic entangling gates with a noisy ancilla","issued":{"date-parts":[[2022,12,21]]},"note":"arxivid:2212.11196\narxiv_version_number:1"},{"id":"arxiv:2503.10935","type":"article-journal","author":[{"given":"Nitish","family":"Mehta"},{"given":"James D.","family":"Teoh"},{"given":"Taewan","family":"Noh"},{"given":"Ankur","family":"Agrawal"},{"given":"Amos","family":"Anderson"},{"given":"Beau","family":"Birdsall"},{"given":"Avadh","family":"Brahmbhatt"},{"given":"Winfred","family":"Byrd"},{"given":"Marc","family":"Cacioppo"},{"given":"Anthony","family":"Cabrera"},{"given":"Leo","family":"Carroll"},{"given":"Jonathan","family":"Chen"},{"given":"Tzu-Chiao","family":"Chien"},{"given":"Richard","family":"Chamberlain"},{"given":"Jacob C.","family":"Curtis"},{"given":"Doreen","family":"Danso"},{"given":"Sanjana Renganatha","family":"Desigan"},{"given":"Francesco","family":"D'Acounto"},{"given":"Bassel Heiba","family":"Elfeky"},{"given":"S. M.","family":"Farzaneh"},{"given":"Chase","family":"Foley"},{"given":"Benjamin","family":"Gudlewski"},{"given":"Hannah","family":"Hastings"},{"given":"Robert","family":"Johnson"},{"given":"Nishaad","family":"Khedkar"},{"given":"Trevor","family":"Keen"},{"given":"Anup","family":"Kumar"},{"given":"Cihan","family":"Kurter"},{"given":"Kamila","family":"Krawczuk"},{"given":"Eric","family":"Langstengel"},{"given":"Richard D.","family":"Li"},{"given":"Gangqiang","family":"Liu"},{"given":"Hanyi","family":"Lu"},{"given":"Pinlei","family":"Lu"},{"given":"Luke","family":"Mastalli-Kelly"},{"given":"Adam","family":"Maines"},{"given":"Michael","family":"Maxwell"},{"given":"Heather","family":"McCarrick"},{"given":"Mona","family":"Mirzaei"},{"given":"Anirudh","family":"Narla"},{"given":"Omar","family":"Rashad"},{"given":"Erik","family":"Reikes"},{"given":"Mizanur","family":"Rahman"},{"given":"Rurik","family":"Primiani"},{"given":"Michael","family":"Schwaller"},{"given":"Ali","family":"Sabbah"},{"given":"Tali","family":"Shemma"},{"given":"Ruby A.","family":"Shi"},{"given":"Sitakanta","family":"Satapathy"},{"given":"Dean","family":"Stolpe"},{"given":"Jonathan","family":"Strenczewilk"},{"given":"Doug","family":"Szperka"},{"given":"Iu-Wei","family":"Sze"},{"given":"David","family":"Sweeney"},{"given":"Preetham","family":"Tikkireddi"},{"given":"Chin-Lun","family":"Tsung"},{"given":"Daren Vet","family":"Sam"},{"given":"Daniel K.","family":"Weiss"},{"given":"Zhibo","family":"Yang"},{"given":"Liuqi","family":"Yu"},{"given":"Teng","family":"Zhang"},{"given":"Olivier","family":"Boireau"},{"given":"Stephen","family":"Horton"},{"given":"Sean","family":"Weinberg"},{"given":"Jose","family":"Aumentado"},{"given":"Bryan","family":"Cord"},{"given":"Chan U","family":"Lei"},{"given":"Joseph O.","family":"Yuan"},{"given":"Shantanu O.","family":"Mundhada"},{"given":"Kevin S.","family":"Chou"},{"given":"S. Harvey","family":"Moseleley"},{"given":"Robert J.","family":"Schoelkopf"}],"title":"Bias-preserving and error-detectable entangling operations in a superconducting dual-rail system","issued":{"date-parts":[[2025,5,6]]},"note":"arxivid:2503.10935\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0512104","type":"article-journal","author":[{"given":"C. R.","family":"Myers"},{"given":"R.","family":"Laflamme"}],"title":"Linear Optics Quantum Computation: an Overview","issued":{"date-parts":[[2005,12,13]]},"note":"arxivid:quant-ph/0512104\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0512071","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/revmodphys.79.135","page":"135-174","source":"Crossref","title":"Linear optical quantum computing with photonic qubits","volume":"79","author":[{"given":"Pieter","family":"Kok","sequence":"first","affiliation":[]},{"given":"W. J.","family":"Munro","sequence":"additional","affiliation":[]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]},{"given":"T. C.","family":"Ralph","sequence":"additional","affiliation":[]},{"given":"Jonathan P.","family":"Dowling","sequence":"additional","affiliation":[]},{"given":"G. J.","family":"Milburn","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2007,1,24]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.79.135","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","note":"arxivid:quant-ph/0512071"},{"id":"arxiv:1907.06331","type":"journal-article","publisher":"AIP Publishing","issue":"4","abstract":"<jats:p>Photons have been a flagship system for studying quantum mechanics, advancing quantum information science, and developing quantum technologies. Quantum entanglement, teleportation, quantum key distribution, and early quantum computing demonstrations were pioneered in this technology because photons represent a naturally mobile and low-noise system with quantum-limited detection readily available. The quantum states of individual photons can be manipulated with very high precision using interferometry, an experimental staple that has been under continuous development since the 19th century. The complexity of photonic quantum computing devices and protocol realizations has raced ahead as both underlying technologies and theoretical schemes have continued to develop. Today, photonic quantum computing represents an exciting path to medium- and large-scale processing. It promises to put aside its reputation for requiring excessive resource overheads due to inefficient two-qubit gates. Instead, the ability to generate large numbers of photons—and the development of integrated platforms, improved sources and detectors, novel noise-tolerant theoretical approaches, and more—have solidified it as a leading contender for both quantum information processing and quantum networking. Our concise review provides a flyover of some key aspects of the field, with a focus on experiment. Apart from being a short and accessible introduction, its many references to in-depth articles and longer specialist reviews serve as a launching point for deeper study of the field.</jats:p>","DOI":"10.1063/1.5115814","source":"Crossref","title":"Photonic quantum information processing: A concise review","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-5318-3790","authenticated-orcid":false,"given":"Sergei","family":"Slussarenko","sequence":"first","affiliation":[{"name":"Centre for Quantum Dynamics and Centre for Quantum Computation and Communication Technology, Griffith University , Brisbane, Queensland 4111, Australia"}]},{"ORCID":"https://orcid.org/0000-0003-3324-6579","authenticated-orcid":false,"given":"Geoff J.","family":"Pryde","sequence":"additional","affiliation":[{"name":"Centre for Quantum Dynamics and Centre for Quantum Computation and Communication Technology, Griffith University , Brisbane, Queensland 4111, Australia"}]}],"container-title":"Applied Physics Reviews","original-title":[],"language":"en","issued":{"date-parts":[[2019,10,14]]},"URL":"http://dx.doi.org/10.1063/1.5115814","ISSN":["1931-9401"],"page":"041303","note":"arxivid:1907.06331"},{"id":"arxiv:2307.08737","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Quantum error correction with erasure qubits promises significant advantages over standard error correction due to favorable thresholds for erasure errors. To realize this advantage in practice requires a qubit for which nearly all errors are such erasure errors, and the ability to check for erasure errors without dephasing the qubit. We demonstrate that a “dual-rail qubit” consisting of a pair of resonantly coupled transmons can form a highly coherent erasure qubit, where transmon <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:msub><a:mi>T</a:mi><a:mn>1</a:mn></a:msub></a:math> errors are converted into erasure errors and residual dephasing is strongly suppressed, leading to millisecond-scale coherence within the qubit subspace. We show that single-qubit gates are limited primarily by erasure errors, with erasure probability <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:msub><c:mi>p</c:mi><c:mtext>erasure</c:mtext></c:msub><c:mo>=</c:mo><c:mn>2.19</c:mn><c:mo stretchy=\"false\">(</c:mo><c:mn>2</c:mn><c:mo stretchy=\"false\">)</c:mo><c:mo>×</c:mo><c:msup><c:mn>10</c:mn><c:mrow><c:mo>−</c:mo><c:mn>3</c:mn></c:mrow></c:msup></c:math> per gate while the residual errors are <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:mo>∼</g:mo><g:mn>40</g:mn></g:math> times lower. We further demonstrate midcircuit detection of erasure errors while introducing <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:mo>&lt;</i:mo><i:mn>0.1</i:mn><i:mo>%</i:mo></i:math> dephasing error per check. Finally, we show that the suppression of transmon noise allows this dual-rail qubit to preserve high coherence over a broad tunable operating range, offering an improved capacity to avoid frequency collisions. This work establishes transmon-based dual-rail qubits as an attractive building block for hardware-efficient quantum error correction.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevx.14.011051","source":"Crossref","title":"Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0001-8270-3233","authenticated-orcid":true,"given":"H.","family":"Levine","sequence":"first","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"A.","family":"Haim","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Institute of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel"}]},{"given":"J. S. C.","family":"Hung","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"N.","family":"Alidoust","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M.","family":"Kalaee","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"L.","family":"DeLorenzo","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-5764-8540","authenticated-orcid":true,"given":"E. A.","family":"Wollack","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"P.","family":"Arrangoiz-Arriola","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-9120-1478","authenticated-orcid":true,"given":"A.","family":"Khalajhedayati","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"R.","family":"Sanil","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"H.","family":"Moradinejad","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"Y.","family":"Vaknin","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel"}]},{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":true,"given":"A.","family":"Kubica","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"D.","family":"Hover","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-9920-493X","authenticated-orcid":true,"given":"S.","family":"Aghaeimeibodi","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J. A.","family":"Alcid","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"C.","family":"Baek","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J.","family":"Barnett","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"K.","family":"Bawdekar","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"P.","family":"Bienias","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"H. A.","family":"Carson","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"C.","family":"Chen","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-1700-7503","authenticated-orcid":true,"given":"L.","family":"Chen","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"H.","family":"Chinkezian","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"E. M.","family":"Chisholm","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"A.","family":"Clifford","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"R.","family":"Cosmic","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"N.","family":"Crisosto","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"A. M.","family":"Dalzell","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"E.","family":"Davis","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J. M.","family":"D’Ewart","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"S.","family":"Diez","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"N.","family":"D’Souza","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-5707-3978","authenticated-orcid":true,"given":"P. T.","family":"Dumitrescu","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"E.","family":"Elkhouly","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M. T.","family":"Fang","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0001-6964-5817","authenticated-orcid":true,"given":"Y.","family":"Fang","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"S.","family":"Flammia","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M. J.","family":"Fling","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"G.","family":"Garcia","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M. K.","family":"Gharzai","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0003-0509-3421","authenticated-orcid":true,"given":"A. V.","family":"Gorshkov","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M. J.","family":"Gray","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"S.","family":"Grimberg","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-5208-0271","authenticated-orcid":true,"given":"A. L.","family":"Grimsmo","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia"}]},{"given":"C. T.","family":"Hann","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"Y.","family":"He","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-5222-1003","authenticated-orcid":true,"given":"S.","family":"Heidel","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"S.","family":"Howell","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M.","family":"Hunt","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J.","family":"Iverson","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"I.","family":"Jarrige","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"L.","family":"Jiang","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA"}]},{"ORCID":"https://orcid.org/0000-0002-8610-2176","authenticated-orcid":true,"given":"W. M.","family":"Jones","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"R.","family":"Karabalin","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0001-6031-4800","authenticated-orcid":true,"given":"P. J.","family":"Karalekas","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3030-1149","authenticated-orcid":true,"given":"A. J.","family":"Keller","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"D.","family":"Lasi","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0003-1954-3198","authenticated-orcid":true,"given":"M.","family":"Lee","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"V.","family":"Ly","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"G.","family":"MacCabe","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"N.","family":"Mahuli","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4694-7262","authenticated-orcid":true,"given":"G.","family":"Marcaud","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M. H.","family":"Matheny","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"S.","family":"McArdle","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0009-0006-8352-1953","authenticated-orcid":true,"given":"G.","family":"McCabe","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"G.","family":"Merton","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"C.","family":"Miles","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"A.","family":"Milsted","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0003-1325-4387","authenticated-orcid":true,"given":"A.","family":"Mishra","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4242-3015","authenticated-orcid":true,"given":"L.","family":"Moncelsi","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M.","family":"Naghiloo","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"K.","family":"Noh","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"E.","family":"Oblepias","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"G.","family":"Ortuno","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J. C.","family":"Owens","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J.","family":"Pagdilao","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"A.","family":"Panduro","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J.-P.","family":"Paquette","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"R. N.","family":"Patel","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"G.","family":"Peairs","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"D. J.","family":"Perello","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"E. C.","family":"Peterson","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"S.","family":"Ponte","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"H.","family":"Putterman","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"G.","family":"Refael","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Department of Physics, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"P.","family":"Reinhold","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"R.","family":"Resnick","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"O. A.","family":"Reyna","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"R.","family":"Rodriguez","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J.","family":"Rose","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0009-0008-2829-6057","authenticated-orcid":true,"given":"A. H.","family":"Rubin","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M.","family":"Runyan","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0001-9923-3354","authenticated-orcid":true,"given":"C. A.","family":"Ryan","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"A.","family":"Sahmoud","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"T.","family":"Scaffidi","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"B.","family":"Shah","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"S.","family":"Siavoshi","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"P.","family":"Sivarajah","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"T.","family":"Skogland","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"C.-J.","family":"Su","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"L. J.","family":"Swenson","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"J.","family":"Sylvia","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"S. M.","family":"Teo","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"A.","family":"Tomada","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"G.","family":"Torlai","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"M.","family":"Wistrom","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"K.","family":"Zhang","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"ORCID":"https://orcid.org/0009-0006-9664-969X","authenticated-orcid":true,"given":"I.","family":"Zuk","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel"}]},{"given":"A. A.","family":"Clerk","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"F. G. S. L.","family":"Brandão","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"A.","family":"Retzker","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel"}]},{"ORCID":"https://orcid.org/0000-0002-1581-9209","authenticated-orcid":true,"given":"O.","family":"Painter","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Thomas J. Watson, Sr., Laboratory of Applied Physics and Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2024,3,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.14.011051","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011051","note":"arxivid:2307.08737"},{"id":"arxiv:2212.12077","type":"journal-article","publisher":"Proceedings of the National Academy of Sciences","issue":"41","abstract":"<jats:p>\n            The design of quantum hardware that reduces and mitigates errors is essential for practical quantum error correction (QEC) and useful quantum computation. To this end, we introduce the circuit-Quantum Electrodynamics (QED) dual-rail qubit in which our physical qubit is encoded in the single-photon subspace,\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\">\n                <mml:mrow>\n                  <mml:mo stretchy=\"false\">{</mml:mo>\n                  <mml:mo>|</mml:mo>\n                  <mml:mn>01</mml:mn>\n                  <mml:mo>〉</mml:mo>\n                  <mml:mo>,</mml:mo>\n                  <mml:mo>|</mml:mo>\n                  <mml:mn>10</mml:mn>\n                  <mml:mo>〉</mml:mo>\n                  <mml:mo stretchy=\"false\">}</mml:mo>\n                </mml:mrow>\n              </mml:math>\n            </jats:inline-formula>\n            , of two superconducting microwave cavities. The dominant photon loss errors can be detected and converted into erasure errors, which are in general much easier to correct. In contrast to linear optics, a circuit-QED implementation of the dual-rail code offers unique capabilities. Using just one additional transmon ancilla per dual-rail qubit, we describe how to perform a gate-based set of universal operations that includes state preparation, logical readout, and parametrizable single and two-qubit gates. Moreover, first-order hardware errors in the cavities and the transmon can be detected and converted to erasure errors in all operations, leaving background Pauli errors that are orders of magnitude smaller. Hence, the dual-rail cavity qubit exhibits a favorable hierarchy of error rates and is expected to perform well below the relevant QEC thresholds with today’s coherence times.\n          </jats:p>","DOI":"10.1073/pnas.2221736120","source":"Crossref","title":"Dual-rail encoding with superconducting cavities","volume":"120","author":[{"ORCID":"https://orcid.org/0009-0009-3086-3513","authenticated-orcid":false,"given":"James D.","family":"Teoh","sequence":"first","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0000-0001-6457-6971","authenticated-orcid":false,"given":"Patrick","family":"Winkel","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"Harshvardhan K.","family":"Babla","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0000-0003-1248-7568","authenticated-orcid":false,"given":"Benjamin J.","family":"Chapman","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0000-0002-6711-4585","authenticated-orcid":false,"given":"Jahan","family":"Claes","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0000-0003-1111-4460","authenticated-orcid":false,"given":"Stijn J.","family":"de Graaf","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"John W. O.","family":"Garmon","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"William D.","family":"Kalfus","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"Yao","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"Aniket","family":"Maiti","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0009-0002-4587-5620","authenticated-orcid":false,"given":"Kaavya","family":"Sahay","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0009-0002-7965-6135","authenticated-orcid":false,"given":"Neel","family":"Thakur","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"Takahiro","family":"Tsunoda","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0000-0002-7785-8297","authenticated-orcid":false,"given":"Sophia H.","family":"Xue","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0000-0002-0272-5481","authenticated-orcid":false,"given":"Luigi","family":"Frunzio","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":false,"given":"Steven M.","family":"Girvin","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]},{"given":"Robert J.","family":"Schoelkopf","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511"},{"name":"Department of Physics, Yale University, New Haven, CT 06511"},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06511"}]}],"container-title":"Proceedings of the National Academy of Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,6]]},"URL":"http://dx.doi.org/10.1073/pnas.2221736120","ISSN":["0027-8424","1091-6490"],"container-title-short":"Proc. Natl. Acad. Sci. U.S.A.","page":"e2221736120","note":"updated-by:{\"updated\":{\"date-parts\":[[2023,12,18]],\"date-time\":\"2023-12-18T00:00:00Z\",\"timestamp\":1702857600000},\"DOI\":\"10.1073/pnas.2320127120\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nalternative-id:10.1073/pnas.2221736120\narxivid:2212.12077"},{"id":"arxiv:2504.12099","type":"journal-article","publisher":"Springer Science and Business Media LLC","DOI":"10.1038/s41567-026-03211-9","source":"Crossref","title":"Logical multi-qubit entanglement with dual-rail superconducting qubits","author":[{"given":"Wenhui","family":"Huang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0009-0009-9220-0977","authenticated-orcid":false,"given":"Xuandong","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Jiawei","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Zechen","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Peisheng","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Yongqi","family":"Liang","sequence":"additional","affiliation":[]},{"given":"Yiting","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Daxiong","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Zilin","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Yuzhe","family":"Xiong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-4555-1275","authenticated-orcid":false,"given":"Xiaohan","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Jiajian","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Libo","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4193-3280","authenticated-orcid":false,"given":"Ji","family":"Chu","sequence":"additional","affiliation":[]},{"given":"Weijie","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Ji","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Song","family":"Liu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4040-1426","authenticated-orcid":false,"given":"Jingjing","family":"Niu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9475-6452","authenticated-orcid":false,"given":"Jiawei","family":"Qiu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3633-6000","authenticated-orcid":false,"given":"Ziyu","family":"Tao","sequence":"additional","affiliation":[]},{"given":"Yuxuan","family":"Zhou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0002-8869-271X","authenticated-orcid":false,"given":"Xiayu","family":"Linpeng","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8120-6259","authenticated-orcid":false,"given":"Youpeng","family":"Zhong","sequence":"additional","affiliation":[]},{"given":"Dapeng","family":"Yu","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,6]]},"URL":"http://dx.doi.org/10.1038/s41567-026-03211-9","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:3211\narxivid:2504.12099"},{"id":"arxiv:2307.03169","type":"article-journal","author":[{"given":"Kevin S.","family":"Chou"},{"given":"Tali","family":"Shemma"},{"given":"Heather","family":"McCarrick"},{"given":"Tzu-Chiao","family":"Chien"},{"given":"James D.","family":"Teoh"},{"given":"Patrick","family":"Winkel"},{"given":"Amos","family":"Anderson"},{"given":"Jonathan","family":"Chen"},{"given":"Jacob","family":"Curtis"},{"given":"Stijn J.","family":"de Graaf"},{"given":"John W. O.","family":"Garmon"},{"given":"Benjamin","family":"Gudlewski"},{"given":"William D.","family":"Kalfus"},{"given":"Trevor","family":"Keen"},{"given":"Nishaad","family":"Khedkar"},{"given":"Chan U","family":"Lei"},{"given":"Gangqiang","family":"Liu"},{"given":"Pinlei","family":"Lu"},{"given":"Yao","family":"Lu"},{"given":"Aniket","family":"Maiti"},{"given":"Luke","family":"Mastalli-Kelly"},{"given":"Nitish","family":"Mehta"},{"given":"Shantanu O.","family":"Mundhada"},{"given":"Anirudh","family":"Narla"},{"given":"Taewan","family":"Noh"},{"given":"Takahiro","family":"Tsunoda"},{"given":"Sophia H.","family":"Xue"},{"given":"Joseph O.","family":"Yuan"},{"given":"Luigi","family":"Frunzio"},{"given":"Jose","family":"Aumentado"},{"given":"Shruti","family":"Puri"},{"given":"Steven M.","family":"Girvin"},{"given":"S. Harvey","family":"Moseley"},{"given":"Robert J.","family":"Schoelkopf"}],"title":"Demonstrating a superconducting dual-rail cavity qubit with erasure-detected logical measurements","issued":{"date-parts":[[2023,10,13]]},"note":"arxivid:2307.03169\narxiv_version_number:3"},{"id":"arxiv:2404.17570","type":"article-journal","author":[{"given":"Koen","family":"Alexander"},{"given":"Andrea","family":"Bahgat"},{"given":"Avishai","family":"Benyamini"},{"given":"Dylan","family":"Black"},{"given":"Damien","family":"Bonneau"},{"given":"Stanley","family":"Burgos"},{"given":"Ben","family":"Burridge"},{"given":"Geoff","family":"Campbell"},{"given":"Gabriel","family":"Catalano"},{"given":"Alex","family":"Ceballos"},{"given":"Chia-Ming","family":"Chang"},{"given":"CJ","family":"Chung"},{"given":"Fariba","family":"Danesh"},{"given":"Tom","family":"Dauer"},{"given":"Michael","family":"Davis"},{"given":"Eric","family":"Dudley"},{"given":"Ping","family":"Er-Xuan"},{"given":"Josep","family":"Fargas"},{"given":"Alessandro","family":"Farsi"},{"given":"Colleen","family":"Fenrich"},{"given":"Jonathan","family":"Frazer"},{"given":"Masaya","family":"Fukami"},{"given":"Yogeeswaran","family":"Ganesan"},{"given":"Gary","family":"Gibson"},{"given":"Mercedes","family":"Gimeno-Segovia"},{"given":"Sebastian","family":"Goeldi"},{"given":"Patrick","family":"Goley"},{"given":"Ryan","family":"Haislmaier"},{"given":"Sami","family":"Halimi"},{"given":"Paul","family":"Hansen"},{"given":"Sam","family":"Hardy"},{"given":"Jason","family":"Horng"},{"given":"Matthew","family":"House"},{"given":"Hong","family":"Hu"},{"given":"Mehdi","family":"Jadidi"},{"given":"Henrik","family":"Johansson"},{"given":"Thomas","family":"Jones"},{"given":"Vimal","family":"Kamineni"},{"given":"Nicholas","family":"Kelez"},{"given":"Ravi","family":"Koustuban"},{"given":"George","family":"Kovall"},{"given":"Peter","family":"Krogen"},{"given":"Nikhil","family":"Kumar"},{"given":"Yong","family":"Liang"},{"given":"Nicholas","family":"LiCausi"},{"given":"Dan","family":"Llewellyn"},{"given":"Kimberly","family":"Lokovic"},{"given":"Michael","family":"Lovelady"},{"given":"Vitor","family":"Manfrinato"},{"given":"Ann","family":"Melnichuk"},{"given":"Mario","family":"Souza"},{"given":"Gabriel","family":"Mendoza"},{"given":"Brad","family":"Moores"},{"given":"Shaunak","family":"Mukherjee"},{"given":"Joseph","family":"Munns"},{"given":"Francois-Xavier","family":"Musalem"},{"given":"Faraz","family":"Najafi"},{"given":"Jeremy L.","family":"O'Brien"},{"given":"J. Elliott","family":"Ortmann"},{"given":"Sunil","family":"Pai"},{"given":"Bryan","family":"Park"},{"given":"Hsuan-Tung","family":"Peng"},{"given":"Nicholas","family":"Penthorn"},{"given":"Brennan","family":"Peterson"},{"given":"Matt","family":"Poush"},{"given":"Geoff J.","family":"Pryde"},{"given":"Tarun","family":"Ramprasad"},{"given":"Gareth","family":"Ray"},{"given":"Angelita","family":"Rodriguez"},{"given":"Brian","family":"Roxworthy"},{"given":"Terry","family":"Rudolph"},{"given":"Dylan J.","family":"Saunders"},{"given":"Pete","family":"Shadbolt"},{"given":"Deesha","family":"Shah"},{"given":"Hyungki","family":"Shin"},{"given":"Jake","family":"Smith"},{"given":"Ben","family":"Sohn"},{"given":"Young-Ik","family":"Sohn"},{"given":"Gyeongho","family":"Son"},{"given":"Chris","family":"Sparrow"},{"given":"Matteo","family":"Staffaroni"},{"given":"Camille","family":"Stavrakas"},{"given":"Vijay","family":"Sukumaran"},{"given":"Davide","family":"Tamborini"},{"given":"Mark G.","family":"Thompson"},{"given":"Khanh","family":"Tran"},{"given":"Mark","family":"Triplet"},{"given":"Maryann","family":"Tung"},{"given":"Alexey","family":"Vert"},{"given":"Mihai D.","family":"Vidrighin"},{"given":"Ilya","family":"Vorobeichik"},{"given":"Peter","family":"Weigel"},{"given":"Mathhew","family":"Wingert"},{"given":"Jamie","family":"Wooding"},{"given":"Xinran","family":"Zhou"}],"title":"A manufacturable platform for photonic quantum computing","issued":{"date-parts":[[2024,4,26]]},"note":"arxivid:2404.17570\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0405112","type":"article-journal","author":[{"given":"Marcus","family":"Silva"}],"title":"Erasure Thresholds for Efficient Linear Optics Quantum Computation","issued":{"date-parts":[[2004,5,19]]},"note":"arxivid:quant-ph/0405112\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0502101","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.72.032307","source":"Crossref","title":"Thresholds for linear optics quantum computing with photon loss at the detectors","volume":"72","author":[{"given":"Marcus","family":"Silva","sequence":"first","affiliation":[]},{"given":"Martin","family":"Rötteler","sequence":"additional","affiliation":[]},{"given":"Christof","family":"Zalka","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,9,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.72.032307","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032307","note":"arxivid:quant-ph/0502101"},{"id":"arxiv:1605.09278","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevlett.117.100501","source":"Crossref","title":"Universal Quantum Computing with Arbitrary Continuous-Variable Encoding","volume":"117","author":[{"given":"Hoi-Kwan","family":"Lau","sequence":"first","affiliation":[]},{"given":"Martin B.","family":"Plenio","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,8,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.117.100501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"100501","note":"arxivid:1605.09278"},{"id":"arxiv:quant-ph/0303008","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.91.037903","source":"Crossref","title":"Deterministic Linear Optics Quantum Computation with Single Photon Qubits","volume":"91","author":[{"given":"N.","family":"Yoran","sequence":"first","affiliation":[]},{"given":"B.","family":"Reznik","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2003,7,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.91.037903","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"037903","note":"arxivid:quant-ph/0303008"},{"id":"arxiv:quant-ph/0402005","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.93.040503","source":"Crossref","title":"Optical Quantum Computation Using Cluster States","volume":"93","author":[{"given":"Michael A.","family":"Nielsen","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2004,7,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.93.040503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"040503","note":"arxivid:quant-ph/0402005"},{"id":"arxiv:quant-ph/0405157","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.95.010501","source":"Crossref","title":"Resource-Efficient Linear Optical Quantum Computation","volume":"95","author":[{"given":"Daniel E.","family":"Browne","sequence":"first","affiliation":[]},{"given":"Terry","family":"Rudolph","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,6,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.95.010501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"010501","note":"arxivid:quant-ph/0405157"},{"id":"arxiv:2505.03691","type":"article-journal","author":[{"given":"Basudha","family":"Srivastava"},{"given":"Yinzi","family":"Xiao"},{"given":"Anton Frisk","family":"Kockum"},{"given":"Ben","family":"Criger"},{"given":"Mats","family":"Granath"}],"title":"Sequential decoding of the XYZ$^2$ hexagonal stabilizer code","issued":{"date-parts":[[2025,5,6]]},"note":"arxivid:2505.03691\narxiv_version_number:1"},{"id":"arxiv:2005.10910","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.127.010504","source":"Crossref","title":"Designing Codes around Interactions: The Case of a Spin","volume":"127","author":[{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":true,"given":"Jonathan A.","family":"Gross","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2021,7,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.127.010504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"010504","note":"arxivid:2005.10910"},{"id":"arxiv:quant-ph/0512209","type":"article-journal","author":[{"given":"Rolando D.","family":"Somma"}],"title":"Quantum Computation, Complexity, and Many-Body Physics","issued":{"date-parts":[[2005,12,22]]},"note":"arxivid:quant-ph/0512209\narxiv_version_number:1"},{"id":"arxiv:1505.04990","type":"article-journal","author":[{"given":"Michael R.","family":"Geller"},{"given":"John M.","family":"Martinis"},{"given":"Andrew T.","family":"Sornborger"},{"given":"Phillip C.","family":"Stancil"},{"given":"Emily J.","family":"Pritchett"},{"given":"Hao","family":"You"},{"given":"Andrei","family":"Galiautdinov"}],"title":"Universal quantum simulation with prethreshold superconducting qubits: Single-excitation subspace method","issued":{"date-parts":[[2015,9,15]]},"note":"arxivid:1505.04990\narxiv_version_number:2"},{"id":"arxiv:1811.04069","type":"journal-article","publisher":"Royal Society of Chemistry (RSC)","issue":"22","abstract":"<p>We investigate how digital quantum computers may be used to calculate molecular vibrational properties, such as energy levels and spectral information.</p>","DOI":"10.1039/c9sc01313j","page":"5725-5735","source":"Crossref","title":"Digital quantum simulation of molecular vibrations","volume":"10","author":[{"given":"Sam","family":"McArdle","sequence":"first","affiliation":[{"name":"Department of Materials"},{"name":"University of Oxford"},{"name":"Oxford OX1 3PH"},{"name":"UK"}]},{"given":"Alexander","family":"Mayorov","sequence":"additional","affiliation":[{"name":"Department of Materials"},{"name":"University of Oxford"},{"name":"Oxford OX1 3PH"},{"name":"UK"},{"name":"Department of Chemistry"}]},{"given":"Xiao","family":"Shan","sequence":"additional","affiliation":[{"name":"Physical and Theoretical Chemical Laboratory"},{"name":"University of Oxford"},{"name":"Oxford OX1 3QZ"},{"name":"UK"}]},{"given":"Simon","family":"Benjamin","sequence":"additional","affiliation":[{"name":"Department of Materials"},{"name":"University of Oxford"},{"name":"Oxford OX1 3PH"},{"name":"UK"}]},{"ORCID":"https://orcid.org/0000-0003-0205-6545","authenticated-orcid":false,"given":"Xiao","family":"Yuan","sequence":"additional","affiliation":[{"name":"Department of Materials"},{"name":"University of Oxford"},{"name":"Oxford OX1 3PH"},{"name":"UK"}]}],"container-title":"Chemical Science","original-title":[],"language":"en","issued":{"date-parts":[[2019]]},"URL":"http://dx.doi.org/10.1039/c9sc01313j","ISSN":["2041-6520","2041-6539"],"container-title-short":"Chem. Sci.","note":"arxivid:1811.04069"},{"id":"arxiv:1812.10495","type":"journal-article","publisher":"American Chemical Society (ACS)","issue":"13","DOI":"10.1021/acs.jpclett.9b01117","page":"3586-3591","source":"Crossref","title":"Quantum Algorithm for Calculating Molecular Vibronic Spectra","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0001-8510-8480","authenticated-orcid":true,"given":"Nicolas P. D.","family":"Sawaya","sequence":"first","affiliation":[{"name":"Intel Laboratories, Santa Clara, California 95054, United States"}]},{"ORCID":"https://orcid.org/0000-0002-8792-5641","authenticated-orcid":true,"given":"Joonsuk","family":"Huh","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Sungkyunkwan University, Suwon, Gyeonggi-do 16419, Republic of Korea"},{"name":"SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, Suwon, Gyeonggi-do 16419, Republic of Korea"}]}],"container-title":"The Journal of Physical Chemistry Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,6,11]]},"URL":"http://dx.doi.org/10.1021/acs.jpclett.9b01117","ISSN":["1948-7185","1948-7185"],"container-title-short":"J. Phys. Chem. Lett.","note":"alternative-id:10.1021/acs.jpclett.9b01117\narxivid:1812.10495"},{"id":"arxiv:1903.05068","type":"journal-article","publisher":"IOP Publishing","issue":"4","abstract":"<jats:title>Abstract</jats:title><jats:p>In this paper I propose a new method of encoding discrete variables into Ising model qubits for quantum optimisation. The new method is based on the physics of domain walls in one-dimensional Ising spin chains. I find that these encodings and the encoding of arbitrary two variable interactions is possible with only two body Ising terms Following on from similar results for the ‘one hot’ method of encoding discrete variables (Hadfield<jats:italic>et al</jats:italic>2019<jats:italic>Algorithms</jats:italic><jats:bold>12</jats:bold>34) I also demonstrate that it is possible to construct two body mixer terms which do not leave the logical subspace, an important consideration for optimising using the quantum alternating operator ansatz. I additionally discuss how, since the couplings in the domain wall encoding only need to be ferromagnetic and therefore could in principle be much stronger than anti-ferromagnetic couplers, application specific quantum annealers for discrete problems based on this construction may be beneficial. Finally, I compare embedding for synthetic scheduling and colouring problems with the domain wall and one hot encodings on two graphs which are relevant for quantum annealing, the chimera graph and the Pegasus graph. For every case I examine I find a similar or better performance from the domain wall encoding as compared to one hot, but this advantage is highly dependent on the structure of the problem. For encoding some problems, I find an advantage similar to the one found by embedding in a Pegasus graph compared to embedding in a chimera graph.</jats:p>","DOI":"10.1088/2058-9565/ab33c2","page":"045004","source":"Crossref","title":"Domain wall encoding of discrete variables for quantum annealing and QAOA","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-1293-0761","authenticated-orcid":false,"given":"Nicholas","family":"Chancellor","sequence":"first","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2019,8,6]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ab33c2","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1903.05068"},{"id":"arxiv:2302.07357","type":"article-journal","author":[{"given":"Márton","family":"Karácsony"},{"given":"László","family":"Oroszlány"},{"given":"Zoltán","family":"Zimborás"}],"title":"Efficient qudit based scheme for photonic quantum computing","issued":{"date-parts":[[2023,2,14]]},"note":"arxivid:2302.07357\narxiv_version_number:1"},{"id":"arxiv:2306.11621","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.133.240603","source":"Crossref","title":"Quantum Error-Correcting Codes with a Covariant Encoding","volume":"133","author":[{"ORCID":"https://orcid.org/0009-0000-5608-7668","authenticated-orcid":true,"given":"Aurélie","family":"Denys","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02kvxyf05","id-type":"ROR","asserted-by":"publisher"}],"name":"Inria Paris"},{"id":[{"id":"https://ror.org/01yjzw780","id-type":"ROR","asserted-by":"publisher"}],"name":"Quandela"}]},{"ORCID":"https://orcid.org/0000-0002-6707-1458","authenticated-orcid":true,"given":"Anthony","family":"Leverrier","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02kvxyf05","id-type":"ROR","asserted-by":"publisher"}],"name":"Inria Paris"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.240603","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240603","note":"update-to:{\"DOI\":\"10.1103/physrevlett.133.240603\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2025,2,5]],\"date-time\":\"2025-02-05T00:00:00Z\",\"timestamp\":1738713600000}}\nupdated-by:{\"DOI\":\"10.1103/physrevlett.133.240603\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2025,2,5]],\"date-time\":\"2025-02-05T00:00:00Z\",\"timestamp\":1738713600000}}\narxivid:2306.11621"},{"id":"arxiv:quant-ph/9811011","type":"article-journal","author":[{"given":"J.","family":"Steinbach"},{"given":"J.","family":"Twamley"}],"title":"Motional Quantum Error Correction","issued":{"date-parts":[[1998,11,9]]},"note":"arxivid:quant-ph/9811011\narxiv_version_number:2"},{"id":"arxiv:1602.00008","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.6.031006","source":"Crossref","title":"New Class of Quantum Error-Correcting Codes for a Bosonic Mode","volume":"6","author":[{"given":"Marios H.","family":"Michael","sequence":"first","affiliation":[]},{"given":"Matti","family":"Silveri","sequence":"additional","affiliation":[]},{"given":"R. T.","family":"Brierley","sequence":"additional","affiliation":[]},{"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]},{"given":"Juha","family":"Salmilehto","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2016,7,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.6.031006","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031006","note":"arxivid:1602.00008"},{"id":"arxiv:1512.07605","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.94.012311","source":"Crossref","title":"Quantum error correction against photon loss using NOON states","volume":"94","author":[{"given":"Marcel","family":"Bergmann","sequence":"first","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,7,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.94.012311","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012311","note":"arxivid:1512.07605"},{"id":"doi:10.1088/0305-4470/4/3/009","type":"journal-article","publisher":"IOP Publishing","issue":"3","DOI":"10.1088/0305-4470/4/3/009","page":"313-323","source":"Crossref","title":"Some properties of coherent spin states","volume":"4","author":[{"given":"J M","family":"Radcliffe","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: General Physics","original-title":[],"issued":{"date-parts":[[1971,5,1]]},"URL":"http://dx.doi.org/10.1088/0305-4470/4/3/009","ISSN":["0022-3689"],"container-title-short":"J. Phys. A: Gen. Phys."},{"id":"doi:10.1103/PhysRevA.6.2211","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.6.2211","page":"2211-2237","source":"Crossref","title":"Atomic Coherent States in Quantum Optics","volume":"6","author":[{"given":"F. T.","family":"Arecchi","sequence":"first","affiliation":[]},{"given":"Eric","family":"Courtens","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Gilmore","sequence":"additional","affiliation":[]},{"given":"Harry","family":"Thomas","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1972,12,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.6.2211","ISSN":["0556-2791"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:2104.10581","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"9","abstract":"<jats:title>Abstract</jats:title><jats:p>Collective spin operators for symmetric multi-quDit (namely identical <jats:italic>D</jats:italic>-level atom) systems generate a U(<jats:italic>D</jats:italic>) symmetry. We explore generalizations to arbitrary <jats:italic>D</jats:italic> of SU(2)-spin coherent states and their adaptation to parity (multi-component Schrödinger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric <jats:italic>N</jats:italic>-quDit states, expressed in the last two cases in terms of collective U(<jats:italic>D</jats:italic>)-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin–Meshkov–Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU(<jats:italic>D</jats:italic>) and relate it to pairwise <jats:italic>D</jats:italic>-level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting <jats:italic>D</jats:italic>-level atom models.</jats:p>","DOI":"10.1007/s11128-021-03218-6","source":"Crossref","title":"Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0002-2566-9590","authenticated-orcid":false,"given":"Manuel","family":"Calixto","sequence":"first","affiliation":[]},{"given":"Alberto","family":"Mayorgas","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7118-4500","authenticated-orcid":false,"given":"Julio","family":"Guerrero","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2021,9]]},"URL":"http://dx.doi.org/10.1007/s11128-021-03218-6","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","page":"304","note":"alternative-id:3218\narxivid:2104.10581"},{"id":"doi:10.1088/0305-4470/26/2/018","type":"journal-article","publisher":"IOP Publishing","issue":"2","DOI":"10.1088/0305-4470/26/2/018","page":"313-327","source":"Crossref","title":"Coherent states of SU(N) groups","volume":"26","author":[{"given":"D M","family":"Gitman","sequence":"first","affiliation":[]},{"given":"A L","family":"Shelepin","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[1993,1,21]]},"URL":"http://dx.doi.org/10.1088/0305-4470/26/2/018","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen."},{"id":"arxiv:2508.20647","type":"article-journal","author":[{"given":"Rabsan Galib","family":"Ahmed"},{"given":"Adithi","family":"Udupa"},{"given":"Giulia","family":"Ferrini"}],"title":"Multimode rotationally symmetric bosonic codes from group-theoretic construction","issued":{"date-parts":[[2026,3,25]]},"note":"arxivid:2508.20647\narxiv_version_number:3"},{"id":"arxiv:1510.06117","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevlett.116.150501","source":"Crossref","title":"Hardware-Efficient and Fully Autonomous Quantum Error Correction in Superconducting Circuits","volume":"116","author":[{"given":"Eliot","family":"Kapit","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,4,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.116.150501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"150501","note":"arxivid:1510.06117"},{"id":"arxiv:2302.06707","type":"article-journal","author":[{"given":"Ziqian","family":"Li"},{"given":"Tanay","family":"Roy"},{"given":"David Rodriguez","family":"Perez"},{"given":"Kan-Heng","family":"Lee"},{"given":"Eliot","family":"Kapit"},{"given":"David I.","family":"Schuster"}],"title":"Autonomous error correction of a single logical qubit using two transmons","issued":{"date-parts":[[2023,2,13]]},"note":"arxivid:2302.06707\narxiv_version_number:1"},{"id":"doi:10.1103/PhysRevLett.115.203601","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.115.203601","source":"Crossref","title":"Fast Quantum Nondemolition Readout by Parametric Modulation of Longitudinal Qubit-Oscillator Interaction","volume":"115","author":[{"given":"Nicolas","family":"Didier","sequence":"first","affiliation":[]},{"given":"Jérôme","family":"Bourassa","sequence":"additional","affiliation":[]},{"given":"Alexandre","family":"Blais","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2015,11,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.115.203601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"203601"},{"id":"arxiv:quant-ph/0702075","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.75.042316","source":"Crossref","title":"Protecting an optical qubit against photon loss","volume":"75","author":[{"given":"Wojciech","family":"Wasilewski","sequence":"first","affiliation":[]},{"given":"Konrad","family":"Banaszek","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,4,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.042316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042316","note":"arxivid:quant-ph/0702075"},{"id":"arxiv:2008.12576","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2021.3102873","page":"6644-6652","source":"Crossref","title":"Trade-Offs on Number and Phase Shift Resilience in Bosonic Quantum Codes","volume":"67","author":[{"ORCID":"https://orcid.org/0000-0003-1115-0074","authenticated-orcid":false,"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2021,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3102873","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2008.12576"},{"id":"arxiv:2211.08448","type":"article-journal","author":[{"given":"ChunJun","family":"Cao"},{"given":"Gong","family":"Cheng"},{"given":"Brian","family":"Swingle"}],"title":"Large $N$ Matrix Quantum Mechanics as a Quantum Memory","issued":{"date-parts":[[2022,11,15]]},"note":"arxivid:2211.08448\narxiv_version_number:1"},{"id":"arxiv:2008.12869","type":"journal-article","publisher":"Stichting SciPost","issue":"5","abstract":"<jats:p>\n                    In recent years quantum error correction (QEC) has become an\nimportant part of AdS/CFT. Unfortunately, there are no field-theoretic\narguments about why QEC holds in known holographic systems. The purpose\nof this paper is to fill this gap by studying the error correcting\nproperties of the fermionic sector of various large\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>N</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mi>N</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    theories. Specifically we examine\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>SU(N)</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mrow>\n                            <mml:mi>S</mml:mi>\n                            <mml:mi>U</mml:mi>\n                            <mml:mrow>\n                              <mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo>\n                              <mml:mi>N</mml:mi>\n                              <mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo>\n                            </mml:mrow>\n                          </mml:mrow>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    matrix quantum mechanics and 3-rank tensor\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>O(N)^3</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mrow>\n                            <mml:mi>O</mml:mi>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo>\n                                <mml:mi>N</mml:mi>\n                                <mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo>\n                              </mml:mrow>\n                              <mml:mn>3</mml:mn>\n                            </mml:msup>\n                          </mml:mrow>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    theories. Both of these theories contain large gauge groups. We argue\nthat gauge singlet states indeed form a quantum error correcting code.\nOur considerations are based purely on large\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>N</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mi>N</mml:mi>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    analysis and do not appeal to a particular form of Hamiltonian or\nholography.\n                  </jats:p>","DOI":"10.21468/scipostphys.11.5.094","source":"Crossref","title":"Quantum error correction and large $N$","volume":"11","author":[{"given":"Alexey","family":"Milekhin","sequence":"first","affiliation":[{"name":"Princeton University"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2021,11,19]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.11.5.094","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"094","note":"arxivid:2008.12869"},{"id":"arxiv:1801.07271","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2018.2873764","page":"2563-2582","source":"Crossref","title":"Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes","volume":"65","author":[{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":false,"given":"Kyungjoo","family":"Noh","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":false,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":false,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2019,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2018.2873764","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1801.07271"},{"id":"arxiv:2205.00341","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Bosonic qubits encoded in continuous-variable systems provide a promising alternative to two-level qubits for quantum computation and communication. So far, photon loss has been the dominant source of errors in bosonic qubits, but the significant reduction of photon loss in recent bosonic qubit experiments suggests that dephasing errors should also be considered. However, a detailed understanding of the combined photon loss and dephasing channel is lacking. Here, we show that, unlike its constituent parts, the combined loss-dephasing channel is non-degradable, pointing towards a richer structure of this channel. We provide bounds for the capacity of the loss-dephasing channel and use numerical optimization to find optimal single-mode codes for a wide range of error rates.</jats:p>","DOI":"10.22331/q-2022-09-29-821","page":"821","source":"Crossref","title":"Quantum capacity and codes for the bosonic loss-dephasing channel","volume":"6","author":[{"given":"Peter","family":"Leviant","sequence":"first","affiliation":[{"name":"Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel"}]},{"given":"Qian","family":"Xu","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"Serge","family":"Rosenblum","sequence":"additional","affiliation":[{"name":"Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,29]]},"URL":"http://dx.doi.org/10.22331/q-2022-09-29-821","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2205.00341"},{"id":"arxiv:2108.02766","type":"article-journal","author":[{"given":"Zhaoyou","family":"Wang"},{"given":"Taha","family":"Rajabzadeh"},{"given":"Nathan","family":"Lee"},{"given":"Amir H.","family":"Safavi-Naeini"}],"title":"Automated discovery of autonomous quantum error correction schemes","issued":{"date-parts":[[2021,8,5]]},"note":"arxivid:2108.02766\narxiv_version_number:1"},{"id":"arxiv:2212.11651","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.131.050601","source":"Crossref","title":"Approximate Autonomous Quantum Error Correction with Reinforcement Learning","volume":"131","author":[{"ORCID":"https://orcid.org/0000-0001-9165-3995","authenticated-orcid":true,"given":"Yexiong","family":"Zeng","sequence":"first","affiliation":[{"name":"Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan"},{"name":"Quantum Computing Center, RIKEN, Wakoshi, Saitama 351-0198, Japan"}]},{"given":"Zheng-Yang","family":"Zhou","sequence":"additional","affiliation":[{"name":"Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan"}]},{"ORCID":"https://orcid.org/0000-0003-4134-809X","authenticated-orcid":true,"given":"Enrico","family":"Rinaldi","sequence":"additional","affiliation":[{"name":"Quantinuum K.K., Otemachi Financial City Grand Cube 3F, 1-9-2 Otemachi, Chiyoda-ku, Tokyo, Japan"},{"name":"Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan"},{"name":"Quantum Computing Center, RIKEN, Wakoshi, Saitama 351-0198, Japan"},{"name":"Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA"},{"name":"Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wakoshi, Saitama 351-0198, Japan"}]},{"ORCID":"https://orcid.org/0000-0001-9686-9277","authenticated-orcid":true,"given":"Clemens","family":"Gneiting","sequence":"additional","affiliation":[{"name":"Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan"},{"name":"Quantum Computing Center, RIKEN, Wakoshi, Saitama 351-0198, Japan"}]},{"ORCID":"https://orcid.org/0000-0003-3682-7432","authenticated-orcid":true,"given":"Franco","family":"Nori","sequence":"additional","affiliation":[{"name":"Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan"},{"name":"Quantum Computing Center, RIKEN, Wakoshi, Saitama 351-0198, Japan"},{"name":"Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.131.050601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050601","note":"arxivid:2212.11651"},{"id":"arxiv:2303.04702","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/prxquantum.4.040334","source":"Crossref","title":"Closest Lattice Point Decoding for Multimode Gottesman-Kitaev-Preskill Codes","volume":"4","author":[{"ORCID":"https://orcid.org/0009-0002-6102-5730","authenticated-orcid":true,"given":"Mao","family":"Lin","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,1]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.040334","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040334","note":"update-to:{\"updated\":{\"date-parts\":[[2024,1,23]],\"date-time\":\"2024-01-23T00:00:00Z\",\"timestamp\":1705968000000},\"DOI\":\"10.1103/prxquantum.4.040334\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2024,1,23]],\"date-time\":\"2024-01-23T00:00:00Z\",\"timestamp\":1705968000000},\"DOI\":\"10.1103/prxquantum.4.040334\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2303.04702"},{"id":"arxiv:2411.01265","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevlett.134.060601","source":"Crossref","title":"Neural-Network-Based Design of Approximate Gottesman-Kitaev-Preskill Code","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0001-9165-3995","authenticated-orcid":true,"given":"Yexiong","family":"Zeng","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"}]},{"ORCID":"https://orcid.org/0000-0003-1766-8245","authenticated-orcid":true,"given":"Wei","family":"Qin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/012tb2g32","id-type":"ROR","asserted-by":"publisher"}],"name":"Tianjin University"},{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/012tb2g32","id-type":"ROR","asserted-by":"publisher"}],"name":"Tianjin University"}]},{"ORCID":"https://orcid.org/0000-0002-7308-2823","authenticated-orcid":true,"given":"Ye-Hong","family":"Chen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/011xvna82","id-type":"ROR","asserted-by":"publisher"}],"name":"Fuzhou University"},{"id":[{"id":"https://ror.org/011xvna82","id-type":"ROR","asserted-by":"publisher"}],"name":"Fuzhou University"},{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"}]},{"ORCID":"https://orcid.org/0000-0001-9686-9277","authenticated-orcid":true,"given":"Clemens","family":"Gneiting","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"}]},{"ORCID":"https://orcid.org/0000-0003-3682-7432","authenticated-orcid":true,"given":"Franco","family":"Nori","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/00jmfr291","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Michigan, Ann Arbor"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.060601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"060601","note":"arxivid:2411.01265"},{"id":"arxiv:2411.12952","type":"article-journal","author":[{"given":"Xuanhui","family":"Mao"},{"given":"Qian","family":"Xu"},{"given":"Liang","family":"Jiang"}],"title":"Optimized four-qubit quantum error correcting code for amplitude damping channel","issued":{"date-parts":[[2025,5,18]]},"note":"arxivid:2411.12952\narxiv_version_number:2"},{"id":"doi:10.1103/PhysRevA.51.1698","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.51.1698","page":"1698-1701","source":"Crossref","title":"Nonclassical properties of correlated two-mode Schrödinger cat states","volume":"51","author":[{"given":"Christopher C.","family":"Gerry","sequence":"first","affiliation":[]},{"given":"Rainer","family":"Grobe","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1995,2,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.51.1698","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:1801.05897","type":"journal-article","publisher":"IOP Publishing","issue":"3","DOI":"10.1088/2058-9565/ab1e69","page":"035007","source":"Crossref","title":"Pair-cat codes: autonomous error-correction with low-order nonlinearity","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":false,"given":"Victor V","family":"Albert","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5952-0743","authenticated-orcid":false,"given":"Shantanu O","family":"Mundhada","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6247-1145","authenticated-orcid":false,"given":"Alexander","family":"Grimm","sequence":"additional","affiliation":[]},{"given":"Steven","family":"Touzard","sequence":"additional","affiliation":[]},{"given":"Michel H","family":"Devoret","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":false,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2019,6,12]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ab1e69","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1801.05897"},{"id":"doi:10.1088/0305-4470/9/9/011","type":"journal-article","publisher":"IOP Publishing","issue":"9","DOI":"10.1088/0305-4470/9/9/011","page":"1507-1512","source":"Crossref","title":"Charged bosons and the coherent state","volume":"9","author":[{"given":"D","family":"Bhaumik","sequence":"first","affiliation":[]},{"given":"K","family":"Bhaumik","sequence":"additional","affiliation":[]},{"given":"B","family":"Dutta-Roy","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[1976,9]]},"URL":"http://dx.doi.org/10.1088/0305-4470/9/9/011","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen."},{"id":"doi:10.1007/BF01646483","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01646483","page":"41-55","source":"Crossref","title":"New “Coherent” States associated with non-compact groups","volume":"21","author":[{"given":"A. O.","family":"Barut","sequence":"first","affiliation":[]},{"given":"L.","family":"Girardello","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1971,3]]},"URL":"http://dx.doi.org/10.1007/BF01646483","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF01646483"},{"id":"doi:10.1103/PhysRevLett.57.827","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.57.827","page":"827-830","source":"Crossref","title":"Generation of Pair Coherent States and Squeezing via the Competition of Four-Wave Mixing and Amplified Spontaneous Emission","volume":"57","author":[{"given":"G. S.","family":"Agarwal","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1986,8,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.57.827","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"doi:10.1364/JOSAB.5.001940","type":"journal-article","publisher":"Optica Publishing Group","issue":"9","DOI":"10.1364/josab.5.001940","page":"1940","source":"Crossref","title":"Nonclassical statistics of fields in pair coherent states","volume":"5","author":[{"given":"G. S.","family":"Agarwal","sequence":"first","affiliation":[]}],"container-title":"Journal of the Optical Society of America B","original-title":[],"language":"en","issued":{"date-parts":[[1988,9,1]]},"URL":"http://dx.doi.org/10.1364/JOSAB.5.001940","ISSN":["0740-3224","1520-8540"],"container-title-short":"J. Opt. Soc. Am. B"},{"id":"arxiv:2208.06913","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.106.062422","source":"Crossref","title":"Construction of bias-preserving operations for pair-cat codes","volume":"106","author":[{"ORCID":"https://orcid.org/0000-0002-5625-6481","authenticated-orcid":true,"given":"Ming","family":"Yuan","sequence":"first","affiliation":[]},{"given":"Qian","family":"Xu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.106.062422","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062422","note":"arxivid:2208.06913"},{"id":"arxiv:2209.11643","type":"article-journal","author":[{"given":"Jeffrey M.","family":"Gertler"},{"given":"Sean","family":"van Geldern"},{"given":"Shruti","family":"Shirol"},{"given":"Liang","family":"Jiang"},{"given":"Chen","family":"Wang"}],"title":"Experimental Realization and Characterization of Stabilized Pair Coherent States","issued":{"date-parts":[[2022,9,23]]},"note":"arxivid:2209.11643\narxiv_version_number:1"},{"id":"doi:10.1080/713821735","type":"journal-article","publisher":"Informa UK Limited","issue":"3","DOI":"10.1080/713821735","page":"345-355","source":"Crossref","title":"Binomial States of the Quantized Radiation Field","volume":"32","author":[{"given":"D.","family":"Stoler","sequence":"first","affiliation":[{"name":"a  AT&amp;T Bell Laboratories, Summit, New Jersey 07901, U.S.A."}]},{"given":"B.E.A.","family":"Saleh","sequence":"additional","affiliation":[{"name":"b  Department of Electrical and Computer Engineering, Univerisity of Wisconsin, Madison, Wisconsin 53706, U.S.A."}]},{"given":"M.C.","family":"Teich","sequence":"additional","affiliation":[{"name":"c  Columbia Radiation Laboratory, Department of Electrical Engineering, New York 10027, U.S.A."}]}],"container-title":"Optica Acta: International Journal of Optics","original-title":[],"language":"en","issued":{"date-parts":[[1985,3]]},"URL":"http://dx.doi.org/10.1080/713821735","ISSN":["0030-3909"],"container-title-short":"Optica Acta: International Journal of Optics","note":"alternative-id:10.1080/713821735"},{"id":"arxiv:2507.08585","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Encoding a qubit in the continuous degrees of freedom of a quantum system, such as bosonic modes, is a powerful alternative to modern quantum error correction (QEC). Among the most prominent bosonic QEC codes, binomial codes provide protection against loss and dephasing errors by encoding logical states in finite superpositions of Fock states with binomially weighted coefficients. While much attention has been given to their error-correcting capabilities and integration into fault-tolerant architectures, efficient methods for generating arbitrary binomial codewords remain scarce. In this work, we propose a scheme for generating these codewords by exploiting nonlinear multiphoton interactions between a continuous-variable bosonic mode (oscillator) and a two-level system (spin/qubit). Our proposed scheme assumes the ability to prepare the oscillator in an arbitrary Fock state and the qubit in an arbitrary superposition of its basis states and access to arbitrarily high multiphoton interactions. To enhance the experimental feasibility of our scheme, we further demonstrate how to reduce the required order parameter of multiphoton interactions by a factor of two for a special class of code states.</jats:p>","DOI":"10.1103/58jr-l1x6","source":"Crossref","title":"Arbitrary high-fidelity binomial codes from multiphoton spin-boson interactions","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-8400-7861","authenticated-orcid":true,"given":"Pradip","family":"Laha","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/023b0x485","id-type":"ROR","asserted-by":"publisher"}],"name":"Johannes Gutenberg-Universität Mainz"}]},{"ORCID":"https://orcid.org/0000-0001-9445-0771","authenticated-orcid":true,"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/023b0x485","id-type":"ROR","asserted-by":"publisher"}],"name":"Johannes Gutenberg-Universität Mainz"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,5]]},"URL":"http://dx.doi.org/10.1103/58jr-l1x6","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013237","note":"arxivid:2507.08585"},{"id":"arxiv:2408.12968","type":"article-journal","author":[{"given":"Yuki","family":"Tanaka"},{"given":"Yuichiro","family":"Mori"},{"given":"Yuta","family":"Shingu"},{"given":"Aiko","family":"Yamaguchi"},{"given":"Tsuyoshi","family":"Yamamoto"},{"given":"Yuichiro","family":"Matsuzaki"}],"title":"Single-qubit rotations on a binomial code without ancillary qubits","issued":{"date-parts":[[2024,8,23]]},"note":"arxivid:2408.12968\narxiv_version_number:1"},{"id":"arxiv:2412.08870","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Bosonic encodings of quantum information offer hardware-efficient, noise-biased approaches to quantum error correction relative to qubit register encodings. Implementations have focused in particular on error correction of stored, idle quantum information, whereas quantum algorithms are likely to desire high duty cycles of active control. Error-transparent operations are one way to preserve error rates during operations, but, to the best of our knowledge, only phase gates have so far been given an explicitly error-transparent formulation for binomial encodings. Here, we introduce the concept of 'parity nested' operations, and show how these operations can be designed to achieve continuous amplitude-mixing logical gates for binomial encodings that are fully error-transparent to the photon loss channel. For a binomial encoding that protects against <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>l</mml:mi></mml:math> photon losses, the construction requires <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230A;</mml:mo></mml:mrow><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">l/2</mml:mtext><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230B;</mml:mo></mml:mrow><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">&amp;#xA0;+ 1</mml:mtext></mml:mrow></mml:mrow></mml:math> orders of generalized squeezing in the parity nested operation to fully preserve this protection. We further show that error-transparency to all the correctable photon jumps, but not the no-jump errors, can be achieved with just a single order of squeezing. Finally, we comment on possible approaches to experimental realization of this concept.</jats:p>","DOI":"10.22331/q-2025-10-21-1890","page":"1890","source":"Crossref","title":"A Mathematical Structure for Amplitude-Mixing Error-Transparent Gates for Binomial Codes","volume":"9","author":[{"given":"Owen C.","family":"Wetherbee","sequence":"first","affiliation":[{"name":"Department of Physics, Cornell University, Ithaca, NY, 14853, USA"}]},{"given":"Saswata","family":"Roy","sequence":"additional","affiliation":[{"name":"Department of Physics, Cornell University, Ithaca, NY, 14853, USA"}]},{"given":"Baptiste","family":"Royer","sequence":"additional","affiliation":[{"name":"Département de Physique and Institut Quantique, Université de Sherbrooke, Sherbrooke J1K 2R1, QC, Canada"}]},{"given":"Valla","family":"Fatemi","sequence":"additional","affiliation":[{"name":"School of Applied and Engineering Physics, Cornell University, Ithaca, NY, 14853, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,21]]},"URL":"http://dx.doi.org/10.22331/q-2025-10-21-1890","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2412.08870"},{"id":"arxiv:2402.04093","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum measurements are ubiquitous in quantum information processing tasks, but errors can render their outputs unreliable. Here, we present a scheme that implements a robust projective measurement through measuring code-inspired observables. Namely, given a projective POVM, a classical code, and a constraint on the number of measurement outcomes each observable can have, we construct commuting observables whose measurement is equivalent to the projective measurement in the noiseless setting. Moreover, we can correct <jats:italic>t</jats:italic> errors on the classical outcomes of the observables’ measurement if the classical code corrects <jats:italic>t</jats:italic> errors. Since our scheme does not require the encoding of quantum data onto a quantum error correction code, it can help construct robust measurements for near-term quantum algorithms that do not use quantum error correction. Moreover, our scheme works for any projective POVM, and hence can allow robust syndrome extraction procedures in non-stabilizer quantum error correction codes.</jats:p>","DOI":"10.1038/s41534-024-00904-y","source":"Crossref","title":"Robust projective measurements through measuring code-inspired observables","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0003-1115-0074","authenticated-orcid":false,"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,26]]},"URL":"http://dx.doi.org/10.1038/s41534-024-00904-y","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"104","note":"alternative-id:904\narxivid:2402.04093"},{"id":"arxiv:1712.05832","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7","DOI":"10.1038/s41567-018-0115-y","page":"705-710","source":"Crossref","title":"On-demand quantum state transfer and entanglement between remote microwave cavity memories","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0002-1708-3688","authenticated-orcid":false,"given":"Christopher J.","family":"Axline","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3229-0236","authenticated-orcid":false,"given":"Luke D.","family":"Burkhart","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4303-0819","authenticated-orcid":false,"given":"Wolfgang","family":"Pfaff","sequence":"additional","affiliation":[]},{"given":"Mengzhen","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7185-0473","authenticated-orcid":false,"given":"Kevin","family":"Chou","sequence":"additional","affiliation":[]},{"given":"Philippe","family":"Campagne-Ibarcq","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8141-1842","authenticated-orcid":false,"given":"Philip","family":"Reinhold","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0272-5481","authenticated-orcid":false,"given":"Luigi","family":"Frunzio","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":false,"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":false,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"M. H.","family":"Devoret","sequence":"additional","affiliation":[]},{"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2018,4,23]]},"URL":"http://dx.doi.org/10.1038/s41567-018-0115-y","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nature Phys","note":"alternative-id:115\narxivid:1712.05832"},{"id":"arxiv:1805.09072","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","DOI":"10.1038/s41567-018-0414-3","page":"503-508","source":"Crossref","title":"Quantum error correction and universal gate set operation on a binomial bosonic logical qubit","volume":"15","author":[{"given":"L.","family":"Hu","sequence":"first","affiliation":[]},{"given":"Y.","family":"Ma","sequence":"additional","affiliation":[]},{"given":"W.","family":"Cai","sequence":"additional","affiliation":[]},{"given":"X.","family":"Mu","sequence":"additional","affiliation":[]},{"given":"Y.","family":"Xu","sequence":"additional","affiliation":[]},{"given":"W.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Y.","family":"Wu","sequence":"additional","affiliation":[]},{"given":"H.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Y. P.","family":"Song","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2484-7292","authenticated-orcid":false,"given":"C.-L.","family":"Zou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":false,"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"given":"L-M.","family":"Duan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4419-5674","authenticated-orcid":false,"given":"L.","family":"Sun","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,2,11]]},"URL":"http://dx.doi.org/10.1038/s41567-018-0414-3","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:414\narxivid:1805.09072"},{"id":"arxiv:1810.04690","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.124.120501","source":"Crossref","title":"Demonstration of Controlled-Phase Gates between Two Error-Correctable Photonic Qubits","volume":"124","author":[{"ORCID":"https://orcid.org/0000-0002-2603-0721","authenticated-orcid":true,"given":"Y.","family":"Xu","sequence":"first","affiliation":[]},{"given":"Y.","family":"Ma","sequence":"additional","affiliation":[]},{"given":"W.","family":"Cai","sequence":"additional","affiliation":[]},{"given":"X.","family":"Mu","sequence":"additional","affiliation":[]},{"given":"W.","family":"Dai","sequence":"additional","affiliation":[]},{"given":"W.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"L.","family":"Hu","sequence":"additional","affiliation":[]},{"given":"X.","family":"Li","sequence":"additional","affiliation":[]},{"given":"J.","family":"Han","sequence":"additional","affiliation":[]},{"given":"H.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Y. P.","family":"Song","sequence":"additional","affiliation":[]},{"given":"Zhen-Biao","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Shi-Biao","family":"Zheng","sequence":"additional","affiliation":[]},{"given":"L.","family":"Sun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.124.120501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120501","note":"arxivid:1810.04690"},{"id":"arxiv:1907.12327","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","DOI":"10.1038/s41567-020-0931-8","page":"822-826","source":"Crossref","title":"Error-corrected gates on an encoded qubit","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0002-8141-1842","authenticated-orcid":false,"given":"Philip","family":"Reinhold","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1984-3206","authenticated-orcid":false,"given":"Serge","family":"Rosenblum","sequence":"additional","affiliation":[]},{"given":"Wen-Long","family":"Ma","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0272-5481","authenticated-orcid":false,"given":"Luigi","family":"Frunzio","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":false,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Robert J.","family":"Schoelkopf","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,8]]},"URL":"http://dx.doi.org/10.1038/s41567-020-0931-8","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:931\narxivid:1907.12327"},{"id":"arxiv:2211.09319","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7955","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum error correction (QEC) aims to protect logical qubits from noises by using the redundancy of a large Hilbert space, which allows errors to be detected and corrected in real time<jats:sup>1</jats:sup>. In most QEC codes<jats:sup>2–8</jats:sup>, a logical qubit is encoded in some discrete variables, for example photon numbers, so that the encoded quantum information can be unambiguously extracted after processing. Over the past decade, repetitive QEC has been demonstrated with various discrete-variable-encoded scenarios<jats:sup>9–17</jats:sup>. However, extending the lifetimes of thus-encoded logical qubits beyond the best available physical qubit still remains elusive, which represents a break-even point for judging the practical usefulness of QEC. Here we demonstrate a QEC procedure in a circuit quantum electrodynamics architecture<jats:sup>18</jats:sup>, where the logical qubit is binomially encoded in photon-number states of a microwave cavity<jats:sup>8</jats:sup>, dispersively coupled to an auxiliary superconducting qubit. By applying a pulse featuring a tailored frequency comb to the auxiliary qubit, we can repetitively extract the error syndrome with high fidelity and perform error correction with feedback control accordingly, thereby exceeding the break-even point by about 16% lifetime enhancement. Our work illustrates the potential of hardware-efficient discrete-variable encodings for fault-tolerant quantum computation<jats:sup>19</jats:sup>.</jats:p>","DOI":"10.1038/s41586-023-05784-4","page":"56-60","source":"Crossref","title":"Beating the break-even point with a discrete-variable-encoded logical qubit","volume":"616","author":[{"given":"Zhongchu","family":"Ni","sequence":"first","affiliation":[]},{"given":"Sai","family":"Li","sequence":"additional","affiliation":[]},{"given":"Xiaowei","family":"Deng","sequence":"additional","affiliation":[]},{"given":"Yanyan","family":"Cai","sequence":"additional","affiliation":[]},{"given":"Libo","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1510-772X","authenticated-orcid":false,"given":"Weiting","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3964-4714","authenticated-orcid":false,"given":"Zhen-Biao","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Haifeng","family":"Yu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4674-2806","authenticated-orcid":false,"given":"Fei","family":"Yan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2340-4556","authenticated-orcid":false,"given":"Song","family":"Liu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2484-7292","authenticated-orcid":false,"given":"Chang-Ling","family":"Zou","sequence":"additional","affiliation":[]},{"given":"Luyan","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Shi-Biao","family":"Zheng","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2603-0721","authenticated-orcid":false,"given":"Yuan","family":"Xu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0672-8233","authenticated-orcid":false,"given":"Dapeng","family":"Yu","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,22]]},"URL":"http://dx.doi.org/10.1038/s41586-023-05784-4","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:5784\narxivid:2211.09319"},{"id":"arxiv:2211.09116","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7955","DOI":"10.1038/s41586-023-05782-6","page":"50-55","source":"Crossref","title":"Real-time quantum error correction beyond break-even","volume":"616","author":[{"ORCID":"https://orcid.org/0000-0001-5108-4011","authenticated-orcid":false,"given":"V. V.","family":"Sivak","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5179-0215","authenticated-orcid":false,"given":"A.","family":"Eickbusch","sequence":"additional","affiliation":[]},{"given":"B.","family":"Royer","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4921-1410","authenticated-orcid":false,"given":"S.","family":"Singh","sequence":"additional","affiliation":[]},{"given":"I.","family":"Tsioutsios","sequence":"additional","affiliation":[]},{"given":"S.","family":"Ganjam","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4280-1845","authenticated-orcid":false,"given":"A.","family":"Miano","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9317-3135","authenticated-orcid":false,"given":"B. L.","family":"Brock","sequence":"additional","affiliation":[]},{"given":"A. Z.","family":"Ding","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0272-5481","authenticated-orcid":false,"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":false,"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2883-9575","authenticated-orcid":false,"given":"M. H.","family":"Devoret","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,22]]},"URL":"http://dx.doi.org/10.1038/s41586-023-05782-6","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:5782\narxivid:2211.09116"},{"id":"arxiv:2111.07965","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.3.030301","source":"Crossref","title":"Robust Preparation of Wigner-Negative States with Optimized SNAP-Displacement Sequences","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0003-4593-2859","authenticated-orcid":true,"given":"Marina","family":"Kudra","sequence":"first","affiliation":[]},{"given":"Mikael","family":"Kervinen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3238-8164","authenticated-orcid":true,"given":"Ingrid","family":"Strandberg","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1145-7279","authenticated-orcid":true,"given":"Shahnawaz","family":"Ahmed","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5679-045X","authenticated-orcid":true,"given":"Marco","family":"Scigliuzzo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0290-1000","authenticated-orcid":true,"given":"Amr","family":"Osman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3098-3266","authenticated-orcid":true,"given":"Daniel Pérez","family":"Lozano","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4233-3279","authenticated-orcid":true,"given":"Mats O.","family":"Tholén","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8199-5510","authenticated-orcid":true,"given":"Riccardo","family":"Borgani","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8534-6577","authenticated-orcid":true,"given":"David B.","family":"Haviland","sequence":"additional","affiliation":[]},{"given":"Giulia","family":"Ferrini","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4521-710X","authenticated-orcid":true,"given":"Jonas","family":"Bylander","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2534-3021","authenticated-orcid":true,"given":"Anton Frisk","family":"Kockum","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2355-0449","authenticated-orcid":true,"given":"Fernando","family":"Quijandría","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1222-3506","authenticated-orcid":true,"given":"Per","family":"Delsing","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7238-693X","authenticated-orcid":true,"given":"Simone","family":"Gasparinetti","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,1]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.030301","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030301","note":"arxivid:2111.07965"},{"id":"arxiv:2501.04460","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/l43p-py55","source":"Crossref","title":"Experimental Demonstration of Entanglement Pumping with Bosonic Logical Qubits","volume":"136","author":[{"ORCID":"https://orcid.org/0009-0009-6880-7058","authenticated-orcid":true,"given":"Jie","family":"Zhou","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"ORCID":"https://orcid.org/0009-0002-5101-1749","authenticated-orcid":true,"given":"Chuanlong","family":"Ma","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"ORCID":"https://orcid.org/0009-0006-9563-6352","authenticated-orcid":true,"given":"Yifang","family":"Xu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"ORCID":"https://orcid.org/0000-0002-0650-0861","authenticated-orcid":true,"given":"Weizhou","family":"Cai","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04c4dkn09","id-type":"ROR","asserted-by":"publisher"}],"name":"CAS Key Laboratory of Quantum Information"}]},{"ORCID":"https://orcid.org/0009-0008-1786-2316","authenticated-orcid":true,"given":"Hongwei","family":"Huang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"ORCID":"https://orcid.org/0000-0001-7080-2306","authenticated-orcid":true,"given":"Lida","family":"Sun","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"ORCID":"https://orcid.org/0000-0001-7156-2017","authenticated-orcid":true,"given":"Guangming","family":"Xue","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04nqf9k60","id-type":"ROR","asserted-by":"publisher"}],"name":"Beijing Academy of Quantum Information Sciences"},{"name":"Hefei National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-4860-8803","authenticated-orcid":true,"given":"Ziyue","family":"Hua","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"given":"Haifeng","family":"Yu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04nqf9k60","id-type":"ROR","asserted-by":"publisher"}],"name":"Beijing Academy of Quantum Information Sciences"},{"name":"Hefei National Laboratory"}]},{"given":"Weiting","family":"Wang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"}]},{"given":"Chang-Ling","family":"Zou","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04c4dkn09","id-type":"ROR","asserted-by":"publisher"}],"name":"CAS Key Laboratory of Quantum Information"},{"name":"Hefei National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0003-4419-5674","authenticated-orcid":true,"given":"Luyan","family":"Sun","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"},{"name":"Hefei National Laboratory"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,16]]},"URL":"http://dx.doi.org/10.1103/l43p-py55","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110801","note":"arxivid:2501.04460"},{"id":"arxiv:2310.15546","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.133.050602","source":"Crossref","title":"Robust and Deterministic Preparation of Bosonic Logical States in a Trapped Ion","volume":"133","author":[{"ORCID":"https://orcid.org/0000-0001-9641-1514","authenticated-orcid":true,"given":"V. G.","family":"Matsos","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/03rtykw43","id-type":"ROR","asserted-by":"publisher"}],"name":"ARC Centre of Excellence for Engineered Quantum Systems"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0002-6763-7132","authenticated-orcid":true,"given":"C. H.","family":"Valahu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/03rtykw43","id-type":"ROR","asserted-by":"publisher"}],"name":"ARC Centre of Excellence for Engineered Quantum Systems"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"T.","family":"Navickas","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/03rtykw43","id-type":"ROR","asserted-by":"publisher"}],"name":"ARC Centre of Excellence for Engineered Quantum Systems"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"A. D.","family":"Rao","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/03rtykw43","id-type":"ROR","asserted-by":"publisher"}],"name":"ARC Centre of Excellence for Engineered Quantum Systems"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"M. J.","family":"Millican","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/03rtykw43","id-type":"ROR","asserted-by":"publisher"}],"name":"ARC Centre of Excellence for Engineered Quantum Systems"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"ORCID":"https://orcid.org/0009-0005-0507-9226","authenticated-orcid":true,"given":"X. C.","family":"Kolesnikow","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/03rtykw43","id-type":"ROR","asserted-by":"publisher"}],"name":"ARC Centre of Excellence for Engineered Quantum Systems"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"M. J.","family":"Biercuk","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/03rtykw43","id-type":"ROR","asserted-by":"publisher"}],"name":"ARC Centre of Excellence for Engineered Quantum Systems"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0003-4520-6419","authenticated-orcid":true,"given":"T. R.","family":"Tan","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/03rtykw43","id-type":"ROR","asserted-by":"publisher"}],"name":"ARC Centre of Excellence for Engineered Quantum Systems"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"},{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.050602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050602","note":"arxivid:2310.15546"},{"id":"arxiv:1901.08071","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevx.10.011058","source":"Crossref","title":"Quantum Computing with Rotation-Symmetric Bosonic Codes","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0002-5208-0271","authenticated-orcid":true,"given":"Arne L.","family":"Grimsmo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3199-5744","authenticated-orcid":true,"given":"Joshua","family":"Combes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":true,"given":"Ben Q.","family":"Baragiola","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.10.011058","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011058","note":"arxivid:1901.08071"},{"id":"doi:10.1103/PhysRev.58.1098","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrev.58.1098","page":"1098-1113","source":"Crossref","title":"Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet","volume":"58","author":[{"given":"T.","family":"Holstein","sequence":"first","affiliation":[]},{"given":"H.","family":"Primakoff","sequence":"additional","affiliation":[]}],"container-title":"Physical Review","original-title":[],"language":"en","issued":{"date-parts":[[1940,12,15]]},"URL":"http://dx.doi.org/10.1103/PhysRev.58.1098","ISSN":["0031-899X"],"container-title-short":"Phys. Rev."},{"id":"doi:10.2307/3212170","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"3","abstract":"<jats:p>The concepts of distribution operator, stochastic independence, convergence in distribution and normal distribution are formulated for pairs of canonically conjugate quantum-mechanical momentum and position operators. It is shown that if the sequence (<jats:italic>p<jats:sub>n</jats:sub></jats:italic>, <jats:italic>q<jats:sub>n</jats:sub></jats:italic>), <jats:italic>n</jats:italic> = 1, 2, ··· is stochastically independent and identically distributed with finite covariance and zero mean then the sequence of pairs of canonical observables\n<jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0021900200035555_eqn1\" /></jats:disp-formula>\nconverges in distribution to a normal limit distribution.</jats:p>","DOI":"10.2307/3212170","page":"454-469","source":"Crossref","title":"A quantum-mechanical central limit theorem","volume":"8","author":[{"given":"C. D.","family":"Cushen","sequence":"first","affiliation":[]},{"given":"R. L.","family":"Hudson","sequence":"additional","affiliation":[]}],"container-title":"Journal of Applied Probability","original-title":[],"language":"en","issued":{"date-parts":[[1971,9]]},"URL":"http://dx.doi.org/10.2307/3212170","ISSN":["0021-9002","1475-6072"],"container-title-short":"Journal of Applied Probability","note":"alternative-id:S0021900200035555"},{"id":"doi:10.1103/RevModPhys.63.375","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/revmodphys.63.375","page":"375-558","source":"Crossref","title":"Boson realizations of Lie algebras with applications to nuclear physics","volume":"63","author":[{"given":"Abraham","family":"Klein","sequence":"first","affiliation":[]},{"given":"E. R.","family":"Marshalek","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[1991,4,1]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.63.375","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys."},{"id":"manual:-L.-Li-private-communication-","type":"document","author":[{"family":"Li","given":"L."}],"genre":"private communication,","issued":"2018"},{"id":"arxiv:2203.09234","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Autonomous quantum error correction has gained considerable attention to avoid complicated measurements and feedback. Despite its simplicity compared with the conventional measurement-based quantum error correction, it is still a far from practical technique because of significant hardware overhead. We propose an autonomous quantum error correction scheme for a rotational symmetric bosonic code in a four-photon Kerr parametric oscillator. Our scheme is the simplest possible error correction scheme that can surpass the break-even point—it requires only a single continuous microwave tone. We also introduce an unconditional reset scheme that requires one more continuous microwave tone in addition to that for the error correction. The key properties underlying this simplicity are protected quasienergy states of a four-photon Kerr parametric oscillator and the degeneracy in its quasienergy level structure. These properties eliminate the need for state-by-state correction in the Fock basis. Our schemes greatly reduce the complexity of autonomous quantum error correction and thus may accelerate the use of the bosonic code for practical quantum computation.</jats:p>","DOI":"10.1038/s41534-022-00553-z","source":"Crossref","title":"Autonomous quantum error correction in a four-photon Kerr parametric oscillator","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-8113-0821","authenticated-orcid":false,"given":"Sangil","family":"Kwon","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2895-698X","authenticated-orcid":false,"given":"Shohei","family":"Watabe","sequence":"additional","affiliation":[]},{"given":"Jaw-Shen","family":"Tsai","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,22]]},"URL":"http://dx.doi.org/10.1038/s41534-022-00553-z","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"40","note":"alternative-id:553\narxivid:2203.09234"},{"id":"arxiv:quant-ph/0601066","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.73.052306","source":"Crossref","title":"Noise thresholds for optical cluster-state quantum computation","volume":"73","author":[{"given":"Christopher M.","family":"Dawson","sequence":"first","affiliation":[]},{"given":"Henry L.","family":"Haselgrove","sequence":"additional","affiliation":[]},{"given":"Michael A.","family":"Nielsen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,5,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.052306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052306","note":"arxivid:quant-ph/0601066"},{"id":"arxiv:quant-ph/0410199","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7029","DOI":"10.1038/nature03350","page":"39-44","source":"Crossref","title":"Quantum computing with realistically noisy devices","volume":"434","author":[{"given":"E.","family":"Knill","sequence":"first","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2005,3]]},"URL":"http://dx.doi.org/10.1038/nature03350","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature03350\narxivid:quant-ph/0410199"},{"id":"arxiv:quant-ph/0312190","type":"article-journal","author":[{"given":"E.","family":"Knill"}],"title":"Scalable Quantum Computation in the Presence of Large Detected-Error Rates","issued":{"date-parts":[[2004,12,9]]},"note":"arxivid:quant-ph/0312190\narxiv_version_number:2"},{"id":"arxiv:2505.08670","type":"article-journal","author":[{"given":"Adithi","family":"Udupa"},{"given":"Timo","family":"Hillmann"},{"given":"Rabsan Galib","family":"Ahmed"},{"given":"Andrea","family":"Smirne"},{"given":"Giulia","family":"Ferrini"}],"title":"Performance of rotation-symmetric bosonic codes in the presence of random telegraph noise","issued":{"date-parts":[[2025,7,14]]},"note":"arxivid:2505.08670\narxiv_version_number:2"},{"id":"arxiv:2406.04157","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Bosonic codes encode quantum information into a single infinite-dimensional physical system endowed with error correction capabilities. This reduces the need for complex management of many physical constituents compared with standard approaches employing multiple physical qubits. Recent discussions of bosonic codes centre around correcting only boson-loss errors, with phase errors either actively suppressed or deferred to subsequent layers of encoding with standard qubit codes. Rotationally symmetric bosonic (RSB) codes, which include the well-known cat and binomial codes, are capable of simultaneous correction of loss and phase errors, offering an alternate route that deals with arbitrary errors already at the base layer. Here, we investigate the robustness of such codes, moving away from the more idealistic past studies towards a circuit-level noise analysis closer to the practical situation where every physical component in the device is potentially faulty. We extend the concept of fault tolerance to the case of RSB codes, and then examine the performance of two known error correction circuits under circuit-level noise. Our analysis reveals a significantly more stringent noise threshold for fault-tolerant operation than found in past works; nevertheless, we show how, through waiting-time optimization and the use of squeezing, we can restore the noise requirements to a regime achievable with near-term quantum hardware. While our focus here is on cat codes for concreteness, a similar analysis applies for general RSB codes.</jats:p>","DOI":"10.22331/q-2025-07-23-1810","page":"1810","source":"Crossref","title":"Circuit-level fault tolerance of cat codes","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0002-4628-6492","authenticated-orcid":false,"given":"Long D. H.","family":"My","sequence":"first","affiliation":[{"name":"Yale-NUS College, Singapore"},{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore"}]},{"ORCID":"https://orcid.org/0009-0000-7872-9393","authenticated-orcid":false,"given":"Shushen","family":"Qin","sequence":"additional","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore"}]},{"ORCID":"https://orcid.org/0000-0003-2397-840X","authenticated-orcid":false,"given":"Hui Khoon","family":"Ng","sequence":"additional","affiliation":[{"name":"Yale-NUS College, Singapore"},{"name":"Centre for Quantum Technologies, National University of Singapore, Singapore"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,23]]},"URL":"http://dx.doi.org/10.22331/q-2025-07-23-1810","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2406.04157"},{"id":"arxiv:2311.16089","type":"article-journal","author":[{"given":"Saurabh","family":"Totey"},{"given":"Akira","family":"Kyle"},{"given":"Steven","family":"Liu"},{"given":"Pratik J.","family":"Barge"},{"given":"Noah","family":"Lordi"},{"given":"Joshua","family":"Combes"}],"title":"The performance of random bosonic rotation codes","issued":{"date-parts":[[2023,11,29]]},"note":"arxivid:2311.16089\narxiv_version_number:2"},{"id":"doi:10.1088/0305-4470/19/18/030","type":"journal-article","publisher":"IOP Publishing","issue":"18","DOI":"10.1088/0305-4470/19/18/030","page":"3849-3862","source":"Crossref","title":"Phase in quantum optics","volume":"19","author":[{"given":"S M","family":"Barnett","sequence":"first","affiliation":[]},{"given":"D T","family":"Pegg","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[1986,12,21]]},"URL":"http://dx.doi.org/10.1088/0305-4470/19/18/030","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen."},{"id":"arxiv:2108.01009","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.3.020334","source":"Crossref","title":"Performance of Teleportation-Based Error-Correction Circuits for Bosonic Codes with Noisy Measurements","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-1476-0647","authenticated-orcid":true,"given":"Timo","family":"Hillmann","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2355-0449","authenticated-orcid":true,"given":"Fernando","family":"Quijandría","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5208-0271","authenticated-orcid":true,"given":"Arne L.","family":"Grimsmo","sequence":"additional","affiliation":[]},{"given":"Giulia","family":"Ferrini","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,13]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.020334","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020334","note":"arxivid:2108.01009"},{"id":"arxiv:2412.15134","type":"journal-article","publisher":"American Physical Society (APS)","DOI":"10.1103/vb9n-g6gx","source":"Crossref","title":"Use of faulty states in cat-code error correction","author":[{"family":"Anonymous","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,9]]},"URL":"http://dx.doi.org/10.1103/vb9n-g6gx","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","note":"arxivid:2412.15134"},{"id":"doi:10.1007/978-88-7642-378-9","type":"book","publisher":"Edizioni della Normale","DOI":"10.1007/978-88-7642-378-9","source":"Crossref","title":"Probabilistic and Statistical Aspects of Quantum Theory","author":[{"given":"Alexander","family":"Holevo","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2011]]},"ISBN":["9788876423758","9788876423789"],"URL":"http://dx.doi.org/10.1007/978-88-7642-378-9","note":"publisher-location:Pisa"},{"id":"arxiv:2412.04865","type":"article-journal","author":[{"given":"Christophe H.","family":"Valahu"},{"given":"Matthew P.","family":"Stafford"},{"given":"Zixin","family":"Huang"},{"given":"Vassili G.","family":"Matsos"},{"given":"Maverick J.","family":"Millican"},{"given":"Teerawat","family":"Chalermpusitarak"},{"given":"Nicolas C.","family":"Menicucci"},{"given":"Joshua","family":"Combes"},{"given":"Ben Q.","family":"Baragiola"},{"given":"Ting Rei","family":"Tan"}],"title":"Quantum-Enhanced Multi-Parameter Sensing in a Single Mode","issued":{"date-parts":[[2025,9,30]]},"note":"arxivid:2412.04865\narxiv_version_number:2"},{"id":"arxiv:2206.01751","type":"article-journal","author":[{"given":"Allan D. C.","family":"Tosta"},{"given":"Thiago O.","family":"Maciel"},{"given":"Leandro","family":"Aolita"}],"title":"Grand Unification of continuous-variable codes","issued":{"date-parts":[[2022,6,3]]},"note":"arxivid:2206.01751\narxiv_version_number:1"},{"id":"arxiv:2511.06108","type":"article-journal","author":[{"given":"Nir","family":"Gutman"},{"given":"Eliya","family":"Blumenthal"},{"given":"Shay","family":"Hacohen-Gourgy"},{"given":"Ariel","family":"Orda"},{"given":"Ido","family":"Kaminer"}],"title":"Squeezed-vacuum bosonic codes","issued":{"date-parts":[[2025,11,8]]},"note":"arxivid:2511.06108\narxiv_version_number:1"},{"id":"arxiv:2504.01664","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Inspired by recent advances in the manipulation of superconducting circuits coupled to mechanical modes in the quantum regime, we propose a protocol for generating superpositions of orthogonally squeezed states in a quantum harmonic oscillator. The protocol relies on a quadratic coupling between the oscillator and a qubit, and is conceptually similar to methods used for preparing cat states in qubit-oscillator systems. We numerically evaluate the robustness of the state-preparation scheme in the presence of decoherence, considering environmental coupling for both the harmonic oscillator and the qubit. As a potential application, we consider a quantum error-correcting code based on conditionally squeezed states and analyze its error correction properties.</jats:p>","DOI":"10.1103/jrrb-hymw","source":"Crossref","title":"Preparation of conditionally squeezed states in qubit-oscillator systems","volume":"8","author":[{"ORCID":"https://orcid.org/0009-0002-8486-0501","authenticated-orcid":true,"given":"Marius K.","family":"Hope","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00nqqgg17","id-type":"ROR","asserted-by":"publisher"}],"name":"National Security Authority"},{"id":[{"id":"https://ror.org/05ecg5h20","id-type":"ROR","asserted-by":"publisher"}],"name":"University of South-Eastern Norway"}]},{"ORCID":"https://orcid.org/0000-0002-9726-025X","authenticated-orcid":true,"given":"Jonas","family":"Lidal","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00nqqgg17","id-type":"ROR","asserted-by":"publisher"}],"name":"National Security Authority"}]},{"ORCID":"https://orcid.org/0000-0002-6132-109X","authenticated-orcid":true,"given":"Francesco","family":"Massel","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05ecg5h20","id-type":"ROR","asserted-by":"publisher"}],"name":"University of South-Eastern Norway"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2026,2,26]]},"URL":"http://dx.doi.org/10.1103/jrrb-hymw","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"L012046","note":"arxivid:2504.01664"},{"id":"doi:10.1142/9038","type":"book","publisher":"WORLD SCIENTIFIC","DOI":"10.1142/9038","source":"Crossref","title":"Quantum Mechanics","author":[{"given":"Leslie E","family":"Ballentine","sequence":"first","affiliation":[{"name":"Simon Fraser UniversityCanada"}]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2013,11,11]]},"ISBN":["9789814578578","9789814578592"],"URL":"http://dx.doi.org/10.1142/9038","note":"edition-number:2\nalternative-id:10.1142/9038"},{"id":"arxiv:2211.05714","type":"chapter","author":[{"family":"Albert","given":"Victor V.","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST and University of Maryland - 4254 Stadium Dr. College Park, MD 20740, USA"}]}],"editor":[{"family":"Bramati","given":"Alberto","affiliation":[{"name":"Sorbonne Université, Laboratoire Kastler Brossel, Paris, France"}]},{"family":"Carusotto","given":"Iacopo","affiliation":[{"name":"Pitaevskii BEC Center, INO-CNR, Povo, Italy"}]},{"family":"Ciuti","given":"Cristiano","affiliation":[{"name":"Université Paris Cité – CNRS, Laboratoire Matériaux et Phénomènes Quantiques, Paris, France"}]}],"issued":{"date-parts":[[2025,1,21]]},"abstract":"Bosonic or continuous-variable coding is a field concerned with robust quantum information processing and communication with electromagnetic signals or mechanical modes. I review bosonic quantum memories, characterizing them as either bosonic stabilizer or bosonic Fock-state codes. I then enumerate various applications of bosonic encodings, four of which circumvent no-go theorems due to the intrinsic infinite-dimensionality of bosonic systems.","collection-title":"International School of Physics “Enrico Fermi”","container-title":"Quantum Fluids of Light and Matter","DOI":"10.3254/ENFI250007","ISBN":"9781643685748","ISSN":"0074784X, 18798195","medium":"BA","page":"79–107","publisher":"IOS Press","publisher-place":"NL","title":"Bosonic coding: introduction and use cases","URL":"https://doi.org/10.3254/ENFI250007","volume":"209","note":"container:{\"date-parts\":[[2025,1,21]]}\narxivid:2211.05714"},{"id":"arxiv:math-ph/0210005","type":"article-journal","author":[{"given":"A. M.","family":"Perelomov"}],"title":"On the completeness of a system of coherent states","issued":{"date-parts":[[2002,10,1]]},"note":"arxivid:math-ph/0210005\narxiv_version_number:1"},{"id":"doi:10.1016/0034-4877(71)90006-1","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/0034-4877(71)90006-1","page":"221-228","source":"Crossref","title":"On the completeness of the coherent states","volume":"2","author":[{"given":"V.","family":"Bargmann","sequence":"first","affiliation":[]},{"given":"P.","family":"Butera","sequence":"additional","affiliation":[]},{"given":"L.","family":"Girardello","sequence":"additional","affiliation":[]},{"given":"John R.","family":"Klauder","sequence":"additional","affiliation":[]}],"container-title":"Reports on Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1971,12]]},"URL":"http://dx.doi.org/10.1016/0034-4877(71)90006-1","ISSN":["0034-4877"],"container-title-short":"Reports on Mathematical Physics","note":"alternative-id:0034487771900061"},{"id":"doi:10.1103/PhysRevB.18.6744","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevb.18.6744","page":"6744-6751","source":"Crossref","title":"Discrete coherent states on the von Neumann lattice","volume":"18","author":[{"given":"M.","family":"Boon","sequence":"first","affiliation":[]},{"given":"J.","family":"Zak","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[1978,12,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.18.6744","ISSN":["0163-1829"],"container-title-short":"Phys. Rev. B"},{"id":"doi:10.1007/978-94-007-0196-0","type":"book","publisher":"Springer Netherlands","DOI":"10.1007/978-94-007-0196-0","source":"Crossref","title":"Coherent States and Applications in Mathematical Physics","author":[{"given":"Monique","family":"Combescure","sequence":"first","affiliation":[]},{"given":"Didier","family":"Robert","sequence":"additional","affiliation":[]}],"container-title":"Theoretical and Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2012]]},"ISBN":["9789400701953","9789400701960"],"URL":"http://dx.doi.org/10.1007/978-94-007-0196-0","ISSN":["1864-5879","1864-5887"],"note":"publisher-location:Dordrecht"},{"id":"doi:10.1103/PhysRev.40.749","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrev.40.749","page":"749-759","source":"Crossref","title":"On the Quantum Correction For Thermodynamic Equilibrium","volume":"40","author":[{"given":"E.","family":"Wigner","sequence":"first","affiliation":[]}],"container-title":"Physical Review","original-title":[],"language":"en","issued":{"date-parts":[[1932,6,1]]},"URL":"http://dx.doi.org/10.1103/PhysRev.40.749","ISSN":["0031-899X"],"container-title-short":"Phys. Rev."},{"id":"doi:10.1103/PhysRevA.15.449","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.15.449","page":"449-450","source":"Crossref","title":"Wigner function as the expectation value of a parity operator","volume":"15","author":[{"given":"Antoine","family":"Royer","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1977,2,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.15.449","ISSN":["0556-2791"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:2407.18394","type":"article-journal","author":[{"given":"Jack","family":"Davis"},{"given":"Nicolas","family":"Fabre"},{"given":"Ulysse","family":"Chabaud"}],"title":"Identifying quantum resources in encoded computations","issued":{"date-parts":[[2024,7,25]]},"note":"arxivid:2407.18394\narxiv_version_number:1"},{"id":"doi:10.1103/PhysRevA.37.3028","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physreva.37.3028","page":"3028-3038","source":"Crossref","title":"Gaussian pure states in quantum mechanics and the symplectic group","volume":"37","author":[{"given":"R.","family":"Simon","sequence":"first","affiliation":[]},{"given":"E. C. G.","family":"Sudarshan","sequence":"additional","affiliation":[]},{"given":"N.","family":"Mukunda","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1988,4,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.37.3028","ISSN":["0556-2791"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:quant-ph/0410100","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/revmodphys.77.513","page":"513-577","source":"Crossref","title":"Quantum information with continuous variables","volume":"77","author":[{"given":"Samuel L.","family":"Braunstein","sequence":"first","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2005,6,29]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.77.513","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","note":"arxivid:quant-ph/0410100"},{"id":"arxiv:quant-ph/0503237","type":"article-journal","author":[{"given":"Alessandro","family":"Ferraro"},{"given":"Stefano","family":"Olivares"},{"given":"Matteo G. A.","family":"Paris"}],"title":"Gaussian states in continuous variable quantum information","issued":{"date-parts":[[2005,3,31]]},"note":"arxivid:quant-ph/0503237\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0701051","type":"journal-article","publisher":"IOP Publishing","issue":"31","DOI":"10.1088/1751-8113/40/31/027","page":"9551-9576","source":"Crossref","title":"Canonical and micro-canonical typical entanglement of continuous variable systems","volume":"40","author":[{"given":"A","family":"Serafini","sequence":"first","affiliation":[]},{"given":"O C O","family":"Dahlsten","sequence":"additional","affiliation":[]},{"given":"D","family":"Gross","sequence":"additional","affiliation":[]},{"given":"M B","family":"Plenio","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2007,7,19]]},"URL":"http://dx.doi.org/10.1088/1751-8113/40/31/027","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(07)41148-9\narxivid:quant-ph/0701051"},{"id":"arxiv:0801.4604","type":"journal-article","publisher":"Elsevier BV","issue":"1-4","DOI":"10.1016/j.physrep.2007.04.005","page":"1-111","source":"Crossref","title":"Quantum information with Gaussian states","volume":"448","author":[{"given":"X","family":"WANG","sequence":"first","affiliation":[]},{"given":"T","family":"HIROSHIMA","sequence":"additional","affiliation":[]},{"given":"A","family":"TOMITA","sequence":"additional","affiliation":[]},{"given":"M","family":"HAYASHI","sequence":"additional","affiliation":[]}],"container-title":"Physics Reports","original-title":[],"language":"en","issued":{"date-parts":[[2007,8]]},"URL":"http://dx.doi.org/10.1016/j.physrep.2007.04.005","ISSN":["0370-1573"],"container-title-short":"Physics Reports","note":"alternative-id:S0370157307001822\narxivid:0801.4604"},{"id":"arxiv:1110.3234","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/revmodphys.84.621","page":"621-669","source":"Crossref","title":"Gaussian quantum information","volume":"84","author":[{"given":"Christian","family":"Weedbrook","sequence":"first","affiliation":[]},{"given":"Stefano","family":"Pirandola","sequence":"additional","affiliation":[]},{"given":"Raúl","family":"García-Patrón","sequence":"additional","affiliation":[]},{"given":"Nicolas J.","family":"Cerf","sequence":"additional","affiliation":[]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[]},{"given":"Jeffrey H.","family":"Shapiro","sequence":"additional","affiliation":[]},{"given":"Seth","family":"Lloyd","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2012,5,1]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.84.621","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","note":"arxivid:1110.3234"},{"id":"arxiv:2010.15518","type":"journal-article","publisher":"Stichting SciPost","issue":"3","abstract":"<jats:p>We show that bosonic and fermionic Gaussian states (also known as\n``squeezed coherent states’’) can be uniquely characterized by their\nlinear complex structure <jats:inline-formula><jats:alternatives><jats:tex-math>J</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>J</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nwhich is a linear map on the classical phase space. This extends\nconventional Gaussian methods based on covariance matrices and provides\na unified framework to treat bosons and fermions simultaneously. Pure\nGaussian states can be identified with the triple\n<jats:inline-formula><jats:alternatives><jats:tex-math>(G,\\Omega,J)</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo stretchy=\"false\" form=\"prefix\">(</mml:mo><mml:mi>G</mml:mi><mml:mo>,</mml:mo><mml:mi>Ω</mml:mi><mml:mo>,</mml:mo><mml:mi>J</mml:mi><mml:mo stretchy=\"false\" form=\"postfix\">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>\nof compatible Kähler structures, consisting of a positive definite\nmetric <jats:inline-formula><jats:alternatives><jats:tex-math>G</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>G</mml:mi></mml:math></jats:alternatives></jats:inline-formula>,\na symplectic form <jats:inline-formula><jats:alternatives><jats:tex-math>\\Omega</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>Ω</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nand a linear complex structure <jats:inline-formula><jats:alternatives><jats:tex-math>J</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>J</mml:mi></mml:math></jats:alternatives></jats:inline-formula>\nwith <jats:inline-formula><jats:alternatives><jats:tex-math>J^2=-\\mathbb{1}</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mstyle mathvariant=\"double-struck\"><mml:mn>1</mml:mn></mml:mstyle></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>.\nMixed Gaussian states can also be identified with such a triple, but\nwith <jats:inline-formula><jats:alternatives><jats:tex-math>J^2\\neq -\\mathbb{1}</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:msup><mml:mi>J</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≠</mml:mo><mml:mo>−</mml:mo><mml:mstyle mathvariant=\"double-struck\"><mml:mn>1</mml:mn></mml:mstyle></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>.\nWe apply these methods to show how computations involving Gaussian\nstates can be reduced to algebraic operations of these objects, leading\nto many known and some unknown identities. We apply these methods to the\nstudy of (A) entanglement and complexity, (B) dynamics of stable\nsystems, (C) dynamics of driven systems. From this, we compile a\ncomprehensive list of mathematical structures and formulas to compare\nbosonic and fermionic Gaussian states side-by-side.</jats:p>","DOI":"10.21468/scipostphyscore.4.3.025","source":"Crossref","title":"Bosonic and fermionic Gaussian states from Kähler structures","volume":"4","author":[{"given":"Lucas","family":"Hackl","sequence":"first","affiliation":[{"name":"University of Melbourne"},{"name":"University of Copenhagen"}]},{"given":"Eugenio","family":"Bianchi","sequence":"additional","affiliation":[{"name":"Pennsylvania State University"}]}],"container-title":"SciPost Physics Core","original-title":[],"issued":{"date-parts":[[2021,9,22]]},"URL":"http://dx.doi.org/10.21468/SciPostPhysCore.4.3.025","ISSN":["2666-9366"],"container-title-short":"SciPost Phys. Core","page":"025","note":"arxivid:2010.15518"},{"id":"arxiv:2102.05748","type":"article-journal","author":[{"given":"Jonatan Bohr","family":"Brask"}],"title":"Gaussian states and operations -- a quick reference","issued":{"date-parts":[[2022,3,30]]},"note":"arxivid:2102.05748\narxiv_version_number:2"},{"id":"arxiv:2409.11628","type":"article-journal","author":[{"given":"Tommaso","family":"Guaita"},{"given":"Lucas","family":"Hackl"},{"given":"Thomas","family":"Quella"}],"title":"Representation theory of Gaussian unitary transformations for bosonic and fermionic systems","issued":{"date-parts":[[2024,9,18]]},"note":"arxivid:2409.11628\narxiv_version_number:1"},{"id":"doi:10.1016/0034-4877(74)90007-X","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0034-4877(74)90007-x","page":"249-252","source":"Crossref","title":"When is the wigner quasi-probability density non-negative?","volume":"6","author":[{"given":"R.L.","family":"Hudson","sequence":"first","affiliation":[]}],"container-title":"Reports on Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1974,10]]},"URL":"http://dx.doi.org/10.1016/0034-4877(74)90007-X","ISSN":["0034-4877"],"container-title-short":"Reports on Mathematical Physics","note":"alternative-id:003448777490007X"},{"id":"doi:10.1063/1.525607","type":"journal-article","publisher":"AIP Publishing","issue":"1","abstract":"<jats:p>It is shown that, for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form. This result generalizes the one obtained by Hudson [Rep. Math. Phys. 6, 249 (1974)] for one-dimensional systems.</jats:p>","DOI":"10.1063/1.525607","page":"97-100","source":"Crossref","title":"When is the Wigner function of multidimensional systems nonnegative?","volume":"24","author":[{"given":"Francisco","family":"Soto","sequence":"first","affiliation":[{"name":"Laboratoire de Chimie Quantique, Institut de Biologie Physico-Chimique, 13, rue Pierre et Marie Curie, 75005 Paris, France"}]},{"given":"Pierre","family":"Claverie","sequence":"additional","affiliation":[{"name":"Laboratoire de Chimie Quantique, Institut de Biologie Physico-Chimique, 13, rue Pierre et Marie Curie, 75005 Paris, France"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1983,1,1]]},"URL":"http://dx.doi.org/10.1063/1.525607","ISSN":["0022-2488","1089-7658"]},{"id":"arxiv:1612.05080","type":"journal-article","publisher":"AIP Publishing","issue":"4","abstract":"<jats:p>We prove a generalization of the quantum de Finetti theorem when the local space is an infinite-dimensional Fock space. In particular, instead of considering the action of the permutation group on n copies of that space, we consider the action of the unitary group U(n) on the creation operators of the n modes and define a natural generalization of the symmetric subspace as the space of states invariant under unitaries in U(n). Our first result is a complete characterization of this subspace, which turns out to be spanned by a family of generalized coherent states related to the special unitary group SU(p, q) of signature (p, q). More precisely, this construction yields a unitary representation of the noncompact simple real Lie group SU(p, q). We therefore find a dual unitary representation of the pair of groups U(n) and SU(p, q) on an n(p + q)-mode Fock space. The (Gaussian) SU(p, q) coherent states resolve the identity on the symmetric subspace, which implies a Gaussian de Finetti theorem stating that tracing over a few modes of a unitary-invariant state yields a state close to a mixture of Gaussian states. As an application of this de Finetti theorem, we show that the n × n upper-left submatrix of an n × n Haar-invariant unitary matrix is close in total variation distance to a matrix of independent normal variables if n3 = O(m).</jats:p>","DOI":"10.1063/1.5007334","source":"Crossref","title":"<i>SU</i>(<i>p</i>,<i>q</i>) coherent states and a Gaussian de Finetti theorem","volume":"59","author":[{"ORCID":"https://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[{"name":"Inria , Paris, France"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2018,4,1]]},"URL":"http://dx.doi.org/10.1063/1.5007334","ISSN":["0022-2488","1089-7658"],"page":"042202","note":"arxivid:1612.05080"},{"id":"preset:Weyl50","type":"book","author":[{"family":"Weyl","given":"H."}],"title":"\\emph{The Theory of Groups and Quantum Mechanics}","publisher":"Courier Corporation","issued":"1950"},{"id":"doi:10.1201/9781315118727","type":"reference-book","publisher":"CRC Press","DOI":"10.1201/9781315118727","source":"Crossref","title":"Quantum Continuous Variables","author":[{"given":"Alessio","family":"Serafini","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2017,7,20]]},"ISBN":["9781315118727"],"URL":"http://dx.doi.org/10.1201/9781315118727","note":"publisher-location:Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017] |\nedition-number:1"},{"id":"doi:10.1103/PhysRev.177.1857","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrev.177.1857","page":"1857-1881","source":"Crossref","title":"Ordered Expansions in Boson Amplitude Operators","volume":"177","author":[{"given":"K. E.","family":"Cahill","sequence":"first","affiliation":[]},{"given":"R. J.","family":"Glauber","sequence":"additional","affiliation":[]}],"container-title":"Physical Review","original-title":[],"language":"en","issued":{"date-parts":[[1969,1,25]]},"URL":"http://dx.doi.org/10.1103/PhysRev.177.1857","ISSN":["0031-899X"],"container-title-short":"Phys. Rev."},{"id":"doi:10.1007/3-7643-7575-2","type":"book","publisher":"Birkhäuser Basel","DOI":"10.1007/3-7643-7575-2","source":"Crossref","title":"Symplectic Geometry and Quantum Mechanics","author":[{"given":"Maurice","family":"de Gosson","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2006]]},"ISBN":["9783764375744","9783764375751"],"URL":"http://dx.doi.org/10.1007/3-7643-7575-2","note":"publisher-location:Basel"},{"id":"arxiv:2502.09514","type":"article-journal","author":[{"given":"Ansgar G.","family":"Burchards"}],"title":"Continuous-Variable Quantum MacWilliams Identities","issued":{"date-parts":[[2025,3,27]]},"note":"arxivid:2502.09514\narxiv_version_number:2"},{"id":"doi:10.1103/PhysicsPhysiqueFizika.1.","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physicsphysiquefizika.1.49","page":"49-61","source":"Crossref","title":"Quantum mechanical phase and time operator","volume":"1","author":[{"given":"Leonard","family":"Susskind","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Glogower","sequence":"additional","affiliation":[]}],"container-title":"Physics Physique Fizika","original-title":[],"language":"en","issued":{"date-parts":[[1964,7,1]]},"URL":"http://dx.doi.org/10.1103/PhysicsPhysiqueFizika.1.49","ISSN":["0554-128X"],"container-title-short":"Physics Physique Fizika"},{"id":"doi:10.1016/0003-4916(91)90037-9","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/0003-4916(91)90037-9","page":"479-505","source":"Crossref","title":"Operators of the phase. Fundamentals","volume":"209","author":[{"given":"János","family":"Bergou","sequence":"first","affiliation":[]},{"given":"Berthold-Georg","family":"Englert","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[1991,8]]},"URL":"http://dx.doi.org/10.1016/0003-4916(91)90037-9","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:0003491691900379"},{"id":"arxiv:quant-ph/0109066","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.65.052316","source":"Crossref","title":"Quantum encodings in spin systems and harmonic oscillators","volume":"65","author":[{"given":"Stephen D.","family":"Bartlett","sequence":"first","affiliation":[]},{"given":"Hubert","family":"de Guise","sequence":"additional","affiliation":[]},{"given":"Barry C.","family":"Sanders","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,5,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.052316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052316","note":"arxivid:quant-ph/0109066"},{"id":"manual:-E.-M.-Stein-Real-variable-me","type":"article-journal","author":[{"family":"Stein","given":"E.M."}],"title":"Real variable methods, orthogonality, and oscillatory integrals","volume":"43","container-title":"Princeton Math. Series","issued":"1993"},{"id":"doi:10.1016/0034-4877(79)90049-1","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0034-4877(79)90049-1","page":"27-39","source":"Crossref","title":"Completely positive quasi-free maps of the CCR-algebra","volume":"15","author":[{"given":"B.","family":"Demoen","sequence":"first","affiliation":[]},{"given":"P.","family":"Vanheuverzwijn","sequence":"additional","affiliation":[]},{"given":"A.","family":"Verbeure","sequence":"additional","affiliation":[]}],"container-title":"Reports on Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1979,2]]},"URL":"http://dx.doi.org/10.1016/0034-4877(79)90049-1","ISSN":["0034-4877"],"container-title-short":"Reports on Mathematical Physics","note":"alternative-id:0034487779900491"},{"id":"arxiv:quant-ph/0505151","type":"article-journal","author":[{"given":"J.","family":"Eisert"},{"given":"M. M.","family":"Wolf"}],"title":"Gaussian quantum channels","issued":{"date-parts":[[2005,5,20]]},"note":"arxivid:quant-ph/0505151\narxiv_version_number:1"},{"id":"arxiv:0707.0604","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.100.070505","source":"Crossref","title":"Not-So-Normal Mode Decomposition","volume":"100","author":[{"given":"Michael M.","family":"Wolf","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,2,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.100.070505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070505","note":"arxivid:0707.0604"},{"id":"arxiv:0804.0511","type":"journal-article","publisher":"IOP Publishing","issue":"8","DOI":"10.1088/1367-2630/10/8/083030","page":"083030","source":"Crossref","title":"Multi-mode bosonic Gaussian channels","volume":"10","author":[{"given":"F","family":"Caruso","sequence":"first","affiliation":[]},{"given":"J","family":"Eisert","sequence":"additional","affiliation":[]},{"given":"V","family":"Giovannetti","sequence":"additional","affiliation":[]},{"given":"A S","family":"Holevo","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2008,8,20]]},"URL":"http://dx.doi.org/10.1088/1367-2630/10/8/083030","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:0804.0511"},{"id":"arxiv:0809.3273","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.102.050503","source":"Crossref","title":"Direct and Reverse Secret-Key Capacities of a Quantum Channel","volume":"102","author":[{"given":"Stefano","family":"Pirandola","sequence":"first","affiliation":[]},{"given":"Raul","family":"García-Patrón","sequence":"additional","affiliation":[]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[]},{"given":"Seth","family":"Lloyd","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,2,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.102.050503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050503","note":"arxivid:0809.3273"},{"id":"arxiv:1004.0196","type":"journal-article","publisher":"AIP Publishing","issue":"4","abstract":"<jats:p>We obtain explicit expressions for the Choi–Jamiolkowski (CJ) forms and operators defining a general bosonic Gaussian channel. The four principal cases are considered in Sec. III; in Sec. IV, we give a decomposition of Gaussian CJ form into product of these four principal types and provide a necessary and sufficient condition for the existence of the bounded CJ operator.</jats:p>","DOI":"10.1063/1.3581879","source":"Crossref","title":"The Choi–Jamiolkowski forms of quantum Gaussian channels","volume":"52","author":[{"given":"A. S.","family":"Holevo","sequence":"first","affiliation":[{"name":"Steklov Mathematical Institute , Russian Academy of Sciences, Gubkina St. 8, 119991 Moscow, Russia"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,4,1]]},"URL":"http://dx.doi.org/10.1063/1.3581879","ISSN":["0022-2488","1089-7658"],"page":"042202","note":"arxivid:1004.0196"},{"id":"arxiv:1009.1108","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.84.022306","source":"Crossref","title":"Optimal unitary dilation for bosonic Gaussian channels","volume":"84","author":[{"given":"Filippo","family":"Caruso","sequence":"first","affiliation":[]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]},{"given":"Vittorio","family":"Giovannetti","sequence":"additional","affiliation":[]},{"given":"Alexander S.","family":"Holevo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,8,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.022306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022306","note":"arxivid:1009.1108"},{"id":"arxiv:1012.4266","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.84.042311","source":"Crossref","title":"Operator-sum representation for bosonic Gaussian channels","volume":"84","author":[{"given":"J. Solomon","family":"Ivan","sequence":"first","affiliation":[]},{"given":"Krishna Kumar","family":"Sabapathy","sequence":"additional","affiliation":[]},{"given":"R.","family":"Simon","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,10,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.042311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042311","note":"arxivid:1012.4266"},{"id":"arxiv:2507.04932","type":"article-journal","author":[{"given":"Ju-Yeon","family":"Gyhm"},{"given":"Dario","family":"Rosa"},{"given":"Dominik","family":"Šafránek"}],"title":"Gaussian Open Quantum Dynamics and Isomorphism to Superconformal Symmetry","issued":{"date-parts":[[2025,7,7]]},"note":"arxivid:2507.04932\narxiv_version_number:1"},{"id":"arxiv:2205.05736","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1038/s41566-023-01190-4","page":"525-530","source":"Crossref","title":"Exact solution for the quantum and private capacities of bosonic dephasing channels","volume":"17","author":[{"ORCID":"https://orcid.org/0000-0003-3290-3557","authenticated-orcid":false,"given":"Ludovico","family":"Lami","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3916-4462","authenticated-orcid":false,"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"Nature Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2023,4,6]]},"URL":"http://dx.doi.org/10.1038/s41566-023-01190-4","ISSN":["1749-4885","1749-4893"],"container-title-short":"Nat. Photon.","note":"alternative-id:1190\narxivid:2205.05736"},{"id":"arxiv:quant-ph/0105058","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.64.062301","source":"Crossref","title":"Achievable rates for the Gaussian quantum channel","volume":"64","author":[{"given":"Jim","family":"Harrington","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2001,11,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.64.062301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062301","note":"arxivid:quant-ph/0105058"},{"id":"arxiv:1510.08863","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. Here, by constructing an upper bound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed ‘teleportation stretching’, we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rate-loss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of point-to-point quantum communications and provide precise and general benchmarks for quantum repeaters.</jats:p>","DOI":"10.1038/ncomms15043","source":"Crossref","title":"Fundamental limits of repeaterless quantum communications","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0001-6165-5615","authenticated-orcid":false,"given":"Stefano","family":"Pirandola","sequence":"first","affiliation":[]},{"given":"Riccardo","family":"Laurenza","sequence":"additional","affiliation":[]},{"given":"Carlo","family":"Ottaviani","sequence":"additional","affiliation":[]},{"given":"Leonardo","family":"Banchi","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2017,4,26]]},"URL":"http://dx.doi.org/10.1038/ncomms15043","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"15043","note":"alternative-id:BFncomms15043\narxivid:1510.08863"},{"id":"arxiv:quant-ph/9912067","type":"article-journal","author":[{"given":"A. S.","family":"Holevo"},{"given":"R. F.","family":"Werner"}],"title":"Evaluating capacities of Bosonic Gaussian channels","issued":{"date-parts":[[1999,12,14]]},"note":"arxivid:quant-ph/9912067\narxiv_version_number:1"},{"id":"arxiv:1310.0129","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2014.2330313","page":"4987-4998","source":"Crossref","title":"The Squashed Entanglement of a Quantum Channel","volume":"60","author":[{"given":"Masahiro","family":"Takeoka","sequence":"first","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3916-4462","authenticated-orcid":false,"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2330313","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1310.0129"},{"id":"arxiv:1504.06390","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/ncomms6235","source":"Crossref","title":"Fundamental rate-loss tradeoff for optical quantum key distribution","volume":"5","author":[{"given":"Masahiro","family":"Takeoka","sequence":"first","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2014,10,24]]},"URL":"http://dx.doi.org/10.1038/ncomms6235","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"5235","note":"alternative-id:BFncomms6235\narxivid:1504.06390"},{"id":"arxiv:1511.08710","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/18/6/063005","page":"063005","source":"Crossref","title":"Assessing the performance of quantum repeaters for all phase-insensitive Gaussian bosonic channels","volume":"18","author":[{"given":"K","family":"Goodenough","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2023-2768","authenticated-orcid":false,"given":"D","family":"Elkouss","sequence":"additional","affiliation":[]},{"given":"S","family":"Wehner","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2016,6,3]]},"URL":"http://dx.doi.org/10.1088/1367-2630/18/6/063005","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1511.08710"},{"id":"arxiv:1602.08898","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2017.2648825","page":"1792-1817","source":"Crossref","title":"Converse Bounds for Private Communication Over Quantum Channels","volume":"63","author":[{"ORCID":"https://orcid.org/0000-0002-3916-4462","authenticated-orcid":false,"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Marco","family":"Tomamichel","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0428-3429","authenticated-orcid":false,"given":"Mario","family":"Berta","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2017,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2017.2648825","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1602.08898"},{"id":"arxiv:1609.02169","type":"proceedings-article","publisher":"SPIE","DOI":"10.1117/12.2244899","page":"999609","source":"Crossref","title":"Secret key capacity of the thermal-loss channel: improving the lower bound","volume":"9996","author":[{"given":"Carlo","family":"Ottaviani","sequence":"additional","affiliation":[{"name":"Univ. of York (United Kingdom)"}]},{"given":"Riccardo","family":"Laurenza","sequence":"additional","affiliation":[{"name":"Univ. of York (United Kingdom)"}]},{"given":"Thomas P. W.","family":"Cope","sequence":"additional","affiliation":[{"name":"Univ. of York (United Kingdom)"}]},{"given":"Gaetana","family":"Spedalieri","sequence":"additional","affiliation":[{"name":"Univ. of York (United Kingdom)"}]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[{"name":"Univ. of York (United Kingdom)"}]},{"given":"Stefano","family":"Pirandola","sequence":"additional","affiliation":[{"name":"Univ. of York (United Kingdom)"}]}],"event":"SPIE Security + Defence","container-title":"SPIE Proceedings","original-title":[],"editor":[{"given":"Mark T.","family":"Gruneisen","sequence":"first","affiliation":[{"name":"Air Force Research Lab. (United States)"}]},{"given":"Miloslav","family":"Dusek","sequence":"additional","affiliation":[{"name":"Palacký Univ. Olomouc (Czech Republic)"}]},{"given":"John G.","family":"Rarity","sequence":"additional","affiliation":[{"name":"Univ. of Bristol (United Kingdom)"}]}],"issued":{"date-parts":[[2016,10,24]]},"URL":"http://dx.doi.org/10.1117/12.2244899","ISSN":["0277-786X"],"note":"arxivid:1609.02169"},{"id":"arxiv:1711.09909","type":"journal-article","publisher":"IOP Publishing","issue":"3","DOI":"10.1088/2058-9565/aac394","page":"035009","source":"Crossref","title":"Theory of channel simulation and bounds for private communication","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-6165-5615","authenticated-orcid":false,"given":"Stefano","family":"Pirandola","sequence":"first","affiliation":[]},{"given":"Samuel L","family":"Braunstein","sequence":"additional","affiliation":[]},{"given":"Riccardo","family":"Laurenza","sequence":"additional","affiliation":[]},{"given":"Carlo","family":"Ottaviani","sequence":"additional","affiliation":[]},{"given":"Thomas P W","family":"Cope","sequence":"additional","affiliation":[]},{"given":"Gaetana","family":"Spedalieri","sequence":"additional","affiliation":[]},{"given":"Leonardo","family":"Banchi","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2018,5,31]]},"URL":"http://dx.doi.org/10.1088/2058-9565/aac394","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"updated-by:{\"DOI\":\"10.1088/2058-9565/aadf3b\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2018,9,28]],\"date-time\":\"2018-09-28T00:00:00Z\",\"timestamp\":1538092800000}}\narxivid:1711.09909"},{"id":"arxiv:1801.08102","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.97.062310","source":"Crossref","title":"Energy-constrained two-way assisted private and quantum capacities of quantum channels","volume":"97","author":[{"given":"Noah","family":"Davis","sequence":"first","affiliation":[]},{"given":"Maksim E.","family":"Shirokov","sequence":"additional","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,6,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.062310","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062310","note":"arxivid:1801.08102"},{"id":"arxiv:1807.05402","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1140/epjd/e2018-90351-0","source":"Crossref","title":"Improving the lower bound to the secret-key capacity of the thermal amplifier channel","volume":"73","author":[{"given":"Gan","family":"Wang","sequence":"first","affiliation":[]},{"given":"Carlo","family":"Ottaviani","sequence":"additional","affiliation":[]},{"given":"Hong","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Stefano","family":"Pirandola","sequence":"additional","affiliation":[]}],"container-title":"The European Physical Journal D","original-title":[],"language":"en","issued":{"date-parts":[[2019,1]]},"URL":"http://dx.doi.org/10.1140/epjd/e2018-90351-0","ISSN":["1434-6060","1434-6079"],"container-title-short":"Eur. Phys. J. D","page":"17","note":"alternative-id:90351\narxivid:1807.05402"},{"id":"arxiv:2303.12867","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1038/s41566-024-01595-9","page":"329-334","source":"Crossref","title":"Maximum tolerable excess noise in continuous-variable quantum key distribution and improved lower bound on two-way capacities","volume":"19","author":[{"ORCID":"https://orcid.org/0000-0002-7900-5167","authenticated-orcid":false,"given":"Francesco Anna","family":"Mele","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3290-3557","authenticated-orcid":false,"given":"Ludovico","family":"Lami","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7636-9002","authenticated-orcid":false,"given":"Vittorio","family":"Giovannetti","sequence":"additional","affiliation":[]}],"container-title":"Nature Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,17]]},"URL":"http://dx.doi.org/10.1038/s41566-024-01595-9","ISSN":["1749-4885","1749-4893"],"container-title-short":"Nat. Photon.","note":"alternative-id:1595\narxivid:2303.12867"},{"id":"arxiv:2309.17066","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2024.3450501","page":"8844-8869","source":"Crossref","title":"Optical Fibers With Memory Effects and Their Quantum Communication Capacities","volume":"70","author":[{"ORCID":"https://orcid.org/0000-0002-7900-5167","authenticated-orcid":false,"given":"Francesco Anna","family":"Mele","sequence":"first","affiliation":[{"name":"NEST, Scuola Normale Superiore and Istituto Nanoscienze, Pisa, Italy"}]},{"ORCID":"https://orcid.org/0000-0002-5064-8695","authenticated-orcid":false,"given":"Giacomo De","family":"Palma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, Bologna, Italy"}]},{"ORCID":"https://orcid.org/0000-0003-0802-8000","authenticated-orcid":false,"given":"Marco","family":"Fanizza","sequence":"additional","affiliation":[{"name":"F&#x00ED;sica Te&#x00F2;rica: Informaci&#x00F3; i Fen&#x00F2;mens Qu&#x00E0;ntics, Departament de F&#x00ED;sica, Universitat Aut&#x00F2;noma de Barcelona, Barcelona, Spain"}]},{"ORCID":"https://orcid.org/0000-0002-7636-9002","authenticated-orcid":false,"given":"Vittorio","family":"Giovannetti","sequence":"additional","affiliation":[{"name":"NEST, Scuola Normale Superiore and Istituto Nanoscienze, Pisa, Italy"}]},{"ORCID":"https://orcid.org/0000-0003-3290-3557","authenticated-orcid":false,"given":"Ludovico","family":"Lami","sequence":"additional","affiliation":[{"name":"QuSoft, Amsterdam, The Netherlands"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2024,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3450501","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2309.17066"},{"id":"arxiv:2205.09711","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum capacity, as the key figure of merit for a given quantum channel, upper bounds the channel&amp;apos;s ability in transmitting quantum information. Identifying different types of channels, evaluating the corresponding quantum capacity, and finding the capacity-approaching coding scheme are the major tasks in quantum communication theory. Quantum channel in discrete variables has been discussed enormously based on various error models, while error model in the continuous variable channel has been less studied due to the infinite dimensional problem. In this paper, we investigate a general continuous variable quantum erasure channel. By defining an effective subspace of the continuous variable system, we find a continuous variable random coding model. We then derive the quantum capacity of the continuous variable erasure channel in the framework of decoupling theory. The discussion in this paper fills the gap of a quantum erasure channel in continuous variable setting and sheds light on the understanding of other types of continuous variable quantum channels.</jats:p>","DOI":"10.22331/q-2023-03-06-939","page":"939","source":"Crossref","title":"Information transmission with continuous variable quantum erasure channels","volume":"7","author":[{"given":"Changchun","family":"Zhong","sequence":"first","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA"}]},{"given":"Changhun","family":"Oh","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,6]]},"URL":"http://dx.doi.org/10.22331/q-2023-03-06-939","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2205.09711"},{"id":"arxiv:2510.01424","type":"article-journal","author":[{"given":"Adam","family":"Taylor"},{"given":"Michael","family":"Hanks"},{"given":"Hyukjoon","family":"Kwon"},{"given":"M. S.","family":"Kim"}],"title":"Optimal Quantum Information Transmission Under a Continuous-Variable Erasure Channel","issued":{"date-parts":[[2025,10,1]]},"note":"arxivid:2510.01424\narxiv_version_number:1"},{"id":"arxiv:1903.12615","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.125.080503","source":"Crossref","title":"Encoding an Oscillator into Many Oscillators","volume":"125","author":[{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":true,"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.080503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080503","note":"arxivid:1903.12615"},{"id":"arxiv:2212.11970","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Bosonic encoding of quantum information into harmonic oscillators is a hardware efficient approach to battle noise. In this regard, oscillator-to-oscillator codes not only provide an additional opportunity in bosonic encoding, but also extend the applicability of error correction to continuous-variable states ubiquitous in quantum sensing and communication. In this work, we derive the optimal oscillator-to-oscillator codes among the general family of Gottesman-Kitaev-Preskill (GKP)-stablizer codes for homogeneous noise. We prove that an arbitrary GKP-stabilizer code can be reduced to a generalized GKP two-mode-squeezing (TMS) code. The optimal encoding to minimize the geometric mean error can be constructed from GKP-TMS codes with an optimized GKP lattice and TMS gains. For single-mode data and ancilla, this optimal code design problem can be efficiently solved, and we further provide numerical evidence that a hexagonal GKP lattice is optimal and strictly better than the previously adopted square lattice. For the multimode case, general GKP lattice optimization is challenging. In the two-mode data and ancilla case, we identify the D4 lattice—a 4-dimensional dense-packing lattice—to be superior to a product of lower dimensional lattices. As a by-product, the code reduction allows us to prove a universal no-threshold-theorem for arbitrary oscillators-to-oscillators codes based on Gaussian encoding, even when the ancilla are not GKP states.</jats:p>","DOI":"10.22331/q-2023-08-16-1082","page":"1082","source":"Crossref","title":"Optimal encoding of oscillators into more oscillators","volume":"7","author":[{"given":"Jing","family":"Wu","sequence":"first","affiliation":[{"name":"James C. Wyant College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA"}]},{"given":"Anthony J.","family":"Brady","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"https://orcid.org/0000-0002-9554-3846","authenticated-orcid":false,"given":"Quntao","family":"Zhuang","sequence":"additional","affiliation":[{"name":"Ming Hsieh Department of Electrical and Computer Engineering & Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA"},{"name":"James C. Wyant College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA"},{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,16]]},"URL":"http://dx.doi.org/10.22331/q-2023-08-16-1082","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2212.11970"},{"id":"arxiv:2502.05524","type":"article-journal","author":[{"given":"Francesco Anna","family":"Mele"},{"given":"Giovanni","family":"Barbarino"},{"given":"Vittorio","family":"Giovannetti"},{"given":"Marco","family":"Fanizza"}],"title":"Achievable rates in non-asymptotic bosonic quantum communication","issued":{"date-parts":[[2025,3,4]]},"note":"arxivid:2502.05524\narxiv_version_number:2"},{"id":"arxiv:2503.13207","type":"article-journal","author":[{"given":"Francesco Anna","family":"Mele"},{"given":"Giovanni","family":"Barbarino"},{"given":"Vittorio","family":"Giovannetti"},{"given":"Marco","family":"Fanizza"}],"title":"Non-asymptotic quantum communication on lossy transmission lines with memory","issued":{"date-parts":[[2025,3,17]]},"note":"arxivid:2503.13207\narxiv_version_number:1"},{"id":"arxiv:2510.07121","type":"article-journal","author":[{"given":"Tobias","family":"Rippchen"},{"given":"Ludovico","family":"Lami"},{"given":"Gerardo","family":"Adesso"},{"given":"Mario","family":"Berta"}],"title":"Fundamental Quality Bound on Optical Quantum Communication","issued":{"date-parts":[[2026,2,26]]},"note":"arxivid:2510.07121\narxiv_version_number:2"},{"id":"preset:Berazin12","type":"document","author":[{"family":"Berazin","given":"F.A."}],"title":"\\emph{The Method of Second Quantization}","volume":"24","issued":"2012"},{"id":"doi:10.1063/1.1665805","type":"journal-article","publisher":"AIP Publishing","issue":"8","abstract":"<jats:p>We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.</jats:p>","DOI":"10.1063/1.1665805","page":"1772-1780","source":"Crossref","title":"Linear Canonical Transformations and Their Unitary Representations","volume":"12","author":[{"given":"M.","family":"Moshinsky","sequence":"first","affiliation":[{"name":"Instituto de Física, Universidad de México, México, D.F."}]},{"given":"C.","family":"Quesne","sequence":"additional","affiliation":[{"name":"Instituto de Física, Universidad de México, México, D.F."}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1971,8,1]]},"URL":"http://dx.doi.org/10.1063/1.1665805","ISSN":["0022-2488","1089-7658"]},{"id":"manual:-M.-Wagner.-Unitary-transform","type":"book","author":[{"family":"Wagner","given":"M."}],"title":"Unitary transformations in solid state physics","publisher":"North Holland","issued":"1986","publisher-place":"Netherlands"},{"id":"arxiv:2403.04596","type":"journal-article","publisher":"Canadian Science Publishing","issue":"10","abstract":"<jats:p> In this tutorial, we summarize four important matrix decompositions commonly used in quantum optics, namely the Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson decompositions. The first two of these decompositions are specialized versions of the singular-value decomposition when applied to symmetric or symplectic matrices. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group. The last one instead gives the symplectic diagonalization of real, positive definite matrices of even size. While proofs of the existence of these decompositions exist in the literature, we review explicit constructions to implement these decompositions using standard linear algebra packages and functionalities such as singular-value, polar, Schur, and QR decompositions, and matrix square roots and inverses. </jats:p>","DOI":"10.1139/cjp-2024-0070","page":"497-507","source":"Crossref","title":"Matrix decompositions in quantum optics: Takagi/Autonne, Bloch–Messiah/Euler, Iwasawa, and Williamson","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0002-1635-9443","authenticated-orcid":false,"given":"Martin","family":"Houde","sequence":"first","affiliation":[{"name":"École Polytechnique de Montréal","place":["Montréal, Canada"]}]},{"ORCID":"https://orcid.org/0000-0003-0344-6385","authenticated-orcid":false,"given":"Will","family":"McCutcheon","sequence":"additional","affiliation":[{"name":"Institute of Photonics and Quantum Sciences, Heriot-Watt University","place":["Edinburgh, UK"]}]},{"ORCID":"https://orcid.org/0000-0002-0175-1688","authenticated-orcid":false,"given":"Nicolás","family":"Quesada","sequence":"additional","affiliation":[{"name":"École Polytechnique de Montréal","place":["Montréal, Canada"]}]}],"container-title":"Canadian Journal of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,1]]},"URL":"http://dx.doi.org/10.1139/cjp-2024-0070","ISSN":["0008-4204","1208-6045"],"container-title-short":"Can. J. Phys.","note":"alternative-id:10.1139/cjp-2024-0070\narxivid:2403.04596"},{"id":"arxiv:2502.12882","type":"article-journal","author":[{"given":"Youngrong","family":"Lim"},{"given":"Changhun","family":"Oh"}],"title":"Classical algorithms for estimating expectation values in linear-optical circuits","issued":{"date-parts":[[2025,11,14]]},"note":"arxivid:2502.12882\narxiv_version_number:3"},{"id":"arxiv:2010.14363","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.3.033018","source":"Crossref","title":"Classical simulation of Gaussian quantum circuits with non-Gaussian input states","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0003-0135-9819","authenticated-orcid":true,"given":"Ulysse","family":"Chabaud","sequence":"first","affiliation":[]},{"given":"Giulia","family":"Ferrini","sequence":"additional","affiliation":[]},{"given":"Frédéric","family":"Grosshans","sequence":"additional","affiliation":[]},{"given":"Damian","family":"Markham","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2021,7,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.3.033018","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033018","note":"arxivid:2010.14363"},{"id":"arxiv:2403.19059","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.110.042402","source":"Crossref","title":"Classical simulation of non-Gaussian bosonic circuits","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0003-2405-1083","authenticated-orcid":true,"given":"Beatriz","family":"Dias","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"},{"id":[{"id":"https://ror.org/04xrcta15","id-type":"ROR","asserted-by":"publisher"}],"name":"Munich Center for Quantum Science and Technology"}]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"},{"id":[{"id":"https://ror.org/04xrcta15","id-type":"ROR","asserted-by":"publisher"}],"name":"Munich Center for Quantum Science and Technology"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.042402","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042402","note":"arxivid:2403.19059"},{"id":"arxiv:2407.06290","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","abstract":"<jats:p>We introduce a framework for simulating, on an (<a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mrow><a:mi>n</a:mi><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math>)-qubit quantum computer, the action of a Gaussian bosonic (GB) circuit on a state over <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:msup><c:mn>2</c:mn><c:mi>n</c:mi></c:msup></c:math> modes. Specifically, we encode the initial bosonic state’s expectation values over quadrature operators (and their covariance matrix) as an input qubit state. This is then evolved by a quantum circuit that effectively implements the symplectic propagators induced by the GB gates. We find families of GB circuits and initial states leading to efficient quantum simulations. For this purpose, we introduce a dictionary that maps between GB and qubit gates such that particle- (non-particle-) preserving GB gates lead to real- (imaginary-) time evolutions at the qubit level. For the special case of particle-preserving circuits, we present a bounded-error-quantum-polynomial time (BQP)-complete GB decision problem, indicating that GB evolutions of Gaussian states on exponentially many modes are as powerful as universal quantum computers. We also perform numerical simulations of an interferometer on <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mo>∼</e:mo><e:mn>8</e:mn><e:mo>×</e:mo><e:msup><e:mn>10</e:mn><e:mn>9</e:mn></e:msup></e:math> modes, illustrating the power of our framework.</jats:p>","DOI":"10.1103/physrevlett.134.070604","source":"Crossref","title":"Gate-Based Quantum Simulation of Gaussian Bosonic Circuits on Exponentially Many Modes","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0002-1766-188X","authenticated-orcid":true,"given":"Alice","family":"Barthe","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01ggx4157","id-type":"ROR","asserted-by":"publisher"}],"name":"CERN"},{"id":[{"id":"https://ror.org/01e41cf67","id-type":"ROR","asserted-by":"publisher"}],"name":"Los Alamos National Laboratory"},{"id":[{"id":"https://ror.org/027bh9e22","id-type":"ROR","asserted-by":"publisher"}],"name":"Universiteit Leiden"}]},{"ORCID":"https://orcid.org/0000-0002-2757-3170","authenticated-orcid":true,"given":"M.","family":"Cerezo","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01e41cf67","id-type":"ROR","asserted-by":"publisher"}],"name":"Los Alamos National Laboratory"},{"id":[{"id":"https://ror.org/01qz5mb56","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantum Science Center"}]},{"ORCID":"https://orcid.org/0000-0001-8036-6624","authenticated-orcid":true,"given":"Andrew T.","family":"Sornborger","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01e41cf67","id-type":"ROR","asserted-by":"publisher"}],"name":"Los Alamos National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-8700-4308","authenticated-orcid":true,"given":"Martín","family":"Larocca","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01e41cf67","id-type":"ROR","asserted-by":"publisher"}],"name":"Los Alamos National Laboratory"},{"id":[{"id":"https://ror.org/01e41cf67","id-type":"ROR","asserted-by":"publisher"}],"name":"Los Alamos National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-0693-1952","authenticated-orcid":true,"given":"Diego","family":"García-Martín","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01e41cf67","id-type":"ROR","asserted-by":"publisher"}],"name":"Los Alamos National Laboratory"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.070604","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070604","note":"arxivid:2407.06290"},{"id":"arxiv:quant-ph/9810082","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.82.1784","page":"1784-1787","source":"Crossref","title":"Quantum Computation over Continuous Variables","volume":"82","author":[{"given":"Seth","family":"Lloyd","sequence":"first","affiliation":[]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1999,2,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.82.1784","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9810082"},{"id":"arxiv:2501.13857","type":"article-journal","author":[{"given":"Francesco","family":"Arzani"},{"given":"Robert I.","family":"Booth"},{"given":"Ulysse","family":"Chabaud"}],"title":"Can effective descriptions of bosonic systems be considered complete?","issued":{"date-parts":[[2025,1,23]]},"note":"arxivid:2501.13857\narxiv_version_number:1"},{"id":"arxiv:2004.11002","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Parametrized quantum optical circuits are a class of quantum circuits in which the carriers of quantum information are photons and the gates are optical transformations. Classically optimizing these circuits is challenging due to the infinite dimensionality of the photon number vector space that is associated to each optical mode. Truncating the space dimension is unavoidable, and it can lead to incorrect results if the gates populate photon number states beyond the cutoff. To tackle this issue, we present an algorithm that is orders of magnitude faster than the current state of the art, to recursively compute the exact matrix elements of Gaussian operators and their gradient with respect to a parametrization. These operators, when augmented with a non-Gaussian transformation such as the Kerr gate, achieve universal quantum computation. Our approach brings two advantages: first, by computing the matrix elements of Gaussian operators directly, we don't need to construct them by combining several other operators; second, we can use any variant of the gradient descent algorithm by plugging our gradients into an automatic differentiation framework such as TensorFlow or PyTorch. Our results will find applications in quantum optical hardware research, quantum machine learning, optical data processing, device discovery and device design.</jats:p>","DOI":"10.22331/q-2020-11-30-366","page":"366","source":"Crossref","title":"Fast optimization of parametrized quantum optical circuits","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-6684-8341","authenticated-orcid":false,"given":"Filippo M.","family":"Miatto","sequence":"first","affiliation":[{"name":"Institut Polytechnique de Paris"},{"name":"Télécom Paris, LTCI, 19 Place Marguerite Perey 91120 Palaiseau"}]},{"ORCID":"https://orcid.org/0000-0002-0175-1688","authenticated-orcid":false,"given":"Nicolás","family":"Quesada","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,11,30]]},"URL":"http://dx.doi.org/10.22331/q-2020-11-30-366","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2004.11002"},{"id":"arxiv:1801.06565","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.97.062311","source":"Crossref","title":"Continuous-variable gate decomposition for the Bose-Hubbard model","volume":"97","author":[{"given":"Timjan","family":"Kalajdzievski","sequence":"first","affiliation":[]},{"given":"Christian","family":"Weedbrook","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Rebentrost","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,6,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.062311","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062311","note":"arxivid:1801.06565"},{"id":"arxiv:1811.10651","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.99.022341","source":"Crossref","title":"Exact gate decompositions for photonic quantum computing","volume":"99","author":[{"given":"Timjan","family":"Kalajdzievski","sequence":"first","affiliation":[]},{"given":"Juan Miguel","family":"Arrazola","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,2,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.022341","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022341","note":"arxivid:1811.10651"},{"id":"doi:10.1103/PhysRevD.29.1107","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevd.29.1107","page":"1107-1110","source":"Crossref","title":"Impossibility of naively generalizing squeezed coherent states","volume":"29","author":[{"given":"Robert A.","family":"Fisher","sequence":"first","affiliation":[]},{"given":"Michael Martin","family":"Nieto","sequence":"additional","affiliation":[]},{"given":"Vernon D.","family":"Sandberg","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[1984,3,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.29.1107","ISSN":["0556-2821"],"container-title-short":"Phys. Rev. D"},{"id":"doi:10.1103/physreva.35.1659","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.35.1659","page":"1659-1667","source":"Crossref","title":"Generalized squeezing","volume":"35","author":[{"given":"Samuel L.","family":"Braunstein","sequence":"first","affiliation":[]},{"given":"Robert I.","family":"McLachlan","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1987,2,1]]},"URL":"http://dx.doi.org/10.1103/physreva.35.1659","ISSN":["0556-2791"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:2508.09041","type":"journal-article","publisher":"American Physical Society (APS)","DOI":"10.1103/9vwp-f35c","source":"Crossref","title":"Finite-dimensional approximations of generalized squeezing","author":[{"family":"Anonymous","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,15]]},"URL":"http://dx.doi.org/10.1103/9vwp-f35c","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","note":"arxivid:2508.09041"},{"id":"arxiv:2508.09044","type":"article-journal","author":[{"given":"Felix","family":"Fischer"},{"given":"Daniel","family":"Burgarth"},{"given":"Davide","family":"Lonigro"}],"title":"Self-adjoint realizations of higher-order squeezing operators","issued":{"date-parts":[[2025,8,13]]},"note":"arxivid:2508.09044\narxiv_version_number:2"},{"id":"arxiv:2110.06942","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x039B;</mml:mi></mml:math> on each local quantum number, and if the initial state has low local quantum numbers, then an error at most <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> can be achieved by choosing <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x039B;</mml:mi></mml:math> to scale polylogarithmically with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x039B;</mml:mi></mml:math> that achieves accuracy <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi></mml:math>, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> by truncating local quantum numbers at <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x039B;</mml:mi><mml:mo>=</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext>polylog</mml:mtext></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>.</jats:p>","DOI":"10.22331/q-2022-09-22-816","page":"816","source":"Crossref","title":"Provably accurate simulation of gauge theories and bosonic systems","volume":"6","author":[{"given":"Yu","family":"Tong","sequence":"first","affiliation":[{"name":"Google Quantum AI, Venice, CA, USA"},{"name":"Department of Mathematics, University of California, Berkeley, CA, USA"}]},{"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, MD, USA"}]},{"given":"Jarrod R.","family":"McClean","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Venice, CA, USA"}]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA"},{"name":"AWS Center for Quantum Computing, Pasadena, CA, USA"}]},{"given":"Yuan","family":"Su","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Venice, CA, USA"},{"name":"Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,22]]},"URL":"http://dx.doi.org/10.22331/q-2022-09-22-816","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2110.06942"},{"id":"arxiv:2510.08546","type":"article-journal","author":[{"given":"Alex","family":"Maltesson"},{"given":"Ludvig","family":"Rodung"},{"given":"Niklas","family":"Budinger"},{"given":"Giulia","family":"Ferrini"},{"given":"Cameron","family":"Calcluth"}],"title":"Equivalence of continuous- and discrete-variable gate-based quantum computers with finite energy","issued":{"date-parts":[[2025,10,9]]},"note":"arxivid:2510.08546\narxiv_version_number:1"},{"id":"arxiv:2510.08545","type":"article-journal","author":[{"given":"Ulysse","family":"Chabaud"},{"given":"Sevag","family":"Gharibian"},{"given":"Saeed","family":"Mehraban"},{"given":"Arsalan","family":"Motamedi"},{"given":"Hamid Reza","family":"Naeij"},{"given":"Dorian","family":"Rudolph"},{"given":"Dhruva","family":"Sambrani"}],"title":"Energy, Bosons and Computational Complexity","issued":{"date-parts":[[2025,10,9]]},"note":"arxivid:2510.08545\narxiv_version_number:1"},{"id":"arxiv:2410.04274","type":"document","categories":["continuous-variable quantum computing","infinite-dimensional quantum systems","stellar rank","Hamiltonian complexity","Theory of computation → Quantum information theory","Theory of computation → Quantum complexity theory"],"language":"en","author":[{"family":"Chabaud","given":"Ulysse"},{"family":"Joseph","given":"Michael"},{"family":"Mehraban","given":"Saeed"},{"family":"Motamedi","given":"Arsalan"}],"contributor":[{"family":"Meka","given":"Raghu"}],"issued":{"date-parts":[[2025]]},"abstract":"In recent years, quantum computing involving physical systems with continuous degrees of freedom, such as the bosonic quantum states of light, has attracted significant interest. However, a well-defined quantum complexity theory for these bosonic computations over infinite-dimensional Hilbert spaces is missing. In this work, we lay the foundations for such a research program. We introduce natural complexity classes and problems based on bosonic generalizations of BQP, the local Hamiltonian problem, and QMA. We uncover several relationships and subtle differences between standard Boolean classical and discrete-variable quantum complexity classes, and identify outstanding open problems. Our main contributions include the following:  \r\n1) Bosonic computations. We show that the power of Gaussian computations up to logspace reductions is equivalent to bounded-error quantum logspace (BQL, characterized by the problem of inverting well-conditioned matrices). More generally, we define classes of continuous-variable quantum polynomial time computations with a bounded probability of error (CVBQP) based on gates generated by polynomial bosonic Hamiltonians and particle-number measurements. Due to the infinite-dimensional Hilbert space, it is not a priori clear whether a decidable upper bound can be obtained for these classes. We identify complete problems for these classes, and we demonstrate a BQP lower bound and an EXPSPACE upper bound by proving bounds on the average energy throughout the computation. We further show that the problem of computing expectation values of polynomial bosonic observables at the output of bosonic quantum circuits using Gaussian and cubic phase gates is in PSPACE. \r\n2) Bosonic ground energy problems. We prove that the problem of deciding whether the spectrum of a bosonic Hamiltonian is bounded from below is co-NP-hard. Furthermore, we show that the problem of finding the minimum energy of a bosonic Hamiltonian critically depends on the non-Gaussian stellar rank of the family of energy-constrained states one optimizes over: for zero stellar rank, i.e., optimizing over Gaussian states, it is NP-complete; for polynomially-bounded stellar rank, it is in QMA; for unbounded stellar rank, it is RE-hard, i.e., undecidable.","DOI":"10.4230/LIPICS.ITCS.2025.33","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"Bosonic Quantum Computational Complexity","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.33","note":"copyright:Creative Commons Attribution 4.0 International license\narxivid:2410.04274"},{"id":"arxiv:1907.11009","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevlett.124.063605","source":"Crossref","title":"Stellar Representation of Non-Gaussian Quantum States","volume":"124","author":[{"given":"Ulysse","family":"Chabaud","sequence":"first","affiliation":[]},{"given":"Damian","family":"Markham","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8170-9668","authenticated-orcid":true,"given":"Frédéric","family":"Grosshans","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,2,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.124.063605","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"063605","note":"arxivid:1907.11009"},{"id":"arxiv:2504.19319","type":"article-journal","author":[{"given":"Francesco Anna","family":"Mele"},{"given":"Salvatore Francesco Emanuele","family":"Oliviero"},{"given":"Varun","family":"Upreti"},{"given":"Ulysse","family":"Chabaud"}],"title":"The symplectic rank of non-Gaussian quantum states","issued":{"date-parts":[[2025,4,27]]},"note":"arxivid:2504.19319\narxiv_version_number:1"},{"id":"arxiv:2504.05860","type":"article-journal","author":[{"given":"Andreas Bock","family":"Michelsen"},{"given":"Frederik K.","family":"Marqversen"},{"given":"Michael","family":"Kastoryano"}],"title":"Functional matrix product state simulation of continuous variable quantum circuits","issued":{"date-parts":[[2026,3,25]]},"note":"arxivid:2504.05860\narxiv_version_number:2"},{"id":"arxiv:2503.03600","type":"article-journal","author":[{"given":"Varun","family":"Upreti"},{"given":"Dorian","family":"Rudolph"},{"given":"Ulysse","family":"Chabaud"}],"title":"Bounding the computational power of bosonic systems","issued":{"date-parts":[[2025,9,4]]},"note":"arxivid:2503.03600\narxiv_version_number:3"},{"id":"arxiv:quant-ph/0208022","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.90.117901","source":"Crossref","title":"Quantum Teleportation of Optical Quantum Gates","volume":"90","author":[{"given":"Stephen D.","family":"Bartlett","sequence":"first","affiliation":[]},{"given":"William J.","family":"Munro","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2003,3,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.90.117901","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"117901","note":"arxivid:quant-ph/0208022"},{"id":"arxiv:2507.01146","type":"article-journal","author":[{"given":"Shraddha","family":"Singh"}],"title":"Quantum Computing in Discrete- and Continuous-Variable Architectures","issued":{"date-parts":[[2025,7,1]]},"note":"arxivid:2507.01146\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0505063","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.73.012719","source":"Crossref","title":"Smooth controllability of infinite-dimensional quantum-mechanical systems","volume":"73","author":[{"given":"Re-Bing","family":"Wu","sequence":"first","affiliation":[]},{"given":"Tzyh-Jong","family":"Tarn","sequence":"additional","affiliation":[]},{"given":"Chun-Wen","family":"Li","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,1,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.012719","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012719","note":"arxivid:quant-ph/0505063"},{"id":"arxiv:quant-ph/0511044","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/revmodphys.81.299","page":"299-332","source":"Crossref","title":"Continuous-variable optical quantum-state tomography","volume":"81","author":[{"given":"A. I.","family":"Lvovsky","sequence":"first","affiliation":[]},{"given":"M. G.","family":"Raymer","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,16]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.81.299","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","note":"arxivid:quant-ph/0511044"},{"id":"arxiv:2405.07246","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Continuous-variable (CV) quantum information processing is a promising candidate for large-scale fault-tolerant quantum computation. However, analysis of CV quantum process relies mostly on direct computation of the evolution of operators in the Heisenberg picture, and the features of CV space has yet to be thoroughly investigated in an intuitive manner. One key ingredient for further exploration of CV quantum computing is the construction of a computational model that brings visual intuition and new tools for the analysis of quantum protocols, especially non-Gaussian processes. In this paper, we delve into a graphical computational model, inspired by a similar model for qubit-based systems called the ZX calculus, which enables the representation of arbitrary CV quantum process as a simple directed graph. We demonstrate the utility of our model as a graphical tool to comprehend CV processes intuitively by showing how equivalences between two distinct quantum processes can be proven as a sequence of diagrammatic transformations for various quantum protocols. We also examine possible applications of our model, such as measurement-based quantum computing, characterization of Gaussian and non-Gaussian processes, and circuit optimization.</jats:p>","DOI":"10.1103/physrevresearch.7.033141","source":"Crossref","title":"ZX graphical calculus for continuous-variable quantum processes","volume":"7","author":[{"ORCID":"https://orcid.org/0009-0005-3516-5643","authenticated-orcid":true,"given":"Hironari","family":"Nagayoshi","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]},{"ORCID":"https://orcid.org/0000-0002-3722-3676","authenticated-orcid":true,"given":"Warit","family":"Asavanant","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"},{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0009-0001-8869-5649","authenticated-orcid":true,"given":"Ryuhoh","family":"Ide","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]},{"ORCID":"https://orcid.org/0000-0002-8231-7241","authenticated-orcid":true,"given":"Kosuke","family":"Fukui","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]},{"given":"Atsushi","family":"Sakaguchi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Quantum Computing"}]},{"given":"Jun-ichi","family":"Yoshikawa","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"given":"Akira","family":"Furusawa","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"},{"id":[{"id":"https://ror.org/01sjwvz98","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Quantum Computing"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.7.033141","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033141","note":"arxivid:2405.07246"},{"id":"arxiv:2406.02905","type":"article-journal","author":[{"given":"Razin A.","family":"Shaikh"},{"given":"Lia","family":"Yeh"},{"given":"Stefano","family":"Gogioso"}],"title":"The Focked-up ZX Calculus: Picturing Continuous-Variable Quantum Computation","issued":{"date-parts":[[2024,6,5]]},"note":"arxivid:2406.02905\narxiv_version_number:1"},{"id":"arxiv:2402.17693","type":"article-journal","author":[{"given":"Nicolas","family":"Heurtel"}],"title":"A Complete Graphical Language for Linear Optical Circuits with Finite-Photon-Number Sources and Detectors","issued":{"date-parts":[[2024,11,17]]},"note":"arxivid:2402.17693\narxiv_version_number:2"},{"id":"arxiv:1007.0725","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.83.042335","source":"Crossref","title":"Graphical calculus for Gaussian pure states","volume":"83","author":[{"given":"Nicolas C.","family":"Menicucci","sequence":"first","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,4,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.042335","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042335","note":"arxivid:1007.0725"},{"id":"arxiv:2410.17069","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>Bosonic codes offer a hardware-efficient approach to encoding and protecting quantum information with a single continuous-variable bosonic system. However, previous quantum gates lack analytical methods for decomposing quantum circuits and require complex implementation techniques. In this paper, we introduce a universal quantum gate set composed of only one type of gate element, which we call the quantum lattice gate. We develop a systematic analytical framework for engineering bosonic code states based on Floquet Hamiltonian engineering, where the target Hamiltonian is constructed directly from the given target state(s), and apply our method to single code state preparation, code space embedding, and transformation. We also explore the application of our method to autonomous quantum error correction against single-photon loss with four-legged cat codes. Our proposal is particularly well-suited for superconducting circuit architectures with Josephson junctions, where the full nonlinearity of the Josephson junction potential is harnessed as a quantum resource and the quantum lattice gate can be implemented on a sub-nanosecond timescale.</jats:p>","DOI":"10.1038/s42005-025-02354-0","source":"Crossref","title":"Engineering bosonic codes with quantum lattice gates","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-5558-3106","authenticated-orcid":false,"given":"Lingzhen","family":"Guo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2824-769X","authenticated-orcid":false,"given":"Tangyou","family":"Huang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0641-440X","authenticated-orcid":false,"given":"Lei","family":"Du","sequence":"additional","affiliation":[]}],"container-title":"Communications Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,29]]},"URL":"http://dx.doi.org/10.1038/s42005-025-02354-0","ISSN":["2399-3650"],"container-title-short":"Commun Phys","page":"414","note":"alternative-id:2354\narxivid:2410.17069"},{"id":"arxiv:quant-ph/9812018","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.60.937","page":"937-942","source":"Crossref","title":"Quantum teleportation with squeezed vacuum states","volume":"60","author":[{"given":"G. J.","family":"Milburn","sequence":"first","affiliation":[]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1999,8,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.60.937","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9812018"},{"id":"arxiv:1912.09321","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/revmodphys.92.035005","source":"Crossref","title":"Modes and states in quantum optics","volume":"92","author":[{"given":"C.","family":"Fabre","sequence":"first","affiliation":[]},{"given":"N.","family":"Treps","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,10]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.92.035005","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","page":"035005","note":"arxivid:1912.09321"},{"id":"arxiv:2002.11008","type":"journal-article","publisher":"IOP Publishing","issue":"4","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>We review some of the recent efforts in devising and engineering bosonic qubits for superconducting devices, with emphasis on the Gottesman–Kitaev–Preskill (GKP) qubit. We present some new results on decoding repeated GKP error correction using finitely-squeezed GKP ancilla qubits, exhibiting differences with previously studied stochastic error models. We discuss circuit-QED ways to realize CZ gates between GKP qubits and we discuss different scenarios for using GKP and regular qubits as building blocks in a scalable superconducting surface code architecture.</jats:p>","DOI":"10.1088/2058-9565/ab98a5","page":"043001","source":"Crossref","title":"Towards scalable bosonic quantum error correction","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0003-0218-6614","authenticated-orcid":false,"given":"B M","family":"Terhal","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6120-9930","authenticated-orcid":false,"given":"J","family":"Conrad","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3445-0179","authenticated-orcid":false,"given":"C","family":"Vuillot","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2020,7,9]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ab98a5","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2002.11008"},{"id":"arxiv:2008.13471","type":"journal-article","publisher":"IOP Publishing","issue":"3","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>The unique features of quantum theory offer a powerful new paradigm for information processing. Translating these mathematical abstractions into useful algorithms and applications requires quantum systems with significant complexity and sufficiently low error rates. Such quantum systems must be made from robust hardware that can coherently store, process, and extract the encoded information, as well as possess effective quantum error correction (QEC) protocols to detect and correct errors. Circuit quantum electrodynamics (cQED) provides a promising hardware platform for implementing robust quantum devices. In particular, bosonic encodings in cQED that use multi-photon states of superconducting cavities to encode information have shown success in realizing hardware-efficient QEC. Here, we review recent developments in the theory and implementation of QEC with bosonic codes and report the progress made toward realizing fault-tolerant quantum information processing with cQED devices.</jats:p>","DOI":"10.1088/2058-9565/abe989","page":"033001","source":"Crossref","title":"Quantum information processing with bosonic qubits in circuit QED","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-1445-2086","authenticated-orcid":false,"given":"Atharv","family":"Joshi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":false,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6360-9679","authenticated-orcid":false,"given":"Yvonne Y","family":"Gao","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2021,4,1]]},"URL":"http://dx.doi.org/10.1088/2058-9565/abe989","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2008.13471"},{"id":"arxiv:2010.08699","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.fmre.2020.12.006","page":"50-67","source":"Crossref","title":"Bosonic quantum error correction codes in superconducting quantum circuits","volume":"1","author":[{"given":"Weizhou","family":"Cai","sequence":"first","affiliation":[]},{"given":"Yuwei","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Weiting","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Chang-Ling","family":"Zou","sequence":"additional","affiliation":[]},{"given":"Luyan","family":"Sun","sequence":"additional","affiliation":[]}],"container-title":"Fundamental Research","original-title":[],"language":"en","issued":{"date-parts":[[2021,1]]},"URL":"http://dx.doi.org/10.1016/j.fmre.2020.12.006","ISSN":["2667-3258"],"container-title-short":"Fundamental Research","note":"alternative-id:S2667325820300145\narxivid:2010.08699"},{"id":"arxiv:2103.09445","type":"article-journal","author":[{"given":"Kyungjoo","family":"Noh"}],"title":"Quantum Computation and Communication in Bosonic Systems","issued":{"date-parts":[[2021,3,17]]},"note":"arxivid:2103.09445\narxiv_version_number:1"},{"id":"doi:10.1142/p489","type":"edited-book","publisher":"PUBLISHED BY IMPERIAL COLLEGE PRESS AND DISTRIBUTED BY WORLD SCIENTIFIC PUBLISHING CO.","DOI":"10.1142/p489","source":"Crossref","title":"Quantum Information with Continuous Variables of Atoms and Light","author":[{"given":"N J","family":"Cerf","sequence":"first","affiliation":[{"name":"Université Libre de BruxellesBelgium"}]},{"given":"G","family":"Leuchs","sequence":"additional","affiliation":[{"name":"Universität Erlangen-NürnbergGermany"}]},{"given":"E S","family":"Polzik","sequence":"additional","affiliation":[{"name":"Niels Bohr InstituteDenmark"}]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2007,2]]},"ISBN":["9781860947605","9781860948169"],"URL":"http://dx.doi.org/10.1142/p489","note":"alternative-id:10.1142/p489"},{"id":"arxiv:2311.08445","type":"article-journal","author":[{"given":"Anton Frisk","family":"Kockum"},{"given":"Ariadna","family":"Soro"},{"given":"Laura","family":"García-Álvarez"},{"given":"Pontus","family":"Vikstål"},{"given":"Tom","family":"Douce"},{"given":"Göran","family":"Johansson"},{"given":"Giulia","family":"Ferrini"}],"title":"Lecture notes on quantum computing","issued":{"date-parts":[[2025,4,15]]},"note":"arxivid:2311.08445\narxiv_version_number:3"},{"id":"doi:10.1103/PhysRev.155.1428","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrev.155.1428","page":"1428-1431","source":"Crossref","title":"Charge Superselection Rule","volume":"155","author":[{"given":"Yakir","family":"Aharonov","sequence":"first","affiliation":[]},{"given":"Leonard","family":"Susskind","sequence":"additional","affiliation":[]}],"container-title":"Physical Review","original-title":[],"language":"en","issued":{"date-parts":[[1967,3,25]]},"URL":"http://dx.doi.org/10.1103/PhysRev.155.1428","ISSN":["0031-899X"],"container-title-short":"Phys. Rev."},{"id":"doi:10.1103/PhysRevA.55.3195","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.55.3195","page":"3195-3203","source":"Crossref","title":"Optical coherence: A convenient fiction","volume":"55","author":[{"given":"Klaus","family":"Mølmer","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,4,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.55.3195","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:quant-ph/0306076","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.68.042329","source":"Crossref","title":"Photon-number superselection and the entangled coherent-state representation","volume":"68","author":[{"given":"Barry C.","family":"Sanders","sequence":"first","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Terry","family":"Rudolph","sequence":"additional","affiliation":[]},{"given":"Peter L.","family":"Knight","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,10,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.042329","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042329","note":"arxivid:quant-ph/0306076"},{"id":"arxiv:quant-ph/0507214","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"01","abstract":"<jats:p> A controversy that has arisen many times over in disparate contexts is whether quantum coherences between eigenstates of certain quantities are fact or fiction. We present a pedagogical introduction to the debate in the form of a hypothetical dialogue between proponents from each of the two camps: a factist and a fictionist. A resolution of the debate can be achieved, we argue, by recognizing that quantum states do not only contain information about the intrinsic properties of a system but about its extrinsic properties as well, that is, about its relation to other systems external to it. Specifically, the coherent quantum state of the factist is the appropriate description of the relation of the system to one reference frame, while the incoherent quantum state of the fictionist is the appropriate description of the relation of the system to another, uncorrelated, reference frame. The two views, we conclude, are alternative but equally valid paradigms of description. </jats:p>","DOI":"10.1142/s0219749906001591","page":"17-43","source":"Crossref","title":"DIALOGUE CONCERNING TWO VIEWS ON QUANTUM COHERENCE: FACTIST AND FICTIONIST","volume":"04","author":[{"given":"STEPHEN D.","family":"BARTLETT","sequence":"first","affiliation":[{"name":"School of Physics, The University of Sydney, New South Wales 2006, Australia"}]},{"given":"TERRY","family":"RUDOLPH","sequence":"additional","affiliation":[{"name":"Optics Section, Blackett Laboratory, Imperial College, London, London SW7 2BW, United Kingdom"},{"name":"Institute for Mathematical Sciences, Imperial College London, London SW7 2BW, United Kingdom"}]},{"given":"ROBERT W.","family":"SPEKKENS","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario N2L 2Y5, Canada"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2006,2]]},"URL":"http://dx.doi.org/10.1142/S0219749906001591","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749906001591\narxivid:quant-ph/0507214"},{"id":"arxiv:1112.1778","type":"journal-article","publisher":"IOP Publishing","issue":"24","DOI":"10.1088/1751-8113/45/24/244002","page":"244002","source":"Crossref","title":"Review of entangled coherent states","volume":"45","author":[{"given":"Barry C","family":"Sanders","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2012,5,30]]},"URL":"http://dx.doi.org/10.1088/1751-8113/45/24/244002","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1112.1778"},{"id":"arxiv:2501.03943","type":"journal-article","publisher":"Optica Publishing Group","issue":"2","abstract":"<jats:p>Since the early days of quantum information science, diverse strategies have been proposed to encode qubits and implement quantum gates across various physical platforms. Modes and bosonic states offer a flexible framework for quantum information processing, from single-photon optics to superconducting circuits, combining massless and massive particles. Universal gate sets have been established for specific encodings—such as those based on Fock states or the field’s quadratures. However, a general framework clarifying how different bosonic resources—such as mode structure, particle-number statistics, Gaussian and non-Gaussian states and operations—combine to enable universality and potential quantum advantage irrespectively of the encoding remains lacking. A main difficulty lies in the coexistence of distinct paradigms that appear to combine loosely in bosonic quantum information: discrete-variable schemes—where information is encoded in qubits or effective finite-dimensional systems—and continuous-variable schemes—which rely on infinite-dimensional Hilbert spaces associated with the field’s quadratures. Despite the success of each approach, they remain seemingly incompatible in many aspects—in particular in terms of physical resource analysis for universality—due to their underlying mathematical differences. In this work, we apply a superselection-rule-compliant framework for bosonic quantum information that unifies these paradigms. This representation explicitly incorporates the phase reference and enforces total particle number conservation, allowing for a consistent treatment of bosonic quantum information protocols and resources across both discrete and continuous regimes. We derive necessary and sufficient conditions expressed in terms of Gaussian and non-Gaussian resources for universality that apply to any encoding, without resorting to specific mappings. Our results show that continuous variable encodings are not fundamentally distinct from discrete ones, but rather emerge as subspaces within a general bosonic structure. Importantly, in such a widely studied framework, non-Gaussian properties of bosonic states and operations can be washed out, providing evidence that the continuous variables regime is not the most suitable to fully assess the quantum properties of bosons.</jats:p>","DOI":"10.1364/opticaq.581218","page":"148","source":"Crossref","title":"Unified framework for bosonic quantum information encoding, resources, and universality from superselection rules","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-6911-452X","authenticated-orcid":true,"given":"Eloi","family":"Descamps","sequence":"first","affiliation":[{"name":"Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques"}]},{"given":"Astghik","family":"Saharyan","sequence":"additional","affiliation":[{"name":"Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques"}]},{"given":"Adrien","family":"Chivet","sequence":"additional","affiliation":[{"name":"Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques"},{"name":"Département de Mathématiques et Applications de l’Ecole Normale Supérieure - PSL"}]},{"given":"Arne","family":"Keller","sequence":"additional","affiliation":[{"name":"Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques"},{"name":"Université Paris-Saclay"}]},{"ORCID":"https://orcid.org/0000-0002-7579-7742","authenticated-orcid":true,"given":"Pérola","family":"Milman","sequence":"additional","affiliation":[{"name":"Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques"}]}],"container-title":"Optica Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,1]]},"URL":"http://dx.doi.org/10.1364/OPTICAQ.581218","ISSN":["2837-6714"],"container-title-short":"Optica Quantum","note":"arxivid:2501.03943"},{"id":"arxiv:2507.13245","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/f1b2-c3qs","source":"Crossref","title":"Resources for bosonic metrology: Quantum-enhanced precision from a superselection rule perspective","volume":"113","author":[{"ORCID":"https://orcid.org/0000-0002-1580-4103","authenticated-orcid":true,"given":"Astghik","family":"Saharyan","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05f82e368","id-type":"ROR","asserted-by":"publisher"}],"name":"Université Paris Cité"},{"id":[{"id":"https://ror.org/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/02p3et738","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire Matériaux et Phénomènes Quantiques"}]},{"ORCID":"https://orcid.org/0000-0002-6911-452X","authenticated-orcid":true,"given":"Eloi","family":"Descamps","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05f82e368","id-type":"ROR","asserted-by":"publisher"}],"name":"Université Paris Cité"},{"id":[{"id":"https://ror.org/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/02p3et738","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire Matériaux et Phénomènes Quantiques"}]},{"ORCID":"https://orcid.org/0000-0002-6934-7198","authenticated-orcid":true,"given":"Arne","family":"Keller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05f82e368","id-type":"ROR","asserted-by":"publisher"}],"name":"Université Paris Cité"},{"id":[{"id":"https://ror.org/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/02p3et738","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire Matériaux et Phénomènes Quantiques"},{"id":[{"id":"https://ror.org/03xjwb503","id-type":"ROR","asserted-by":"publisher"}],"name":"Université Paris-Saclay"}]},{"ORCID":"https://orcid.org/0000-0002-7579-7742","authenticated-orcid":true,"given":"Pérola","family":"Milman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05f82e368","id-type":"ROR","asserted-by":"publisher"}],"name":"Université Paris Cité"},{"id":[{"id":"https://ror.org/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/02p3et738","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire Matériaux et Phénomènes Quantiques"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2026,2,18]]},"URL":"http://dx.doi.org/10.1103/f1b2-c3qs","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022428","note":"arxivid:2507.13245"},{"id":"doi:10.1007/BF01330618","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7-8","DOI":"10.1007/bf01330618","page":"531-535","source":"Crossref","title":"Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrk�rperproblem","volume":"94","author":[{"given":"P.","family":"Jordan","sequence":"first","affiliation":[]}],"container-title":"Zeitschrift f�r Physik","original-title":[],"language":"de","issued":{"date-parts":[[1935,7]]},"URL":"http://dx.doi.org/10.1007/BF01330618","ISSN":["1434-6001","1434-601X"],"container-title-short":"Z. Physik","note":"alternative-id:BF01330618"},{"id":"manual:-J.-Schwinger.-On-Angular-Mom","type":"book","author":[{"family":"Schwinger","given":"J."}],"title":"On Angular Momentum","publisher":"Courier Dover Publications","issued":"2015"},{"id":"doi:10.1007/978-3-662-04589-3_4","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-662-04589-3_4","page":"149-181","source":"Crossref","title":"Angular Momentum","author":[{"given":"Julian","family":"Schwinger","sequence":"first","affiliation":[]}],"container-title":"Quantum Mechanics","original-title":[],"language":"en","issued":{"date-parts":[[2001]]},"ISBN":["9783642074677","9783662045893"],"URL":"http://dx.doi.org/10.1007/978-3-662-04589-3_4","note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:quant-ph/9711021","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.80.4088","page":"4088-4091","source":"Crossref","title":"Analog Quantum Error Correction","volume":"80","author":[{"given":"Seth","family":"Lloyd","sequence":"first","affiliation":[]},{"given":"Jean-Jacques E.","family":"Slotine","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1998,5,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.80.4088","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9711021"},{"id":"arxiv:quant-ph/9711049","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.80.4084","page":"4084-4087","source":"Crossref","title":"Error Correction for Continuous Quantum Variables","volume":"80","author":[{"given":"Samuel L.","family":"Braunstein","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1998,5,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.80.4084","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9711049"},{"id":"arxiv:2210.16188","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum computers often manipulate physical qubits encoded on two-level quantum systems. Bosonic qubit codes depart from this idea by encoding information in a well-chosen subspace of an infinite-dimensional Fock space. This larger physical space provides a natural protection against experimental imperfections and allows bosonic codes to circumvent no-go results that apply to states constrained by a 2-dimensional Hilbert space. A bosonic qubit is usually defined in a single bosonic mode but it makes sense to look for multimode versions that could exhibit better performance.In this work, building on the observation that the cat code lives in the span of coherent states indexed by a finite subgroup of the complex numbers, we consider a two-mode generalisation living in the span of 24 coherent states indexed by the binary tetrahedral group <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math> of the quaternions. The resulting <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math>-qutrit naturally inherits the algebraic properties of the group <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math> and appears to be quite robust in the low-loss regime. We initiate its study and identify stabilisers as well as some logical operators for this bosonic code.</jats:p>","DOI":"10.22331/q-2023-06-05-1032","page":"1032","source":"Crossref","title":"The <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>T</mml:mi></mml:math>-qutrit, a two-mode bosonic qutrit","volume":"7","author":[{"given":"Aurélie","family":"Denys","sequence":"first","affiliation":[{"name":"Inria Paris, France"}]},{"ORCID":"https://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"additional","affiliation":[{"name":"Inria Paris, France"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,5]]},"URL":"http://dx.doi.org/10.22331/q-2023-06-05-1032","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2210.16188"},{"id":"doi:10.1016/0031-8914(74)90215-8","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0031-8914(74)90215-8","page":"597-615","source":"Crossref","title":"Even and odd coherent states and excitations of a singular oscillator","volume":"72","author":[{"given":"V.V.","family":"Dodonov","sequence":"first","affiliation":[]},{"given":"I.A.","family":"Malkin","sequence":"additional","affiliation":[]},{"given":"V.I.","family":"Man'ko","sequence":"additional","affiliation":[]}],"container-title":"Physica","original-title":[],"language":"en","issued":{"date-parts":[[1974,3]]},"URL":"http://dx.doi.org/10.1016/0031-8914(74)90215-8","ISSN":["0031-8914"],"container-title-short":"Physica","note":"alternative-id:0031891474902158"},{"id":"doi:10.1007/BF02581033","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1007/bf02581033","page":"477-525","source":"Crossref","title":"Crystallized schrödinger cat states","volume":"16","author":[{"given":"O.","family":"Castaños","sequence":"first","affiliation":[]},{"given":"R.","family":"López-Peña","sequence":"additional","affiliation":[]},{"given":"V. I.","family":"Man’ko","sequence":"additional","affiliation":[]}],"container-title":"Journal of Russian Laser Research","original-title":[],"language":"en","issued":{"date-parts":[[1995,11]]},"URL":"http://dx.doi.org/10.1007/BF02581033","ISSN":["1071-2836","1573-8760"],"container-title-short":"J Russ Laser Res","note":"alternative-id:BF02581033"},{"id":"arxiv:1207.0679","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.111.120501","source":"Crossref","title":"Hardware-Efficient Autonomous Quantum Memory Protection","volume":"111","author":[{"given":"Zaki","family":"Leghtas","sequence":"first","affiliation":[]},{"given":"Gerhard","family":"Kirchmair","sequence":"additional","affiliation":[]},{"given":"Brian","family":"Vlastakis","sequence":"additional","affiliation":[]},{"given":"Robert J.","family":"Schoelkopf","sequence":"additional","affiliation":[]},{"given":"Michel H.","family":"Devoret","sequence":"additional","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2013,9,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.111.120501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120501","note":"arxivid:1207.0679"},{"id":"doi:10.1103/PhysRevLett.60.1836","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.60.1836","page":"1836-1839","source":"Crossref","title":"Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions","volume":"60","author":[{"given":"M.","family":"Wolinsky","sequence":"first","affiliation":[]},{"given":"H. J.","family":"Carmichael","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1988,5,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.60.1836","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"doi:10.1103/PhysRevA.50.4330","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.50.4330","page":"4330-4338","source":"Crossref","title":"Transient macroscopic quantum superposition states in degenerate parametric oscillation: Calculations in the large-quantum-noise limit using the positive<i>P</i>representation","volume":"50","author":[{"given":"L.","family":"Krippner","sequence":"first","affiliation":[]},{"given":"W. J.","family":"Munro","sequence":"additional","affiliation":[]},{"given":"M. D.","family":"Reid","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1994,11,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.50.4330","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"doi:10.1103/PhysRevA.49.490","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.49.490","page":"490-498","source":"Crossref","title":"Generation of mixtures of Schrödinger-cat states from a competitive two-photon process","volume":"49","author":[{"given":"Edwin E.","family":"Hach III","sequence":"first","affiliation":[]},{"given":"Christopher C.","family":"Gerry","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1994,1,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.49.490","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"doi:10.1103/PhysRevA.49.2785","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.49.2785","page":"2785-2799","source":"Crossref","title":"Generation of nonclassical light by dissipative two-photon processes","volume":"49","author":[{"given":"L.","family":"Gilles","sequence":"first","affiliation":[]},{"given":"B. M.","family":"Garraway","sequence":"additional","affiliation":[]},{"given":"P. L.","family":"Knight","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1994,4,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.49.2785","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:1312.2017","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1088/1367-2630/16/4/045014","page":"045014","source":"Crossref","title":"Dynamically protected cat-qubits: a new paradigm for universal quantum computation","volume":"16","author":[{"given":"Mazyar","family":"Mirrahimi","sequence":"first","affiliation":[]},{"given":"Zaki","family":"Leghtas","sequence":"additional","affiliation":[]},{"given":"Victor V","family":"Albert","sequence":"additional","affiliation":[]},{"given":"Steven","family":"Touzard","sequence":"additional","affiliation":[]},{"given":"Robert J","family":"Schoelkopf","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Michel H","family":"Devoret","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2014,4,22]]},"URL":"http://dx.doi.org/10.1088/1367-2630/16/4/045014","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1312.2017"},{"id":"arxiv:1902.02323","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.100.052301","source":"Crossref","title":"Conversion of Gaussian states to non-Gaussian states using photon-number-resolving detectors","volume":"100","author":[{"given":"Daiqin","family":"Su","sequence":"first","affiliation":[]},{"given":"Casey R.","family":"Myers","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3107-6844","authenticated-orcid":true,"given":"Krishna Kumar","family":"Sabapathy","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,11,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.100.052301","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052301","note":"arxivid:1902.02323"},{"id":"arxiv:1503.00194","type":"journal-article","publisher":"American Physical Society (APS)","issue":"14","DOI":"10.1103/physrevlett.116.140502","source":"Crossref","title":"Holonomic Quantum Control with Continuous Variable Systems","volume":"116","author":[{"given":"Victor V.","family":"Albert","sequence":"first","affiliation":[]},{"given":"Chi","family":"Shu","sequence":"additional","affiliation":[]},{"given":"Stefan","family":"Krastanov","sequence":"additional","affiliation":[]},{"given":"Chao","family":"Shen","sequence":"additional","affiliation":[]},{"given":"Ren-Bao","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Zhen-Biao","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Robert J.","family":"Schoelkopf","sequence":"additional","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"given":"Michel H.","family":"Devoret","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,4,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.116.140502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"140502","note":"arxivid:1503.00194"},{"id":"arxiv:2310.20578","type":"article-journal","author":[{"given":"Qian","family":"Xu"},{"given":"Pei","family":"Zeng"},{"given":"Daohong","family":"Xu"},{"given":"Liang","family":"Jiang"}],"title":"Fault-Tolerant Operation of Bosonic Qubits with Discrete-Variable Ancillae","issued":{"date-parts":[[2023,10,31]]},"note":"arxivid:2310.20578\narxiv_version_number:1"},{"id":"arxiv:2411.11313","type":"journal-article","publisher":"IOP Publishing","issue":"3","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Quantum-information processing and computation with bosonic qubits are corruptible by noise channels. Using interferometers and photon-subtraction gadgets (PSGs) accompanied by linear amplification and attenuation, we establish linear-optical methods to mitigate and suppress bosonic noise channels. We first show that by employing amplifying and attenuating PSGs respectively at the input and output of either a thermal or random-displacement channel, probabilistic error cancellation (PEC) can be carried out to mitigate errors in expectation-value estimation. We also derive optimal physical estimators that are properly constrained to improve the sampling accuracy of PEC. Next, we prove that a purely-dephasing channel is <jats:italic>coherently</jats:italic> suppressible using a multimode Mach–Zehnder interferometer and conditional vacuum measurements (vacuum-based Mach–Zehnder scheme or the VMZ scheme). In the limit of infinitely-many ancillas, with nonvanishing success rates, VMZ using either Hadamard or two-design interferometers turns <jats:italic>any</jats:italic> dephasing channel into a phase-space-rotated linear-attenuation channel that can subsequently be inverted with (rotated) linear amplification <jats:italic>without</jats:italic> Kerr nonlinearity. Moreover, for weak central-Gaussian dephasing, the suppression fidelity increases monotonically with the number of ancillas and most optimally with Hadamard interferometers. We demonstrate the performance of these linear-optical mitigation and suppression schemes on common noise channels (and their compositions) and popular bosonic codes. While the theoretical formalism pertains to idling noise channels, we also provide numerical evidence supporting mitigation and suppression capabilities with respect to noise from universal gate operations.</jats:p>","DOI":"10.1088/2058-9565/adc82c","page":"035003","source":"Crossref","title":"Linear-optical protocols for mitigating and suppressing noise in bosonic systems","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0002-1766-6402","authenticated-orcid":true,"given":"Y S","family":"Teo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0576-9145","authenticated-orcid":true,"given":"S U","family":"Shringarpure","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0002-3140-5855","authenticated-orcid":true,"given":"S","family":"Cho","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0348-3397","authenticated-orcid":false,"given":"H","family":"Jeong","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2025,4,22]]},"URL":"http://dx.doi.org/10.1088/2058-9565/adc82c","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2411.11313"},{"id":"arxiv:1311.2534","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7510","DOI":"10.1038/nature13436","page":"444-448","source":"Crossref","title":"Tracking photon jumps with repeated quantum non-demolition parity measurements","volume":"511","author":[{"given":"L.","family":"Sun","sequence":"first","affiliation":[]},{"given":"A.","family":"Petrenko","sequence":"additional","affiliation":[]},{"given":"Z.","family":"Leghtas","sequence":"additional","affiliation":[]},{"given":"B.","family":"Vlastakis","sequence":"additional","affiliation":[]},{"given":"G.","family":"Kirchmair","sequence":"additional","affiliation":[]},{"given":"K. M.","family":"Sliwa","sequence":"additional","affiliation":[]},{"given":"A.","family":"Narla","sequence":"additional","affiliation":[]},{"given":"M.","family":"Hatridge","sequence":"additional","affiliation":[]},{"given":"S.","family":"Shankar","sequence":"additional","affiliation":[]},{"given":"J.","family":"Blumoff","sequence":"additional","affiliation":[]},{"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"given":"M. H.","family":"Devoret","sequence":"additional","affiliation":[]},{"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2014,7]]},"URL":"http://dx.doi.org/10.1038/nature13436","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature13436\narxivid:1311.2534"},{"id":"arxiv:1602.04768","type":"article-journal","author":[{"given":"Nissim","family":"Ofek"},{"given":"Andrei","family":"Petrenko"},{"given":"Reinier","family":"Heeres"},{"given":"Philip","family":"Reinhold"},{"given":"Zaki","family":"Leghtas"},{"given":"Brian","family":"Vlastakis"},{"given":"Yehan","family":"Liu"},{"given":"Luigi","family":"Frunzio"},{"given":"S. M.","family":"Girvin"},{"given":"Liang","family":"Jiang"},{"given":"Mazyar","family":"Mirrahimi"},{"given":"M. H.","family":"Devoret"},{"given":"R. J.","family":"Schoelkopf"}],"title":"Demonstrating Quantum Error Correction that Extends the Lifetime of Quantum Information","issued":{"date-parts":[[2016,2,15]]},"note":"arxivid:1602.04768\narxiv_version_number:1"},{"id":"arxiv:1803.00102","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6399","abstract":"<jats:title>Fault-tolerant quantum coding</jats:title>\n          <jats:p>\n            Noise and imperfections in a quantum system can result in the presence and propagation of errors through the system. A reliable quantum processor will need to be able to correct for these errors and error syndromes. Rosenblum\n            <jats:italic>et al.</jats:italic>\n            used higher quantum states of a superconducting-based quantum circuit to demonstrate a method for the fault-tolerant measurement of an error-correctable logical qubit. Such fault-tolerant measurements will allow more frequent interrogations of the state of the logical qubit, ultimately leading to the implementation of more quantum operations and more complex entangled quantum circuits.\n          </jats:p>\n          <jats:p>\n            <jats:italic>Science</jats:italic>\n            , this issue p.\n            <jats:related-article xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"doi\" issue=\"6399\" page=\"266\" related-article-type=\"in-this-issue\" vol=\"361\" xlink:href=\"10.1126/science.aat3996\">266</jats:related-article>\n          </jats:p>","DOI":"10.1126/science.aat3996","page":"266-270","source":"Crossref","title":"Fault-tolerant detection of a quantum error","volume":"361","author":[{"given":"S.","family":"Rosenblum","sequence":"first","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8141-1842","authenticated-orcid":true,"given":"P.","family":"Reinhold","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."},{"name":"QUANTIC team, INRIA de Paris, 2 Rue Simone Iff, 75012 Paris, France."}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"https://orcid.org/0000-0002-0272-5481","authenticated-orcid":true,"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2469-5068","authenticated-orcid":true,"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2018,7,20]]},"URL":"http://dx.doi.org/10.1126/science.aat3996","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.aat3996\narxivid:1803.00102"},{"id":"arxiv:1904.09474","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.9.041053","source":"Crossref","title":"Repetition Cat Qubits for Fault-Tolerant Quantum Computation","volume":"9","author":[{"given":"Jérémie","family":"Guillaud","sequence":"first","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.041053","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041053","note":"arxivid:1904.09474"},{"id":"arxiv:1905.00450","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"34","abstract":"<jats:p>Noise asymmetry in the quantum operations with cat-qubits is exploited for efficient fault-tolerant quantum error correction.</jats:p>","DOI":"10.1126/sciadv.aay5901","source":"Crossref","title":"Bias-preserving gates with stabilized cat qubits","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-5194-0591","authenticated-orcid":true,"given":"Shruti","family":"Puri","sequence":"first","affiliation":[{"name":"Department of Physics, Yale University, New Haven, CT 06520, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]},{"given":"Lucas","family":"St-Jean","sequence":"additional","affiliation":[{"name":"Institut quantique and D’epartment de Physique, Universit’e de Sherbrooke, 2500 boulevard de l’Universit’e, Sherbrooke, Quebec J1K 2R1, Canada."}]},{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":true,"given":"Jonathan A.","family":"Gross","sequence":"additional","affiliation":[{"name":"Institut quantique and D’epartment de Physique, Universit’e de Sherbrooke, 2500 boulevard de l’Universit’e, Sherbrooke, Quebec J1K 2R1, Canada."}]},{"ORCID":"https://orcid.org/0000-0002-6247-1145","authenticated-orcid":true,"given":"Alexander","family":"Grimm","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."},{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511, USA."}]},{"ORCID":"https://orcid.org/0000-0003-4442-9522","authenticated-orcid":true,"given":"Nicholas E.","family":"Frattini","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."},{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511, USA."}]},{"given":"Pavithran S.","family":"Iyer","sequence":"additional","affiliation":[{"name":"Institute of Quantum Computing, 200 University Of Waterloo, Waterloo, Ontario, Canada."}]},{"ORCID":"https://orcid.org/0000-0002-2027-6448","authenticated-orcid":true,"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[{"name":"Institut quantique and D’epartment de Physique, Universit’e de Sherbrooke, 2500 boulevard de l’Universit’e, Sherbrooke, Quebec J1K 2R1, Canada."}]},{"ORCID":"https://orcid.org/0000-0002-3475-2839","authenticated-orcid":true,"given":"Steven","family":"Touzard","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."},{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511, USA."}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."},{"name":"Department of Applied Physics, Yale University, New Haven, CT 06511, USA."},{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA."}]},{"ORCID":"https://orcid.org/0000-0002-0591-2012","authenticated-orcid":true,"given":"Alexandre","family":"Blais","sequence":"additional","affiliation":[{"name":"Institut quantique and D’epartment de Physique, Universit’e de Sherbrooke, 2500 boulevard de l’Universit’e, Sherbrooke, Quebec J1K 2R1, Canada."},{"name":"Canadian Institute for Advanced Research, Toronto, Ontario, Canada."}]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."},{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia."}]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":true,"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[{"name":"Department of Physics, Yale University, New Haven, CT 06520, USA."},{"name":"Yale Quantum Institute, Yale University, New Haven, CT 06520, USA."}]}],"container-title":"Science Advances","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,21]]},"URL":"http://dx.doi.org/10.1126/sciadv.aay5901","ISSN":["2375-2548"],"container-title-short":"Sci. Adv.","page":"eaay5901","note":"alternative-id:10.1126/sciadv.aay5901\narxivid:1905.00450"},{"id":"arxiv:2009.10756","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.103.042413","source":"Crossref","title":"Error rates and resource overheads of repetition cat qubits","volume":"103","author":[{"given":"Jérémie","family":"Guillaud","sequence":"first","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,4,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.042413","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042413","note":"arxivid:2009.10756"},{"id":"arxiv:2212.11927","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Bosonic cat qubits stabilized by two-photon driven dissipation benefit from exponential suppression of bit-flip errors and an extensive set of gates preserving this protection. These properties make them promising building blocks of a hardware-efficient and fault-tolerant quantum processor. In this paper, we propose a performance optimization of the repetition cat code architecture using fast but noisy CNOT gates for stabilizer measurements. This optimization leads to high thresholds for the physical figure of merit, given as the ratio between intrinsic single-photon loss rate of the bosonic mode and the engineered two-photon loss rate, as well as an improved scaling below threshold of the required overhead, to reach an expected level of logical error rate. Relying on the specific error models for cat qubit operations, this optimization exploits fast parity measurements, using accelerated low-fidelity CNOT gates, combined with fast ancilla parity-check qubits. The significant enhancement in the performance is explained by: 1- the highly asymmetric error model of cat qubit CNOT gates with a major component on control (ancilla) qubits, and 2- the robustness of the repetition cat code error correction performance in presence of the leakage induced by fast operations. In order to demonstrate these performances, we develop a method to sample the repetition code under circuit-level noise that also takes into account cat qubit state leakage.</jats:p>","DOI":"10.22331/q-2023-12-06-1198","page":"1198","source":"Crossref","title":"High-performance repetition cat code using fast noisy operations","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-5229-7155","authenticated-orcid":false,"given":"Francois-Marie Le","family":"Régent","sequence":"first","affiliation":[{"name":"Alice&Bob, 53 boulevard du Général Martial Valin, 75015 Paris"},{"name":"Laboratoire de Physique de l&apos;Ecole Normale Supérieure, Ecole normale supérieure, MINES Paris, Université PSL, Sorbonne Université, CNRS, Inria, 75005 Paris"}]},{"given":"Camille","family":"Berdou","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique de l&apos;Ecole Normale Supérieure, Ecole normale supérieure, MINES Paris, Université PSL, Sorbonne Université, CNRS, Inria, 75005 Paris"}]},{"ORCID":"https://orcid.org/0000-0002-9172-1537","authenticated-orcid":false,"given":"Zaki","family":"Leghtas","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique de l&apos;Ecole Normale Supérieure, Ecole normale supérieure, MINES Paris, Université PSL, Sorbonne Université, CNRS, Inria, 75005 Paris"}]},{"ORCID":"https://orcid.org/0000-0001-6507-9344","authenticated-orcid":false,"given":"Jérémie","family":"Guillaud","sequence":"additional","affiliation":[{"name":"Alice&Bob, 53 boulevard du Général Martial Valin, 75015 Paris"}]},{"ORCID":"https://orcid.org/0000-0001-9471-6031","authenticated-orcid":false,"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique de l&apos;Ecole Normale Supérieure, Ecole normale supérieure, MINES Paris, Université PSL, Sorbonne Université, CNRS, Inria, 75005 Paris"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,6]]},"URL":"http://dx.doi.org/10.22331/q-2023-12-06-1198","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2212.11927"},{"id":"arxiv:2012.04108","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.3.010329","source":"Crossref","title":"Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"given":"Patricio","family":"Arrangoiz-Arriola","sequence":"additional","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0665-7161","authenticated-orcid":true,"given":"Connor T.","family":"Hann","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Iverson","sequence":"additional","affiliation":[]},{"given":"Harald","family":"Putterman","sequence":"additional","affiliation":[]},{"given":"Thomas C.","family":"Bohdanowicz","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"Andrew","family":"Keller","sequence":"additional","affiliation":[]},{"given":"Gil","family":"Refael","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Amir H.","family":"Safavi-Naeini","sequence":"additional","affiliation":[]},{"given":"Oskar","family":"Painter","sequence":"additional","affiliation":[]},{"given":"Fernando G.S.L.","family":"Brandão","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,23]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.010329","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010329","note":"arxivid:2012.04108"},{"id":"arxiv:0707.0327","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.100.030503","source":"Crossref","title":"Fault-Tolerant Linear Optical Quantum Computing with Small-Amplitude Coherent States","volume":"100","author":[{"given":"A. P.","family":"Lund","sequence":"first","affiliation":[]},{"given":"T. C.","family":"Ralph","sequence":"additional","affiliation":[]},{"given":"H. L.","family":"Haselgrove","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,1,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.100.030503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030503","note":"arxivid:0707.0327"},{"id":"arxiv:2508.03796","type":"article-journal","author":[{"given":"Harshvardhan K.","family":"Babla"},{"given":"James D.","family":"Teoh"},{"given":"Jahan","family":"Claes"},{"given":"Daniel K.","family":"Weiss"},{"given":"Shraddha","family":"Singh"},{"given":"Robert J.","family":"Schoelkopf"},{"given":"Shruti","family":"Puri"}],"title":"Fault-tolerant Fusion-based Quantum Computing with the Four-legged Cat Code","issued":{"date-parts":[[2025,8,7]]},"note":"arxivid:2508.03796\narxiv_version_number:2"},{"id":"arxiv:1409.6759","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.90.062344","source":"Crossref","title":"Dissipation-induced continuous quantum error correction for superconducting circuits","volume":"90","author":[{"given":"Joachim","family":"Cohen","sequence":"first","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.062344","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062344","note":"arxivid:1409.6759"},{"id":"arxiv:2205.09767","type":"article-journal","author":[{"given":"Simon","family":"Lieu"},{"given":"Yu-Jie","family":"Liu"},{"given":"Alexey V.","family":"Gorshkov"}],"title":"Candidate for a passively protected quantum memory in two dimensions","issued":{"date-parts":[[2023,3,2]]},"note":"arxivid:2205.09767\narxiv_version_number:3"},{"id":"arxiv:2409.13025","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8052","DOI":"10.1038/s41586-025-08642-7","page":"927-934","source":"Crossref","title":"Hardware-efficient quantum error correction via concatenated bosonic qubits","volume":"638","author":[{"ORCID":"https://orcid.org/0000-0002-5841-181X","authenticated-orcid":false,"given":"Harald","family":"Putterman","sequence":"first","affiliation":[]},{"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0665-7161","authenticated-orcid":false,"given":"Connor T.","family":"Hann","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2369-1580","authenticated-orcid":false,"given":"Gregory S.","family":"MacCabe","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9920-493X","authenticated-orcid":false,"given":"Shahriar","family":"Aghaeimeibodi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9033-3527","authenticated-orcid":false,"given":"Rishi N.","family":"Patel","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1954-3198","authenticated-orcid":false,"given":"Menyoung","family":"Lee","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8610-2176","authenticated-orcid":false,"given":"William M.","family":"Jones","sequence":"additional","affiliation":[]},{"given":"Hesam","family":"Moradinejad","sequence":"additional","affiliation":[]},{"given":"Roberto","family":"Rodriguez","sequence":"additional","affiliation":[]},{"given":"Neha","family":"Mahuli","sequence":"additional","affiliation":[]},{"given":"Jefferson","family":"Rose","sequence":"additional","affiliation":[]},{"given":"John Clai","family":"Owens","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8270-3233","authenticated-orcid":false,"given":"Harry","family":"Levine","sequence":"additional","affiliation":[]},{"given":"Emma","family":"Rosenfeld","sequence":"additional","affiliation":[]},{"given":"Philip","family":"Reinhold","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4242-3015","authenticated-orcid":false,"given":"Lorenzo","family":"Moncelsi","sequence":"additional","affiliation":[]},{"given":"Joshua Ari","family":"Alcid","sequence":"additional","affiliation":[]},{"given":"Nasser","family":"Alidoust","sequence":"additional","affiliation":[]},{"given":"Patricio","family":"Arrangoiz-Arriola","sequence":"additional","affiliation":[]},{"given":"James","family":"Barnett","sequence":"additional","affiliation":[]},{"given":"Przemyslaw","family":"Bienias","sequence":"additional","affiliation":[]},{"given":"Hugh A.","family":"Carson","sequence":"additional","affiliation":[]},{"given":"Cliff","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Li","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Harutiun","family":"Chinkezian","sequence":"additional","affiliation":[]},{"given":"Eric M.","family":"Chisholm","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5646-4272","authenticated-orcid":false,"given":"Ming-Han","family":"Chou","sequence":"additional","affiliation":[]},{"given":"Aashish","family":"Clerk","sequence":"additional","affiliation":[]},{"given":"Andrew","family":"Clifford","sequence":"additional","affiliation":[]},{"given":"R.","family":"Cosmic","sequence":"additional","affiliation":[]},{"given":"Ana Valdes","family":"Curiel","sequence":"additional","affiliation":[]},{"given":"Erik","family":"Davis","sequence":"additional","affiliation":[]},{"given":"Laura","family":"DeLorenzo","sequence":"additional","affiliation":[]},{"given":"J. Mitchell","family":"D’Ewart","sequence":"additional","affiliation":[]},{"given":"Art","family":"Diky","sequence":"additional","affiliation":[]},{"given":"Nathan","family":"D’Souza","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5707-3978","authenticated-orcid":false,"given":"Philipp T.","family":"Dumitrescu","sequence":"additional","affiliation":[]},{"given":"Shmuel","family":"Eisenmann","sequence":"additional","affiliation":[]},{"given":"Essam","family":"Elkhouly","sequence":"additional","affiliation":[]},{"given":"Glen","family":"Evenbly","sequence":"additional","affiliation":[]},{"given":"Michael T.","family":"Fang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6964-5817","authenticated-orcid":false,"given":"Yawen","family":"Fang","sequence":"additional","affiliation":[]},{"given":"Matthew J.","family":"Fling","sequence":"additional","affiliation":[]},{"given":"Warren","family":"Fon","sequence":"additional","affiliation":[]},{"given":"Gabriel","family":"Garcia","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0509-3421","authenticated-orcid":false,"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[]},{"given":"Julia A.","family":"Grant","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2778-9166","authenticated-orcid":false,"given":"Mason J.","family":"Gray","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9188-0562","authenticated-orcid":false,"given":"Sebastian","family":"Grimberg","sequence":"additional","affiliation":[]},{"given":"Arne L.","family":"Grimsmo","sequence":"additional","affiliation":[]},{"given":"Arbel","family":"Haim","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Hand","sequence":"additional","affiliation":[]},{"given":"Yuan","family":"He","sequence":"additional","affiliation":[]},{"given":"Mike","family":"Hernandez","sequence":"additional","affiliation":[]},{"given":"David","family":"Hover","sequence":"additional","affiliation":[]},{"given":"Jimmy S. C.","family":"Hung","sequence":"additional","affiliation":[]},{"given":"Matthew","family":"Hunt","sequence":"additional","affiliation":[]},{"given":"Joe","family":"Iverson","sequence":"additional","affiliation":[]},{"given":"Ignace","family":"Jarrige","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3379-229X","authenticated-orcid":false,"given":"Jean-Christophe","family":"Jaskula","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Mahmoud","family":"Kalaee","sequence":"additional","affiliation":[]},{"given":"Rassul","family":"Karabalin","sequence":"additional","affiliation":[]},{"given":"Peter J.","family":"Karalekas","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3030-1149","authenticated-orcid":false,"given":"Andrew J.","family":"Keller","sequence":"additional","affiliation":[]},{"given":"Amirhossein","family":"Khalajhedayati","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":false,"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Hanho","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Catherine","family":"Leroux","sequence":"additional","affiliation":[]},{"given":"Simon","family":"Lieu","sequence":"additional","affiliation":[]},{"given":"Victor","family":"Ly","sequence":"additional","affiliation":[]},{"given":"Keven Villegas","family":"Madrigal","sequence":"additional","affiliation":[]},{"given":"Guillaume","family":"Marcaud","sequence":"additional","affiliation":[]},{"given":"Gavin","family":"McCabe","sequence":"additional","affiliation":[]},{"given":"Cody","family":"Miles","sequence":"additional","affiliation":[]},{"given":"Ashley","family":"Milsted","sequence":"additional","affiliation":[]},{"given":"Joaquin","family":"Minguzzi","sequence":"additional","affiliation":[]},{"given":"Anurag","family":"Mishra","sequence":"additional","affiliation":[]},{"given":"Biswaroop","family":"Mukherjee","sequence":"additional","affiliation":[]},{"given":"Mahdi","family":"Naghiloo","sequence":"additional","affiliation":[]},{"given":"Eric","family":"Oblepias","sequence":"additional","affiliation":[]},{"given":"Gerson","family":"Ortuno","sequence":"additional","affiliation":[]},{"given":"Jason","family":"Pagdilao","sequence":"additional","affiliation":[]},{"given":"Nicola","family":"Pancotti","sequence":"additional","affiliation":[]},{"given":"Ashley","family":"Panduro","sequence":"additional","affiliation":[]},{"given":"JP","family":"Paquette","sequence":"additional","affiliation":[]},{"given":"Minje","family":"Park","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9958-9975","authenticated-orcid":false,"given":"Gregory A.","family":"Peairs","sequence":"additional","affiliation":[]},{"given":"David","family":"Perello","sequence":"additional","affiliation":[]},{"given":"Eric C.","family":"Peterson","sequence":"additional","affiliation":[]},{"given":"Sophia","family":"Ponte","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3198-800X","authenticated-orcid":false,"given":"Johnson","family":"Qiao","sequence":"additional","affiliation":[]},{"given":"Gil","family":"Refael","sequence":"additional","affiliation":[]},{"given":"Rachel","family":"Resnick","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Retzker","sequence":"additional","affiliation":[]},{"given":"Omar A.","family":"Reyna","sequence":"additional","affiliation":[]},{"given":"Marc","family":"Runyan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9923-3354","authenticated-orcid":false,"given":"Colm A.","family":"Ryan","sequence":"additional","affiliation":[]},{"given":"Abdulrahman","family":"Sahmoud","sequence":"additional","affiliation":[]},{"given":"Ernesto","family":"Sanchez","sequence":"additional","affiliation":[]},{"given":"Rohan","family":"Sanil","sequence":"additional","affiliation":[]},{"given":"Krishanu","family":"Sankar","sequence":"additional","affiliation":[]},{"given":"Yuki","family":"Sato","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Scaffidi","sequence":"additional","affiliation":[]},{"given":"Salome","family":"Siavoshi","sequence":"additional","affiliation":[]},{"given":"Prasahnt","family":"Sivarajah","sequence":"additional","affiliation":[]},{"given":"Trenton","family":"Skogland","sequence":"additional","affiliation":[]},{"given":"Chun-Ju","family":"Su","sequence":"additional","affiliation":[]},{"given":"Loren J.","family":"Swenson","sequence":"additional","affiliation":[]},{"given":"Stephanie M.","family":"Teo","sequence":"additional","affiliation":[]},{"given":"Astrid","family":"Tomada","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8478-4436","authenticated-orcid":false,"given":"Giacomo","family":"Torlai","sequence":"additional","affiliation":[]},{"given":"E. Alex","family":"Wollack","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0555-9263","authenticated-orcid":false,"given":"Yufeng","family":"Ye","sequence":"additional","affiliation":[]},{"given":"Jessica A.","family":"Zerrudo","sequence":"additional","affiliation":[]},{"given":"Kailing","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Fernando G. S. L.","family":"Brandão","sequence":"additional","affiliation":[]},{"given":"Matthew H.","family":"Matheny","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1581-9209","authenticated-orcid":false,"given":"Oskar","family":"Painter","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,26]]},"URL":"http://dx.doi.org/10.1038/s41586-025-08642-7","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:8642\narxivid:2409.13025"},{"id":"arxiv:2112.05099","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevd.106.105007","source":"Crossref","title":"Renormalization group and approximate error correction","volume":"106","author":[{"ORCID":"https://orcid.org/0000-0002-1289-6831","authenticated-orcid":true,"given":"Keiichiro","family":"Furuya","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3446-5933","authenticated-orcid":true,"given":"Nima","family":"Lashkari","sequence":"additional","affiliation":[]},{"given":"Mudassir","family":"Moosa","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2022,11,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.106.105007","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"105007","note":"arxivid:2112.05099"},{"id":"arxiv:2401.17795","type":"journal-article","publisher":"Oxford University Press (OUP)","issue":"8","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>We demonstrate that quantum error correction is realized by the renormalization group in scalar field theories. We construct q-level states by using coherent states in the IR region. By acting on them the inverse of the unitary operator U that describes the renormalization group flow of the ground state, we encode them into states in the UV region. We find the situations in which the Knill–Laflamme condition is satisfied for operators that create coherent states. We verify this to the first order in the perturbation theory. This result suggests a general relationship between the renormalization group and quantum error correction and should give insights into understanding the role played by them in the gauge/gravity correspondence.</jats:p>","DOI":"10.1093/ptep/ptae102","source":"Crossref","title":"Quantum Error Correction Realized by the Renormalization Group in Scalar Field Theories","volume":"2024","author":[{"ORCID":"https://orcid.org/0000-0002-7304-9564","authenticated-orcid":false,"given":"Takaaki","family":"Kuwahara","sequence":"first","affiliation":[{"name":"Department of Physics, Kyoto University , Sakyo-ku, Kyoto 606-8502 , Japan"}]},{"ORCID":"https://orcid.org/0009-0007-8924-2573","authenticated-orcid":false,"given":"Ryota","family":"Nasu","sequence":"additional","affiliation":[{"name":"Graduate School of Science and Technology, Shizuoka University , 836 Ohya, Suruga-ku, Shizuoka 422-8529 , Japan"}]},{"ORCID":"https://orcid.org/0000-0003-1750-6625","authenticated-orcid":false,"given":"Gota","family":"Tanaka","sequence":"additional","affiliation":[{"name":"Institute for Mathematical Informatics, Meiji Gakuin University , 1518 Kamikuratacho, Totsuka-ku, Yokohama 244-8539 , Japan"}]},{"ORCID":"https://orcid.org/0000-0002-2971-1135","authenticated-orcid":false,"given":"Asato","family":"Tsuchiya","sequence":"additional","affiliation":[{"name":"Graduate School of Science and Technology, Shizuoka University , 836 Ohya, Suruga-ku, Shizuoka 422-8529 , Japan"},{"name":"Department of Physics, Shizuoka University , 836 Ohya, Suruga-ku, Shizuoka 422-8529 , Japan"}]}],"container-title":"Progress of Theoretical and Experimental Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,12]]},"URL":"http://dx.doi.org/10.1093/ptep/ptae102","ISSN":["2050-3911"],"page":"083B01","note":"arxivid:2401.17795"},{"id":"arxiv:2201.02570","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.106.022431","source":"Crossref","title":"Quantum error correction using squeezed Schrödinger cat states","volume":"106","author":[{"ORCID":"https://orcid.org/0000-0003-2013-1676","authenticated-orcid":true,"given":"David S.","family":"Schlegel","sequence":"first","affiliation":[]},{"given":"Fabrizio","family":"Minganti","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8984-6584","authenticated-orcid":true,"given":"Vincenzo","family":"Savona","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.106.022431","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022431","note":"arxivid:2201.02570"},{"id":"arxiv:2210.13406","type":"article-journal","author":[{"given":"Qian","family":"Xu"},{"given":"Guo","family":"Zheng"},{"given":"Yu-Xin","family":"Wang"},{"given":"Peter","family":"Zoller"},{"given":"Aashish A.","family":"Clerk"},{"given":"Liang","family":"Jiang"}],"title":"Autonomous quantum error correction and fault-tolerant quantum computation with squeezed cat qubits","issued":{"date-parts":[[2022,10,24]]},"note":"arxivid:2210.13406\narxiv_version_number:1"},{"id":"arxiv:2210.13359","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.107.032423","source":"Crossref","title":"Quantum error correction with dissipatively stabilized squeezed-cat qubits","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0002-1476-0647","authenticated-orcid":true,"given":"Timo","family":"Hillmann","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2355-0449","authenticated-orcid":true,"given":"Fernando","family":"Quijandría","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.032423","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032423","note":"arxivid:2210.13359"},{"id":"arxiv:2407.18087","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>We introduce a novel reservoir engineering approach for stabilizing multicomponent Schrödinger’s cat manifolds. The fundamental principle of the method lies in the destructive interference at crossings of gain and loss Hamiltonian terms in the coupling of an oscillator to a zero-temperature auxiliary system, which are nonlinear with respect to the oscillator’s energy. The nature of these gain and loss terms is found to determine the rotational symmetry, energy distributions, and degeneracy of the resulting stabilized manifolds. Considering these systems as bosonic error-correction codes, we analyze their properties with respect to a variety of errors, including both autonomous and passive error correction, where we find that our formalism gives straightforward insights into the nature of the correction. We give example implementations using the anharmonic laser-ion coupling of a trapped ion outside the Lamb-Dicke regime as well as nonlinear superconducting circuits. Beyond the dissipative stabilization of standard cat manifolds and novel rotation symmetric codes, we demonstrate that our formalism allows for the stabilization of bosonic codes linked to cat states through unitary transformations, such as quadrature-squeezed cats. Our work establishes a design approach for creating and utilizing codes using nonlinearity, providing access to novel quantum states and processes across a range of physical systems.</jats:p>","DOI":"10.1103/d557-9lr4","source":"Crossref","title":"Stabilization of Cat-State Manifolds Using Nonlinear Reservoir Engineering","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0001-7164-0265","authenticated-orcid":true,"given":"Ivan","family":"Rojkov","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"},{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"}]},{"ORCID":"https://orcid.org/0009-0005-2528-7866","authenticated-orcid":true,"given":"Matteo","family":"Simoni","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"},{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"}]},{"ORCID":"https://orcid.org/0000-0002-6421-6308","authenticated-orcid":true,"given":"Elias","family":"Zapusek","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"},{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"}]},{"ORCID":"https://orcid.org/0000-0002-5217-3064","authenticated-orcid":true,"given":"Florentin","family":"Reiter","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"},{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"},{"id":[{"id":"https://ror.org/0083ncs46","id-type":"ROR","asserted-by":"publisher"}],"name":"Fraunhofer Institute for Applied Solid State Physics IAF"}]},{"ORCID":"https://orcid.org/0000-0002-4093-1550","authenticated-orcid":true,"given":"Jonathan","family":"Home","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"},{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,13]]},"URL":"http://dx.doi.org/10.1103/d557-9lr4","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011056","note":"arxivid:2407.18087"},{"id":"arxiv:1707.06244","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.120.073603","source":"Crossref","title":"Slowing Quantum Decoherence by Squeezing in Phase Space","volume":"120","author":[{"given":"H.","family":"Le Jeannic","sequence":"first","affiliation":[]},{"given":"A.","family":"Cavaillès","sequence":"additional","affiliation":[]},{"given":"K.","family":"Huang","sequence":"additional","affiliation":[]},{"given":"R.","family":"Filip","sequence":"additional","affiliation":[]},{"given":"J.","family":"Laurat","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2018,2,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.120.073603","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"073603","note":"arxivid:1707.06244"},{"id":"arxiv:2212.01271","type":"article-journal","author":[{"given":"Xiaozhou","family":"Pan"},{"given":"Jonathan","family":"Schwinger"},{"given":"Ni-Ni","family":"Huang"},{"given":"Pengtao","family":"Song"},{"given":"Weipin","family":"Chua"},{"given":"Fumiya","family":"Hanamura"},{"given":"Atharv","family":"Joshi"},{"given":"Fernando","family":"Valadares"},{"given":"Radim","family":"Filip"},{"given":"Yvonne Y.","family":"Gao"}],"title":"Protecting the quantum interference of cat states by phase-space compression","issued":{"date-parts":[[2022,12,2]]},"note":"arxivid:2212.01271\narxiv_version_number:1"},{"id":"arxiv:2502.07892","type":"article-journal","author":[{"given":"Rémi","family":"Rousseau"},{"given":"Diego","family":"Ruiz"},{"given":"Emanuele","family":"Albertinale"},{"given":"Pol","family":"d'Avezac"},{"given":"Danielius","family":"Banys"},{"given":"Ugo","family":"Blandin"},{"given":"Nicolas","family":"Bourdaud"},{"given":"Giulio","family":"Campanaro"},{"given":"Gil","family":"Cardoso"},{"given":"Nathanael","family":"Cottet"},{"given":"Charlotte","family":"Cullip"},{"given":"Samuel","family":"Deléglise"},{"given":"Louise","family":"Devanz"},{"given":"Adam","family":"Devulder"},{"given":"Antoine","family":"Essig"},{"given":"Pierre","family":"Février"},{"given":"Adrien","family":"Gicquel"},{"given":"Élie","family":"Gouzien"},{"given":"Antoine","family":"Gras"},{"given":"Jérémie","family":"Guillaud"},{"given":"Efe","family":"Gümüş"},{"given":"Mattis","family":"Hallén"},{"given":"Anissa","family":"Jacob"},{"given":"Paul","family":"Magnard"},{"given":"Antoine","family":"Marquet"},{"given":"Salim","family":"Miklass"},{"given":"Théau","family":"Peronnin"},{"given":"Stéphane","family":"Polis"},{"given":"Felix","family":"Rautschke"},{"given":"Ulysse","family":"Réglade"},{"given":"Julien","family":"Roul"},{"given":"Jeremy","family":"Stevens"},{"given":"Jeanne","family":"Solard"},{"given":"Alexandre","family":"Thomas"},{"given":"Jean-Loup","family":"Ville"},{"given":"Pierre","family":"Wan-Fat"},{"given":"Raphaël","family":"Lescanne"},{"given":"Zaki","family":"Leghtas"},{"given":"Joachim","family":"Cohen"},{"given":"Sébastien","family":"Jezouin"},{"given":"Anil","family":"Murani"}],"title":"Enhancing dissipative cat qubit protection by squeezing","issued":{"date-parts":[[2025,2,28]]},"note":"arxivid:2502.07892\narxiv_version_number:2"},{"id":"arxiv:2004.09322","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7845","DOI":"10.1038/s41586-021-03257-0","page":"243-248","source":"Crossref","title":"Protecting a bosonic qubit with autonomous quantum error correction","volume":"590","author":[{"given":"Jeffrey M.","family":"Gertler","sequence":"first","affiliation":[]},{"given":"Brian","family":"Baker","sequence":"additional","affiliation":[]},{"given":"Juliang","family":"Li","sequence":"additional","affiliation":[]},{"given":"Shruti","family":"Shirol","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5047-631X","authenticated-orcid":false,"given":"Jens","family":"Koch","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4488-7056","authenticated-orcid":false,"given":"Chen","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,10]]},"URL":"http://dx.doi.org/10.1038/s41586-021-03257-0","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:3257\narxivid:2004.09322"},{"id":"arxiv:quant-ph/9809037","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.59.2631","page":"2631-2634","source":"Crossref","title":"Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping","volume":"59","author":[{"given":"P. T.","family":"Cochrane","sequence":"first","affiliation":[]},{"given":"G. J.","family":"Milburn","sequence":"additional","affiliation":[]},{"given":"W. J.","family":"Munro","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1999,4,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.59.2631","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9809037"},{"id":"arxiv:1901.05358","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.103.062427","source":"Crossref","title":"Phase-engineered bosonic quantum codes","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0001-8796-2453","authenticated-orcid":true,"given":"Linshu","family":"Li","sequence":"first","affiliation":[{"name":"Yale Quantum Institute, Department of Applied Physics and Department of Physics, Yale University, New Haven, Connecticut 06511, USA"}]},{"given":"Dylan J.","family":"Young","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Department of Applied Physics and Department of Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"JILA, NIST, and Department of Physics, University of Colorado, 440 UCB, Boulder, Colorado 80309, USA"}]},{"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Department of Applied Physics and Department of Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Department of Applied Physics and Department of Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"}]},{"given":"Chang-Ling","family":"Zou","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Department of Applied Physics and Department of Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui 230026, China"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Yale Quantum Institute, Department of Applied Physics and Department of Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Pritzker School of Molecular Engineering, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,6,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.062427","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062427","note":"arxivid:1901.05358"},{"id":"arxiv:2211.03689","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.107.042407","source":"Crossref","title":"Two-photon driven Kerr quantum oscillator with multiple spectral degeneracies","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0003-2003-7030","authenticated-orcid":true,"given":"Diego","family":"Ruiz","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6136-6688","authenticated-orcid":true,"given":"Ronan","family":"Gautier","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6507-9344","authenticated-orcid":true,"given":"Jérémie","family":"Guillaud","sequence":"additional","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,4,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.042407","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042407","note":"arxivid:2211.03689"},{"id":"arxiv:2503.11624","type":"article-journal","author":[{"given":"Adrià","family":"Labay-Mora"},{"given":"Alberto","family":"Mercurio"},{"given":"Vincenzo","family":"Savona"},{"given":"Gian Luca","family":"Giorgi"},{"given":"Fabrizio","family":"Minganti"}],"title":"Chiral cat code: Enhanced error correction induced by higher-order nonlinearities","issued":{"date-parts":[[2025,10,19]]},"note":"arxivid:2503.11624\narxiv_version_number:2"},{"id":"doi:10.1103/PhysRevLett.77.4728","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.77.4728","page":"4728-4731","source":"Crossref","title":"Quantum Reservoir Engineering with Laser Cooled Trapped Ions","volume":"77","author":[{"given":"J. F.","family":"Poyatos","sequence":"first","affiliation":[]},{"given":"J. I.","family":"Cirac","sequence":"additional","affiliation":[]},{"given":"P.","family":"Zoller","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1996,12,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.77.4728","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett."},{"id":"arxiv:2407.17299","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.111.012617","source":"Crossref","title":"Bit-flip errors in dissipative cat qubits: Second-order perturbation theory","volume":"111","author":[{"ORCID":"https://orcid.org/0009-0008-9377-0894","authenticated-orcid":true,"given":"Kirill S.","family":"Dubovitskii","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02rx3b187","id-type":"ROR","asserted-by":"publisher"}],"name":"Université Grenoble Alpes"},{"id":[{"id":"https://ror.org/04px4e658","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"},{"id":[{"id":"https://ror.org/02mc6qk71","id-type":"ROR","asserted-by":"publisher"}],"name":"LPMMC"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.111.012617","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012617","note":"arxivid:2407.17299"},{"id":"doi:10.1088/0954-8998/3/6/005","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/0954-8998/3/6/005","page":"359-383","source":"Crossref","title":"Displaced Fock states and their connection to quasiprobabilities","volume":"3","author":[{"given":"A","family":"Wunsche","sequence":"first","affiliation":[]}],"container-title":"Quantum Optics: Journal of the European Optical Society Part B","original-title":[],"issued":{"date-parts":[[1991,12]]},"URL":"http://dx.doi.org/10.1088/0954-8998/3/6/005","ISSN":["0954-8998"],"container-title-short":"Quantum Opt."},{"id":"arxiv:1311.1920","type":"journal-article","publisher":"American Association of Physics Teachers (AAPT)","issue":"8","abstract":"<jats:p>In the coherent state of the harmonic oscillator, the probability density is that of the ground state subjected to an oscillation along a classical trajectory. Senitzky and others pointed out that there are states of the harmonic oscillator corresponding to an identical oscillatory displacement of the probability density of any energy eigenstate. These generalizations of the coherent state are rarely discussed, yet they furnish an interesting set of quantum states of light that combine features of number states and coherent states. Here, we give an elementary account of the quantum optics of generalized coherent states.</jats:p>","DOI":"10.1119/1.4876963","page":"742-748","source":"Crossref","title":"Generalized coherent states","volume":"82","author":[{"given":"T. G.","family":"Philbin","sequence":"first","affiliation":[{"name":"University of Exeter Physics and Astronomy Department, , Stocker Road, Exeter EX4 4QL, United Kingdom"}]}],"container-title":"American Journal of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,8,1]]},"URL":"http://dx.doi.org/10.1119/1.4876963","ISSN":["0002-9505","1943-2909"],"note":"arxivid:1311.1920"},{"id":"arxiv:2507.18714","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/q981-pd5j","source":"Crossref","title":"Nonperturbative Switching Rates in Bistable Open Quantum Systems: From Driven Kerr Oscillators to Dissipative Cat Qubits","volume":"136","author":[{"ORCID":"https://orcid.org/0009-0000-3288-7994","authenticated-orcid":true,"given":"Léon","family":"Carde","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05a0dhs15","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire de Physique de l’École Normale Supérieure"},{"id":[{"id":"https://ror.org/05qrz8r95","id-type":"ROR","asserted-by":"publisher"}],"name":"Alice and Bob"}]},{"ORCID":"https://orcid.org/0000-0001-6136-6688","authenticated-orcid":true,"given":"Ronan","family":"Gautier","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qrz8r95","id-type":"ROR","asserted-by":"publisher"}],"name":"Alice and Bob"}]},{"given":"Nicolas","family":"Didier","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qrz8r95","id-type":"ROR","asserted-by":"publisher"}],"name":"Alice and Bob"}]},{"ORCID":"https://orcid.org/0000-0003-1770-5181","authenticated-orcid":true,"given":"Alexandru","family":"Petrescu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05a0dhs15","id-type":"ROR","asserted-by":"publisher"}],"name":"Laboratoire de Physique de l’École Normale Supérieure"}]},{"ORCID":"https://orcid.org/0009-0005-1014-5259","authenticated-orcid":true,"given":"Joachim","family":"Cohen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qrz8r95","id-type":"ROR","asserted-by":"publisher"}],"name":"Alice and Bob"}]},{"ORCID":"https://orcid.org/0000-0002-7257-5853","authenticated-orcid":true,"given":"Alexander","family":"McDonald","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00kybxq39","id-type":"ROR","asserted-by":"publisher"}],"name":"Université de Sherbrooke"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,9]]},"URL":"http://dx.doi.org/10.1103/q981-pd5j","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"100402","note":"arxivid:2507.18714"},{"id":"arxiv:1605.09408","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Photonic cat states stored in high-Q resonators show great promise for hardware efficient universal quantum computing. We propose an approach to efficiently prepare such cat states in a Kerr-nonlinear resonator by the use of a two-photon drive. Significantly, we show that this preparation is robust against single-photon loss. An outcome of this observation is that a two-photon drive can eliminate undesirable phase evolution induced by a Kerr nonlinearity. By exploiting the concept of transitionless quantum driving, we moreover demonstrate how non-adiabatic initialization of cat states is possible. Finally, we present a universal set of quantum logical gates that can be performed on the engineered eigenspace of such a two-photon driven resonator and discuss a possible realization using superconducting circuits. The robustness of the engineered subspace to higher-order circuit nonlinearities makes this implementation favorable for scalable quantum computation.</jats:p>","DOI":"10.1038/s41534-017-0019-1","source":"Crossref","title":"Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving","volume":"3","author":[{"given":"Shruti","family":"Puri","sequence":"first","affiliation":[]},{"given":"Samuel","family":"Boutin","sequence":"additional","affiliation":[]},{"given":"Alexandre","family":"Blais","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2017,4,19]]},"URL":"http://dx.doi.org/10.1038/s41534-017-0019-1","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"18","note":"alternative-id:19\narxivid:1605.09408"},{"id":"arxiv:1510.02566","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The dynamics of nonlinear systems qualitatively change depending on their parameters, which is called bifurcation. A quantum-mechanical nonlinear oscillator can yield a quantum superposition of two oscillation states, known as a Schrödinger cat state, via quantum adiabatic evolution through its bifurcation point. Here we propose a quantum computer comprising such quantum nonlinear oscillators, instead of quantum bits, to solve hard combinatorial optimization problems. The nonlinear oscillator network finds optimal solutions via quantum adiabatic evolution, where nonlinear terms are increased slowly, in contrast to conventional adiabatic quantum computation or quantum annealing, where quantum fluctuation terms are decreased slowly. As a result of numerical simulations, it is concluded that quantum superposition and quantum fluctuation work effectively to find optimal solutions. It is also notable that the present computer is analogous to neural computers, which are also networks of nonlinear components. Thus, the present scheme will open new possibilities for quantum computation, nonlinear science and artificial intelligence.</jats:p>","DOI":"10.1038/srep21686","source":"Crossref","title":"Bifurcation-based adiabatic quantum computation with a nonlinear oscillator network","volume":"6","author":[{"given":"Hayato","family":"Goto","sequence":"first","affiliation":[]}],"container-title":"Scientific Reports","original-title":[],"language":"en","issued":{"date-parts":[[2016,2,22]]},"URL":"http://dx.doi.org/10.1038/srep21686","ISSN":["2045-2322"],"container-title-short":"Sci Rep","page":"21686","note":"alternative-id:BFsrep21686\narxivid:1510.02566"},{"id":"arxiv:2107.09198","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.128.110502","source":"Crossref","title":"Stabilizing a Bosonic Qubit Using Colored Dissipation","volume":"128","author":[{"ORCID":"https://orcid.org/0000-0002-5841-181X","authenticated-orcid":true,"given":"Harald","family":"Putterman","sequence":"first","affiliation":[]},{"given":"Joseph","family":"Iverson","sequence":"additional","affiliation":[]},{"given":"Qian","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Oskar","family":"Painter","sequence":"additional","affiliation":[]},{"given":"Fernando G. S. L.","family":"Brandão","sequence":"additional","affiliation":[]},{"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,3,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.128.110502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110502","note":"arxivid:2107.09198"},{"id":"arxiv:2112.05545","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.3.020339","source":"Crossref","title":"Combined Dissipative and Hamiltonian Confinement of Cat Qubits","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-6136-6688","authenticated-orcid":true,"given":"Ronan","family":"Gautier","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5909-437X","authenticated-orcid":true,"given":"Alain","family":"Sarlette","sequence":"additional","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,20]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.020339","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020339","note":"arxivid:2112.05545"},{"id":"arxiv:2208.04928","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.4.020337","source":"Crossref","title":"Critical Schrödinger Cat Qubit","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0003-4497-1813","authenticated-orcid":true,"given":"Luca","family":"Gravina","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4850-1130","authenticated-orcid":true,"given":"Fabrizio","family":"Minganti","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8984-6584","authenticated-orcid":true,"given":"Vincenzo","family":"Savona","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,7]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.020337","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020337","note":"arxivid:2208.04928"},{"id":"arxiv:quant-ph/0306004","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.68.042319","source":"Crossref","title":"Quantum computation with optical coherent states","volume":"68","author":[{"given":"T. C.","family":"Ralph","sequence":"first","affiliation":[]},{"given":"A.","family":"Gilchrist","sequence":"additional","affiliation":[]},{"given":"G. J.","family":"Milburn","sequence":"additional","affiliation":[]},{"given":"W. J.","family":"Munro","sequence":"additional","affiliation":[]},{"given":"S.","family":"Glancy","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,10,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.042319","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042319","note":"arxivid:quant-ph/0306004"},{"id":"arxiv:quant-ph/0109077","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.65.042305","source":"Crossref","title":"Efficient quantum computation using coherent states","volume":"65","author":[{"given":"H.","family":"Jeong","sequence":"first","affiliation":[]},{"given":"M. S.","family":"Kim","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,3,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.042305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042305","note":"arxivid:quant-ph/0109077"},{"id":"doi:10.1103/PhysRevLett.57.13","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.57.13","page":"13-16","source":"Crossref","title":"Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion","volume":"57","author":[{"given":"B.","family":"Yurke","sequence":"first","affiliation":[]},{"given":"D.","family":"Stoler","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1986,7,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.57.13","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"arxiv:2410.23363","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Dissipative cat qubits are a promising physical platform for quantum computing, since their large noise bias can enable more hardware-efficient quantum error correction. In this work we theoretically study the long-term prospects of a hybrid cat-transmon quantum computing architecture where dissipative cat qubits play the role of data qubits, and error syndromes are measured using ancillary transmon qubits. The cat qubits’ noise bias enables more hardware-efficient quantum error correction, and the use of transmons allows for practical, high-fidelity syndrome measurement. While correction of the dominant cat <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>Z</a:mi></a:math> errors with a repetition code has recently been demonstrated in experiment, here we show how the architecture can be scaled beyond a repetition code. In particular, we propose a cat-transmon entangling gate that enables the correction of residual cat <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>X</c:mi></c:math> errors in a thin rectangular surface code, so that logical error can be arbitrarily suppressed by increasing code distance. We numerically estimate logical memory performance, finding significant overhead reductions in comparison to architectures without biased noise. For example, with current state-of-the-art coherence, physical error rates of <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:msup><e:mn>10</e:mn><e:mrow><e:mo>−</e:mo><e:mn>3</e:mn></e:mrow></e:msup></e:math> and noise biases in the range <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:msup><g:mn>10</g:mn><g:mn>3</g:mn></g:msup></g:math>–<i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:msup><i:mn>10</i:mn><i:mn>4</i:mn></i:msup></i:math> are achievable. With this level of performance, the qubit overhead required to reach algorithmically relevant logical error rates with the cat-transmon architecture matches that of an unbiased-noise architecture with physical error rates in the range <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:msup><k:mn>10</k:mn><k:mrow><k:mo>−</k:mo><k:mn>5</k:mn></k:mrow></k:msup></k:math>–<m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:msup><m:mn>10</m:mn><m:mrow><m:mo>−</m:mo><m:mn>4</m:mn></m:mrow></m:msup></m:math>.</jats:p>","DOI":"10.1103/75x7-5ysv","source":"Crossref","title":"Hybrid Cat-Transmon Architecture for Scalable, Hardware-Efficient Quantum Error Correction","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-0665-7161","authenticated-orcid":true,"given":"Connor T.","family":"Hann","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-5841-181X","authenticated-orcid":true,"given":"Harald","family":"Putterman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-3488-1083","authenticated-orcid":true,"given":"Matthew H.","family":"Matheny","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Joseph K.","family":"Iverson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0003-2321-1321","authenticated-orcid":true,"given":"Michael T.","family":"Fang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-1581-9209","authenticated-orcid":true,"given":"Oskar","family":"Painter","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"given":"Fernando G.S.L.","family":"Brandão","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,11]]},"URL":"http://dx.doi.org/10.1103/75x7-5ysv","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030305","note":"arxivid:2410.23363"},{"id":"arxiv:2108.12225","type":"article-journal","author":[{"given":"Jacob","family":"Hastrup"},{"given":"Ulrik Lund","family":"Andersen"}],"title":"All-optical cat-code quantum error correction","issued":{"date-parts":[[2021,8,27]]},"note":"arxivid:2108.12225\narxiv_version_number:1"},{"id":"arxiv:1412.4633","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6224","abstract":"<jats:title>A way to induce quantum stability</jats:title>\n          <jats:p>\n            Dynamical systems, whether classical or quantum, usually require a method to stabilize performance and maintain the required state. For instance, communication between computers requires error correction codes to ensure that information is transferred correctly. In a quantum system, however, the very act of measuring it can perturb it. Leghtas\n            <jats:italic>et al.</jats:italic>\n            show that engineering the interaction between a quantum system and its environment can induce stability for the delicate quantum states, a process that could simplify quantum information processing.\n          </jats:p>\n          <jats:p>\n            <jats:italic>Science</jats:italic>\n            , this issue p.\n            <jats:related-article xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"doi\" issue=\"6224\" page=\"853\" related-article-type=\"in-this-issue\" vol=\"347\" xlink:href=\"10.1126/science.aaa2085\">853</jats:related-article>\n          </jats:p>","DOI":"10.1126/science.aaa2085","page":"853-857","source":"Crossref","title":"Confining the state of light to a quantum manifold by engineered two-photon loss","volume":"347","author":[{"given":"Z.","family":"Leghtas","sequence":"first","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"S.","family":"Touzard","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"I. M.","family":"Pop","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"A.","family":"Kou","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"B.","family":"Vlastakis","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"A.","family":"Petrenko","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"K. M.","family":"Sliwa","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"A.","family":"Narla","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"S.","family":"Shankar","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"M. J.","family":"Hatridge","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"M.","family":"Reagor","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."},{"name":"Institut National de Recherche en Informatique et en Automatique (INRIA) Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France."}]},{"given":"M. H.","family":"Devoret","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, CT 06520, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2015,2,20]]},"URL":"http://dx.doi.org/10.1126/science.aaa2085","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.aaa2085\narxivid:1412.4633"},{"id":"arxiv:1705.02401","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.8.021005","source":"Crossref","title":"Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation","volume":"8","author":[{"given":"S.","family":"Touzard","sequence":"first","affiliation":[]},{"given":"A.","family":"Grimm","sequence":"additional","affiliation":[]},{"given":"Z.","family":"Leghtas","sequence":"additional","affiliation":[]},{"given":"S. O.","family":"Mundhada","sequence":"additional","affiliation":[]},{"given":"P.","family":"Reinhold","sequence":"additional","affiliation":[]},{"given":"C.","family":"Axline","sequence":"additional","affiliation":[]},{"given":"M.","family":"Reagor","sequence":"additional","affiliation":[]},{"given":"K.","family":"Chou","sequence":"additional","affiliation":[]},{"given":"J.","family":"Blumoff","sequence":"additional","affiliation":[]},{"given":"K. M.","family":"Sliwa","sequence":"additional","affiliation":[]},{"given":"S.","family":"Shankar","sequence":"additional","affiliation":[]},{"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[]},{"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"given":"M. H.","family":"Devoret","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2018,4,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.8.021005","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021005","note":"arxivid:1705.02401"},{"id":"arxiv:1907.12131","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7820","DOI":"10.1038/s41586-020-2587-z","page":"205-209","source":"Crossref","title":"Stabilization and operation of a Kerr-cat qubit","volume":"584","author":[{"ORCID":"https://orcid.org/0000-0002-6247-1145","authenticated-orcid":false,"given":"A.","family":"Grimm","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4442-9522","authenticated-orcid":false,"given":"N. E.","family":"Frattini","sequence":"additional","affiliation":[]},{"given":"S.","family":"Puri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5952-0743","authenticated-orcid":false,"given":"S. O.","family":"Mundhada","sequence":"additional","affiliation":[]},{"given":"S.","family":"Touzard","sequence":"additional","affiliation":[]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":false,"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1311-9508","authenticated-orcid":false,"given":"S.","family":"Shankar","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2883-9575","authenticated-orcid":false,"given":"M. H.","family":"Devoret","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,12]]},"URL":"http://dx.doi.org/10.1038/s41586-020-2587-z","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:2587\narxivid:1907.12131"},{"id":"arxiv:1907.11729","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","DOI":"10.1038/s41567-020-0824-x","page":"509-513","source":"Crossref","title":"Exponential suppression of bit-flips in a qubit encoded in an oscillator","volume":"16","author":[{"given":"Raphaël","family":"Lescanne","sequence":"first","affiliation":[]},{"given":"Marius","family":"Villiers","sequence":"additional","affiliation":[]},{"given":"Théau","family":"Peronnin","sequence":"additional","affiliation":[]},{"given":"Alain","family":"Sarlette","sequence":"additional","affiliation":[]},{"given":"Matthieu","family":"Delbecq","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9848-3658","authenticated-orcid":false,"given":"Benjamin","family":"Huard","sequence":"additional","affiliation":[]},{"given":"Takis","family":"Kontos","sequence":"additional","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9172-1537","authenticated-orcid":false,"given":"Zaki","family":"Leghtas","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,16]]},"URL":"http://dx.doi.org/10.1038/s41567-020-0824-x","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:824\narxivid:1907.11729"},{"id":"arxiv:2204.09128","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.4.020350","source":"Crossref","title":"One Hundred Second Bit-Flip Time in a Two-Photon Dissipative Oscillator","volume":"4","author":[{"given":"C.","family":"Berdou","sequence":"first","affiliation":[]},{"given":"A.","family":"Murani","sequence":"additional","affiliation":[]},{"given":"U.","family":"Réglade","sequence":"additional","affiliation":[]},{"given":"W.C.","family":"Smith","sequence":"additional","affiliation":[]},{"given":"M.","family":"Villiers","sequence":"additional","affiliation":[]},{"given":"J.","family":"Palomo","sequence":"additional","affiliation":[]},{"given":"M.","family":"Rosticher","sequence":"additional","affiliation":[]},{"given":"A.","family":"Denis","sequence":"additional","affiliation":[]},{"given":"P.","family":"Morfin","sequence":"additional","affiliation":[]},{"given":"M.","family":"Delbecq","sequence":"additional","affiliation":[]},{"given":"T.","family":"Kontos","sequence":"additional","affiliation":[]},{"given":"N.","family":"Pankratova","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5250-8103","authenticated-orcid":true,"given":"F.","family":"Rautschke","sequence":"additional","affiliation":[]},{"given":"T.","family":"Peronnin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2586-2796","authenticated-orcid":true,"given":"L.-A.","family":"Sellem","sequence":"additional","affiliation":[]},{"given":"P.","family":"Rouchon","sequence":"additional","affiliation":[]},{"given":"A.","family":"Sarlette","sequence":"additional","affiliation":[]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"given":"P.","family":"Campagne-Ibarcq","sequence":"additional","affiliation":[]},{"given":"S.","family":"Jezouin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6025-1993","authenticated-orcid":true,"given":"R.","family":"Lescanne","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9172-1537","authenticated-orcid":true,"given":"Z.","family":"Leghtas","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,23]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.020350","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020350","note":"arxivid:2204.09128"},{"id":"arxiv:2307.06617","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8013","DOI":"10.1038/s41586-024-07294-3","page":"778-783","source":"Crossref","title":"Quantum control of a cat qubit with bit-flip times exceeding ten seconds","volume":"629","author":[{"given":"U.","family":"Réglade","sequence":"first","affiliation":[]},{"given":"A.","family":"Bocquet","sequence":"additional","affiliation":[]},{"given":"R.","family":"Gautier","sequence":"additional","affiliation":[]},{"given":"J.","family":"Cohen","sequence":"additional","affiliation":[]},{"given":"A.","family":"Marquet","sequence":"additional","affiliation":[]},{"given":"E.","family":"Albertinale","sequence":"additional","affiliation":[]},{"given":"N.","family":"Pankratova","sequence":"additional","affiliation":[]},{"given":"M.","family":"Hallén","sequence":"additional","affiliation":[]},{"given":"F.","family":"Rautschke","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2586-2796","authenticated-orcid":false,"given":"L.-A.","family":"Sellem","sequence":"additional","affiliation":[]},{"given":"P.","family":"Rouchon","sequence":"additional","affiliation":[]},{"given":"A.","family":"Sarlette","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9471-6031","authenticated-orcid":false,"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9039-3810","authenticated-orcid":false,"given":"P.","family":"Campagne-Ibarcq","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6025-1993","authenticated-orcid":false,"given":"R.","family":"Lescanne","sequence":"additional","affiliation":[]},{"given":"S.","family":"Jezouin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9172-1537","authenticated-orcid":false,"given":"Z.","family":"Leghtas","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,6]]},"URL":"http://dx.doi.org/10.1038/s41586-024-07294-3","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"updated-by:{\"updated\":{\"date-parts\":[[2024,5,17]],\"date-time\":\"2024-05-17T00:00:00Z\",\"timestamp\":1715904000000},\"DOI\":\"10.1038/s41586-024-07565-z\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nalternative-id:7294\narxivid:2307.06617"},{"id":"arxiv:2307.06761","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Cat qubits, for which logical <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mo stretchy=\"false\">|</a:mo><a:mn>0</a:mn><a:mo stretchy=\"false\">⟩</a:mo></a:math> and <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mo stretchy=\"false\">|</e:mo><e:mn>1</e:mn><e:mo stretchy=\"false\">⟩</e:mo></e:math> are coherent states <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:mo stretchy=\"false\">|</i:mo><i:mo>±</i:mo><i:mi>α</i:mi><i:mo stretchy=\"false\">⟩</i:mo></i:math> of a harmonic mode, offer a promising route towards quantum error correction. Using dissipation to our advantage so that photon pairs of the harmonic mode are exchanged with single photons of its environment, it is possible to stabilize the logical states and exponentially increase the bit-flip time of the cat qubit with the photon number <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:mo stretchy=\"false\">|</m:mo><m:mi>α</m:mi><m:msup><m:mo stretchy=\"false\">|</m:mo><m:mn>2</m:mn></m:msup></m:math>. A large two-photon dissipation rate <q:math xmlns:q=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><q:msub><q:mi>κ</q:mi><q:mn>2</q:mn></q:msub></q:math> ensures fast qubit manipulation and short error-correction cycles, which are instrumental to correct the remaining phase-flip errors in a repetition code of cat qubits. Here, we introduce and operate an autoparametric superconducting circuit that couples a mode containing the cat qubit to a lossy mode whose frequency is set at twice that of the cat mode. This passive coupling does not require a parametric pump, and it reaches a rate <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><s:msub><s:mi>κ</s:mi><s:mn>2</s:mn></s:msub><s:mo>/</s:mo><s:mn>2</s:mn><s:mi>π</s:mi><s:mo>≈</s:mo><s:mn>2</s:mn><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mi>MHz</s:mi></s:math>. With such a strong two-photon dissipation, bit-flip errors of the autoparametric cat qubit are prevented for a characteristic time up to 0.3 s with only a mild impact on phase-flip errors. In addition, we illustrate how the phase of a quantum superposition between <u:math xmlns:u=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><u:mo stretchy=\"false\">|</u:mo><u:mi>α</u:mi><u:mo stretchy=\"false\">⟩</u:mo></u:math> and <y:math xmlns:y=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><y:mo stretchy=\"false\">|</y:mo><y:mo>−</y:mo><y:mi>α</y:mi><y:mo stretchy=\"false\">⟩</y:mo></y:math> can be arbitrarily changed by driving the harmonic mode while keeping the engineered dissipation active.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevx.14.021019","source":"Crossref","title":"Autoparametric Resonance Extending the Bit-Flip Time of a Cat Qubit up to 0.3 s","volume":"14","author":[{"given":"A.","family":"Marquet","sequence":"first","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"},{"name":"Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France"}]},{"ORCID":"https://orcid.org/0009-0009-5192-6413","authenticated-orcid":true,"given":"A.","family":"Essig","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"}]},{"given":"J.","family":"Cohen","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"}]},{"ORCID":"https://orcid.org/0000-0003-1300-8186","authenticated-orcid":true,"given":"N.","family":"Cottet","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"}]},{"given":"A.","family":"Murani","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"}]},{"given":"E.","family":"Albertinale","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"}]},{"ORCID":"https://orcid.org/0009-0002-8596-3887","authenticated-orcid":true,"given":"S.","family":"Dupouy","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"},{"name":"Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France"}]},{"ORCID":"https://orcid.org/0000-0002-2927-1037","authenticated-orcid":true,"given":"A.","family":"Bienfait","sequence":"additional","affiliation":[{"name":"Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France"}]},{"given":"T.","family":"Peronnin","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"}]},{"given":"S.","family":"Jezouin","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"}]},{"given":"R.","family":"Lescanne","sequence":"additional","affiliation":[{"name":"Alice & Bob, 49 Bd du Général Martial Valin, 75015 Paris, France"}]},{"ORCID":"https://orcid.org/0000-0002-9848-3658","authenticated-orcid":true,"given":"B.","family":"Huard","sequence":"additional","affiliation":[{"name":"Ecole Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.14.021019","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021019","note":"arxivid:2307.06761"},{"id":"arxiv:2404.16697","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>The Kerr-cat qubit is a bosonic qubit in which multiphoton Schrödinger cat states are stabilized by applying a two-photon drive to an oscillator with a Kerr nonlinearity. The suppressed bit-flip rate with increasing cat size makes this qubit a promising candidate to implement quantum error correction codes tailored for noise-biased qubits. However, achieving strong light-matter interactions necessary for stabilizing and controlling this qubit has traditionally required strong microwave drives that heat the qubit and degrade its performance. In contrast, increasing the coupling to the drive port removes the need for strong drives at the expense of large Purcell decay. By integrating an effective band-block filter on chip, we overcome this trade-off and realize a Kerr-cat qubit in a scalable 2D superconducting circuit with high coherence. This filter provides 30 dB of isolation at the qubit frequency with negligible attenuation at the frequencies required for stabilization and readout. We experimentally demonstrate quantum nondemolition readout fidelity of 99.6% for a cat with eight photons. Also, to have high-fidelity universal control over this qubit, we combine fast Rabi oscillations with a new demonstration of the <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>X</a:mi><a:mo stretchy=\"false\">(</a:mo><a:mi>π</a:mi><a:mo>/</a:mo><a:mn>2</a:mn><a:mo stretchy=\"false\">)</a:mo></a:math> gate through phase modulation of the stabilization drive. Finally, the lifetime in this architecture is examined as a function of the cat size of up to ten photons in the oscillator, achieving a bit-flip time higher than 1 ms and only a linear increase in the phase-flip rate, in good agreement with the theoretical analysis of the circuit. Our qubit shows promise as a building block for fault-tolerant quantum processors with a small footprint.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevx.14.041049","source":"Crossref","title":"High-Coherence Kerr-Cat Qubit in 2D Architecture","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0002-6452-7237","authenticated-orcid":true,"given":"Ahmed","family":"Hajr","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"}]},{"ORCID":"https://orcid.org/0000-0003-4522-7017","authenticated-orcid":true,"given":"Bingcheng","family":"Qing","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"}]},{"ORCID":"https://orcid.org/0000-0003-4158-8544","authenticated-orcid":true,"given":"Ke","family":"Wang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"}]},{"given":"Gerwin","family":"Koolstra","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"}]},{"given":"Zahra","family":"Pedramrazi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"},{"id":[{"id":"https://ror.org/02jbv0t02","id-type":"ROR","asserted-by":"publisher"}],"name":"Lawrence Berkeley National Laboratory"}]},{"given":"Ziqi","family":"Kang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"}]},{"ORCID":"https://orcid.org/0000-0001-8358-8521","authenticated-orcid":true,"given":"Larry","family":"Chen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"}]},{"ORCID":"https://orcid.org/0000-0001-8770-1763","authenticated-orcid":true,"given":"Long B.","family":"Nguyen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"}]},{"given":"Christian","family":"Jünger","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"},{"id":[{"id":"https://ror.org/02jbv0t02","id-type":"ROR","asserted-by":"publisher"}],"name":"Lawrence Berkeley National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-3377-9415","authenticated-orcid":true,"given":"Noah","family":"Goss","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"}]},{"given":"Irwin","family":"Huang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/022kthw22","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Rochester"}]},{"ORCID":"https://orcid.org/0000-0003-3274-5464","authenticated-orcid":true,"given":"Bibek","family":"Bhandari","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0452jzg20","id-type":"ROR","asserted-by":"publisher"}],"name":"Chapman University"}]},{"given":"Nicholas E.","family":"Frattini","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0001-7216-1581","authenticated-orcid":true,"given":"Justin","family":"Dressel","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0452jzg20","id-type":"ROR","asserted-by":"publisher"}],"name":"Chapman University"}]},{"ORCID":"https://orcid.org/0000-0002-9646-7013","authenticated-orcid":true,"given":"Andrew N.","family":"Jordan","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/022kthw22","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Rochester"},{"id":[{"id":"https://ror.org/0452jzg20","id-type":"ROR","asserted-by":"publisher"}],"name":"Chapman University"},{"id":[{"id":"https://ror.org/0452jzg20","id-type":"ROR","asserted-by":"publisher"}],"name":"Chapman University"}]},{"ORCID":"https://orcid.org/0000-0001-8074-5130","authenticated-orcid":true,"given":"David I.","family":"Santiago","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02jbv0t02","id-type":"ROR","asserted-by":"publisher"}],"name":"Lawrence Berkeley National Laboratory"}]},{"given":"Irfan","family":"Siddiqi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California at Berkeley"},{"id":[{"id":"https://ror.org/02jbv0t02","id-type":"ROR","asserted-by":"publisher"}],"name":"Lawrence Berkeley National Laboratory"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.14.041049","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041049","note":"arxivid:2404.16697"},{"id":"arxiv:2209.03934","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>By applying a microwave drive to a specially designed Josephson circuit, we have realized a double-well model system: a Kerr oscillator submitted to a squeezing force. We have observed, for the first time, the spectroscopic fingerprint of a quantum double-well Hamiltonian when its barrier height is increased: a pairwise level kissing (coalescence) corresponding to the exponential reduction of tunnel splitting in the excited states as they sink under the barrier. The discrete levels in the wells also manifest themselves in the activation time across the barrier which, instead of increasing smoothly as a function of the barrier height, presents steps each time a pair of excited states is captured by the wells. This experiment illustrates the quantum regime of Arrhenius’s law, whose observation is made possible here by the unprecedented combination of low dissipation, time-resolved state control, 98.5% quantum nondemolition single shot measurement fidelity, and complete microwave control over all Hamiltonian parameters in the quantum regime. Direct applications to quantum computation and simulation are discussed.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevx.14.031040","source":"Crossref","title":"Observation of Pairwise Level Degeneracies and the Quantum Regime of the Arrhenius Law in a Double-Well Parametric Oscillator","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0003-4442-9522","authenticated-orcid":true,"given":"Nicholas E.","family":"Frattini","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-3706-306X","authenticated-orcid":true,"given":"Rodrigo G.","family":"Cortiñas","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-1736-3807","authenticated-orcid":true,"given":"Jayameenakshi","family":"Venkatraman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"given":"Xu","family":"Xiao","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"given":"Qile","family":"Su","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-2790-2421","authenticated-orcid":true,"given":"Chan U.","family":"Lei","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0003-1248-7568","authenticated-orcid":true,"given":"Benjamin J.","family":"Chapman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"given":"Vidul R.","family":"Joshi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":true,"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"given":"Robert J.","family":"Schoelkopf","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-2883-9575","authenticated-orcid":true,"given":"Michel H.","family":"Devoret","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.14.031040","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031040","note":"arxivid:2209.03934"},{"id":"arxiv:2409.17556","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Cat qubits, a type of bosonic qubit encoded in a harmonic oscillator, can exhibit an exponential noise bias against bit-flip errors with increasing mean photon number. Here, we focus on cat qubits stabilized by two-photon dissipation, where pairs of photons are added and removed from a harmonic oscillator by an auxiliary, lossy buffer mode. This process requires a large loss rate and strong nonlinearities of the buffer mode that must not degrade the coherence and linearity of the oscillator. In this work, we show how to overcome this challenge by coloring the loss environment of the buffer mode with a multipole filter and optimizing the circuit to take into account additional inductances in the buffer mode. Using these techniques, we achieve near-ideal enhancement of cat-qubit bit-flip times with increasing photon number, reaching over 0.1 s with a mean photon number of only 4. Concurrently, our cat qubit remains highly phase coherent, with phase-flip times corresponding to an effective lifetime of <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mrow><a:msub><a:mrow><a:mi>T</a:mi></a:mrow><a:mrow><a:mn>1</a:mn><a:mo>,</a:mo><a:mi>eff</a:mi></a:mrow></a:msub><a:mo>≃</a:mo><a:mn>70</a:mn><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi mathvariant=\"normal\">μ</a:mi><a:mi mathvariant=\"normal\">s</a:mi></a:mrow></a:math>, comparable with the bare oscillator lifetime. We achieve this performance even in the presence of an ancilla transmon, used for reading out the cat-qubit states, by engineering a tunable oscillator-ancilla dispersive coupling. Furthermore, the low nonlinearity of the harmonic oscillator mode allows us to perform pulsed cat-qubit stabilization, an important control primitive, where the stabilization can remain off for a significant fraction (e.g., two-thirds) of a <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mn>3</e:mn><e:mtext> </e:mtext><e:mtext> </e:mtext><e:mrow><e:mi mathvariant=\"normal\">μ</e:mi><e:mi mathvariant=\"normal\">s</e:mi></e:mrow></e:math> cycle without degrading bit-flip times. These advances are important for the realization of scalable error correction with cat qubits, where large noise bias and low phase-flip error rate enable the use of hardware-efficient outer error-correcting codes.</jats:p>","DOI":"10.1103/physrevx.15.011070","source":"Crossref","title":"Preserving Phase Coherence and Linearity in Cat Qubits with Exponential Bit-Flip Suppression","volume":"15","author":[{"ORCID":"https://orcid.org/0000-0002-5841-181X","authenticated-orcid":true,"given":"Harald","family":"Putterman","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0001-9033-3527","authenticated-orcid":true,"given":"Rishi N.","family":"Patel","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0001-9958-9975","authenticated-orcid":true,"given":"Gregory A.","family":"Peairs","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0003-2369-1580","authenticated-orcid":true,"given":"Gregory S.","family":"MacCabe","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0003-1954-3198","authenticated-orcid":true,"given":"Menyoung","family":"Lee","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-9920-493X","authenticated-orcid":true,"given":"Shahriar","family":"Aghaeimeibodi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0003-0665-7161","authenticated-orcid":true,"given":"Connor T.","family":"Hann","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Ignace","family":"Jarrige","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Guillaume","family":"Marcaud","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Yuan","family":"He","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Hesam","family":"Moradinejad","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0009-0006-7362-4481","authenticated-orcid":true,"given":"John Clai","family":"Owens","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-3143-0797","authenticated-orcid":true,"given":"Thomas","family":"Scaffidi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Patricio","family":"Arrangoiz-Arriola","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0003-4665-8839","authenticated-orcid":true,"given":"Joe","family":"Iverson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0001-8270-3233","authenticated-orcid":true,"given":"Harry","family":"Levine","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Fernando G. S. L.","family":"Brandão","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"given":"Matthew H.","family":"Matheny","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Oskar","family":"Painter","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.15.011070","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011070","note":"arxivid:2409.17556"},{"id":"arxiv:2203.03222","type":"journal-article","publisher":"Stichting SciPost","abstract":"<jats:p>These are the lecture notes from the 2019 Les Houches Summer School on “Quantum Information Machines”. After a brief introduction to quantum error correction and bosonic codes, we focus on the case of cat qubits stabilized by a nonlinear multi-photon driven dissipation process. We argue that such a system can be seen as a self-correcting qubit where bit-flip errors are robustly and exponentially suppressed. Next, we provide some experimental directions to engineer such a multi-photon driven dissipation process with superconducting circuits. Finally, we analyze various logical gates that can be implemented without re-introducing bit-flip errors. This set of bias-preserving gates pave the way towards a hardware-efficient and fault-tolerant quantum processor.</jats:p>","DOI":"10.21468/scipostphyslectnotes.72","source":"Crossref","title":"Quantum computation with cat qubits","author":[{"given":"Jérémie","family":"Guillaud","sequence":"first","affiliation":[{"name":"Alice and Bob"}]},{"given":"Joachim","family":"Cohen","sequence":"additional","affiliation":[{"name":"Université de Sherbrooke"}]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[{"name":"Physics Laboratory of the École Normale Supérieure"}]}],"container-title":"SciPost Physics Lecture Notes","original-title":[],"issued":{"date-parts":[[2023,6,16]]},"URL":"http://dx.doi.org/10.21468/SciPostPhysLectNotes.72","ISSN":["2590-1990"],"container-title-short":"SciPost Phys. Lect. Notes","page":"72","note":"arxivid:2203.03222"},{"id":"manual:-S.-Puri-href-https://physinf","type":"document","author":[{"family":"Puri","given":"S."}],"title":"\\href{https://physinfo.fr/houches2019/files/Puri_v1.pdf}{QEC when the noise is biased}","issued":"2019"},{"id":"doi:10.1093/acprof:oso/9780198509141","type":"monograph","publisher":"Oxford University Press","DOI":"10.1093/acprof:oso/9780198509141.001.0001","source":"Crossref","title":"Exploring the Quantum","author":[{"given":"Serge","family":"Haroche","sequence":"first","affiliation":[]},{"given":"Jean-Michel","family":"Raimond","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2006,8,10]]},"ISBN":["9780198509141"],"URL":"http://dx.doi.org/10.1093/acprof:oso/9780198509141.001.0001"},{"id":"doi:10.1142/9781860948169_0009","type":"book-chapter","publisher":"PUBLISHED BY IMPERIAL COLLEGE PRESS AND DISTRIBUTED BY WORLD SCIENTIFIC PUBLISHING CO.","DOI":"10.1142/9781860948169_0009","page":"159-179","source":"Crossref","title":"Schrödinger Cat States for Quantum Information Processing","author":[{"given":"Hyunseok","family":"Jeong","sequence":"first","affiliation":[{"name":"Centre for Quantum Computer Technology, Department of Physics, University of Queensland, St Lucia, Qld 4072, Australia"}]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[{"name":"Centre for Quantum Computer Technology, Department of Physics, University of Queensland, St Lucia, Qld 4072, Australia"}]}],"container-title":"Quantum Information with Continuous Variables of Atoms and Light","original-title":[],"language":"en","issued":{"date-parts":[[2007,2]]},"ISBN":["9781860947605","9781860948169"],"URL":"http://dx.doi.org/10.1142/9781860948169_0009","note":"alternative-id:10.1142/9781860948169_0009\n10.1142/p489"},{"id":"doi:10.1002/9783527695805","type":"monograph","publisher":"Wiley","DOI":"10.1002/9783527695805","source":"Crossref","title":"A Guide to Experiments in Quantum Optics","author":[{"given":"Hans‐A.","family":"Bachor","sequence":"first","affiliation":[]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2019,7,5]]},"ISBN":["9783527411931","9783527695805"],"URL":"http://dx.doi.org/10.1002/9783527695805","note":"edition-number:1\nalternative-id:10.1002/9783527695805"},{"id":"arxiv:2501.16425","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/p8fy-qdc8","source":"Crossref","title":"Viewing protected superconducting qubits through the lens of the cat qubit","volume":"112","author":[{"given":"Simon","family":"Lieu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"given":"Emma L.","family":"Rosenfeld","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0003-0665-7161","authenticated-orcid":true,"given":"Connor T.","family":"Hann","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,7]]},"URL":"http://dx.doi.org/10.1103/p8fy-qdc8","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022414","note":"arxivid:2501.16425"},{"id":"arxiv:1804.11293","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.98.042118","source":"Crossref","title":"Spectral theory of Liouvillians for dissipative phase transitions","volume":"98","author":[{"given":"Fabrizio","family":"Minganti","sequence":"first","affiliation":[]},{"given":"Alberto","family":"Biella","sequence":"additional","affiliation":[]},{"given":"Nicola","family":"Bartolo","sequence":"additional","affiliation":[]},{"given":"Cristiano","family":"Ciuti","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,10,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.98.042118","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042118","note":"arxivid:1804.11293"},{"id":"arxiv:2008.02816","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.125.240405","source":"Crossref","title":"Symmetry Breaking and Error Correction in Open Quantum Systems","volume":"125","author":[{"given":"Simon","family":"Lieu","sequence":"first","affiliation":[]},{"given":"Ron","family":"Belyansky","sequence":"additional","affiliation":[]},{"given":"Jeremy T.","family":"Young","sequence":"additional","affiliation":[]},{"given":"Rex","family":"Lundgren","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0509-3421","authenticated-orcid":true,"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.240405","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240405","note":"update-to:{\"DOI\":\"10.1103/physrevlett.125.240405\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2021,7,23]],\"date-time\":\"2021-07-23T00:00:00Z\",\"timestamp\":1626998400000}}\nupdated-by:{\"DOI\":\"10.1103/physrevlett.125.240405\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2021,7,23]],\"date-time\":\"2021-07-23T00:00:00Z\",\"timestamp\":1626998400000}}\narxivid:2008.02816"},{"id":"arxiv:quant-ph/0511098","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.73.060302","source":"Crossref","title":"Quantum error correction via robust probe modes","volume":"73","author":[{"given":"Fumiko","family":"Yamaguchi","sequence":"first","affiliation":[]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]},{"given":"William J.","family":"Munro","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,6,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.060302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"060302","note":"arxivid:quant-ph/0511098"},{"id":"arxiv:1807.09334","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.9.041009","source":"Crossref","title":"Stabilized Cat in a Driven Nonlinear Cavity: A Fault-Tolerant Error Syndrome Detector","volume":"9","author":[{"given":"Shruti","family":"Puri","sequence":"first","affiliation":[]},{"given":"Alexander","family":"Grimm","sequence":"additional","affiliation":[]},{"given":"Philippe","family":"Campagne-Ibarcq","sequence":"additional","affiliation":[]},{"given":"Alec","family":"Eickbusch","sequence":"additional","affiliation":[]},{"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"given":"Gabrielle","family":"Roberts","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Mazyar","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"given":"Michel H.","family":"Devoret","sequence":"additional","affiliation":[]},{"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2019,10,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.041009","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041009","note":"arxivid:1807.09334"},{"id":"arxiv:2505.16618","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>\n                    While 2-level systems, aka qubits, are a natural choice to perform a logical quantum computation, the situation is less clear at the physical level. Encoding information in higher-dimensional physical systems can indeed provide a first level of redundancy and error correction that simplifies the overall fault-tolerant architecture. A challenge then is to ensure universal control over the encoded qubits. Here, we explore an approach where information is encoded in an irreducible representation of a finite subgroup of\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mi>U</mml:mi>\n                      <mml:mo stretchy=\"false\">(</mml:mo>\n                      <mml:mn>2</mml:mn>\n                      <mml:mo stretchy=\"false\">)</mml:mo>\n                    </mml:math>\n                    through an inverse quantum Fourier transform. We illustrate this idea by applying it to the real Pauli group\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mo fence=\"false\" stretchy=\"false\">&amp;#x27E8;</mml:mo>\n                      <mml:mi>X</mml:mi>\n                      <mml:mo>,</mml:mo>\n                      <mml:mi>Z</mml:mi>\n                      <mml:mo fence=\"false\" stretchy=\"false\">&amp;#x27E9;</mml:mo>\n                    </mml:math>\n                    in the bosonic setting. The resulting two-mode Fourier cat code displays good error correction properties and admits an experimentally-friendly universal gate set that we discuss in detail.\n                  </jats:p>","DOI":"10.22331/q-2026-02-09-2000","page":"2000","source":"Crossref","title":"Bosonic quantum Fourier codes","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0002-6707-1458","authenticated-orcid":false,"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[{"name":"Inria Paris, France"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,2,9]]},"URL":"http://dx.doi.org/10.22331/q-2026-02-09-2000","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2505.16618"},{"id":"arxiv:quant-ph/0109155","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.65.030302","source":"Crossref","title":"Optimizing completely positive maps using semidefinite programming","volume":"65","author":[{"given":"Koenraad","family":"Audenaert","sequence":"first","affiliation":[]},{"given":"Bart","family":"De Moor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,2,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.030302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"030302","note":"arxivid:quant-ph/0109155"},{"id":"arxiv:quant-ph/0307138","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.94.080501","source":"Crossref","title":"Iterative Optimization of Quantum Error Correcting Codes","volume":"94","author":[{"given":"M.","family":"Reimpell","sequence":"first","affiliation":[]},{"given":"R. F.","family":"Werner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,3,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.94.080501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080501","note":"arxivid:quant-ph/0307138"},{"id":"arxiv:0706.3400","type":"article-journal","author":[{"given":"Andrew S.","family":"Fletcher"}],"title":"Channel-Adapted Quantum Error Correction","issued":{"date-parts":[[2007,6,22]]},"note":"arxivid:0706.3400\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0007106","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.63.012305","source":"Crossref","title":"Quantum teleportation and Bell’s inequality using single-particle entanglement","volume":"63","author":[{"given":"Hai-Woong","family":"Lee","sequence":"first","affiliation":[]},{"given":"Jaewan","family":"Kim","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,12,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.63.012305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012305","note":"arxivid:quant-ph/0007106"},{"id":"arxiv:quant-ph/0205044","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.66.032307","source":"Crossref","title":"Nondeterministic gates for photonic single-rail quantum logic","volume":"66","author":[{"given":"A. P.","family":"Lund","sequence":"first","affiliation":[]},{"given":"T. C.","family":"Ralph","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,9,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.66.032307","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032307","note":"arxivid:quant-ph/0205044"},{"id":"doi:10.1002/1521-3978(200212)50:12-1","type":"journal-article","publisher":"Wiley","issue":"12","DOI":"10.1002/1521-3978(200212)50:12<1177::aid-prop1177>3.0.co;2-t","page":"1177-1372","source":"Crossref","title":"Quantum Communication with Continuous Variables","volume":"50","author":[{"given":"P.","family":"van Loock","sequence":"first","affiliation":[]}],"container-title":"Fortschritte der Physik","original-title":[],"issued":{"date-parts":[[2002,12]]},"URL":"http://dx.doi.org/10.1002/1521-3978(200212)50:12<1177::AID-PROP1177>3.0.CO;2-T","ISSN":["0015-8208","1521-3978"],"container-title-short":"Fortschr. Phys."},{"id":"doi:10.1103/PhysRev.47.777","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrev.47.777","page":"777-780","source":"Crossref","title":"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?","volume":"47","author":[{"given":"A.","family":"Einstein","sequence":"first","affiliation":[{"name":"Institute for Advanced Study, Princeton, New Jersey"}]},{"given":"B.","family":"Podolsky","sequence":"additional","affiliation":[{"name":"Institute for Advanced Study, Princeton, New Jersey"}]},{"given":"N.","family":"Rosen","sequence":"additional","affiliation":[{"name":"Institute for Advanced Study, Princeton, New Jersey"}]}],"container-title":"Physical Review","original-title":[],"language":"en","issued":{"date-parts":[[1935,5,15]]},"URL":"http://dx.doi.org/10.1103/PhysRev.47.777","ISSN":["0031-899X"],"container-title-short":"Phys. Rev."},{"id":"doi:10.1038/nature02858","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7007","DOI":"10.1038/nature02858","page":"430-433","source":"Crossref","title":"Demonstration of a quantum teleportation network for continuous variables","volume":"431","author":[{"given":"Hidehiro","family":"Yonezawa","sequence":"first","affiliation":[]},{"given":"Takao","family":"Aoki","sequence":"additional","affiliation":[]},{"given":"Akira","family":"Furusawa","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2004,9]]},"URL":"http://dx.doi.org/10.1038/nature02858","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature02858"},{"id":"arxiv:quant-ph/0311015","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.92.177903","source":"Crossref","title":"Tripartite Quantum State Sharing","volume":"92","author":[{"given":"Andrew M.","family":"Lance","sequence":"first","affiliation":[]},{"given":"Thomas","family":"Symul","sequence":"additional","affiliation":[]},{"given":"Warwick P.","family":"Bowen","sequence":"additional","affiliation":[]},{"given":"Barry C.","family":"Sanders","sequence":"additional","affiliation":[]},{"given":"Ping Koy","family":"Lam","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2004,4,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.92.177903","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"177903","note":"arxivid:quant-ph/0311015"},{"id":"arxiv:quant-ph/0210132","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevlett.90.167903","source":"Crossref","title":"Experimental Demonstration of Tripartite Entanglement and Controlled Dense Coding for Continuous Variables","volume":"90","author":[{"given":"Jietai","family":"Jing","sequence":"first","affiliation":[]},{"given":"Jing","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Ying","family":"Yan","sequence":"additional","affiliation":[]},{"given":"Fagang","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Changde","family":"Xie","sequence":"additional","affiliation":[]},{"given":"Kunchi","family":"Peng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2003,4,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.90.167903","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"167903","note":"arxivid:quant-ph/0210132"},{"id":"arxiv:quant-ph/9906021","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevlett.84.3482","page":"3482-3485","source":"Crossref","title":"Multipartite Entanglement for Continuous Variables: A Quantum Teleportation Network","volume":"84","author":[{"given":"P.","family":"van Loock","sequence":"first","affiliation":[]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2000,4,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.84.3482","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9906021"},{"id":"arxiv:0901.1488","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.103.070501","source":"Crossref","title":"Quantum Teamwork for Unconditional Multiparty Communication with Gaussian States","volume":"103","author":[{"given":"Jing","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Gerardo","family":"Adesso","sequence":"additional","affiliation":[]},{"given":"Changde","family":"Xie","sequence":"additional","affiliation":[]},{"given":"Kunchi","family":"Peng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.103.070501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070501","note":"arxivid:0901.1488"},{"id":"arxiv:0903.3233","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.79.062318","source":"Crossref","title":"Quantum computing with continuous-variable clusters","volume":"79","author":[{"given":"Mile","family":"Gu","sequence":"first","affiliation":[]},{"given":"Christian","family":"Weedbrook","sequence":"additional","affiliation":[]},{"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,6,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.062318","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062318","note":"arxivid:0903.3233"},{"id":"arxiv:quant-ph/0405064","type":"article-journal","author":[{"given":"Richard L.","family":"Barnes"}],"title":"Stabilizer Codes for Continuous-variable Quantum Error Correction","issued":{"date-parts":[[2004,5,13]]},"note":"arxivid:quant-ph/0405064\narxiv_version_number:1"},{"id":"doi:10.1103/PhysRevA.81.062305","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.81.062305","source":"Crossref","title":"Five-wave-packet linear optics quantum-error-correcting code","volume":"81","author":[{"given":"Thomas A.","family":"Walker","sequence":"first","affiliation":[]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,6,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.062305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062305"},{"id":"arxiv:0811.3128","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.102.120501","source":"Crossref","title":"No-Go Theorem for Gaussian Quantum Error Correction","volume":"102","author":[{"given":"Julien","family":"Niset","sequence":"first","affiliation":[]},{"given":"Jaromír","family":"Fiurášek","sequence":"additional","affiliation":[]},{"given":"Nicolas J.","family":"Cerf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.102.120501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120501","note":"arxivid:0811.3128"},{"id":"arxiv:1810.00047","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.99.032344","source":"Crossref","title":"Quantum error correction with the toric Gottesman-Kitaev-Preskill code","volume":"99","author":[{"given":"Christophe","family":"Vuillot","sequence":"first","affiliation":[]},{"given":"Hamed","family":"Asasi","sequence":"additional","affiliation":[]},{"given":"Yang","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,3,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.032344","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032344","note":"arxivid:1810.00047"},{"id":"arxiv:quant-ph/0204052","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.89.137903","source":"Crossref","title":"Distilling Gaussian States with Gaussian Operations is Impossible","volume":"89","author":[{"given":"J.","family":"Eisert","sequence":"first","affiliation":[]},{"given":"S.","family":"Scheel","sequence":"additional","affiliation":[]},{"given":"M. B.","family":"Plenio","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2002,9,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.89.137903","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"137903","note":"arxivid:quant-ph/0204052"},{"id":"arxiv:quant-ph/0204085","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.66.032316","source":"Crossref","title":"Characterization of Gaussian operations and distillation of Gaussian states","volume":"66","author":[{"given":"Géza","family":"Giedke","sequence":"first","affiliation":[]},{"given":"J.","family":"Ignacio Cirac","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,9,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.66.032316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032316","note":"arxivid:quant-ph/0204085"},{"id":"arxiv:0811.3616","type":"article-journal","author":[{"given":"Peter","family":"van Loock"}],"title":"A note on quantum error correction with continuous variables","issued":{"date-parts":[[2008,11,21]]},"note":"arxivid:0811.3616\narxiv_version_number:1"},{"id":"arxiv:0710.2868","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.77.060304","source":"Crossref","title":"Maximally multipartite entangled states","volume":"77","author":[{"given":"Paolo","family":"Facchi","sequence":"first","affiliation":[]},{"given":"Giuseppe","family":"Florio","sequence":"additional","affiliation":[]},{"given":"Giorgio","family":"Parisi","sequence":"additional","affiliation":[]},{"given":"Saverio","family":"Pascazio","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,6,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.060304","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"060304","note":"arxivid:0710.2868"},{"id":"arxiv:0908.0114","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.80.062311","source":"Crossref","title":"Gaussian maximally multipartite-entangled states","volume":"80","author":[{"given":"Paolo","family":"Facchi","sequence":"first","affiliation":[]},{"given":"Giuseppe","family":"Florio","sequence":"additional","affiliation":[]},{"given":"Cosmo","family":"Lupo","sequence":"additional","affiliation":[]},{"given":"Stefano","family":"Mancini","sequence":"additional","affiliation":[]},{"given":"Saverio","family":"Pascazio","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,12,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.062311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062311","note":"arxivid:0908.0114"},{"id":"arxiv:1912.06463","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.2.030343","source":"Crossref","title":"Cluster States from Gaussian States: Essential Diagnostic Tools for Continuous-Variable One-Way Quantum Computing","volume":"2","author":[{"given":"Carlos","family":"González-Arciniegas","sequence":"first","affiliation":[]},{"given":"Paulo","family":"Nussenzveig","sequence":"additional","affiliation":[]},{"given":"Marcelo","family":"Martinelli","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3386-9661","authenticated-orcid":true,"given":"Olivier","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,9,13]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.030343","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030343","note":"arxivid:1912.06463"},{"id":"arxiv:0711.0820","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.78.052121","source":"Crossref","title":"Anyon statistics with continuous variables","volume":"78","author":[{"given":"Jing","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Changde","family":"Xie","sequence":"additional","affiliation":[]},{"given":"Kunchi","family":"Peng","sequence":"additional","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,11,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.052121","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052121","note":"arxivid:0711.0820"},{"id":"arxiv:1302.3428","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/revmodphys.87.307","page":"307-346","source":"Crossref","title":"Quantum error correction for quantum memories","volume":"87","author":[{"given":"Barbara M.","family":"Terhal","sequence":"first","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,7]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.87.307","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","note":"arxivid:1302.3428"},{"id":"doi:10.1038/27850","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6688","DOI":"10.1038/27850","page":"47-49","source":"Crossref","title":"Quantum error correction for communication with linear optics","volume":"394","author":[{"given":"Samuel L.","family":"Braunstein","sequence":"first","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[1998,7,1]]},"URL":"http://dx.doi.org/10.1038/27850","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BF27850"},{"id":"doi:10.1103/PhysRevA.73.032318","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.73.032318","source":"Crossref","title":"Continuous-variable Gaussian analog of cluster states","volume":"73","author":[{"given":"Jing","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,3,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.032318","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032318"},{"id":"arxiv:quant-ph/0605198","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.97.110501","source":"Crossref","title":"Universal Quantum Computation with Continuous-Variable Cluster States","volume":"97","author":[{"given":"Nicolas C.","family":"Menicucci","sequence":"first","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]},{"given":"Mile","family":"Gu","sequence":"additional","affiliation":[]},{"given":"Christian","family":"Weedbrook","sequence":"additional","affiliation":[]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[]},{"given":"Michael A.","family":"Nielsen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,9,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.97.110501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110501","note":"arxivid:quant-ph/0605198"},{"id":"arxiv:1007.3434","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.83.062314","source":"Crossref","title":"Temporal-mode continuous-variable cluster states using linear optics","volume":"83","author":[{"given":"Nicolas C.","family":"Menicucci","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,6,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.062314","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062314","note":"arxivid:1007.3434"},{"id":"arxiv:1306.3366","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","DOI":"10.1038/nphoton.2013.287","page":"982-986","source":"Crossref","title":"Ultra-large-scale continuous-variable cluster states multiplexed in the time domain","volume":"7","author":[{"given":"Shota","family":"Yokoyama","sequence":"first","affiliation":[]},{"given":"Ryuji","family":"Ukai","sequence":"additional","affiliation":[]},{"given":"Seiji C.","family":"Armstrong","sequence":"additional","affiliation":[]},{"given":"Chanond","family":"Sornphiphatphong","sequence":"additional","affiliation":[]},{"given":"Toshiyuki","family":"Kaji","sequence":"additional","affiliation":[]},{"given":"Shigenari","family":"Suzuki","sequence":"additional","affiliation":[]},{"given":"Jun-ichi","family":"Yoshikawa","sequence":"additional","affiliation":[]},{"given":"Hidehiro","family":"Yonezawa","sequence":"additional","affiliation":[]},{"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]},{"given":"Akira","family":"Furusawa","sequence":"additional","affiliation":[]}],"container-title":"Nature Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2013,11,17]]},"URL":"http://dx.doi.org/10.1038/nphoton.2013.287","ISSN":["1749-4885","1749-4893"],"container-title-short":"Nature Photon","note":"alternative-id:BFnphoton2013287\narxivid:1306.3366"},{"id":"arxiv:1311.2957","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.112.120505","source":"Crossref","title":"Experimental Realization of Multipartite Entanglement of 60 Modes of a Quantum Optical Frequency Comb","volume":"112","author":[{"given":"Moran","family":"Chen","sequence":"first","affiliation":[]},{"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]},{"given":"Olivier","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,3,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.112.120505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120505","note":"arxivid:1311.2957"},{"id":"arxiv:1606.06688","type":"journal-article","publisher":"AIP Publishing","issue":"6","abstract":"<jats:p>In recent quantum optical continuous-variable experiments, the number of fully inseparable light modes has drastically increased by introducing a multiplexing scheme either in the time domain or in the frequency domain. Here, modifying the time-domain multiplexing experiment reported in the work of Yokoyama et al. [Nat. Photonics 7, 982 (2013)], we demonstrate the successive generation of fully inseparable light modes for more than one million modes. The resulting multi-mode state is useful as a dual-rail continuous variable cluster state. We circumvent the previous problem of optical phase drifts, which has limited the number of fully inseparable light modes to around ten thousands, by continuous feedback control of the optical system.</jats:p>","DOI":"10.1063/1.4962732","source":"Crossref","title":"Invited Article: Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing","volume":"1","author":[{"ORCID":"https://orcid.org/0000-0001-5344-4949","authenticated-orcid":false,"given":"Jun-ichi","family":"Yoshikawa","sequence":"first","affiliation":[{"name":"The University of Tokyo 1 Department of Applied Physics, School of Engineering, , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"}]},{"given":"Shota","family":"Yokoyama","sequence":"additional","affiliation":[{"name":"The University of Tokyo 1 Department of Applied Physics, School of Engineering, , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"},{"name":"University of New South Wales 2 Centre for Quantum Computation and Communication Technology, School of Engineering and Information Technology, , Canberra, Australian Capital Territory 2600, Australia"}]},{"given":"Toshiyuki","family":"Kaji","sequence":"additional","affiliation":[{"name":"The University of Tokyo 1 Department of Applied Physics, School of Engineering, , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"}]},{"given":"Chanond","family":"Sornphiphatphong","sequence":"additional","affiliation":[{"name":"The University of Tokyo 1 Department of Applied Physics, School of Engineering, , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"}]},{"given":"Yu","family":"Shiozawa","sequence":"additional","affiliation":[{"name":"The University of Tokyo 1 Department of Applied Physics, School of Engineering, , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"}]},{"given":"Kenzo","family":"Makino","sequence":"additional","affiliation":[{"name":"The University of Tokyo 1 Department of Applied Physics, School of Engineering, , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"}]},{"given":"Akira","family":"Furusawa","sequence":"additional","affiliation":[{"name":"The University of Tokyo 1 Department of Applied Physics, School of Engineering, , 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"}]}],"container-title":"APL Photonics","original-title":[],"language":"en","issued":{"date-parts":[[2016,9,1]]},"URL":"http://dx.doi.org/10.1063/1.4962732","ISSN":["2378-0967"],"page":"060801","note":"arxivid:1606.06688"},{"id":"doi:10.1038/s41586-024-08406-9","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8052","DOI":"10.1038/s41586-024-08406-9","page":"912-919","source":"Crossref","title":"Scaling and networking a modular photonic quantum computer","volume":"638","author":[{"given":"H.","family":"Aghaee Rad","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0009-0003-6940-8709","authenticated-orcid":false,"given":"T.","family":"Ainsworth","sequence":"additional","affiliation":[]},{"given":"R. N.","family":"Alexander","sequence":"additional","affiliation":[]},{"given":"B.","family":"Altieri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9571-8466","authenticated-orcid":false,"given":"M. F.","family":"Askarani","sequence":"additional","affiliation":[]},{"given":"R.","family":"Baby","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6324-8754","authenticated-orcid":false,"given":"L.","family":"Banchi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":false,"given":"B. Q.","family":"Baragiola","sequence":"additional","affiliation":[]},{"given":"J. E.","family":"Bourassa","sequence":"additional","affiliation":[]},{"given":"R. S.","family":"Chadwick","sequence":"additional","affiliation":[]},{"given":"I.","family":"Charania","sequence":"additional","affiliation":[]},{"given":"H.","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3969-5797","authenticated-orcid":false,"given":"M. J.","family":"Collins","sequence":"additional","affiliation":[]},{"given":"P.","family":"Contu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0005-9950-6996","authenticated-orcid":false,"given":"N.","family":"D’Arcy","sequence":"additional","affiliation":[]},{"given":"G.","family":"Dauphinais","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7212-3355","authenticated-orcid":false,"given":"R.","family":"De Prins","sequence":"additional","affiliation":[]},{"given":"D.","family":"Deschenes","sequence":"additional","affiliation":[]},{"given":"I.","family":"Di Luch","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0009-5948-2650","authenticated-orcid":false,"given":"S.","family":"Duque","sequence":"additional","affiliation":[]},{"given":"P.","family":"Edke","sequence":"additional","affiliation":[]},{"given":"S. E.","family":"Fayer","sequence":"additional","affiliation":[]},{"given":"S.","family":"Ferracin","sequence":"additional","affiliation":[]},{"given":"H.","family":"Ferretti","sequence":"additional","affiliation":[]},{"given":"J.","family":"Gefaell","sequence":"additional","affiliation":[]},{"given":"S.","family":"Glancy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8907-4177","authenticated-orcid":false,"given":"C.","family":"González-Arciniegas","sequence":"additional","affiliation":[]},{"given":"T.","family":"Grainge","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0002-8116-6581","authenticated-orcid":false,"given":"Z.","family":"Han","sequence":"additional","affiliation":[]},{"given":"J.","family":"Hastrup","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0346-2342","authenticated-orcid":false,"given":"L. G.","family":"Helt","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1476-0647","authenticated-orcid":false,"given":"T.","family":"Hillmann","sequence":"additional","affiliation":[]},{"given":"J.","family":"Hundal","sequence":"additional","affiliation":[]},{"given":"S.","family":"Izumi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-4591-3686","authenticated-orcid":false,"given":"T.","family":"Jaeken","sequence":"additional","affiliation":[]},{"given":"M.","family":"Jonas","sequence":"additional","affiliation":[]},{"given":"S.","family":"Kocsis","sequence":"additional","affiliation":[]},{"given":"I.","family":"Krasnokutska","sequence":"additional","affiliation":[]},{"given":"M. V.","family":"Larsen","sequence":"additional","affiliation":[]},{"given":"P.","family":"Laskowski","sequence":"additional","affiliation":[]},{"given":"F.","family":"Laudenbach","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5208-6729","authenticated-orcid":false,"given":"J.","family":"Lavoie","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0005-6379-8341","authenticated-orcid":false,"given":"M.","family":"Li","sequence":"additional","affiliation":[]},{"given":"E.","family":"Lomonte","sequence":"additional","affiliation":[]},{"given":"C. E.","family":"Lopetegui","sequence":"additional","affiliation":[]},{"given":"B.","family":"Luey","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1983-3059","authenticated-orcid":false,"given":"A. P.","family":"Lund","sequence":"additional","affiliation":[]},{"given":"C.","family":"Ma","sequence":"additional","affiliation":[]},{"given":"L. S.","family":"Madsen","sequence":"additional","affiliation":[]},{"given":"D. H.","family":"Mahler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-9379-5231","authenticated-orcid":false,"given":"L.","family":"Mantilla Calderón","sequence":"additional","affiliation":[]},{"given":"M.","family":"Menotti","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6684-8341","authenticated-orcid":false,"given":"F. M.","family":"Miatto","sequence":"additional","affiliation":[]},{"given":"B.","family":"Morrison","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1351-2959","authenticated-orcid":false,"given":"P. J.","family":"Nadkarni","sequence":"additional","affiliation":[]},{"given":"T.","family":"Nakamura","sequence":"additional","affiliation":[]},{"given":"L.","family":"Neuhaus","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4025-5994","authenticated-orcid":false,"given":"Z.","family":"Niu","sequence":"additional","affiliation":[]},{"given":"R.","family":"Noro","sequence":"additional","affiliation":[]},{"given":"K.","family":"Papirov","sequence":"additional","affiliation":[]},{"given":"A.","family":"Pesah","sequence":"additional","affiliation":[]},{"given":"D. S.","family":"Phillips","sequence":"additional","affiliation":[]},{"given":"W. N.","family":"Plick","sequence":"additional","affiliation":[]},{"given":"T.","family":"Rogalsky","sequence":"additional","affiliation":[]},{"given":"F.","family":"Rortais","sequence":"additional","affiliation":[]},{"given":"J.","family":"Sabines-Chesterking","sequence":"additional","affiliation":[]},{"given":"S.","family":"Safavi-Bayat","sequence":"additional","affiliation":[]},{"given":"E.","family":"Sazhaev","sequence":"additional","affiliation":[]},{"given":"M.","family":"Seymour","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6473-938X","authenticated-orcid":false,"given":"K.","family":"Rezaei Shad","sequence":"additional","affiliation":[]},{"given":"M.","family":"Silverman","sequence":"additional","affiliation":[]},{"given":"S. A.","family":"Srinivasan","sequence":"additional","affiliation":[]},{"given":"M.","family":"Stephan","sequence":"additional","affiliation":[]},{"given":"Q. Y.","family":"Tang","sequence":"additional","affiliation":[]},{"given":"J. F.","family":"Tasker","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1766-6402","authenticated-orcid":false,"given":"Y. S.","family":"Teo","sequence":"additional","affiliation":[]},{"given":"R. B.","family":"Then","sequence":"additional","affiliation":[]},{"given":"J. E.","family":"Tremblay","sequence":"additional","affiliation":[]},{"given":"I.","family":"Tzitrin","sequence":"additional","affiliation":[]},{"given":"V. D.","family":"Vaidya","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":false,"given":"M.","family":"Vasmer","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3268-6986","authenticated-orcid":false,"given":"Z.","family":"Vernon","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0009-5626-691X","authenticated-orcid":false,"given":"L. F. S. S. M.","family":"Villalobos","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4343-3087","authenticated-orcid":false,"given":"B. W.","family":"Walshe","sequence":"additional","affiliation":[]},{"given":"R.","family":"Weil","sequence":"additional","affiliation":[]},{"given":"X.","family":"Xin","sequence":"additional","affiliation":[]},{"given":"X.","family":"Yan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0619-071X","authenticated-orcid":false,"given":"Y.","family":"Yao","sequence":"additional","affiliation":[]},{"given":"M.","family":"Zamani Abnili","sequence":"additional","affiliation":[]},{"given":"Y.","family":"Zhang","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,22]]},"URL":"http://dx.doi.org/10.1038/s41586-024-08406-9","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:8406"},{"id":"arxiv:0906.3141","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.80.050303","source":"Crossref","title":"Demonstration of a universal one-way quantum quadratic phase gate","volume":"80","author":[{"given":"Yoshichika","family":"Miwa","sequence":"first","affiliation":[]},{"given":"Jun-ichi","family":"Yoshikawa","sequence":"additional","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]},{"given":"Akira","family":"Furusawa","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,11,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.050303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"050303","note":"arxivid:0906.3141"},{"id":"arxiv:0705.4314","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.76.052308","source":"Crossref","title":"Entanglement-assisted quantum error correction with linear optics","volume":"76","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Hari","family":"Krovi","sequence":"additional","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,11,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.052308","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052308","note":"arxivid:0705.4314"},{"id":"arxiv:0804.1404","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.77.064302","source":"Crossref","title":"Optimal entanglement formulas for entanglement-assisted quantum coding","volume":"77","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,6,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.064302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"064302","note":"arxivid:0804.1404"},{"id":"arxiv:0811.3734","type":"article-journal","author":[{"given":"Takao","family":"Aoki"},{"given":"Go","family":"Takahashi"},{"given":"Tadashi","family":"Kajiya"},{"given":"Jun-ichi","family":"Yoshikawa"},{"given":"Samuel L.","family":"Braunstein"},{"given":"Peter","family":"van Loock"},{"given":"Akira","family":"Furusawa"}],"title":"Quantum error correction beyond qubits","issued":{"date-parts":[[2008,11,24]]},"note":"arxivid:0811.3734\narxiv_version_number:1"},{"id":"arxiv:1310.7596","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.112.120504","source":"Crossref","title":"Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States","volume":"112","author":[{"given":"Nicolas C.","family":"Menicucci","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,3,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.112.120504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120504","note":"arxivid:1310.7596"},{"id":"arxiv:2010.02905","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Photonics is the platform of choice to build a modular, easy-to-network quantum computer operating at room temperature. However, no concrete architecture has been presented so far that exploits both the advantages of qubits encoded into states of light and the modern tools for their generation. Here we propose such a design for a scalable fault-tolerant photonic quantum computer informed by the latest developments in theory and technology. Central to our architecture is the generation and manipulation of three-dimensional resource states comprising both bosonic qubits and squeezed vacuum states. The proposal exploits state-of-the-art procedures for the non-deterministic generation of bosonic qubits combined with the strengths of continuous-variable quantum computation, namely the implementation of Clifford gates using easy-to-generate squeezed states. Moreover, the architecture is based on two-dimensional integrated photonic chips used to produce a qubit cluster state in one temporal and two spatial dimensions. By reducing the experimental challenges as compared to existing architectures and by enabling room-temperature quantum computation, our design opens the door to scalable fabrication and operation, which may allow photonics to leap-frog other platforms on the path to a quantum computer with millions of qubits.</jats:p>","DOI":"10.22331/q-2021-02-04-392","page":"392","source":"Crossref","title":"Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0003-4333-5006","authenticated-orcid":false,"given":"J. Eli","family":"Bourassa","sequence":"first","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"Department of Physics, University of Toronto, Toronto, Canada"}]},{"ORCID":"https://orcid.org/0000-0002-7462-4516","authenticated-orcid":false,"given":"Rafael N.","family":"Alexander","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA"},{"name":"Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, VIC 3000, Australia"}]},{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":false,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}]},{"given":"Ashlesha","family":"Patil","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"College of Optical Sciences, University of Arizona, Tucson, Arizona 85719, USA"}]},{"ORCID":"https://orcid.org/0000-0001-5201-3987","authenticated-orcid":false,"given":"Ilan","family":"Tzitrin","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"Department of Physics, University of Toronto, Toronto, Canada"}]},{"ORCID":"https://orcid.org/0000-0003-4164-4307","authenticated-orcid":false,"given":"Takaya","family":"Matsuura","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan"}]},{"given":"Daiqin","family":"Su","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}]},{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":false,"given":"Ben Q.","family":"Baragiola","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, VIC 3000, Australia"}]},{"ORCID":"https://orcid.org/0000-0002-2581-4380","authenticated-orcid":false,"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"College of Optical Sciences, University of Arizona, Tucson, Arizona 85719, USA"}]},{"given":"Guillaume","family":"Dauphinais","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}]},{"ORCID":"https://orcid.org/0000-0003-3107-6844","authenticated-orcid":false,"given":"Krishna K.","family":"Sabapathy","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":false,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"},{"name":"Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, VIC 3000, Australia"}]},{"ORCID":"https://orcid.org/0000-0002-7041-2575","authenticated-orcid":false,"given":"Ish","family":"Dhand","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON, M5G 2C8, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,4]]},"URL":"http://dx.doi.org/10.22331/q-2021-02-04-392","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2010.02905"},{"id":"arxiv:1712.00294","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.8.021054","source":"Crossref","title":"High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction","volume":"8","author":[{"given":"Kosuke","family":"Fukui","sequence":"first","affiliation":[]},{"given":"Akihisa","family":"Tomita","sequence":"additional","affiliation":[]},{"given":"Atsushi","family":"Okamoto","sequence":"additional","affiliation":[]},{"given":"Keisuke","family":"Fujii","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.8.021054","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021054","note":"arxivid:1712.00294"},{"id":"arxiv:2104.03241","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/prxquantum.2.040353","source":"Crossref","title":"Fault-Tolerant Quantum Computation with Static Linear Optics","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0001-5201-3987","authenticated-orcid":true,"given":"Ilan","family":"Tzitrin","sequence":"first","affiliation":[]},{"given":"Takaya","family":"Matsuura","sequence":"additional","affiliation":[]},{"given":"Rafael N.","family":"Alexander","sequence":"additional","affiliation":[]},{"given":"Guillaume","family":"Dauphinais","sequence":"additional","affiliation":[]},{"given":"J. Eli","family":"Bourassa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3107-6844","authenticated-orcid":true,"given":"Krishna K.","family":"Sabapathy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]},{"given":"Ish","family":"Dhand","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,16]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.040353","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040353","note":"arxivid:2104.03241"},{"id":"arxiv:2008.12791","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.102.062411","source":"Crossref","title":"Continuous-variable gate teleportation and bosonic-code error correction","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0003-4343-3087","authenticated-orcid":true,"given":"Blayney W.","family":"Walshe","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":true,"given":"Ben Q.","family":"Baragiola","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7462-4516","authenticated-orcid":true,"given":"Rafael N.","family":"Alexander","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.062411","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062411","note":"arxivid:2008.12791"},{"id":"arxiv:1903.02162","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.100.010301","source":"Crossref","title":"Robust fault tolerance for continuous-variable cluster states with excess antisqueezing","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0003-4343-3087","authenticated-orcid":true,"given":"Blayney W.","family":"Walshe","sequence":"first","affiliation":[]},{"given":"Lucas J.","family":"Mensen","sequence":"additional","affiliation":[]},{"given":"Ben Q.","family":"Baragiola","sequence":"additional","affiliation":[]},{"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.100.010301","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"010301","note":"arxivid:1903.02162"},{"id":"arxiv:2402.05228","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Stabilizer codes are the most widely studied class of quantum error-correcting codes and form the basis of most proposals for a fault-tolerant quantum computer. A stabilizer code is defined by a set of parity-check operators, which are measured in order to infer information about errors that may have occurred. In typical settings, measuring these operators is itself a noisy process and the noise strength scales with the number of qubits involved in a given parity check, or its weight. Hastings has proposed a method for reducing the weights of the parity checks of a stabilizer code, though it has previously only been studied in the asymptotic regime. Here, we instead focus on the regime of small to medium-size codes suitable for quantum computing hardware. We both provide a fully explicit description of Hastings’s method and propose a substantially simplified weight-reduction method that is applicable to the class of quantum product codes. Our simplified method allows us to reduce the check weights of hypergraph- and lifted-product codes to at most six, while preserving the number of logical qubits and at least retaining (in fact, often increasing) the code distance. The price we pay is an increase in the number of physical qubits by a constant factor but we find that our method is much more efficient than Hastings’s method in this regard. We benchmark the performance of our codes on a simulated photonic quantum computing architecture based on Gottesman-Kitaev-Preskill qubits and passive linear optics, finding that our weight-reduction method substantially improves code performance.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.040302","source":"Crossref","title":"Weight-Reduced Stabilizer Codes with Lower Overhead","volume":"5","author":[{"given":"Eric","family":"Sabo","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"given":"Lane G.","family":"Gunderman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"given":"Benjamin","family":"Ide","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":true,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"},{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"},{"name":"University of Waterloo"}]},{"given":"Guillaume","family":"Dauphinais","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,4]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.040302","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040302","note":"arxivid:2402.05228"},{"id":"arxiv:2209.04573","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.4.020342","source":"Crossref","title":"Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison","volume":"4","author":[{"given":"Yijia","family":"Xu","sequence":"first","affiliation":[]},{"given":"Yixu","family":"Wang","sequence":"additional","affiliation":[]},{"given":"En-Jui","family":"Kuo","sequence":"additional","affiliation":[]},{"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,14]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.020342","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020342","note":"arxivid:2209.04573"},{"id":"arxiv:2102.05545","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2021.3126881","page":"1068-1084","source":"Crossref","title":"Oscillator-to-Oscillator Codes Do Not Have a Threshold","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0002-6001-1188","authenticated-orcid":false,"given":"Lisa","family":"Hanggli","sequence":"first","affiliation":[]},{"given":"Robert","family":"Konig","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3126881","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2102.05545"},{"id":"arxiv:2109.14645","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.</jats:p>","DOI":"10.22331/q-2022-02-10-648","page":"648","source":"Crossref","title":"Gottesman-Kitaev-Preskill codes: A lattice perspective","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-6120-9930","authenticated-orcid":false,"given":"Jonathan","family":"Conrad","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany"}]},{"ORCID":"https://orcid.org/0000-0003-3033-1292","authenticated-orcid":false,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany"}]},{"ORCID":"https://orcid.org/0000-0002-4439-6962","authenticated-orcid":false,"given":"Francesco","family":"Arzani","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,10]]},"URL":"http://dx.doi.org/10.22331/q-2022-02-10-648","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2109.14645"},{"id":"doi:10.1007/BF00670008","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf00670008","page":"213-230","source":"Crossref","title":"Modular variables in quantum theory","volume":"2","author":[{"given":"Yakir","family":"Aharonov","sequence":"first","affiliation":[]},{"given":"Hugh","family":"Pendleton","sequence":"additional","affiliation":[]},{"given":"Aage","family":"Petersen","sequence":"additional","affiliation":[]}],"container-title":"International Journal of Theoretical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1969,10]]},"URL":"http://dx.doi.org/10.1007/BF00670008","ISSN":["0020-7748","1572-9575"],"container-title-short":"Int J Theor Phys","note":"alternative-id:BF00670008"},{"id":"doi:10.1515/9781400889921","type":"edited-book","publisher":"Princeton University Press","DOI":"10.1515/9781400889921","source":"Crossref","title":"Mathematical Foundations of Quantum Mechanics","author":[{"given":"John","family":"von Neumann","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"editor":[{"given":"Nicholas A.","family":"Wheeler","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2018,4,11]]},"ISBN":["9781400889921"],"URL":"http://dx.doi.org/10.1515/9781400889921","note":"alternative-id:10.1515/9781400889921"},{"id":"manual:-P.-Cartier-Quantum-mechanica","type":"article-journal","author":[{"family":"Cartier","given":"P."}],"title":"Quantum mechanical commutation relations and theta functions","volume":"9","container-title":"Proc. Sympos. Pure Math","issued":"1966"},{"id":"doi:10.1007/BF01077648","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/bf01077648","page":"292-300","source":"Crossref","title":"Coherent states and theta functions","volume":"6","author":[{"given":"A. M.","family":"Perelomov","sequence":"first","affiliation":[]}],"container-title":"Functional Analysis and Its Applications","original-title":[],"language":"en","issued":{"date-parts":[[1973]]},"URL":"http://dx.doi.org/10.1007/BF01077648","ISSN":["0016-2663","1573-8485"],"container-title-short":"Funct Anal Its Appl","note":"alternative-id:BF01077648"},{"id":"doi:10.1007/978-3-642-61629-7","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-61629-7","source":"Crossref","title":"Generalized Coherent States and Their Applications","author":[{"given":"Askold","family":"Perelomov","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1986]]},"ISBN":["9783642648915","9783642616297"],"URL":"http://dx.doi.org/10.1007/978-3-642-61629-7","note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1103/PhysRevB.12.1118","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.12.1118","page":"1118-1120","source":"Crossref","title":"Proof of completeness of lattice states in the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>k</mml:mi><mml:mi/><mml:mi>q</mml:mi></mml:math>representation","volume":"12","author":[{"given":"H.","family":"Bacry","sequence":"first","affiliation":[]},{"given":"A.","family":"Grossmann","sequence":"additional","affiliation":[]},{"given":"J.","family":"Zak","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[1975,8,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.12.1118","ISSN":["0556-2805"],"container-title-short":"Phys. Rev. B"},{"id":"manual:-I.-M.-Gelfand-Expansion-in-c","type":"chapter","author":[{"family":"Gelfand","given":"I.M."}],"title":"Expansion in characteristic functions of an equation with periodic coefficients","container-title":"Doklady Akademii Nauk SSSR (N.S","volume":"73","issued":"1950","page":"1117–1120"},{"id":"doi:10.1103/PhysRevLett.19.1385","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.19.1385","page":"1385-1387","source":"Crossref","title":"Finite Translations in Solid-State Physics","volume":"19","author":[{"given":"J.","family":"Zak","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1967,12,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.19.1385","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"doi:10.1007/BF02391012","type":"journal-article","publisher":"International Press of Boston","issue":"0","DOI":"10.1007/bf02391012","page":"143-211","source":"Crossref","title":"Sur certains groupes d'opérateurs unitaires","volume":"111","author":[{"given":"André","family":"Weil","sequence":"first","affiliation":[]}],"container-title":"Acta Mathematica","original-title":[],"language":"en","issued":{"date-parts":[[1964]]},"URL":"http://dx.doi.org/10.1007/BF02391012","ISSN":["0001-5962"],"container-title-short":"Acta Math."},{"id":"doi:10.1007/978-1-4612-2016-9","type":"book","publisher":"Birkhäuser Boston","DOI":"10.1007/978-1-4612-2016-9","source":"Crossref","title":"Gabor Analysis and Algorithms","container-title":[],"original-title":[],"language":"en","editor":[{"given":"Hans G.","family":"Feichtinger","sequence":"first","affiliation":[]},{"given":"Thomas","family":"Strohmer","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[1998]]},"ISBN":["9781461273820","9781461220169"],"URL":"http://dx.doi.org/10.1007/978-1-4612-2016-9","note":"publisher-location:Boston, MA"},{"id":"doi:10.1515/9781400882427","type":"edited-book","publisher":"Princeton University Press","DOI":"10.1515/9781400882427","source":"Crossref","title":"Harmonic Analysis in Phase Space. (AM-122)","author":[{"given":"Gerald B.","family":"Folland","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1989,12,31]]},"ISBN":["9781400882427"],"URL":"http://dx.doi.org/10.1515/9781400882427","note":"alternative-id:10.1515/9781400882427"},{"id":"manual:-A.-J.-E.-M.-Janssen-(1988).-","type":"article-journal","author":[{"family":"Janssen","given":"A.J.E.M."}],"title":"The Zak transform: a signal transform for sampled time-continuous signals","volume":"43","container-title":"Philips Journal of Research","issue":"1","issued":"1988","page":"23–69"},{"id":"doi:10.1109/18.179336","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/18.179336","page":"3-6","source":"Crossref","title":"Two theorems on lattice expansions","volume":"39","author":[{"given":"I.","family":"Daubechies","sequence":"first","affiliation":[]},{"given":"A.J.E.M.","family":"Janssen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1993]]},"URL":"http://dx.doi.org/10.1109/18.179336","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:2002.07718","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevx.11.011032","source":"Crossref","title":"Hardware-Encoding Grid States in a Nonreciprocal Superconducting Circuit","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0002-8253-2080","authenticated-orcid":true,"given":"Martin","family":"Rymarz","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4035-9654","authenticated-orcid":true,"given":"Stefano","family":"Bosco","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8707-0532","authenticated-orcid":true,"given":"Alessandro","family":"Ciani","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4332-645X","authenticated-orcid":true,"given":"David P.","family":"DiVincenzo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.11.011032","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011032","note":"arxivid:2002.07718"},{"id":"arxiv:2106.11093","type":"journal-article","publisher":"AIP Publishing","issue":"10","abstract":"<jats:p>We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction, the Zak’s magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant, which is the non-commutative torus of magnetic translations in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori. The main focus of our review is the Kähler structure on the Hilbert space of Bloch waves and its inherent quantum toric geometry. We reveal that the metaplectic group Mp(2,R) of the automorphisms of magnetic translation algebras is represented by the quantum optics squeezing operators.</jats:p>","DOI":"10.1063/5.0063174","source":"Crossref","title":"Bloch waves and non-commutative tori of magnetic translations","volume":"62","author":[{"ORCID":"https://orcid.org/0000-0001-7071-8585","authenticated-orcid":false,"given":"Tekin","family":"Dereli","sequence":"first","affiliation":[{"name":"Department of Physics, Koç University 1 , Sariyer, İstanbul 34450, Turkey"}]},{"ORCID":"https://orcid.org/0000-0001-5151-6144","authenticated-orcid":false,"given":"Todor","family":"Popov","sequence":"additional","affiliation":[{"name":"Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences 2 , bld. Tzarigradsko chaussée, 72, Sofia 1784, Bulgaria"},{"name":"American University in Bulgaria, Svoboda Bachvarova Str. 8 3 , Blagoevgrad 2700, Bulgaria"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,10,1]]},"URL":"http://dx.doi.org/10.1063/5.0063174","ISSN":["0022-2488","1089-7658"],"note":"arxivid:2106.11093"},{"id":"doi:10.1103/PhysRevB.31.2529","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.31.2529","page":"2529-2531","source":"Crossref","title":"Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect","volume":"31","author":[{"given":"F. D. M.","family":"Haldane","sequence":"first","affiliation":[]},{"given":"E. H.","family":"Rezayi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[1985,2,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.31.2529","ISSN":["0163-1829"],"container-title-short":"Phys. Rev. B"},{"id":"doi:10.1007/978-0-8176-4577-9","type":"book","publisher":"Birkhäuser Boston","DOI":"10.1007/978-0-8176-4577-9","source":"Crossref","title":"Tata Lectures on Theta I","container-title":"Modern Birkhäuser Classics","original-title":[],"language":"en","issued":{"date-parts":[[2007]]},"ISBN":["9780817645724"],"URL":"http://dx.doi.org/10.1007/978-0-8176-4577-9","note":"publisher-location:Boston, MA"},{"id":"arxiv:1506.05033","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.93.012315","source":"Crossref","title":"Encoding a qubit into a cavity mode in circuit QED using phase estimation","volume":"93","author":[{"given":"B. M.","family":"Terhal","sequence":"first","affiliation":[]},{"given":"D.","family":"Weigand","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,1,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.012315","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012315","note":"arxivid:1506.05033"},{"id":"arxiv:2504.13497","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/rmlm-vbfd","source":"Crossref","title":"Logical channel for heralded and pure loss with the Gottesman-Kitaev-Preskill code","volume":"112","author":[{"given":"Tom B.","family":"Harris","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"ORCID":"https://orcid.org/0000-0003-4164-4307","authenticated-orcid":true,"given":"Takaya","family":"Matsuura","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN Center for Quantum Computing"}]},{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":true,"given":"Ben Q.","family":"Baragiola","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,14]]},"URL":"http://dx.doi.org/10.1103/rmlm-vbfd","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042417","note":"arxivid:2504.13497"},{"id":"arxiv:2106.12989","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.2.020101","source":"Crossref","title":"Quantum Error Correction with the Gottesman-Kitaev-Preskill Code","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-5208-0271","authenticated-orcid":true,"given":"Arne L.","family":"Grimsmo","sequence":"first","affiliation":[]},{"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,6,23]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.020101","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020101","note":"arxivid:2106.12989"},{"id":"arxiv:2512.03583","type":"article-journal","author":[{"given":"Gui-Zhong","family":"Luo"},{"given":"Matthew","family":"Otten"}],"title":"Energy-Scaled Zero-Noise Extrapolation for Gottesman-Kitaev-Preskill Code","issued":{"date-parts":[[2025,12,3]]},"note":"arxivid:2512.03583\narxiv_version_number:1"},{"id":"arxiv:2009.07941","type":"journal-article","publisher":"American Physical Society (APS)","issue":"26","DOI":"10.1103/physrevlett.125.260509","source":"Crossref","title":"Stabilization of Finite-Energy Gottesman-Kitaev-Preskill States","volume":"125","author":[{"ORCID":"https://orcid.org/0000-0001-9695-8552","authenticated-orcid":true,"given":"Baptiste","family":"Royer","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4921-1410","authenticated-orcid":true,"given":"Shraddha","family":"Singh","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":true,"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.260509","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"260509","note":"arxivid:2009.07941"},{"id":"arxiv:2010.09681","type":"article-journal","author":[{"given":"Brennan","family":"de Neeve"},{"given":"Thanh Long","family":"Nguyen"},{"given":"Tanja","family":"Behrle"},{"given":"Jonathan","family":"Home"}],"title":"Error correction of a logical grid state qubit by dissipative pumping","issued":{"date-parts":[[2020,10,19]]},"note":"arxivid:2010.09681\narxiv_version_number:1"},{"id":"arxiv:1709.08580","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.97.022341","source":"Crossref","title":"Generating grid states from Schrödinger-cat states without postselection","volume":"97","author":[{"given":"Daniel J.","family":"Weigand","sequence":"first","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,2,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.022341","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022341","note":"arxivid:1709.08580"},{"id":"arxiv:1907.12487","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7821","DOI":"10.1038/s41586-020-2603-3","page":"368-372","source":"Crossref","title":"Quantum error correction of a qubit encoded in grid states of an oscillator","volume":"584","author":[{"ORCID":"https://orcid.org/0000-0001-9039-3810","authenticated-orcid":false,"given":"P.","family":"Campagne-Ibarcq","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5179-0215","authenticated-orcid":false,"given":"A.","family":"Eickbusch","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3475-2839","authenticated-orcid":false,"given":"S.","family":"Touzard","sequence":"additional","affiliation":[]},{"given":"E.","family":"Zalys-Geller","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4442-9522","authenticated-orcid":false,"given":"N. E.","family":"Frattini","sequence":"additional","affiliation":[]},{"given":"V. V.","family":"Sivak","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8141-1842","authenticated-orcid":false,"given":"P.","family":"Reinhold","sequence":"additional","affiliation":[]},{"given":"S.","family":"Puri","sequence":"additional","affiliation":[]},{"given":"S.","family":"Shankar","sequence":"additional","affiliation":[]},{"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0272-5481","authenticated-orcid":false,"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[]},{"given":"M.","family":"Mirrahimi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2883-9575","authenticated-orcid":false,"given":"M. H.","family":"Devoret","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,19]]},"URL":"http://dx.doi.org/10.1038/s41586-020-2603-3","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:2603\narxivid:1907.12487"},{"id":"arxiv:1910.03673","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.101.032315","source":"Crossref","title":"Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes","volume":"101","author":[{"ORCID":"https://orcid.org/0000-0001-5201-3987","authenticated-orcid":true,"given":"Ilan","family":"Tzitrin","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4333-5006","authenticated-orcid":true,"given":"J. Eli","family":"Bourassa","sequence":"additional","affiliation":[]},{"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3107-6844","authenticated-orcid":true,"given":"Krishna Kumar","family":"Sabapathy","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.101.032315","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032315","note":"arxivid:1910.03673"},{"id":"arxiv:1910.08301","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.102.032408","source":"Crossref","title":"Equivalence of approximate Gottesman-Kitaev-Preskill codes","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0003-4164-4307","authenticated-orcid":true,"given":"Takaya","family":"Matsuura","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3521-831X","authenticated-orcid":true,"given":"Hayata","family":"Yamasaki","sequence":"additional","affiliation":[]},{"given":"Masato","family":"Koashi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.032408","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032408","note":"arxivid:1910.08301"},{"id":"arxiv:2012.12488","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.104.022408","source":"Crossref","title":"Phase-space methods for representing, manipulating, and correcting Gottesman-Kitaev-Preskill qubits","volume":"104","author":[{"given":"Lucas J.","family":"Mensen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":true,"given":"Ben Q.","family":"Baragiola","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.104.022408","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022408","note":"arxivid:2012.12488"},{"id":"arxiv:2303.03541","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.132.130605","source":"Crossref","title":"Gottesman-Kitaev-Preskill State Preparation Using Periodic Driving","volume":"132","author":[{"ORCID":"https://orcid.org/0009-0005-0507-9226","authenticated-orcid":true,"given":"Xanda C.","family":"Kolesnikow","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]},{"ORCID":"https://orcid.org/0000-0001-6876-5681","authenticated-orcid":true,"given":"Raditya W.","family":"Bomantara","sequence":"additional","affiliation":[{"name":"Department of Physics, Interdisciplinary Research Center for Intelligent Secure Systems, King Fahd University of Petroleum and Minerals, 31261 Dhahran, Saudi Arabia"}]},{"ORCID":"https://orcid.org/0000-0002-8069-7754","authenticated-orcid":true,"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]},{"ORCID":"https://orcid.org/0000-0002-5208-0271","authenticated-orcid":true,"given":"Arne L.","family":"Grimsmo","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"},{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"California Institute of Technology, Pasadena, California 91125, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,3,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.132.130605","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130605","note":"arxivid:2303.03541"},{"id":"arxiv:2410.19610","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations and single- and two-qubit operations as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an approximation to the state in trace distance. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mo stretchy=\"false\">|</a:mo><a:msub><a:mrow><a:mi mathvariant=\"sans-serif\">G</a:mi><a:mi mathvariant=\"sans-serif\">K</a:mi><a:mi mathvariant=\"sans-serif\">P</a:mi></a:mrow><a:mrow><a:mi>κ</a:mi><a:mo>,</a:mo><a:mi mathvariant=\"normal\">Δ</a:mi></a:mrow></a:msub><a:mo stretchy=\"false\">⟩</a:mo></a:math>. Here, <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:msup><i:mi>κ</i:mi><i:mrow><i:mo>−</i:mo><i:mn>2</i:mn></i:mrow></i:msup></i:math> is the variance of the envelope, and <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:msup><k:mi mathvariant=\"normal\">Δ</k:mi><k:mn>2</k:mn></k:msup></k:math> is the variance of the individual peaks. We show that the circuit accepts with constant probability and—conditioned on acceptance—the output state is polynomially close in <n:math xmlns:n=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><n:mo stretchy=\"false\">(</n:mo><n:mi>κ</n:mi><n:mo>,</n:mo><n:mi mathvariant=\"normal\">Δ</n:mi><n:mo stretchy=\"false\">)</n:mo></n:math> to the state <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><s:mo stretchy=\"false\">|</s:mo><s:msub><s:mrow><s:mi mathvariant=\"sans-serif\">G</s:mi><s:mi mathvariant=\"sans-serif\">K</s:mi><s:mi mathvariant=\"sans-serif\">P</s:mi></s:mrow><s:mrow><s:mi>κ</s:mi><s:mo>,</s:mo><s:mi mathvariant=\"normal\">Δ</s:mi></s:mrow></s:msub><s:mo stretchy=\"false\">⟩</s:mo></s:math>. The size of our circuit is linear in <ab:math xmlns:ab=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><ab:mo stretchy=\"false\">(</ab:mo><ab:mi>log</ab:mi><ab:mn>1</ab:mn><ab:mo>/</ab:mo><ab:mi>κ</ab:mi><ab:mo>,</ab:mo><ab:mi>log</ab:mi><ab:mn>1</ab:mn><ab:mo>/</ab:mo><ab:mi mathvariant=\"normal\">Δ</ab:mi><ab:mo stretchy=\"false\">)</ab:mo></ab:math>. To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states.</jats:p>","DOI":"10.1103/4ww5-4yww","source":"Crossref","title":"Complexity of Gottesman-Kitaev-Preskill States","volume":"15","author":[{"ORCID":"https://orcid.org/0009-0005-2828-365X","authenticated-orcid":true,"given":"Lukas","family":"Brenner","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"},{"id":[{"id":"https://ror.org/04xrcta15","id-type":"ROR","asserted-by":"publisher"}],"name":"Munich Center for Quantum Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0001-7401-8076","authenticated-orcid":true,"given":"Libor","family":"Caha","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"},{"id":[{"id":"https://ror.org/04xrcta15","id-type":"ROR","asserted-by":"publisher"}],"name":"Munich Center for Quantum Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0002-7355-4632","authenticated-orcid":true,"given":"Xavier","family":"Coiteux-Roy","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03yjb2x39","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Calgary"},{"id":[{"id":"https://ror.org/03yjb2x39","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Calgary"}]},{"ORCID":"https://orcid.org/0000-0002-0563-1229","authenticated-orcid":true,"given":"Robert","family":"Koenig","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"},{"id":[{"id":"https://ror.org/04xrcta15","id-type":"ROR","asserted-by":"publisher"}],"name":"Munich Center for Quantum Science and Technology"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,19]]},"URL":"http://dx.doi.org/10.1103/4ww5-4yww","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031073","note":"arxivid:2410.19610"},{"id":"arxiv:2506.13643","type":"article-journal","author":[{"given":"Sheron","family":"Blair"},{"given":"Francesco","family":"Arzani"},{"given":"Giulia","family":"Ferrini"},{"given":"Alessandro","family":"Ferraro"}],"title":"Towards fault-tolerant quantum computation with universal continuous-variable gates","issued":{"date-parts":[[2025,6,16]]},"note":"arxivid:2506.13643\narxiv_version_number:1"},{"id":"arxiv:2405.05671","type":"article-journal","author":[{"given":"Frederik","family":"Nathan"},{"given":"Liam","family":"O'Brien"},{"given":"Kyungjoo","family":"Noh"},{"given":"Matthew H.","family":"Matheny"},{"given":"Arne L.","family":"Grimsmo"},{"given":"Liang","family":"Jiang"},{"given":"Gil","family":"Refael"}],"title":"Self-correcting GKP qubit and gates in a driven-dissipative circuit","issued":{"date-parts":[[2025,4,10]]},"note":"arxivid:2405.05671\narxiv_version_number:2"},{"id":"arxiv:1903.00012","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.123.200502","source":"Crossref","title":"All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code","volume":"123","author":[{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":true,"given":"Ben Q.","family":"Baragiola","sequence":"first","affiliation":[]},{"given":"Giacomo","family":"Pantaleoni","sequence":"additional","affiliation":[]},{"given":"Rafael N.","family":"Alexander","sequence":"additional","affiliation":[]},{"given":"Angela","family":"Karanjai","sequence":"additional","affiliation":[]},{"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,11,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.123.200502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200502","note":"arxivid:1903.00012"},{"id":"arxiv:2507.09684","type":"article-journal","author":[{"given":"Jérémie","family":"Boudreault"},{"given":"Ross","family":"Shillito"},{"given":"Jean-Baptiste","family":"Bertrand"},{"given":"Baptiste","family":"Royer"}],"title":"Using a Kerr interaction for GKP magic state preparation","issued":{"date-parts":[[2025,7,13]]},"note":"arxivid:2507.09684\narxiv_version_number:1"},{"id":"arxiv:2507.19713","type":"article-journal","author":[{"given":"Liam","family":"O'Brien"},{"given":"Gil","family":"Refael"},{"given":"Frederik","family":"Nathan"}],"title":"Exponentially robust non-Clifford gate in a driven-dissipative circuit","issued":{"date-parts":[[2025,7,25]]},"note":"arxivid:2507.19713\narxiv_version_number:1"},{"id":"arxiv:2312.07391","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Bosonic codes allow the encoding of a logical qubit in a single component device, utilizing the infinitely large Hilbert space of a harmonic oscillator. In particular, the Gottesman-Kitaev-Preskill code has recently been demonstrated to be correctable well beyond the break-even point of the best passive encoding in the same system. Current approaches to quantum error correction (QEC) for this system are based on protocols that use feedback, but the response is based only on the latest measurement outcome. In our work, we use the recently proposed feedback-GRAPE (gradient-ascent pulse engineering with feedback) method to train a recurrent neural network that provides a QEC scheme based on memory, responding in a non-Markovian way to the full history of previous measurement outcomes, optimizing all subsequent unitary operations. This approach significantly outperforms current strategies and paves the way for more powerful measurement-based QEC protocols.</jats:p>","DOI":"10.1103/physrevlett.134.020601","source":"Crossref","title":"Non-Markovian Feedback for Optimized Quantum Error Correction","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0002-5332-213X","authenticated-orcid":true,"given":"Matteo","family":"Puviani","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"}]},{"ORCID":"https://orcid.org/0000-0002-6212-6245","authenticated-orcid":true,"given":"Sangkha","family":"Borah","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"},{"id":[{"id":"https://ror.org/00f7hpc57","id-type":"ROR","asserted-by":"publisher"}],"name":"Friedrich-Alexander Universität Erlangen-Nürnberg"}]},{"ORCID":"https://orcid.org/0000-0002-7645-125X","authenticated-orcid":true,"given":"Remmy","family":"Zen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"}]},{"ORCID":"https://orcid.org/0000-0003-3338-5130","authenticated-orcid":true,"given":"Jan","family":"Olle","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"}]},{"ORCID":"https://orcid.org/0000-0003-4566-1753","authenticated-orcid":true,"given":"Florian","family":"Marquardt","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"},{"id":[{"id":"https://ror.org/00f7hpc57","id-type":"ROR","asserted-by":"publisher"}],"name":"Friedrich-Alexander Universität Erlangen-Nürnberg"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.020601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020601","note":"arxivid:2312.07391"},{"id":"arxiv:2505.14775","type":"article-journal","author":[{"given":"Frederik K.","family":"Marqversen"},{"given":"Janus H.","family":"Wesenberg"},{"given":"Nikolaj T.","family":"Zinner"},{"given":"Ulrik L.","family":"Andersen"}],"title":"Performance analysis of GKP error correction","issued":{"date-parts":[[2025,5,20]]},"note":"arxivid:2505.14775\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0510107","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.73.012325","source":"Crossref","title":"Error analysis for encoding a qubit in an oscillator","volume":"73","author":[{"given":"S.","family":"Glancy","sequence":"first","affiliation":[]},{"given":"E.","family":"Knill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,1,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.012325","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012325","note":"arxivid:quant-ph/0510107"},{"id":"arxiv:1807.01033","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7745","DOI":"10.1038/s41586-019-0960-6","page":"513-517","source":"Crossref","title":"Encoding a qubit in a trapped-ion mechanical oscillator","volume":"566","author":[{"given":"C.","family":"Flühmann","sequence":"first","affiliation":[]},{"given":"T. L.","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"M.","family":"Marinelli","sequence":"additional","affiliation":[]},{"given":"V.","family":"Negnevitsky","sequence":"additional","affiliation":[]},{"given":"K.","family":"Mehta","sequence":"additional","affiliation":[]},{"given":"J. P.","family":"Home","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2019,2,27]]},"URL":"http://dx.doi.org/10.1038/s41586-019-0960-6","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:960\narxivid:1807.01033"},{"id":"arxiv:1907.06478","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.125.043602","source":"Crossref","title":"Direct Characteristic-Function Tomography of Quantum States of the Trapped-Ion Motional Oscillator","volume":"125","author":[{"ORCID":"https://orcid.org/0000-0003-4065-073X","authenticated-orcid":true,"given":"C.","family":"Flühmann","sequence":"first","affiliation":[]},{"given":"J. P.","family":"Home","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.043602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"043602","note":"arxivid:1907.06478"},{"id":"arxiv:2409.05455","type":"article-journal","author":[{"given":"V. G.","family":"Matsos"},{"given":"C. H.","family":"Valahu"},{"given":"M. J.","family":"Millican"},{"given":"T.","family":"Navickas"},{"given":"X. C.","family":"Kolesnikow"},{"given":"M. J.","family":"Biercuk"},{"given":"T. R.","family":"Tan"}],"title":"Universal Quantum Gate Set for Gottesman-Kitaev-Preskill Logical Qubits","issued":{"date-parts":[[2024,9,9]]},"note":"arxivid:2409.05455\narxiv_version_number:1"},{"id":"arxiv:2310.11400","type":"article-journal","author":[{"given":"Dany","family":"Lachance-Quirion"},{"given":"Marc-Antoine","family":"Lemonde"},{"given":"Jean Olivier","family":"Simoneau"},{"given":"Lucas","family":"St-Jean"},{"given":"Pascal","family":"Lemieux"},{"given":"Sara","family":"Turcotte"},{"given":"Wyatt","family":"Wright"},{"given":"Amélie","family":"Lacroix"},{"given":"Joëlle","family":"Fréchette-Viens"},{"given":"Ross","family":"Shillito"},{"given":"Florian","family":"Hopfmueller"},{"given":"Maxime","family":"Tremblay"},{"given":"Nicholas E.","family":"Frattini"},{"given":"Julien Camirand","family":"Lemyre"},{"given":"Philippe","family":"St-Jean"}],"title":"Autonomous quantum error correction of Gottesman-Kitaev-Preskill states","issued":{"date-parts":[[2023,10,19]]},"note":"arxivid:2310.11400\narxiv_version_number:2"},{"id":"arxiv:2409.15065","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8063","DOI":"10.1038/s41586-025-08899-y","page":"612-618","source":"Crossref","title":"Quantum error correction of qudits beyond break-even","volume":"641","author":[{"ORCID":"https://orcid.org/0000-0001-9317-3135","authenticated-orcid":false,"given":"Benjamin L.","family":"Brock","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4921-1410","authenticated-orcid":false,"given":"Shraddha","family":"Singh","sequence":"additional","affiliation":[]},{"given":"Alec","family":"Eickbusch","sequence":"additional","affiliation":[]},{"given":"Volodymyr V.","family":"Sivak","sequence":"additional","affiliation":[]},{"given":"Andy Z.","family":"Ding","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0272-5481","authenticated-orcid":false,"given":"Luigi","family":"Frunzio","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6470-5494","authenticated-orcid":false,"given":"Steven M.","family":"Girvin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2883-9575","authenticated-orcid":false,"given":"Michel H.","family":"Devoret","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,14]]},"URL":"http://dx.doi.org/10.1038/s41586-025-08899-y","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:8899\narxivid:2409.15065"},{"id":"arxiv:2309.02306","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6680","abstract":"<jats:p>To harness the potential of a quantum computer, quantum information must be protected against error by encoding it into a logical state that is suitable for quantum error correction. The Gottesman-Kitaev-Preskill (GKP) qubit is a promising candidate because the required multiqubit operations are readily available at optical frequency. To date, however, GKP qubits have been demonstrated only at mechanical and microwave frequencies. We realized a GKP state in propagating light at telecommunication wavelength and verified it through homodyne measurements without loss corrections. The generation is based on interference of cat states, followed by homodyne measurements. Our final states exhibit nonclassicality and non-Gaussianity, including the trident shape of faint instances of GKP states. Improvements toward brighter, multipeaked GKP qubits will be the basis for quantum computation with light.</jats:p>","DOI":"10.1126/science.adk7560","page":"289-293","source":"Crossref","title":"Logical states for fault-tolerant quantum computation with propagating light","volume":"383","author":[{"ORCID":"https://orcid.org/0000-0001-5805-5029","authenticated-orcid":true,"given":"Shunya","family":"Konno","sequence":"first","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."}]},{"ORCID":"https://orcid.org/0000-0002-3722-3676","authenticated-orcid":true,"given":"Warit","family":"Asavanant","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."},{"name":"Optical Quantum Computing Research Team, RIKEN Center for Quantum Computing, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan."}]},{"ORCID":"https://orcid.org/0000-0002-2382-4593","authenticated-orcid":true,"given":"Fumiya","family":"Hanamura","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."}]},{"ORCID":"https://orcid.org/0009-0005-3516-5643","authenticated-orcid":true,"given":"Hironari","family":"Nagayoshi","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."}]},{"ORCID":"https://orcid.org/0000-0002-8231-7241","authenticated-orcid":true,"given":"Kosuke","family":"Fukui","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."}]},{"ORCID":"https://orcid.org/0000-0001-6846-5240","authenticated-orcid":true,"given":"Atsushi","family":"Sakaguchi","sequence":"additional","affiliation":[{"name":"Optical Quantum Computing Research Team, RIKEN Center for Quantum Computing, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan."}]},{"ORCID":"https://orcid.org/0009-0001-8869-5649","authenticated-orcid":true,"given":"Ryuhoh","family":"Ide","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."}]},{"ORCID":"https://orcid.org/0000-0003-2144-8229","authenticated-orcid":true,"given":"Fumihiro","family":"China","sequence":"additional","affiliation":[{"name":"Advanced ICR Research Institute, National Institute of Information and Communications Technology, 588-2 Iwaoka, Nishi, Kobe 651-2492, Japan."}]},{"ORCID":"https://orcid.org/0000-0002-6436-1636","authenticated-orcid":true,"given":"Masahiro","family":"Yabuno","sequence":"additional","affiliation":[{"name":"Advanced ICR Research Institute, National Institute of Information and Communications Technology, 588-2 Iwaoka, Nishi, Kobe 651-2492, Japan."}]},{"given":"Shigehito","family":"Miki","sequence":"additional","affiliation":[{"name":"Advanced ICR Research Institute, National Institute of Information and Communications Technology, 588-2 Iwaoka, Nishi, Kobe 651-2492, Japan."},{"name":"Graduate School of Engineering, Kobe University, 1-1 Rokko-dai, Nada, Kobe 657-0013, Japan."}]},{"ORCID":"https://orcid.org/0000-0002-5575-2221","authenticated-orcid":true,"given":"Hirotaka","family":"Terai","sequence":"additional","affiliation":[{"name":"Advanced ICR Research Institute, National Institute of Information and Communications Technology, 588-2 Iwaoka, Nishi, Kobe 651-2492, Japan."}]},{"ORCID":"https://orcid.org/0000-0003-4017-8332","authenticated-orcid":true,"given":"Kan","family":"Takase","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."},{"name":"Optical Quantum Computing Research Team, RIKEN Center for Quantum Computing, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan."}]},{"ORCID":"https://orcid.org/0000-0003-4594-6791","authenticated-orcid":true,"given":"Mamoru","family":"Endo","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."},{"name":"Optical Quantum Computing Research Team, RIKEN Center for Quantum Computing, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan."}]},{"ORCID":"https://orcid.org/0000-0002-5761-8966","authenticated-orcid":true,"given":"Petr","family":"Marek","sequence":"additional","affiliation":[{"name":"Department of Optics, Palacky University, 17. listopadu 1192/12, 77146 Olomouc, Czech Republic."}]},{"ORCID":"https://orcid.org/0000-0003-4114-6068","authenticated-orcid":true,"given":"Radim","family":"Filip","sequence":"additional","affiliation":[{"name":"Department of Optics, Palacky University, 17. listopadu 1192/12, 77146 Olomouc, Czech Republic."}]},{"ORCID":"https://orcid.org/0000-0001-9445-0771","authenticated-orcid":true,"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[{"name":"Institute of Physics, Johannes-Gutenberg University of Mainz, Staudingerweg 7, 55128 Mainz, Germany."}]},{"ORCID":"https://orcid.org/0000-0002-6176-6742","authenticated-orcid":true,"given":"Akira","family":"Furusawa","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan."},{"name":"Optical Quantum Computing Research Team, RIKEN Center for Quantum Computing, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,19]]},"URL":"http://dx.doi.org/10.1126/science.adk7560","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.adk7560\narxivid:2309.02306"},{"id":"doi:10.1038/s41586-025-09044-5","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8068","DOI":"10.1038/s41586-025-09044-5","page":"587-591","source":"Crossref","title":"Integrated photonic source of Gottesman–Kitaev–Preskill qubits","volume":"642","author":[{"given":"M. V.","family":"Larsen","sequence":"first","affiliation":[]},{"given":"J. E.","family":"Bourassa","sequence":"additional","affiliation":[]},{"given":"S.","family":"Kocsis","sequence":"additional","affiliation":[]},{"given":"J. F.","family":"Tasker","sequence":"additional","affiliation":[]},{"given":"R. S.","family":"Chadwick","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8907-4177","authenticated-orcid":false,"given":"C.","family":"González-Arciniegas","sequence":"additional","affiliation":[]},{"given":"J.","family":"Hastrup","sequence":"additional","affiliation":[]},{"given":"C. E.","family":"Lopetegui-González","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6684-8341","authenticated-orcid":false,"given":"F. M.","family":"Miatto","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0007-1177-8300","authenticated-orcid":false,"given":"A.","family":"Motamedi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0007-2988-0637","authenticated-orcid":false,"given":"R.","family":"Noro","sequence":"additional","affiliation":[]},{"given":"G.","family":"Roeland","sequence":"additional","affiliation":[]},{"given":"R.","family":"Baby","sequence":"additional","affiliation":[]},{"given":"H.","family":"Chen","sequence":"additional","affiliation":[]},{"given":"P.","family":"Contu","sequence":"additional","affiliation":[]},{"given":"I.","family":"Di Luch","sequence":"additional","affiliation":[]},{"given":"C.","family":"Drago","sequence":"additional","affiliation":[]},{"given":"M.","family":"Giesbrecht","sequence":"additional","affiliation":[]},{"given":"T.","family":"Grainge","sequence":"additional","affiliation":[]},{"given":"I.","family":"Krasnokutska","sequence":"additional","affiliation":[]},{"given":"M.","family":"Menotti","sequence":"additional","affiliation":[]},{"given":"B.","family":"Morrison","sequence":"additional","affiliation":[]},{"given":"C.","family":"Puviraj","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6473-938X","authenticated-orcid":false,"given":"K.","family":"Rezaei Shad","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7442-9537","authenticated-orcid":false,"given":"B.","family":"Hussain","sequence":"additional","affiliation":[]},{"given":"J.","family":"McMahon","sequence":"additional","affiliation":[]},{"given":"J. E.","family":"Ortmann","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3969-5797","authenticated-orcid":false,"given":"M. J.","family":"Collins","sequence":"additional","affiliation":[]},{"given":"C.","family":"Ma","sequence":"additional","affiliation":[]},{"given":"D. S.","family":"Phillips","sequence":"additional","affiliation":[]},{"given":"M.","family":"Seymour","sequence":"additional","affiliation":[]},{"given":"Q. Y.","family":"Tang","sequence":"additional","affiliation":[]},{"given":"B.","family":"Yang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3268-6986","authenticated-orcid":false,"given":"Z.","family":"Vernon","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7462-4516","authenticated-orcid":false,"given":"R. N.","family":"Alexander","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9485-4132","authenticated-orcid":false,"given":"D. H.","family":"Mahler","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,4]]},"URL":"http://dx.doi.org/10.1038/s41586-025-09044-5","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:9044"},{"id":"arxiv:1904.01351","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.102.012607","source":"Crossref","title":"Generation of a time-frequency grid state with integrated biphoton frequency combs","volume":"102","author":[{"given":"N.","family":"Fabre","sequence":"first","affiliation":[]},{"given":"G.","family":"Maltese","sequence":"additional","affiliation":[]},{"given":"F.","family":"Appas","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0447-5380","authenticated-orcid":true,"given":"S.","family":"Felicetti","sequence":"additional","affiliation":[]},{"given":"A.","family":"Ketterer","sequence":"additional","affiliation":[]},{"given":"A.","family":"Keller","sequence":"additional","affiliation":[]},{"given":"T.","family":"Coudreau","sequence":"additional","affiliation":[]},{"given":"F.","family":"Baboux","sequence":"additional","affiliation":[]},{"given":"M. I.","family":"Amanti","sequence":"additional","affiliation":[]},{"given":"S.","family":"Ducci","sequence":"additional","affiliation":[]},{"given":"P.","family":"Milman","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.012607","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012607","note":"arxivid:1904.01351"},{"id":"arxiv:2310.12618","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.132.170601","source":"Crossref","title":"Gottesman-Kitaev-Preskill Encoding in Continuous Modal Variables of Single Photons","volume":"132","author":[{"ORCID":"https://orcid.org/0000-0002-6911-452X","authenticated-orcid":true,"given":"Éloi","family":"Descamps","sequence":"first","affiliation":[{"name":"Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France"}]},{"given":"Arne","family":"Keller","sequence":"additional","affiliation":[{"name":"Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France"},{"name":"Department de Physique, Université Paris-Saclay, 91405 Orsay Cedex, France"}]},{"ORCID":"https://orcid.org/0000-0002-7579-7742","authenticated-orcid":true,"given":"Pérola","family":"Milman","sequence":"additional","affiliation":[{"name":"Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.132.170601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"170601","note":"update-to:{\"updated\":{\"date-parts\":[[2024,5,17]],\"date-time\":\"2024-05-17T00:00:00Z\",\"timestamp\":1715904000000},\"DOI\":\"10.1103/physrevlett.132.170601\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2024,5,17]],\"date-time\":\"2024-05-17T00:00:00Z\",\"timestamp\":1715904000000},\"DOI\":\"10.1103/physrevlett.132.170601\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2310.12618"},{"id":"arxiv:quant-ph/0008046","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.63.022309","source":"Crossref","title":"Secure quantum key distribution using squeezed states","volume":"63","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2001,1,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.63.022309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022309","note":"arxivid:quant-ph/0008046"},{"id":"manual:-A.-Doherty-M.-Gimeno-Segovia","type":"chapter","author":[{"family":"Doherty","given":"A."},{"family":"Gimeno-Segovia","given":"M."},{"family":"Litinski","given":"D."},{"family":"Nickerson","given":"N."},{"family":"Pant","given":"M."},{"family":"Rudolph","given":"T."},{"family":"Sparrow","given":"C."},{"family":"Psiquantum","given":"Corp"}],"container-title":"GENERATION AND MEASUREMENT OF ENTANGLED SYSTEMS OF PHOTONIC GKP QUBITS. U.S. Patent Application 18/273,753","issued":"2024"},{"id":"arxiv:1507.08966","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.93.075118","source":"Crossref","title":"Formalism for the solution of quadratic Hamiltonians with large cosine terms","volume":"93","author":[{"given":"Sriram","family":"Ganeshan","sequence":"first","affiliation":[]},{"given":"Michael","family":"Levin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2016,2,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.93.075118","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"075118","note":"arxivid:1507.08966"},{"id":"arxiv:1706.03011","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.119.180507","source":"Crossref","title":"Analog Quantum Error Correction with Encoding a Qubit into an Oscillator","volume":"119","author":[{"given":"Kosuke","family":"Fukui","sequence":"first","affiliation":[]},{"given":"Akihisa","family":"Tomita","sequence":"additional","affiliation":[]},{"given":"Atsushi","family":"Okamoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,11,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.119.180507","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180507","note":"arxivid:1706.03011"},{"id":"arxiv:2412.06715","type":"article-journal","author":[{"given":"Guo","family":"Zheng"},{"given":"Wenhao","family":"He"},{"given":"Gideon","family":"Lee"},{"given":"Kyungjoo","family":"Noh"},{"given":"Liang","family":"Jiang"}],"title":"Performance and achievable rates of the Gottesman-Kitaev-Preskill code for pure-loss and amplification channels","issued":{"date-parts":[[2024,12,9]]},"note":"arxivid:2412.06715\narxiv_version_number:1"},{"id":"arxiv:2505.10499","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>The Gottesman-Kitaev-Preskill (GKP) codes are known to achieve optimal rates under displacement noise and pure-loss channels, which establishes theoretical foundations for its optimality. However, such optimal rates are only known to be achieved at a discrete set of noise strengths with the current self-dual symplectic lattice construction. In this work, we develop a new coding strategy using concatenated continuous variable-discrete variable encodings to go beyond past results and establish GKP’s optimal rate over all noise strengths. In particular, for displacement noise, the rate is obtained through a constructive approach by concatenating GKP codes with a quantum polar code and analog decoding. For a pure-loss channel, we prove the existence of capacity-achieving GKP codes through a random coding approach. These results highlight the capability of concatenation-based GKP codes and provides new methods for constructing good GKP lattices.</jats:p>","DOI":"10.1103/56vj-z7h1","source":"Crossref","title":"Achievable Rates for Concatenated Square Gottesman-Kitaev-Preskill Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-4945-6693","authenticated-orcid":true,"given":"Mahadevan","family":"Subramanian","sequence":"first","affiliation":[{"name":"The University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0001-6338-0192","authenticated-orcid":true,"given":"Guo","family":"Zheng","sequence":"additional","affiliation":[{"name":"The University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"The University of Chicago"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,21]]},"URL":"http://dx.doi.org/10.1103/56vj-z7h1","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040342","note":"arxivid:2505.10499"},{"id":"arxiv:2408.04126","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevlett.134.100602","source":"Crossref","title":"Linear-Optical Quantum Computation with Arbitrary Error-Correcting Codes","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0003-4343-3087","authenticated-orcid":true,"given":"Blayney W.","family":"Walshe","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]},{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":true,"given":"Ben Q.","family":"Baragiola","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."},{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"ORCID":"https://orcid.org/0000-0002-5534-6557","authenticated-orcid":true,"given":"Hugo","family":"Ferretti","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]},{"ORCID":"https://orcid.org/0009-0006-3609-5699","authenticated-orcid":true,"given":"José","family":"Gefaell","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]},{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":true,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."},{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Waterloo"}]},{"ORCID":"https://orcid.org/0000-0001-5493-9740","authenticated-orcid":true,"given":"Ryohei","family":"Weil","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]},{"ORCID":"https://orcid.org/0000-0003-4164-4307","authenticated-orcid":true,"given":"Takaya","family":"Matsuura","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."},{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"ORCID":"https://orcid.org/0009-0008-4591-3686","authenticated-orcid":true,"given":"Thomas","family":"Jaeken","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]},{"ORCID":"https://orcid.org/0000-0002-8688-0656","authenticated-orcid":true,"given":"Giacomo","family":"Pantaleoni","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."},{"id":[{"id":"https://ror.org/01sf06y89","id-type":"ROR","asserted-by":"publisher"}],"name":"Macquarie University"}]},{"ORCID":"https://orcid.org/0009-0002-8116-6581","authenticated-orcid":true,"given":"Zhihua","family":"Han","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]},{"ORCID":"https://orcid.org/0000-0002-1476-0647","authenticated-orcid":true,"given":"Timo","family":"Hillmann","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."},{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"ORCID":"https://orcid.org/0000-0001-5201-3987","authenticated-orcid":true,"given":"Ilan","family":"Tzitrin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]},{"ORCID":"https://orcid.org/0000-0002-7462-4516","authenticated-orcid":true,"given":"Rafael N.","family":"Alexander","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu Quantum Technologies Inc."}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.100602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"100602","note":"arxivid:2408.04126"},{"id":"arxiv:2505.06385","type":"article-journal","author":[{"given":"Shantom K.","family":"Borah"},{"given":"Asit K.","family":"Pradhan"},{"given":"Nithin","family":"Raveendran"},{"given":"Michele","family":"Pacenti"},{"given":"Bane","family":"Vasic"}],"title":"Fault Tolerant Decoding of QLDPC-GKP Codes with Circuit Level Soft Information","issued":{"date-parts":[[2025,5,9]]},"note":"arxivid:2505.06385\narxiv_version_number:1"},{"id":"arxiv:2212.11397","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.108.052428","source":"Crossref","title":"Biased Gottesman-Kitaev-Preskill repetition code","volume":"108","author":[{"ORCID":"https://orcid.org/0000-0001-9938-3854","authenticated-orcid":true,"given":"Matthew P.","family":"Stafford","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.108.052428","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052428","note":"arxivid:2212.11397"},{"id":"arxiv:2112.14447","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.104.062434","source":"Crossref","title":"Quantum error correction with the color-Gottesman-Kitaev-Preskill code","volume":"104","author":[{"ORCID":"https://orcid.org/0000-0002-7836-0394","authenticated-orcid":true,"given":"Jiaxuan","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Jian","family":"Zhao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8997-3030","authenticated-orcid":true,"given":"Yu-Chun","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Guo-Ping","family":"Guo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.104.062434","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062434","note":"arxivid:2112.14447"},{"id":"arxiv:2410.12365","type":"article-journal","author":[{"given":"Takaya","family":"Matsuura"},{"given":"Nicolas C.","family":"Menicucci"},{"given":"Hayata","family":"Yamasaki"}],"title":"Continuous-Variable Fault-Tolerant Quantum Computation under General Noise","issued":{"date-parts":[[2024,10,16]]},"note":"arxivid:2410.12365\narxiv_version_number:1"},{"id":"arxiv:2309.16241","type":"article-journal","author":[{"given":"Robert","family":"König"},{"given":"Cambyse","family":"Rouzé"}],"title":"Limitations of local update recovery in stabilizer-GKP codes: a quantum optimal transport approach","issued":{"date-parts":[[2023,9,28]]},"note":"arxivid:2309.16241\narxiv_version_number:1"},{"id":"arxiv:2402.09333","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>A promising route towards fault-tolerant quantum error correction is the concatenation of a Gottesman-Kitaev-Preskill (GKP) code with a qubit code. Development of such concatenated codes requires simulation tools which realistically model noise, while being able to simulate the dynamics of many modes. However, so far, large-scale simulation tools for concatenated GKP codes have been limited to idealized noise models and GKP code implementations. Here, we introduce the Bosonic Pauli+ model (BP+), which can be simulated efficiently for a large number of modes, while capturing the rich dynamics in the bosonic multi-mode Hilbert space. We demonstrate the method by simulating a hybrid surface code, where the data qubits are finite-energy GKP qubits stabilized using the small-Big-small (sBs) protocol, and the syndrome qubits are standard two-level systems. Using BP+, we present logical error rates of such an implementation. Confidence in the accuracy of the method is gained by comparing its predictions with full time evolution simulations for several relevant quantum circuits. While developed specifically for GKP qubits stabilized using the sBs protocol, the mathematical structure of BP+ is generic and may be applicable also to the simulation of concatenations using other bosonic codes.</jats:p>","DOI":"10.22331/q-2024-11-26-1539","page":"1539","source":"Crossref","title":"Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes","volume":"8","author":[{"given":"Florian","family":"Hopfmueller","sequence":"first","affiliation":[{"name":"Nord Quantique, Sherbrooke, Québec, Canada"}]},{"given":"Maxime","family":"Tremblay","sequence":"additional","affiliation":[{"name":"Nord Quantique, Sherbrooke, Québec, Canada"}]},{"given":"Philippe","family":"St-Jean","sequence":"additional","affiliation":[{"name":"Nord Quantique, Sherbrooke, Québec, Canada"}]},{"given":"Baptiste","family":"Royer","sequence":"additional","affiliation":[{"name":"Institut Quantique and Départment de Physique, Université de Sherbrooke, Québec, Canada"}]},{"given":"Marc-Antoine","family":"Lemonde","sequence":"additional","affiliation":[{"name":"Nord Quantique, Sherbrooke, Québec, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,26]]},"URL":"http://dx.doi.org/10.22331/q-2024-11-26-1539","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2402.09333"},{"id":"arxiv:2102.01374","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.131.170603","source":"Crossref","title":"Efficient Concatenated Bosonic Code for Additive Gaussian Noise","volume":"131","author":[{"ORCID":"https://orcid.org/0000-0002-8231-7241","authenticated-orcid":true,"given":"Kosuke","family":"Fukui","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4164-4307","authenticated-orcid":true,"given":"Takaya","family":"Matsuura","sequence":"additional","affiliation":[]},{"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.131.170603","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"170603","note":"arxivid:2102.01374"},{"id":"arxiv:2411.04277","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.111.052445","source":"Crossref","title":"Exploring the quantum capacity of a Gaussian random-displacement channel using Gottesman-Kitaev-Preskill codes and maximum-likelihood decoding","volume":"111","author":[{"ORCID":"https://orcid.org/0009-0002-6102-5730","authenticated-orcid":true,"given":"Mao","family":"Lin","sequence":"first","affiliation":[{"name":"Amazon Braket"}]},{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.111.052445","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052445","note":"arxivid:2411.04277"},{"id":"arxiv:1908.03579","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.101.012316","source":"Crossref","title":"Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code","volume":"101","author":[{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,1,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.101.012316","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012316","note":"arxivid:1908.03579"},{"id":"arxiv:2101.03014","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.2.030325","source":"Crossref","title":"Fault-Tolerant Continuous-Variable Measurement-based Quantum Computation Architecture","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-7019-1506","authenticated-orcid":true,"given":"Mikkel V.","family":"Larsen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8585-0068","authenticated-orcid":true,"given":"Jonas S.","family":"Neergaard-Nielsen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1990-7687","authenticated-orcid":true,"given":"Ulrik L.","family":"Andersen","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,11]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.030325","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030325","note":"arxivid:2101.03014"},{"id":"arxiv:2103.06994","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.3.010315","source":"Crossref","title":"Low-Overhead Fault-Tolerant Quantum Error Correction with the Surface-GKP Code","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":true,"given":"Kyungjoo","family":"Noh","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"additional","affiliation":[]},{"given":"Fernando G.S.L.","family":"Brandão","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,1,28]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.010315","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010315","note":"arxivid:2103.06994"},{"id":"arxiv:2207.04383","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.107.062408","source":"Crossref","title":"Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0002-7836-0394","authenticated-orcid":true,"given":"Jiaxuan","family":"Zhang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8997-3030","authenticated-orcid":true,"given":"Yu-Chun","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Guo-Ping","family":"Guo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.062408","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062408","note":"arxivid:2207.04383"},{"id":"arxiv:2412.02442","type":"thesis","categories":["quantum error correction","quantum information","lattice theory","coding theory","cryptography","Hamiltonian engineering","algebraic geometry","Physics"],"language":"en","author":[{"family":"Conrad","given":"Jonathan"}],"contributor":[{"literal":"Universitätsbibliothek Der FU Berlin"},{"literal":"Universitätsbibliothek Der FU Berlin"}],"issued":{"date-parts":[[2024]]},"abstract":"Quantum error correction is an essential ingredient in the development of quan- tum technologies. Its core subject is to investigate ways to embed quantum Hilbert spaces into a physical system such that this subspace is robust against small imperfections in the physical systems. This task is exceedingly complex: for one, this is due to the vast diversity of possible physical systems with dif- ferent inherent structure to use. For another, every different physical setting also comes with different types of dominant imperfections that need to be protected against. Bred by the complexity of this technological ambition, research on quantum error correction has developed into a large field of research that ranges from questions about the engineering of small systems with a single photon to the creation of macroscopic topological phases of matter and models of complex emergent physics. A quintessential tool in quantum error correction is the stabilizer formalism, which tames complicated quantum systems by enforcing symmetries. A Gottesman-Kitaev-Preskill (GKP) code is a stabilizer code that creates a log- ical subspace within the infinite dimensional Hilbert space of a collection of quantum harmonic oscillators by endowing it with translational symmetries. While practical approaches to GKP codes consider the infinitude of the Hilbert space, as well as the infinitude of the translational symmetry group as obstacles for implementation, in theory these are precisely the features that make the theory of GKP codes particularly rich, well behaved and well-connected to fascinating topics in mathematics. The purpose of this thesis is to explore these connections: to understand the coding theoretic- and practical properties of GKP codes, utilizing its rich mathematical foundation, and to provide a foundation for future research. Along this journey we discover – through the looking glass of GKP codes – how quantum error correction in general fits into a fabulous mathematical world and formulate a series of dreams about possible directions of research.","DOI":"10.17169/REFUBIUM-45505","publisher":"Freie Universität Berlin","title":"The fabulous world of GKP codes","URL":"https://refubium.fu-berlin.de/handle/fub188/45792","note":"copyright:Creative Commons Attribution 4.0 International\narxivid:2412.02442"},{"id":"arxiv:2407.03270","type":"article-journal","author":[{"given":"Jonathan","family":"Conrad"},{"given":"Ansgar G.","family":"Burchards"},{"given":"Steven T.","family":"Flammia"}],"title":"Lattices, Gates, and Curves: GKP codes as a Rosetta stone","issued":{"date-parts":[[2024,7,10]]},"note":"arxivid:2407.03270\narxiv_version_number:2"},{"id":"doi:10.1109/ITW61385.2024.10806993","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw61385.2024.10806993","page":"615-620","source":"Crossref","title":"A Tighter Distance Upper-Bound for Gottesman-Kitaev-Preskill Codes","author":[{"given":"Shobhit","family":"Bhatnagar","sequence":"first","affiliation":[{"name":"IISc Bangalore,Department of Electrical Communication Engineering"}]},{"given":"P. Vijay","family":"Kumar","sequence":"additional","affiliation":[{"name":"IISc Bangalore,Department of Electrical Communication Engineering"}]}],"event":"2024 IEEE Information Theory Workshop (ITW)","container-title":"2024 IEEE Information Theory Workshop (ITW)","original-title":[],"issued":{"date-parts":[[2024,11,24]]},"URL":"http://dx.doi.org/10.1109/ITW61385.2024.10806993"},{"id":"arxiv:1708.07257","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/aac11a","page":"063025","source":"Crossref","title":"Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0003-3132-1088","authenticated-orcid":false,"given":"Kunal","family":"Sharma","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3916-4462","authenticated-orcid":false,"given":"Mark M","family":"Wilde","sequence":"additional","affiliation":[]},{"given":"Sushovit","family":"Adhikari","sequence":"additional","affiliation":[]},{"given":"Masahiro","family":"Takeoka","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2018,6,15]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aac11a","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1708.07257"},{"id":"arxiv:1801.04731","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Thermal attenuator channels model the decoherence of quantum systems interacting with a thermal bath, e.g., a two-level system subject to thermal noise and an electromagnetic signal traveling through a fiber or in free-space. Hence determining the quantum capacity of these channels is an outstanding open problem for quantum computation and communication. Here we derive several upper bounds on the quantum capacity of qubit and bosonic thermal attenuators. We introduce an extended version of such channels which is degradable and hence has a single-letter quantum capacity, bounding that of the original thermal attenuators. Another bound for bosonic attenuators is given by the bottleneck inequality applied to a particular channel decomposition. With respect to previously known bounds we report better results in a broad range of attenuation and noise: we can now approximate the quantum capacity up to a negligible uncertainty for most practical applications, e.g., for low thermal noise.</jats:p>","DOI":"10.1038/s41467-018-06848-0","source":"Crossref","title":"Narrow bounds for the quantum capacity of thermal attenuators","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0002-8972-2936","authenticated-orcid":false,"given":"Matteo","family":"Rosati","sequence":"first","affiliation":[]},{"given":"Andrea","family":"Mari","sequence":"additional","affiliation":[]},{"given":"Vittorio","family":"Giovannetti","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2018,10,18]]},"URL":"http://dx.doi.org/10.1038/s41467-018-06848-0","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"4339","note":"updated-by:{\"updated\":{\"date-parts\":[[2019,1,15]],\"date-time\":\"2019-01-15T00:00:00Z\",\"timestamp\":1547510400000},\"DOI\":\"10.1038/s41467-019-08338-3\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nalternative-id:6848\narxivid:1801.04731"},{"id":"arxiv:2406.06418","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Although the similarity between nonstabilizer states—also known as magic states—in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer Rényi entropy—a recent proposal for a magic measure for multiqubit states. We further provide the magic measure with an operational interpretation as a runtime of a classical simulation algorithm. In addition, we employ our results to prove that implementing a multiqubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.</jats:p>","DOI":"10.1103/prxquantum.6.010330","source":"Crossref","title":"Bridging Magic and Non-Gaussian Resources via Gottesman-Kitaev-Preskill Encoding","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-1677-8696","authenticated-orcid":true,"given":"Oliver","family":"Hahn","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/040wg7k59","id-type":"ROR","asserted-by":"publisher"}],"name":"Chalmers University of Technology, Sweden"},{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]},{"ORCID":"https://orcid.org/0000-0002-7130-6723","authenticated-orcid":true,"given":"Giulia","family":"Ferrini","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/040wg7k59","id-type":"ROR","asserted-by":"publisher"}],"name":"Chalmers University of Technology, Sweden"}]},{"ORCID":"https://orcid.org/0000-0003-3837-8159","authenticated-orcid":true,"given":"Ryuji","family":"Takagi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,18]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.010330","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010330","note":"arxivid:2406.06418"},{"id":"arxiv:2309.07820","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Continuous-variable bosonic systems stand as prominent candidates for implementing quantum computational tasks. While various necessary criteria have been established to assess their resourcefulness, sufficient conditions have remained elusive. We address this gap by focusing on promoting circuits that are otherwise simulatable to computational universality. The class of simulatable, albeit non-Gaussian, circuits that we consider is composed of Gottesman-Kitaev-Preskill (GKP) states, Gaussian operations, and homodyne measurements. Based on these circuits, we first introduce a general framework for mapping a continuous-variable state into a qubit state. Subsequently, we cast existing maps into this framework, including the modular and stabilizer subsystem decompositions. By combining these findings with established results for discrete-variable systems, we formulate a sufficient condition for achieving universal quantum computation. Leveraging this, we evaluate the computational resourcefulness of a variety of states, including Gaussian states, finite-squeezing GKP states, and cat states. Furthermore, our framework reveals that both the stabilizer subsystem decomposition and the modular subsystem decomposition (of position-symmetric states) can be constructed in terms of simulatable operations. This establishes a robust resource-theoretical foundation for employing these techniques to evaluate the logical content of a generic continuous-variable state, which can be of independent interest.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020337","source":"Crossref","title":"Sufficient Condition for Universal Quantum Computation Using Bosonic Circuits","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0001-7654-9356","authenticated-orcid":true,"given":"Cameron","family":"Calcluth","sequence":"first","affiliation":[{"name":"Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology"}]},{"given":"Nicolas","family":"Reichel","sequence":"additional","affiliation":[{"name":"Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-7579-6336","authenticated-orcid":true,"given":"Alessandro","family":"Ferraro","sequence":"additional","affiliation":[{"name":"Centre for Theoretical Atomic, Molecular and Optical Physics, Queen’s University Belfast"},{"name":"Dipartimento di Fisica “Aldo Pontremoli,”, Università degli Studi di Milano"}]},{"ORCID":"https://orcid.org/0000-0002-7130-6723","authenticated-orcid":true,"given":"Giulia","family":"Ferrini","sequence":"additional","affiliation":[{"name":"Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,17]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020337","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020337","note":"arxivid:2309.07820"},{"id":"arxiv:2412.13136","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Classically simulating circuits with bosonic codes is challenging due to the prohibitive cost of simulating quantum systems with many, possibly infinite, energy levels. We propose an algorithm to simulate circuits with encoded Gottesman-Kitaev-Preskill (GKP) states, specifically for odd-dimensional encoded qudits. Our approach is tailored to be especially effective in the most challenging but practically relevant regime, where the codeword states exhibit high (but finite) squeezing. Our algorithm leverages the Zak-Gross Wigner function introduced by Davis, Fabre, and Chabaud, which represents infinitely squeezed encoded stabilizer states positively. The run-time of the algorithm scales with the negativity of the Wigner function, allowing for efficient simulation of certain large-scale circuits—namely, input stabilizer GKP states undergoing generalized GKP-encoded Clifford operations followed by modular measurement—with a high degree of squeezing. For stabilizer GKP states exhibiting 12 dB of squeezing, our algorithm can simulate circuits with up to 1000 modes with less than double the number of samples required for a single input mode, which is in stark contrast to existing simulators. Therefore, this approach holds significant potential for benchmarking early implementations of quantum computing architectures utilizing bosonic codes.</jats:p>","DOI":"10.1103/xmtw-g54f","source":"Crossref","title":"Classical Simulation of Circuits with Realistic Odd-Dimensional Gottesman-Kitaev-Preskill States","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0001-7654-9356","authenticated-orcid":true,"given":"Cameron","family":"Calcluth","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/040wg7k59","id-type":"ROR","asserted-by":"publisher"}],"name":"Chalmers University of Technology"}]},{"ORCID":"https://orcid.org/0000-0003-1677-8696","authenticated-orcid":true,"given":"Oliver","family":"Hahn","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/040wg7k59","id-type":"ROR","asserted-by":"publisher"}],"name":"Chalmers University of Technology"},{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]},{"ORCID":"https://orcid.org/0000-0003-3727-8092","authenticated-orcid":true,"given":"Juani","family":"Bermejo-Vega","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04njjy449","id-type":"ROR","asserted-by":"publisher"}],"name":"Universidad de Granada"}]},{"ORCID":"https://orcid.org/0000-0002-7579-6336","authenticated-orcid":true,"given":"Alessandro","family":"Ferraro","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00wjc7c48","id-type":"ROR","asserted-by":"publisher"}],"name":"Università degli Studi di Milano"},{"id":[{"id":"https://ror.org/00hswnk62","id-type":"ROR","asserted-by":"publisher"}],"name":"Queen’s University Belfast"}]},{"ORCID":"https://orcid.org/0000-0002-7130-6723","authenticated-orcid":true,"given":"Giulia","family":"Ferrini","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/040wg7k59","id-type":"ROR","asserted-by":"publisher"}],"name":"Chalmers University of Technology"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,1]]},"URL":"http://dx.doi.org/10.1103/xmtw-g54f","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"010601","note":"arxivid:2412.13136"},{"id":"arxiv:2109.13018","type":"journal-article","publisher":"American Physical Society (APS)","issue":"21","DOI":"10.1103/physrevlett.128.210502","source":"Crossref","title":"Quantifying Qubit Magic Resource with Gottesman-Kitaev-Preskill Encoding","volume":"128","author":[{"ORCID":"https://orcid.org/0000-0003-1677-8696","authenticated-orcid":true,"given":"Oliver","family":"Hahn","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7579-6336","authenticated-orcid":true,"given":"Alessandro","family":"Ferraro","sequence":"additional","affiliation":[]},{"given":"Lina","family":"Hultquist","sequence":"additional","affiliation":[]},{"given":"Giulia","family":"Ferrini","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3367-8083","authenticated-orcid":true,"given":"Laura","family":"García-Álvarez","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.128.210502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"210502","note":"update-to:{\"updated\":{\"date-parts\":[[2022,5,27]],\"date-time\":\"2022-05-27T00:00:00Z\",\"timestamp\":1653609600000},\"DOI\":\"10.1103/physrevlett.128.210502\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2022,5,27]],\"date-time\":\"2022-05-27T00:00:00Z\",\"timestamp\":1653609600000},\"DOI\":\"10.1103/physrevlett.128.210502\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2109.13018"},{"id":"arxiv:2009.05309","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.103.032409","source":"Crossref","title":"Unsuitability of cubic phase gates for non-Clifford operations on Gottesman-Kitaev-Preskill states","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0003-1333-7430","authenticated-orcid":true,"given":"Jacob","family":"Hastrup","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7019-1506","authenticated-orcid":true,"given":"Mikkel V.","family":"Larsen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8585-0068","authenticated-orcid":true,"given":"Jonas S.","family":"Neergaard-Nielsen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1990-7687","authenticated-orcid":true,"given":"Ulrik L.","family":"Andersen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,3,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.032409","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032409","note":"arxivid:2009.05309"},{"id":"arxiv:2411.00235","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We consider the task of performing shadow tomography of a logical subsystem defined via the Gottesman--Kitaev--Preskill (GKP) error correcting code. Our protocol does not require the input state to be a code state but is implemented by appropriate twirling of the measurement channel, such that the encoded logical tomographic information becomes encoded in the classical shadow. We showcase this protocol for measurements natural in continuous variable (CV) quantum computing. For heterodyne measurement, the protocol yields a probabilistic decomposition of any input state into Gaussian states that simulate the encoded logical information of the input relative to a fixed GKP code where we prove bounds on the Gaussian compressibility of states in this setting. For photon parity measurements, our protocol is equivalent to a Wigner sampling protocol for which we develop the appropriate sampling strategies. Finally, by randomizing over the reference GKP code, we show how Wigner samples of any input state relative to a random GKP codes can be used to estimate any sufficiently bounded observable.</jats:p>","DOI":"10.22331/q-2026-01-19-1973","page":"1973","source":"Crossref","title":"Chasing shadows with Gottesman-Kitaev-Preskill codes","volume":"10","author":[{"given":"Jonathan","family":"Conrad","sequence":"first","affiliation":[{"name":"Institute of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne CH-1015, Switzerland"},{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany"}]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Virginia Tech, Alexandria, USA"},{"name":"Phasecraft Inc., Washington DC, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,19]]},"URL":"http://dx.doi.org/10.22331/q-2026-01-19-1973","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2411.00235"},{"id":"arxiv:2509.14658","type":"article-journal","author":[{"given":"Lukas","family":"Brenner"},{"given":"Beatriz","family":"Dias"},{"given":"Robert","family":"Koenig"}],"title":"Composable logical gate error in approximate quantum error correction: reexamining gate implementations in Gottesman-Kitaev-Preskill codes","issued":{"date-parts":[[2025,12,18]]},"note":"arxivid:2509.14658\narxiv_version_number:3"},{"id":"arxiv:2504.13383","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/c8hk-v1qf","source":"Crossref","title":"Logical channels in approximate Gottesman-Kitaev-Preskill error correction","volume":"112","author":[{"ORCID":"https://orcid.org/0000-0002-2270-3087","authenticated-orcid":true,"given":"Mahnaz","family":"Jafarzadeh","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"ORCID":"https://orcid.org/0000-0001-6120-9930","authenticated-orcid":true,"given":"Jonathan","family":"Conrad","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02s376052","id-type":"ROR","asserted-by":"publisher"}],"name":"EPFL"}]},{"ORCID":"https://orcid.org/0000-0002-7462-4516","authenticated-orcid":true,"given":"Rafael N.","family":"Alexander","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"ORCID":"https://orcid.org/0000-0003-3566-2955","authenticated-orcid":true,"given":"Ben Q.","family":"Baragiola","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"},{"id":[{"id":"https://ror.org/00rnbty21","id-type":"ROR","asserted-by":"publisher"}],"name":"Centre for Quantum Computation and Communication Technology"},{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"},{"id":[{"id":"https://ror.org/0589kgd85","id-type":"ROR","asserted-by":"publisher"}],"name":"Yukawa Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/02kpeqv85","id-type":"ROR","asserted-by":"publisher"}],"name":"Kyoto University"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,8]]},"URL":"http://dx.doi.org/10.1103/c8hk-v1qf","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062413","note":"arxivid:2504.13383"},{"id":"doi:10.1109/TIT.2002.800499","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2002.800499","page":"2201-2214","source":"Crossref","title":"Closest point search in lattices","volume":"48","author":[{"given":"E.","family":"Agrell","sequence":"first","affiliation":[]},{"given":"T.","family":"Eriksson","sequence":"additional","affiliation":[]},{"given":"A.","family":"Vardy","sequence":"additional","affiliation":[]},{"given":"K.","family":"Zeger","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2002,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.800499","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.1007/bf02579403","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf02579403","page":"1-13","source":"Crossref","title":"On Lovász’ lattice reduction and the nearest lattice point problem","volume":"6","author":[{"given":"L.","family":"Babai","sequence":"first","affiliation":[]}],"container-title":"Combinatorica","original-title":[],"language":"en","issued":{"date-parts":[[1986,3]]},"URL":"http://dx.doi.org/10.1007/bf02579403","ISSN":["0209-9683","1439-6912"],"container-title-short":"Combinatorica","note":"alternative-id:BF02579403"},{"id":"arxiv:1309.7062","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"},{"given":"Lucy Liuxuan","family":"Zhang"}],"title":"Fibre bundle framework for unitary quantum fault tolerance","issued":{"date-parts":[[2017,4,25]]},"note":"arxivid:1309.7062\narxiv_version_number:2"},{"id":"arxiv:2308.02913","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.pquantelec.2023.100496","page":"100496","source":"Crossref","title":"Advances in bosonic quantum error correction with Gottesman–Kitaev–Preskill Codes: Theory, engineering and applications","volume":"93","author":[{"given":"Anthony J.","family":"Brady","sequence":"first","affiliation":[]},{"given":"Alec","family":"Eickbusch","sequence":"additional","affiliation":[]},{"given":"Shraddha","family":"Singh","sequence":"additional","affiliation":[]},{"given":"Jing","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Quntao","family":"Zhuang","sequence":"additional","affiliation":[]}],"container-title":"Progress in Quantum Electronics","original-title":[],"language":"en","issued":{"date-parts":[[2024,1]]},"URL":"http://dx.doi.org/10.1016/j.pquantelec.2023.100496","ISSN":["0079-6727"],"container-title-short":"Progress in Quantum Electronics","note":"special_numbering:C\nalternative-id:S0079672723000459\narxivid:2308.02913"},{"id":"arxiv:2507.06943","type":"article-journal","author":[{"given":"Richard A.","family":"Wolf"},{"given":"Pavithran","family":"Iyer"}],"title":"No physics required! A visual-based introduction to GKP qubits for computer scientists","issued":{"date-parts":[[2025,7,9]]},"note":"arxivid:2507.06943\narxiv_version_number:1"},{"id":"arxiv:2312.16298","type":"journal-article","publisher":"IOP Publishing","issue":"16","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Quantum gravity in 4D asymptotically flat spacetimes features spontaneous symmetry breaking due to soft radiation hair, intimately tied to the proliferation of IR divergences. A holographic description via a putative 2D CFT is expected free of such redundancies. In this series of two papers, we address this issue by initiating the study of quantum error correction in celestial CFT (CCFT). In part I we construct a toy model with finite degrees of freedom by revisiting noncommutative geometry in Kleinian hyperkähler spacetimes. The model obeys a Wick algebra that renormalizes in the radial direction and admits an isometric embedding á la Gottesman–Kitaev–Preskill. The code subspace is composed of two-qubit stabilizer states which are robust under soft spacetime fluctuations. Symmetries of the hyperkähler space become discrete and translate into the Clifford group familiar from quantum computation. The construction is then embedded into the incidence relation of twistor space, paving the way for the CCFT regime addressed in follow-up work.</jats:p>","DOI":"10.1088/1361-6382/adf686","page":"165006","source":"Crossref","title":"Celestial quantum error correction: I. Qubits from noncommutative Klein space","volume":"42","author":[{"given":"Alfredo","family":"Guevara","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3548-8574","authenticated-orcid":true,"given":"Yangrui","family":"Hu","sequence":"additional","affiliation":[]}],"container-title":"Classical and Quantum Gravity","original-title":[],"issued":{"date-parts":[[2025,8,13]]},"URL":"http://dx.doi.org/10.1088/1361-6382/adf686","ISSN":["0264-9381","1361-6382"],"container-title-short":"Class. Quantum Grav.","note":"arxivid:2312.16298"},{"id":"arxiv:2412.19653","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>A holographic CFT description of asymptotically flat spacetimes inherits vacuum degeneracies and IR divergences from its gravitational dual. We devise a Quantum Error Correcting (QEC) framework to encode both effects as correctable fluctuations on the CFT dual. The framework is physically motivated by embedding a chain of qudits in the so-called Klein spacetime and then taking a continuum 𝑁 → ∞ limit. At finite 𝑁 the qudit chain 1) enjoys a discrete version of celestial symmetries and 2) supports a Gottesman-Kitaev-Preskill (GKP) code. The limit results in hard states with quantized BMS hair in the celestial torus forming the logical subspace, robust under errors induced by soft radiation. Technically, the construction leverages the recently studied 𝑤<jats:sub>1+∞</jats:sub> hierarchy of soft currents and its realization from a sigma model in twistor space.</jats:p>","DOI":"10.1007/jhep06(2025)121","source":"Crossref","title":"Celestial Quantum Error Correction. Part II. From qudits to celestial CFT","volume":"2025","author":[{"ORCID":"https://orcid.org/0000-0002-8963-6560","authenticated-orcid":false,"given":"Alfredo","family":"Guevara","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00f809463","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Advanced Study"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-3548-8574","authenticated-orcid":false,"given":"Yangrui","family":"Hu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Waterloo"}]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,12]]},"URL":"http://dx.doi.org/10.1007/JHEP06(2025)121","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"121","note":"alternative-id:26437\narxivid:2412.19653"},{"id":"arxiv:2303.02432","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We introduce a new class of random Gottesman-Kitaev-Preskill (GKP) codes derived from the cryptanalysis of the so-called NTRU cryptosystem. The derived codes are <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>g</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi></mml:math> in that they exhibit constant rate and average distance scaling <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x0394;</mml:mi><mml:mo>&amp;#x221D;</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt></mml:math> with high probability, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> is the number of bosonic modes, which is a distance scaling equivalent to that of a GKP code obtained by concatenating single mode GKP codes into a qubit-quantum error correcting code with linear distance. The derived class of NTRU-GKP codes has the additional property that <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>g</mml:mi></mml:math> for a stochastic displacement noise model is equivalent to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>r</mml:mi><mml:mi>y</mml:mi><mml:mi>p</mml:mi><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>g</mml:mi></mml:math> the NTRU cryptosystem, such that every random instance of the code naturally comes with an efficient decoder. This construction highlights how the GKP code bridges aspects of classical error correction, quantum error correction as well as post-quantum cryptography. We underscore this connection by discussing the computational hardness of decoding GKP codes and propose, as a new application, a simple public key quantum communication protocol with security inherited from the NTRU cryptosystem.</jats:p>","DOI":"10.22331/q-2024-07-04-1398","page":"1398","source":"Crossref","title":"Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0001-6120-9930","authenticated-orcid":false,"given":"Jonathan","family":"Conrad","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany"}]},{"ORCID":"https://orcid.org/0000-0003-3033-1292","authenticated-orcid":false,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany"},{"name":"Fraunhofer Heinrich Hertz Institute, Einsteinufer 37, 10587 Berlin, Germany"}]},{"ORCID":"https://orcid.org/0000-0002-5372-4825","authenticated-orcid":false,"given":"Jean-Pierre","family":"Seifert","sequence":"additional","affiliation":[{"name":"Electrical Engineering and Computer Science Department, Technische Universität Berlin, Straße des 17. Juni 135, 10587 Berlin, Germany"},{"name":"Fraunhofer Institute for Secure Information Technology, Rheinstraße 75, 64295 Darmstadt, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,4]]},"URL":"http://dx.doi.org/10.22331/q-2024-07-04-1398","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2303.02432"},{"id":"doi:10.1145/2591796.2591860","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/2591796.2591860","page":"293-302","source":"Crossref","title":"A quantum algorithm for computing the unit group of an arbitrary degree number field","author":[{"given":"Kirsten","family":"Eisenträger","sequence":"first","affiliation":[{"name":"The Pennsylvania State University and Harvard University"}]},{"given":"Sean","family":"Hallgren","sequence":"additional","affiliation":[{"name":"The Pennsylvania State University"}]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[{"name":"University of California, Santa Barbara and California Institute of Technology"}]},{"given":"Fang","family":"Song","sequence":"additional","affiliation":[{"name":"University of Waterloo"}]}],"event":"STOC '14: Symposium on Theory of Computing","container-title":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2014,5,31]]},"URL":"http://dx.doi.org/10.1145/2591796.2591860","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/2591796.2591860\n10.1145/2591796"},{"id":"doi:10.1007/BF01078777","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/bf01078777","page":"94-106","source":"Crossref","title":"Hypergeometric functions and toral manifolds","volume":"23","author":[{"given":"I. M.","family":"Gel'fand","sequence":"first","affiliation":[]},{"given":"A. V.","family":"Zelevinskii","sequence":"additional","affiliation":[]},{"given":"M. M.","family":"Kapranov","sequence":"additional","affiliation":[]}],"container-title":"Functional Analysis and Its Applications","original-title":[],"language":"en","issued":{"date-parts":[[1989]]},"URL":"http://dx.doi.org/10.1007/BF01078777","ISSN":["0016-2663","1573-8485"],"container-title-short":"Funct Anal Its Appl","note":"alternative-id:BF01078777"},{"id":"doi:10.1215/S0012-7094-94-07313-4","type":"journal-article","publisher":"Duke University Press","issue":"2","DOI":"10.1215/s0012-7094-94-07313-4","source":"Crossref","title":"Hypergeometric functions and rings generated by monomials","volume":"73","author":[{"given":"Alan","family":"Adolphson","sequence":"first","affiliation":[]}],"container-title":"Duke Mathematical Journal","original-title":[],"issued":{"date-parts":[[1994,2,1]]},"URL":"http://dx.doi.org/10.1215/S0012-7094-94-07313-4","ISSN":["0012-7094"],"container-title-short":"Duke Math. J."},{"id":"doi:10.1017/9780511777165.005","type":"book-chapter","publisher":"Cambridge University Press","DOI":"10.1017/9780511777165.005","page":"101-121","source":"Crossref","title":"A-Hypergeometric Functions","author":[{"given":"N.","family":"Takayama","sequence":"first","affiliation":[]}],"container-title":"Encyclopedia of Special Functions: The Askey-Bateman Project","original-title":[],"issued":{"date-parts":[[2020,10,15]]},"ISBN":["9780511777165","9781107003736"],"URL":"http://dx.doi.org/10.1017/9780511777165.005","note":"edition-number:1"},{"id":"arxiv:2311.13040","type":"article-journal","author":[{"given":"Zhi","family":"Li"},{"given":"Latham","family":"Boyle"}],"title":"The Penrose Tiling is a Quantum Error-Correcting Code","issued":{"date-parts":[[2024,1,25]]},"note":"arxivid:2311.13040\narxiv_version_number:2"},{"id":"arxiv:2509.16993","type":"article-journal","author":[{"given":"S. B.","family":"Korolev"},{"given":"T. Yu.","family":"Golubeva"}],"title":"Bosonic error correction codes based on states generated via particle number resolving measurements","issued":{"date-parts":[[2025,9,21]]},"note":"arxivid:2509.16993\narxiv_version_number:1"},{"id":"arxiv:2312.16000","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","DOI":"10.1007/s11128-024-04549-w","source":"Crossref","title":"Error correction using squeezed Fock states","volume":"23","author":[{"given":"S. B.","family":"Korolev","sequence":"first","affiliation":[]},{"given":"E. N.","family":"Bashmakova","sequence":"additional","affiliation":[]},{"given":"T. Yu.","family":"Golubeva","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,18]]},"URL":"http://dx.doi.org/10.1007/s11128-024-04549-w","ISSN":["1573-1332"],"container-title-short":"Quantum Inf Process","page":"354","note":"alternative-id:4549\narxivid:2312.16000"},{"id":"arxiv:2506.00300","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/97yt-nzg2","source":"Crossref","title":"Bosonic quantum error correction using squeezed Fock states","volume":"112","author":[{"ORCID":"https://orcid.org/0000-0002-3841-5426","authenticated-orcid":true,"given":"E. N.","family":"Bashmakova","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/023znxa73","id-type":"ROR","asserted-by":"publisher"}],"name":"St. Petersburg State University"}]},{"ORCID":"https://orcid.org/0000-0002-3055-0367","authenticated-orcid":true,"given":"S. B.","family":"Korolev","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/023znxa73","id-type":"ROR","asserted-by":"publisher"}],"name":"St. Petersburg State University"}]},{"ORCID":"https://orcid.org/0000-0002-1139-8531","authenticated-orcid":true,"given":"T. Yu.","family":"Golubeva","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/023znxa73","id-type":"ROR","asserted-by":"publisher"}],"name":"St. Petersburg State University"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,25]]},"URL":"http://dx.doi.org/10.1103/97yt-nzg2","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032434","note":"arxivid:2506.00300"},{"id":"arxiv:2510.04209","type":"article-journal","author":[{"given":"Yexiong","family":"Zeng"},{"given":"Fernando","family":"Quijandría"},{"given":"Clemens","family":"Gneiting"},{"given":"Franco","family":"Nori"}],"title":"Quantum Error Correction with Superpositions of Squeezed Fock States","issued":{"date-parts":[[2025,10,5]]},"note":"arxivid:2510.04209\narxiv_version_number:1"},{"id":"arxiv:2105.14777","type":"journal-article","publisher":"IOP Publishing","issue":"2","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>In this work, we develop the theory of quasi-exact fault-tolerant quantum (QEQ) computation, which uses qubits encoded into quasi-exact quantum error-correction codes (‘quasi codes’). By definition, a quasi code is a parametric approximate code that can become exact by tuning its parameters. The model of QEQ computation lies in between the two well-known ones: the usual noisy quantum computation without error correction and the usual fault-tolerant quantum computation, but closer to the later. Many notions of exact quantum codes need to be adjusted for the quasi setting. Here we develop quasi error-correction theory using quantum instrument, the notions of quasi universality, quasi code distances, and quasi thresholds, etc. We find a wide class of quasi codes which are called valence-bond-solid codes, and we use them as concrete examples to demonstrate QEQ computation.</jats:p>","DOI":"10.1088/1367-2630/ac4737","page":"023019","source":"Crossref","title":"Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes","volume":"24","author":[{"given":"Dong-Sheng","family":"Wang","sequence":"first","affiliation":[]},{"given":"Yun-Jiang","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Ningping","family":"Cao","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2022,2,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ac4737","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:2105.14777"},{"id":"arxiv:quant-ph/0503139","type":"article-journal","author":[{"given":"Claude","family":"Crepeau"},{"given":"Daniel","family":"Gottesman"},{"given":"Adam","family":"Smith"}],"title":"Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes","issued":{"date-parts":[[2005,3,15]]},"note":"arxivid:quant-ph/0503139\narxiv_version_number:1"},{"id":"arxiv:1706.09434","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-020-03689-1","page":"369-432","source":"Crossref","title":"Approximate Quantum Error Correction Revisited: Introducing the Alpha-Bit","volume":"374","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8627-5237","authenticated-orcid":false,"given":"Geoffrey","family":"Penington","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,2,10]]},"URL":"http://dx.doi.org/10.1007/s00220-020-03689-1","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:3689\narxivid:1706.09434"},{"id":"arxiv:0909.1466","type":"article-journal","author":[{"given":"Avraham","family":"Ben-Aroya"},{"given":"Amnon","family":"Ta-Shma"}],"title":"Approximate quantum error correction for correlated noise","issued":{"date-parts":[[2009,9,8]]},"note":"arxivid:0909.1466\narxiv_version_number:1"},{"id":"arxiv:1102.3809","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.107.080501","source":"Crossref","title":"Perturbative Quantum Error Correction","volume":"107","author":[{"given":"Cédric","family":"Bény","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,8,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.107.080501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080501","note":"arxivid:1102.3809"},{"id":"arxiv:0907.3386","type":"journal-article","publisher":"AIP Publishing","issue":"9","abstract":"<jats:p>In a unified framework, we estimate the following quantities of interest in quantum information theory: (1) the minimum-error distinguishability of arbitrary ensembles of mixed quantum states; (2) the approximate reversibility of quantum dynamics in terms of entanglement fidelity (This is referred to as “channel-adapted quantum error recovery” when applied to the composition of an encoding operation and a noise channel.); (3) the maximum overlap between a bipartite pure quantum state and a bipartite mixed-state that may be achieved by applying a local quantum operation to one part of the mixed-state; and (4) the conditional min-entropy of bipartite quantum states. A refined version of the author’s techniques [J. Tyson, J. Math. Phys. 50, 032016 (2009)] for bounding the first quantity is employed to give two-sided estimates of the remaining three quantities. We obtain a closed-form approximate reversal channel. Using a state-dependent Kraus decomposition, our reversal may be interpreted as a quadratically weighted version of that of Barnum and Knill [J. Math. Phys. 43, 2097 (2002)]. The relationship between our reversal and Barnum and Knill’s is therefore similar to the relationship between Holevo’s asymptotically optimal measurement [A. S. Kholevo, Theor. Probab. Appl. 23, 411 (1978)] and the “pretty good” measurement of Belavkin [Stochastics 1, 315 (1975)] and Hausladen and Wootters [J. Mod. Opt. 41, 2385 (1994)]. In particular, we obtain relatively simple reversibility estimates without negative matrix-powers at no cost in tightness of our bounds. Our recovery operation is found to significantly outperform the so-called “transpose channel” in the simple case of depolarizing noise acting on half of a maximally entangled state. Furthermore, our overlap results allow the entangled input state and the output target state to differ, thus obtaining estimates in a somewhat more general setting. Using a result of König et al. [IEEE Trans. Inf. Theory 55, 4337 (2009)], our maximum overlap estimate is used to bound the conditional min-entropy of arbitrary bipartite states. Our primary tool is “small angle” initialization of an abstract generalization of the iterative schemes of Ježek et al. [Phys. Rev. A 65, 060301 (2002)], Ježek et al. [Phys. Rev. A 68, 012305 (2003)], and Reimpell and Werner [Phys. Rev. Lett. 94, 080501 (2005)]. The monotonicity result of Reimpell [Ph.D. thesis, Technishe Universität, 2007] follows in greater generality.</jats:p>","DOI":"10.1063/1.3463451","source":"Crossref","title":"Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates","volume":"51","author":[{"given":"Jon","family":"Tyson","sequence":"first","affiliation":[{"name":"Harvard University Jefferson Laboratory, , Cambridge, Massachusetts 02138, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2010,9,1]]},"URL":"http://dx.doi.org/10.1063/1.3463451","ISSN":["0022-2488","1089-7658"],"page":"092204","note":"arxivid:0907.3386"},{"id":"arxiv:1103.0649","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.84.022333","source":"Crossref","title":"Approximate simulation of quantum channels","volume":"84","author":[{"given":"Cédric","family":"Bény","sequence":"first","affiliation":[]},{"given":"Ognyan","family":"Oreshkov","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,8,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.022333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022333","note":"arxivid:1103.0649"},{"id":"arxiv:2312.16991","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>The robustness of quantum memory against physical noises is measured by two methods: the exact and approximate quantum error correction (QEC) conditions for error recoverability and the decoder-dependent error threshold which assesses if the logical error rate diminishes with system size. Here we unravel their relations and propose a unified framework to extract an intrinsic error threshold from the approximate QEC condition, which could upper bound other decoder-dependent error thresholds. Our proof establishes that relative entropy, effectively measuring deviations from exact QEC conditions, serves as the order parameter delineating the transition from asymptotic recoverability to unrecoverability. Consequently, we establish a unified framework for determining the error threshold across both exact and approximate QEC codes, addressing errors originating from noise channels as well as those from code space imperfections. This result sharpens our comprehension of error thresholds across diverse QEC codes and error models.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevresearch.6.043258","source":"Crossref","title":"Extracting error thresholds through the framework of approximate quantum error correction condition","volume":"6","author":[{"given":"Yuanchen","family":"Zhao","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"},{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Frontier Science Center for Quantum Information"}]},{"ORCID":"https://orcid.org/0000-0002-1295-2362","authenticated-orcid":true,"given":"Dong E.","family":"Liu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Tsinghua University"},{"id":[{"id":"https://ror.org/04nqf9k60","id-type":"ROR","asserted-by":"publisher"}],"name":"Beijing Academy of Quantum Information Sciences"},{"id":[{"id":"https://ror.org/03cve4549","id-type":"ROR","asserted-by":"publisher"}],"name":"Frontier Science Center for Quantum Information"},{"name":"Hefei National Laboratory"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.6.043258","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"043258","note":"arxivid:2312.16991"},{"id":"arxiv:2410.07983","type":"article-journal","author":[{"given":"Mengxin","family":"Du"},{"given":"Chao","family":"Zhang"},{"given":"Yiu-Tung","family":"Poon"},{"given":"Bei","family":"Zeng"}],"title":"Characterizing Quantum Codes via the Coefficients in Knill-Laflamme Conditions","issued":{"date-parts":[[2024,10,10]]},"note":"arxivid:2410.07983\narxiv_version_number:1"},{"id":"arxiv:2504.20847","type":"article-journal","author":[{"given":"Chao","family":"Zhang"},{"given":"Zipeng","family":"Wu"},{"given":"Shilin","family":"Huang"},{"given":"Bei","family":"Zeng"}],"title":"Transversal Gates in Nonadditive Quantum Codes","issued":{"date-parts":[[2025,4,29]]},"note":"arxivid:2504.20847\narxiv_version_number:1"},{"id":"arxiv:2108.04434","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2022.3162264","page":"5234-5247","source":"Crossref","title":"Linear Programming Bounds for Approximate Quantum Error Correction Over Arbitrary Quantum Channels","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0003-1115-0074","authenticated-orcid":false,"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, The University of Sheffield, Sheffield, U.K."}]},{"ORCID":"https://orcid.org/0000-0003-1970-8167","authenticated-orcid":false,"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[{"name":"Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu, Taiwan"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2022.3162264","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2108.04434"},{"id":"arxiv:2112.01858","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.105.062438","source":"Crossref","title":"Nonlinear quantum error correction","volume":"105","author":[{"given":"Maximilian","family":"Reichert","sequence":"first","affiliation":[]},{"given":"Louis W.","family":"Tessler","sequence":"additional","affiliation":[]},{"given":"Marcel","family":"Bergmann","sequence":"additional","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]},{"given":"Tim","family":"Byrnes","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,6,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.062438","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062438","note":"arxivid:2112.01858"},{"id":"arxiv:1210.1281","type":"journal-article","publisher":"Hindawi Limited","abstract":"<jats:p>We review the progress in the last 20–30 years, during which we discovered that there are many new phases of matter that are beyond the traditional Landau symmetry breaking theory.  We discuss new “topological” phenomena, such as topological degeneracy that reveals the existence of those new phases—topologically ordered phases.  Just like zero viscosity defines the superfluid order, the new “topological” phenomena define the topological order at macroscopic level.  More recently, we found that at the microscopical level, topological order is due to long-range quantum entanglements.  Long-range quantum entanglements lead to many amazing emergent phenomena, such as fractional charges and fractional statistics.  Long-range quantum entanglements can even provide a unified origin of light and electrons; light is a fluctuation of long-range entanglements, and electrons are defects in long-range entanglements.</jats:p>","DOI":"10.1155/2013/198710","page":"1-20","source":"Crossref","title":"Topological Order: From Long-Range Entangled Quantum Matter to a Unified Origin of Light and Electrons","volume":"2013","author":[{"given":"Xiao-Gang","family":"Wen","sequence":"first","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada N2L 2Y5"},{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"},{"name":"Institute for Advanced Study, Tsinghua University, Beijing 100084, China"}]}],"container-title":"ISRN Condensed Matter Physics","original-title":[],"language":"en","issued":{"date-parts":[[2013,3,27]]},"URL":"http://dx.doi.org/10.1155/2013/198710","ISSN":["2090-7400"],"container-title-short":"ISRN Condensed Matter Physics","note":"alternative-id:198710\n198710\narxivid:1210.1281"},{"id":"arxiv:1402.5674","type":"article-journal","author":[{"given":"Leonard","family":"Susskind"}],"title":"Computational Complexity and Black Hole Horizons","issued":{"date-parts":[[2014,2,25]]},"note":"arxivid:1402.5674\narxiv_version_number:2"},{"id":"arxiv:1607.05256","type":"article-journal","author":[{"given":"Scott","family":"Aaronson"}],"title":"The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes","issued":{"date-parts":[[2016,7,18]]},"note":"arxivid:1607.05256\narxiv_version_number:1"},{"id":"arxiv:2310.04710","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"11","DOI":"10.1038/s41567-024-02621-x","page":"1798-1803","source":"Crossref","title":"Complexity and order in approximate quantum error-correcting codes","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0001-6948-2464","authenticated-orcid":false,"given":"Jinmin","family":"Yi","sequence":"first","affiliation":[]},{"given":"Weicheng","family":"Ye","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3402-9763","authenticated-orcid":false,"given":"Zi-Wen","family":"Liu","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,3]]},"URL":"http://dx.doi.org/10.1038/s41567-024-02621-x","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2621\narxivid:2310.04710"},{"id":"arxiv:2510.04453","type":"article-journal","author":[{"given":"Jinmin","family":"Yi"},{"given":"Ruizhi","family":"Liu"},{"given":"Zhi","family":"Li"}],"title":"Lovász Meets Lieb-Schultz-Mattis: Complexity in Approximate Quantum Error Correction","issued":{"date-parts":[[2025,10,6]]},"note":"arxivid:2510.04453\narxiv_version_number:1"},{"id":"arxiv:1707.09951","type":"journal-article","publisher":"IOP Publishing","issue":"2","DOI":"10.1088/1367-2630/aaa372","page":"023009","source":"Crossref","title":"An integrity measure to benchmark quantum error correcting memories","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0002-4894-8322","authenticated-orcid":false,"given":"Xiaosi","family":"Xu","sequence":"first","affiliation":[]},{"given":"Niel de","family":"Beaudrap","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8715-6440","authenticated-orcid":false,"given":"Joe","family":"O’Gorman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7766-5348","authenticated-orcid":false,"given":"Simon C","family":"Benjamin","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2018,2,5]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aaa372","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1707.09951"},{"id":"arxiv:0708.3658","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.77.012320","source":"Crossref","title":"Structured near-optimal channel-adapted quantum error correction","volume":"77","author":[{"given":"Andrew S.","family":"Fletcher","sequence":"first","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]},{"given":"Moe Z.","family":"Win","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,1,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.012320","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012320","note":"arxivid:0708.3658"},{"id":"arxiv:2204.03560","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum error correction is believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various constructions of quantum error-correcting codes (QECCs) have been developed, leading to many good code families. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g., <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>n</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> from 7 to 14. We also found new <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>6</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.</jats:p>","DOI":"10.22331/q-2022-10-06-828","page":"828","source":"Crossref","title":"Quantum variational learning for quantum error-correcting codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-5589-7503","authenticated-orcid":false,"given":"Chenfeng","family":"Cao","sequence":"first","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"https://orcid.org/0000-0002-2093-7496","authenticated-orcid":false,"given":"Chao","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"https://orcid.org/0000-0002-9349-1325","authenticated-orcid":false,"given":"Zipeng","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]},{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gdansk, 80-309 Gdansk, Poland"}]},{"ORCID":"https://orcid.org/0000-0003-3989-4948","authenticated-orcid":false,"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,10,6]]},"URL":"http://dx.doi.org/10.22331/q-2022-10-06-828","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2204.03560"},{"id":"arxiv:quant-ph/0606105","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.71.022322","source":"Crossref","title":"Suboptimal quantum-error-correcting procedure based on semidefinite programming","volume":"71","author":[{"given":"Naoki","family":"Yamamoto","sequence":"first","affiliation":[]},{"given":"Shinji","family":"Hara","sequence":"additional","affiliation":[]},{"given":"Koji","family":"Tsumura","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,2,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.022322","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022322","note":"arxivid:quant-ph/0606105"},{"id":"arxiv:0905.3838","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.80.012326","source":"Crossref","title":"Robustness of channel-adapted quantum error correction","volume":"80","author":[{"given":"Gábor","family":"Balló","sequence":"first","affiliation":[]},{"given":"Péter","family":"Gurin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,7,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.012326","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012326","note":"arxivid:0905.3838"},{"id":"arxiv:quant-ph/0702005","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"01","abstract":"<jats:p> We give a short proof that the coherent information is an achievable rate for the transmission of quantum information through a noisy quantum channel. Our method is to produce random codes by performing a unitarily covariant projective measurement on a typical subspace of a tensor power state. We show that, provided the rank of each measurement operator is sufficiently small, the transmitted data will, with high probability, be decoupled from the channel environment. We also show that our construction leads to random codes whose average input is close to a product state and outline a modification yielding unitarily invariant ensembles of maximally entangled codes. </jats:p>","DOI":"10.1142/s1230161208000043","page":"7-19","source":"Crossref","title":"A Decoupling Approach to the Quantum Capacity","volume":"15","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[{"name":"School of Computer Science, McGill University, Montreal, Quebec, Canada"}]},{"given":"Michał","family":"Horodecki","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Poland"},{"name":"National Quantum Information Centre of Gdańsk, Gen. W. Andersa 27, 81 – 824 Sopot, Poland"}]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bristol, UK"}]},{"given":"Jon","family":"Yard","sequence":"additional","affiliation":[{"name":"CNLS (Center for Nonlinear Studies), CCS-3 (Computer, Computational and Statistical Sciences), Los Alamos National Laboratories, Los Alamos, NM, USA"},{"name":"Institute for Quantum Information, Caltech, Pasadena, CA, USA"}]}],"container-title":"Open Systems &amp; Information Dynamics","original-title":[],"language":"en","issued":{"date-parts":[[2008,3]]},"URL":"http://dx.doi.org/10.1142/S1230161208000043","ISSN":["1230-1612","1793-7191"],"container-title-short":"Open Syst. Inf. Dyn.","note":"alternative-id:10.1142/S1230161208000043\narxivid:quant-ph/0702005"},{"id":"arxiv:1004.1641","type":"article-journal","author":[{"given":"Frédéric","family":"Dupuis"}],"title":"The decoupling approach to quantum information theory","issued":{"date-parts":[[2010,4,9]]},"note":"arxivid:1004.1641\narxiv_version_number:1"},{"id":"arxiv:1012.6044","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s00220-014-1990-4","page":"251-284","source":"Crossref","title":"One-Shot Decoupling","volume":"328","author":[{"given":"Frédéric","family":"Dupuis","sequence":"first","affiliation":[]},{"given":"Mario","family":"Berta","sequence":"additional","affiliation":[]},{"given":"Jürg","family":"Wullschleger","sequence":"additional","affiliation":[]},{"given":"Renato","family":"Renner","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,3,21]]},"URL":"http://dx.doi.org/10.1007/s00220-014-1990-4","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1990\narxivid:1012.6044"},{"id":"arxiv:1810.03787","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","DOI":"10.1038/s41567-019-0648-8","page":"1273-1278","source":"Crossref","title":"Quantum convolutional neural networks","volume":"15","author":[{"given":"Iris","family":"Cong","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1247-062X","authenticated-orcid":false,"given":"Soonwon","family":"Choi","sequence":"additional","affiliation":[]},{"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,26]]},"URL":"http://dx.doi.org/10.1038/s41567-019-0648-8","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:648\narxivid:1810.03787"},{"id":"arxiv:2202.00555","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Active quantum error correction is a central ingredient to achieve robust quantum processors. In this paper we investigate the potential of quantum machine learning for quantum error correction in a quantum memory. Specifically, we demonstrate how quantum neural networks, in the form of quantum autoencoders, can be trained to learn optimal strategies for active detection and correction of errors, including spatially correlated computational errors as well as qubit losses. We highlight that the denoising capabilities of quantum autoencoders are not limited to the protection of specific states but extend to the entire logical codespace. We also show that quantum neural networks can be used to discover new logical encodings that are optimally adapted to the underlying noise. Moreover, we find that, even in the presence of moderate noise in the quantum autoencoders themselves, they may still be successfully used to perform beneficial quantum error correction and thereby extend the lifetime of a logical qubit.</jats:p>","DOI":"10.22331/q-2023-03-09-942","page":"942","source":"Crossref","title":"Quantum Error Correction with Quantum Autoencoders","volume":"7","author":[{"given":"David F.","family":"Locher","sequence":"first","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]},{"given":"Lorenzo","family":"Cardarelli","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]},{"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,9]]},"URL":"http://dx.doi.org/10.22331/q-2023-03-09-942","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2202.00555"},{"id":"arxiv:1308.4582","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.89.022316","source":"Crossref","title":"Approximate quantum error correction for generalized amplitude-damping errors","volume":"89","author":[{"given":"Carlo","family":"Cafaro","sequence":"first","affiliation":[]},{"given":"Peter","family":"van Loock","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.89.022316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022316","note":"arxivid:1308.4582"},{"id":"doi:10.1007/BF01212345","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01212345","page":"123-131","source":"Crossref","title":"Sufficient subalgebras and the relative entropy of states of a von Neumann algebra","volume":"105","author":[{"given":"D�nes","family":"Petz","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1986,3]]},"URL":"http://dx.doi.org/10.1007/BF01212345","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF01212345"},{"id":"doi:10.1093/qmath/39.1.97","type":"journal-article","publisher":"Oxford University Press (OUP)","issue":"1","DOI":"10.1093/qmath/39.1.97","page":"97-108","source":"Crossref","title":"SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRAS","volume":"39","author":[{"given":"DÉNES","family":"PETZ","sequence":"first","affiliation":[]}],"container-title":"The Quarterly Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1988]]},"URL":"http://dx.doi.org/10.1093/qmath/39.1.97","ISSN":["0033-5606","1464-3847"],"container-title-short":"Q J Math"},{"id":"arxiv:1810.03150","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.9.031029","source":"Crossref","title":"Fluctuation Theorems for a Quantum Channel","volume":"9","author":[{"given":"Hyukjoon","family":"Kwon","sequence":"first","affiliation":[]},{"given":"M. S.","family":"Kim","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.031029","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031029","note":"arxivid:1810.03150"},{"id":"arxiv:quant-ph/0004088","type":"journal-article","publisher":"AIP Publishing","issue":"5","abstract":"<jats:p>We consider the problem of reversing quantum dynamics, with the goal of preserving an initial state’s quantum entanglement or classical correlation with a reference system. We exhibit an approximate reversal operation, adapted to the initial density operator and the “noise” dynamics to be reversed. We show that its error in preserving either quantum or classical information is no more than twice that of the optimal reversal operation. Applications to quantum algorithms and information transmission are discussed.</jats:p>","DOI":"10.1063/1.1459754","page":"2097-2106","source":"Crossref","title":"Reversing quantum dynamics with near-optimal quantum and classical fidelity","volume":"43","author":[{"given":"H.","family":"Barnum","sequence":"first","affiliation":[{"name":"Los Alamos National Laboratories, Los Alamos, New Mexico 87545"}]},{"given":"E.","family":"Knill","sequence":"additional","affiliation":[{"name":"Los Alamos National Laboratories, Los Alamos, New Mexico 87545"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2002,5,1]]},"URL":"http://dx.doi.org/10.1063/1.1459754","ISSN":["0022-2488","1089-7658"],"note":"arxivid:quant-ph/0004088"},{"id":"arxiv:2401.02022","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/physrevlett.132.250602","source":"Crossref","title":"Near-Optimal Performance of Quantum Error Correction Codes","volume":"132","author":[{"ORCID":"https://orcid.org/0000-0001-6338-0192","authenticated-orcid":true,"given":"Guo","family":"Zheng","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Chicago"}]},{"given":"Wenhao","family":"He","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02v51f717","id-type":"ROR","asserted-by":"publisher"}],"name":"Peking University"},{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"given":"Gideon","family":"Lee","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Chicago"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.132.250602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"250602","note":"arxivid:2401.02022"},{"id":"arxiv:0909.0931","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.81.062342","source":"Crossref","title":"Simple approach to approximate quantum error correction based on the transpose channel","volume":"81","author":[{"given":"Hui Khoon","family":"Ng","sequence":"first","affiliation":[]},{"given":"Prabha","family":"Mandayam","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,6,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.062342","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062342","note":"arxivid:0909.0931"},{"id":"arxiv:2410.23622","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.134.200602","source":"Crossref","title":"Optimality Condition for the Petz Map","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0001-5044-3131","authenticated-orcid":true,"given":"Bikun","family":"Li","sequence":"first","affiliation":[{"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0001-8573-6970","authenticated-orcid":true,"given":"Zhaoyou","family":"Wang","sequence":"additional","affiliation":[{"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0001-6338-0192","authenticated-orcid":true,"given":"Guo","family":"Zheng","sequence":"additional","affiliation":[{"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0001-5912-7354","authenticated-orcid":true,"given":"Yat","family":"Wong","sequence":"additional","affiliation":[{"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"University of Chicago"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.200602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200602","note":"arxivid:2410.23622"},{"id":"arxiv:1410.0664","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-015-2466-x","page":"575-611","source":"Crossref","title":"Quantum Conditional Mutual Information and Approximate Markov Chains","volume":"340","author":[{"given":"Omar","family":"Fawzi","sequence":"first","affiliation":[]},{"given":"Renato","family":"Renner","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2015,9,9]]},"URL":"http://dx.doi.org/10.1007/s00220-015-2466-x","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2466\narxivid:1410.0664"},{"id":"arxiv:1509.07127","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","DOI":"10.1007/s00023-018-0716-0","page":"2955-2978","source":"Crossref","title":"Universal Recovery Maps and Approximate Sufficiency of Quantum Relative Entropy","volume":"19","author":[{"given":"Marius","family":"Junge","sequence":"first","affiliation":[]},{"given":"Renato","family":"Renner","sequence":"additional","affiliation":[]},{"given":"David","family":"Sutter","sequence":"additional","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[]}],"container-title":"Annales Henri Poincaré","original-title":[],"language":"en","issued":{"date-parts":[[2018,8,6]]},"URL":"http://dx.doi.org/10.1007/s00023-018-0716-0","ISSN":["1424-0637","1424-0661"],"container-title-short":"Ann. Henri Poincaré","note":"alternative-id:716\narxivid:1509.07127"},{"id":"arxiv:2408.00857","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevb.110.195107","source":"Crossref","title":"Petz map recovery for long-range entangled quantum many-body states","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0002-3548-8574","authenticated-orcid":true,"given":"Yangrui","family":"Hu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"}]},{"ORCID":"https://orcid.org/0000-0001-8573-000X","authenticated-orcid":true,"given":"Yijian","family":"Zou","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.110.195107","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"195107","note":"arxivid:2408.00857"},{"id":"arxiv:2405.06051","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Reliably transmitting quantum information via a noisy quantum channel is a central challenge in quantum information science. While constructing a decoder is crucial to this goal, little was known about quantum circuit implementations of decoders that reach high communication rates. In this paper, we provide two decoders with explicit quantum circuits capable of recovering quantum information when the decoupling condition is satisfied, i.e., when quantum information is in principle recoverable. These are applicable to both entanglement-assisted and non-assisted settings. By developing a technique that relies on a symmetric structure of the decoders, we show that they are applicable to any noise model. As a consequence, for any noisy channel, our decoders can be used to achieve a communication rate arbitrarily close to the quantum capacity by increasing the number of channel uses. To construct the decoders, we employ the fixed-point amplitude amplification (FPAA) based on the quantum singular value transformation (QSVT), extending a previous approach applicable only to erasure noise. Our constructions offer advantages in the computational cost, largely reducing the circuit complexity compared to previous explicit decoders. Through an investigation of the decoding problem, unique advantages of the QSVT-based FPAA are highlighted.</jats:p>","DOI":"10.22331/q-2026-03-13-2024","page":"2024","source":"Crossref","title":"Explicit decoders using fixed-point amplitude amplification based on QSVT","volume":"10","author":[{"ORCID":"https://orcid.org/0009-0004-2935-9794","authenticated-orcid":false,"given":"Takeru","family":"Utsumi","sequence":"first","affiliation":[{"name":"Graduate School of Arts and Sciences, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan"}]},{"ORCID":"https://orcid.org/0000-0003-1285-6968","authenticated-orcid":false,"given":"Yoshifumi","family":"Nakata","sequence":"additional","affiliation":[{"name":"Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,13]]},"URL":"http://dx.doi.org/10.22331/q-2026-03-13-2024","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2405.06051"},{"id":"arxiv:2502.09177","type":"article-journal","author":[{"given":"Nirupam","family":"Basak"},{"given":"Andrew","family":"Tanggara"},{"given":"Ankith","family":"Mohan"},{"given":"Goutam","family":"Paul"},{"given":"Kishor","family":"Bharti"}],"title":"Approximate Dynamical Quantum Error-Correcting Codes","issued":{"date-parts":[[2025,8,25]]},"note":"arxivid:2502.09177\narxiv_version_number:2"},{"id":"arxiv:2510.08719","type":"article-journal","author":[{"given":"Debjyoti","family":"Biswas"},{"given":"Prabha","family":"Mandayam"}],"title":"Universal syndrome-based recovery for noise-adapted quantum error correction","issued":{"date-parts":[[2025,10,9]]},"note":"arxivid:2510.08719\narxiv_version_number:1"},{"id":"arxiv:2410.00319","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/5n4p-bxhm","source":"Crossref","title":"Quantum Bayes’ Rule and Petz Transpose Map from the Minimum Change Principle","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0002-6814-8840","authenticated-orcid":true,"given":"Ge","family":"Bai","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01mgdzc49","id-type":"ROR","asserted-by":"publisher"}],"name":"Centre for Quantum Technologies"},{"id":[{"id":"https://ror.org/01tgyzw49","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Singapore"},{"id":[{"id":"https://ror.org/00q4vv597","id-type":"ROR","asserted-by":"publisher"}],"name":"The Hong Kong University of Science and Technology (Guangzhou)"}]},{"ORCID":"https://orcid.org/0000-0001-9741-0628","authenticated-orcid":true,"given":"Francesco","family":"Buscemi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04chrp450","id-type":"ROR","asserted-by":"publisher"}],"name":"Nagoya University"}]},{"given":"Valerio","family":"Scarani","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01mgdzc49","id-type":"ROR","asserted-by":"publisher"}],"name":"Centre for Quantum Technologies"},{"id":[{"id":"https://ror.org/01tgyzw49","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Singapore"},{"id":[{"id":"https://ror.org/01tgyzw49","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Singapore"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,28]]},"URL":"http://dx.doi.org/10.1103/5n4p-bxhm","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090203","note":"arxivid:2410.00319"},{"id":"arxiv:1710.03363","type":"article-journal","author":[{"given":"Beni","family":"Yoshida"},{"given":"Alexei","family":"Kitaev"}],"title":"Efficient decoding for the Hayden-Preskill protocol","issued":{"date-parts":[[2017,10,16]]},"note":"arxivid:1710.03363\narxiv_version_number:2"},{"id":"arxiv:2410.00155","type":"article-journal","author":[{"given":"Sourav","family":"Dutta"},{"given":"Aditya","family":"Jain"},{"given":"Prabha","family":"Mandayam"}],"title":"Smallest quantum codes for amplitude damping noise","issued":{"date-parts":[[2026,4,1]]},"note":"arxivid:2410.00155\narxiv_version_number:4"},{"id":"arxiv:2208.00365","type":"article-journal","author":[{"given":"Akshaya","family":"Jayashankar"},{"given":"Prabha","family":"Mandayam"}],"title":"Quantum Error Correction: Noise-adapted Techniques and Applications","issued":{"date-parts":[[2022,7,31]]},"note":"arxivid:2208.00365\narxiv_version_number:1"},{"id":"doi:10.1016/0003-4916(61)90115-4","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/0003-4916(61)90115-4","page":"407-466","source":"Crossref","title":"Two soluble models of an antiferromagnetic chain","volume":"16","author":[{"given":"Elliott","family":"Lieb","sequence":"first","affiliation":[]},{"given":"Theodore","family":"Schultz","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Mattis","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[1961,12]]},"URL":"http://dx.doi.org/10.1016/0003-4916(61)90115-4","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:0003491661901154"},{"id":"arxiv:cond-mat/9911137","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.84.1535","page":"1535-1538","source":"Crossref","title":"Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Systems on a Periodic Lattice","volume":"84","author":[{"given":"Masaki","family":"Oshikawa","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2000,2,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.84.1535","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:cond-mat/9911137"},{"id":"arxiv:2112.06946","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.12.031007","source":"Crossref","title":"Nonzero Momentum Requires Long-Range Entanglement","volume":"12","author":[{"ORCID":"https://orcid.org/0000-0002-4367-4801","authenticated-orcid":true,"given":"Lei","family":"Gioia","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6078-0545","authenticated-orcid":true,"given":"Chong","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.12.031007","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031007","note":"arxivid:2112.06946"},{"id":"arxiv:2405.14929","type":"journal-article","publisher":"American Physical Society (APS)","issue":"21","DOI":"10.1103/2jgh-nrj1","source":"Crossref","title":"Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement","volume":"112","author":[{"given":"Ruizhi","family":"Liu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01e6qks80","id-type":"ROR","asserted-by":"publisher"}],"name":"Dalhousie University"},{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"}]},{"ORCID":"https://orcid.org/0000-0001-6948-2464","authenticated-orcid":true,"given":"Jinmin","family":"Yi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"},{"id":[{"id":"https://ror.org/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Waterloo"}]},{"given":"Shiyu","family":"Zhou","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"}]},{"ORCID":"https://orcid.org/0000-0002-6666-4163","authenticated-orcid":true,"given":"Liujun","family":"Zou","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"},{"id":[{"id":"https://ror.org/01tgyzw49","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Singapore"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,4]]},"URL":"http://dx.doi.org/10.1103/2jgh-nrj1","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"214408","note":"arxivid:2405.14929"},{"id":"arxiv:2310.07770","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We discuss families of approximate quantum error correcting codes which arise as the nearly-degenerate ground states of certain quantum many-body Hamiltonians composed of non-commuting terms. For exact codes, the conditions for error correction can be formulated in terms of the vanishing of a two-sided mutual information in a low-temperature thermofield double state. We consider a notion of distance for approximate codes obtained by demanding that this mutual information instead be small, and we evaluate this mutual information for the SYK model and for a family of low-rank SYK models. After an extrapolation to nearly zero temperature, we find that both kinds of models produce fermionic codes with constant rate as the number, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>, of fermions goes to infinity. For SYK, the distance scales as <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>, and for low-rank SYK, the distance can be arbitrarily close to linear scaling, e.g. <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>.99</mml:mn></mml:mrow></mml:msup></mml:math>, while maintaining a constant rate. We also consider an analog of the no low-energy trivial states property which we dub the no low-energy adiabatically accessible states property and show that these models do have low-energy states that can be prepared adiabatically in a time that does not scale with system size <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>. We discuss a holographic model of these codes in which the large code distance is a consequence of the emergence of a long wormhole geometry in a simple model of quantum gravity.</jats:p>","DOI":"10.22331/q-2024-08-14-1439","page":"1439","source":"Crossref","title":"Approximate Quantum Codes From Long Wormholes","volume":"8","author":[{"given":"Gregory","family":"Bentsen","sequence":"first","affiliation":[{"name":"Department of Physics, Brandeis University, Waltham, MA"}]},{"given":"Phuc","family":"Nguyen","sequence":"additional","affiliation":[{"name":"Department of Physics, Brandeis University, Waltham, MA"}]},{"given":"Brian","family":"Swingle","sequence":"additional","affiliation":[{"name":"Department of Physics, Brandeis University, Waltham, MA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,14]]},"URL":"http://dx.doi.org/10.22331/q-2024-08-14-1439","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2310.07770"},{"id":"arxiv:2510.26758","type":"article-journal","author":[{"given":"Shozab","family":"Qasim"},{"given":"Jason","family":"Pollack"}],"title":"Approximate quantum error correction, eigenstate thermalization and the chaos bound","issued":{"date-parts":[[2025,10,30]]},"note":"arxivid:2510.26758\narxiv_version_number:1"},{"id":"arxiv:1807.06041","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>When the bulk geometry in AdS/CFT contains a black hole, boundary subregions may be sufficient to reconstruct certain bulk operators if and only if the black hole microstate is known, an example of state dependence. Reconstructions exist for any microstate, but no reconstruction works for all microstates. We refine this dichotomy, demonstrating that the same boundary operator can often be used for large subspaces of black hole microstates, corresponding to a constant fraction <jats:italic>α</jats:italic> of the black hole entropy. In the Schrödinger picture, the boundary subregion encodes the <jats:italic>α</jats:italic>-bits (a concept from quantum information) of a bulk region containing the black hole and bounded by extremal surfaces. These results have important consequences for the structure of AdS/CFT and for quantum information. Firstly, they imply that the bulk reconstruction is necessarily only approximate and allow us to place non-perturbative lower bounds on the error when doing so. Second, they provide a simple and tractable limit in which the entanglement wedge is state dependent, but in a highly controlled way. Although the state dependence of operators comes from ordinary quantum error correction, there are clear connections to the Papadodimas-Raju proposal for understanding operators behind black hole horizons. In tensor network toy models of AdS/CFT, we see how state dependence arises from the bulk operator being ‘pushed’ through the black hole itself. Finally, we show that black holes provide the first ‘explicit’ examples of capacity-achieving <jats:italic>α</jats:italic>-bit codes. Unintuitively, Hawking radiation always reveals the <jats:italic>α</jats:italic>-bits of a black hole as soon as possible. In an appendix, we apply a result from the quantum information literature to prove that entanglement wedge reconstruction can be made exact to all orders in 1<jats:italic>/N</jats:italic>.</jats:p>","DOI":"10.1007/jhep12(2019)007","source":"Crossref","title":"Learning the Alpha-bits of black holes","volume":"2019","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8627-5237","authenticated-orcid":false,"given":"Geoffrey","family":"Penington","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,12]]},"URL":"http://dx.doi.org/10.1007/JHEP12(2019)007","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"7","note":"alternative-id:11902\narxivid:1807.06041"},{"id":"arxiv:quant-ph/0606107","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.75.032345","source":"Crossref","title":"Asymmetric quantum error-correcting codes","volume":"75","author":[{"given":"Lev","family":"Ioffe","sequence":"first","affiliation":[]},{"given":"Marc","family":"Mézard","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,3,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.032345","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032345","note":"arxivid:quant-ph/0606107"},{"id":"arxiv:1207.6512","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2013.2272575","page":"6732-6754","source":"Crossref","title":"CSS-Like Constructions of Asymmetric Quantum Codes","volume":"59","author":[{"given":"Martianus Frederic","family":"Ezerman","sequence":"first","affiliation":[]},{"given":"Somphong","family":"Jitman","sequence":"additional","affiliation":[]},{"given":"San","family":"Ling","sequence":"additional","affiliation":[]},{"given":"Dmitrii V.","family":"Pasechnik","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2272575","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1207.6512"},{"id":"arxiv:1705.04087","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","DOI":"10.1007/s11128-017-1748-y","source":"Crossref","title":"Two Gilbert–Varshamov-type existential bounds for asymmetric quantum error-correcting codes","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0002-5085-8879","authenticated-orcid":false,"given":"Ryutaroh","family":"Matsumoto","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,23]]},"URL":"http://dx.doi.org/10.1007/s11128-017-1748-y","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","page":"285","note":"alternative-id:1748\narxivid:1705.04087"},{"id":"arxiv:0803.0764","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"}],"title":"Asymmetric and Symmetric Subsystem BCH Codes and Beyond","issued":{"date-parts":[[2008,3,6]]},"note":"arxivid:0803.0764\narxiv_version_number:1"},{"id":"arxiv:1006.1694","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"03","abstract":"<jats:p> Using the Calderbank–Shor–Steane (CSS) construction, pure q-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for. </jats:p>","DOI":"10.1142/s0219749913500275","page":"1350027","source":"Crossref","title":"PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION","volume":"11","author":[{"given":"MARTIANUS FREDERIC","family":"EZERMAN","sequence":"first","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore 117543, Republic of Singapore"}]},{"given":"SOMPHONG","family":"JITMAN","sequence":"additional","affiliation":[{"name":"Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Republic of Singapore"}]},{"given":"HAN MAO","family":"KIAH","sequence":"additional","affiliation":[{"name":"Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Republic of Singapore"}]},{"given":"SAN","family":"LING","sequence":"additional","affiliation":[{"name":"Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Republic of Singapore"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2013,4]]},"URL":"http://dx.doi.org/10.1142/S0219749913500275","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749913500275\narxivid:1006.1694"},{"id":"arxiv:1509.05032","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.92.062309","source":"Crossref","title":"Reducing the overhead for quantum computation when noise is biased","volume":"92","author":[{"given":"Paul","family":"Webster","sequence":"first","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,12,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.92.062309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062309","note":"arxivid:1509.05032"},{"id":"arxiv:0806.0383","type":"journal-article","publisher":"IOP Publishing","issue":"1","DOI":"10.1088/1367-2630/11/1/013061","page":"013061","source":"Crossref","title":"Fault-tolerant computing with biased-noise superconducting qubits: a case study","volume":"11","author":[{"given":"P","family":"Aliferis","sequence":"first","affiliation":[]},{"given":"F","family":"Brito","sequence":"additional","affiliation":[]},{"given":"D P","family":"DiVincenzo","sequence":"additional","affiliation":[]},{"given":"J","family":"Preskill","sequence":"additional","affiliation":[]},{"given":"M","family":"Steffen","sequence":"additional","affiliation":[]},{"given":"B M","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2009,1,30]]},"URL":"http://dx.doi.org/10.1088/1367-2630/11/1/013061","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:0806.0383"},{"id":"arxiv:2305.02045","type":"article-journal","author":[{"given":"Marco","family":"Fellous-Asiani"},{"given":"Moein","family":"Naseri"},{"given":"Chandan","family":"Datta"},{"given":"Alexander","family":"Streltsov"},{"given":"Michał","family":"Oszmaniec"}],"title":"Scalable noisy quantum circuits for biased-noise qubits","issued":{"date-parts":[[2024,8,19]]},"note":"arxivid:2305.02045\narxiv_version_number:5"},{"id":"arxiv:2510.07532","type":"article-journal","author":[{"given":"Debadrito","family":"Roy"},{"given":"Aryaman Manish","family":"Kolhe"},{"given":"V.","family":"Lalitha"},{"given":"Navin","family":"Kashyap"}],"title":"On Quantum Computation Using Bias-Preserving Gates","issued":{"date-parts":[[2025,10,10]]},"note":"arxivid:2510.07532\narxiv_version_number:2"},{"id":"arxiv:2505.15669","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    In this paper, we derive optimized measurement-free protocols for quantum error correction and the implementation of a universal gate set optimized for an error model that is noise biased. The noise bias is adapted for neutral-atom platforms, where two- and multi-qubit gates are realized with Rydberg interactions and are thus expected to be the dominant source of noise. Careful design of the gates allows us to further reduce the noise model to Pauli-\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>Z</a:mi>\n                    </a:math>\n                    errors. In addition, the presented circuits are robust to arbitrary single-qubit gate errors, and we demonstrate that the break-even point can be significantly improved compared to fully fault-tolerant measurement-free schemes. The obtained logical qubits with their suppressed error rates on logical gate operations can then be used as building blocks in a first step of error correction in order to push the effective error rates below the threshold of a fully fault-tolerant and scalable quantum error-correction scheme.\n                  </jats:p>","DOI":"10.1103/wsnf-prwl","source":"Crossref","title":"Measurement-Free Quantum Error Correction Optimized for Biased Noise","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-0501-1406","authenticated-orcid":true,"given":"Katharina","family":"Brechtelsbauer","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04vnq7t77","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Stuttgart"}]},{"given":"Friederike","family":"Butt","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0009-0001-0180-4225","authenticated-orcid":true,"given":"David F.","family":"Locher","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0001-6905-4662","authenticated-orcid":true,"given":"Santiago Higuera","family":"Quintero","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04vnq7t77","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Stuttgart"}]},{"ORCID":"https://orcid.org/0000-0001-9763-9131","authenticated-orcid":true,"given":"Sebastian","family":"Weber","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04vnq7t77","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Stuttgart"}]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":true,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0002-7233-6828","authenticated-orcid":true,"given":"Hans Peter","family":"Büchler","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04vnq7t77","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Stuttgart"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,1]]},"URL":"http://dx.doi.org/10.1103/wsnf-prwl","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040349","note":"arxivid:2505.15669"},{"id":"arxiv:0708.3969","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.77.062335","source":"Crossref","title":"Asymmetric quantum error correction via code conversion","volume":"77","author":[{"given":"Ashley M.","family":"Stephens","sequence":"first","affiliation":[]},{"given":"Zachary W. E.","family":"Evans","sequence":"additional","affiliation":[]},{"given":"Simon J.","family":"Devitt","sequence":"additional","affiliation":[]},{"given":"Lloyd C. L.","family":"Hollenberg","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,6,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.062335","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062335","note":"arxivid:0708.3969"},{"id":"arxiv:0709.3875","type":"article-journal","author":[{"given":"Z. W. E.","family":"Evans"},{"given":"A. M.","family":"Stephens"},{"given":"J. H.","family":"Cole"},{"given":"L. C. L.","family":"Hollenberg"}],"title":"Error correction optimisation in the presence of X/Z asymmetry","issued":{"date-parts":[[2007,9,25]]},"note":"arxivid:0709.3875\narxiv_version_number:1"},{"id":"arxiv:0710.1301","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.78.052331","source":"Crossref","title":"Fault-tolerant quantum computation against biased noise","volume":"78","author":[{"given":"Panos","family":"Aliferis","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,11,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.052331","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052331","note":"arxivid:0710.1301"},{"id":"arxiv:1403.7755","type":"article-journal","author":[{"given":"Liqi","family":"Wang"},{"given":"Shixin","family":"Zhu"}],"title":"On the Construction of Optimal Asymmetric Quantum Codes","issued":{"date-parts":[[2014,3,30]]},"note":"arxivid:1403.7755\narxiv_version_number:1"},{"id":"arxiv:2010.09626","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.11.031039","source":"Crossref","title":"Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0001-9880-5218","authenticated-orcid":true,"given":"Oscar","family":"Higgott","sequence":"first","affiliation":[]},{"given":"Nikolas P.","family":"Breuckmann","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.11.031039","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031039","note":"arxivid:2010.09626"},{"id":"arxiv:2201.07802","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Various realizations of Kitaev’s surface code perform surprisingly well for biased Pauli noise. Attracted by these potential gains, we study the performance of Clifford-deformed surface codes (CDSCs) obtained from the surface code by the application of single-qubit Clifford operators. We first analyze CDSCs on the <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mn>3</a:mn><a:mo>×</a:mo><a:mn>3</a:mn></a:math> square lattice and find that, depending on the noise bias, their logical error rates can differ by orders of magnitude. To explain the observed behavior, we introduce the effective distance <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><d:msup><d:mi>d</d:mi><d:mo>′</d:mo></d:msup></d:math>, which reduces to the standard distance for unbiased noise. To study CDSC performance in the thermodynamic limit, we focus on random CDSCs. Using the statistical mechanical mapping for quantum codes, we uncover a phase diagram that describes random CDSC families with <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><g:mn>50</g:mn><g:mi mathvariant=\"normal\">%</g:mi></g:math> threshold at infinite bias. In the high-threshold region, we further demonstrate that typical code realizations outperform the thresholds and subthreshold logical error rates, at finite bias, of the best-known translationally invariant codes. We demonstrate the practical relevance of these random CDSC families by constructing a translation-invariant CDSC belonging to a high-performance random CDSC family. We also show that our translation-invariant CDSC outperforms well-known translation-invariant CDSCs, such as the <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><k:mi>X</k:mi><k:mi>Z</k:mi><k:mi>Z</k:mi><k:mi>X</k:mi></k:math> and <n:math xmlns:n=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><n:mi>X</n:mi><n:mi>Y</n:mi></n:math> codes.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.010347","source":"Crossref","title":"Clifford-Deformed Surface Codes","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"first","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"California Institute of Technology, Pasadena, California, 91125, USA"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA"}]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"California Institute of Technology, Pasadena, California, 91125, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,3,19]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.010347","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010347","note":"arxivid:2201.07802"},{"id":"arxiv:1708.08474","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.120.050505","source":"Crossref","title":"Ultrahigh Error Threshold for Surface Codes with Biased Noise","volume":"120","author":[{"given":"David K.","family":"Tuckett","sequence":"first","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2018,1,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.120.050505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050505","note":"arxivid:1708.08474"},{"id":"arxiv:2408.03123","type":"article-journal","author":[{"given":"Zhipeng","family":"Liang"},{"given":"Zhengzhong","family":"Yi"},{"given":"Fusheng","family":"Yang"},{"given":"Jiahan","family":"Chen"},{"given":"Zicheng","family":"Wang"},{"given":"Xuan","family":"Wang"}],"title":"High-dimensional quantum XYZ product codes for biased noise","issued":{"date-parts":[[2025,8,12]]},"note":"arxivid:2408.03123\narxiv_version_number:4"},{"id":"arxiv:2203.16534","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Self-correcting quantum memories demonstrate robust properties that can be exploited to improve active quantum error-correction protocols. Here we propose a cellular automaton decoder for a variation of the color code where the bases of the physical qubits are locally rotated, which we call the XYZ color code. The local transformation means our decoder demonstrates key properties of a two-dimensional fractal code if the noise acting on the system is infinitely biased towards dephasing, namely, no string-like logical operators. As such, in the high-bias limit, our local decoder reproduces the behavior of a partially self-correcting memory. At low error rates, our simulations show that the memory time diverges polynomially with system size without intervention from a global decoder, up to some critical system size that grows as the error rate is lowered. Furthermore, although we find that we cannot reproduce partially self-correcting behavior at finite bias, our numerics demonstrate improved memory times at realistic noise biases. Our results therefore motivate the design of tailored cellular automaton decoders that help to reduce the bandwidth demands of global decoding for realistic noise models.</jats:p>","DOI":"10.22331/q-2023-03-09-940","page":"940","source":"Crossref","title":"A cellular automaton decoder for a noise-bias tailored color code","volume":"7","author":[{"given":"Jonathan F. San","family":"Miguel","sequence":"first","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, CA 94305"}]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, CA 94305"},{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,9]]},"URL":"http://dx.doi.org/10.22331/q-2023-03-09-940","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2203.16534"},{"id":"arxiv:1703.08179","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevapplied.8.064004","source":"Crossref","title":"Tailored Codes for Small Quantum Memories","volume":"8","author":[{"given":"Alan","family":"Robertson","sequence":"first","affiliation":[]},{"given":"Christopher","family":"Granade","sequence":"additional","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2017,12,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.8.064004","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"064004","note":"arxivid:1703.08179"},{"id":"arxiv:2009.07851","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.</jats:p>","DOI":"10.1038/s41467-021-22274-1","source":"Crossref","title":"The XZZX surface code","volume":"12","author":[{"ORCID":"https://orcid.org/0000-0001-5518-7907","authenticated-orcid":false,"given":"J. Pablo","family":"Bonilla Ataides","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3776-2864","authenticated-orcid":false,"given":"David K.","family":"Tuckett","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":false,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":false,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2021,4,12]]},"URL":"http://dx.doi.org/10.1038/s41467-021-22274-1","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"2172","note":"alternative-id:22274\narxivid:2009.07851"},{"id":"arxiv:2203.16486","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevresearch.5.013035","source":"Crossref","title":"Tailored XZZX codes for biased noise","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-8738-9420","authenticated-orcid":true,"given":"Qian","family":"Xu","sequence":"first","affiliation":[]},{"given":"Nam","family":"Mannucci","sequence":"additional","affiliation":[]},{"given":"Alireza","family":"Seif","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":true,"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2023,1,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.5.013035","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013035","note":"arxivid:2203.16486"},{"id":"arxiv:1112.1613","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.85.022313","source":"Crossref","title":"Incoherent dynamics in the toric code subject to disorder","volume":"85","author":[{"given":"Beat","family":"Röthlisberger","sequence":"first","affiliation":[]},{"given":"James R.","family":"Wootton","sequence":"additional","affiliation":[]},{"given":"Robert M.","family":"Heath","sequence":"additional","affiliation":[]},{"given":"Jiannis K.","family":"Pachos","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,2,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.85.022313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022313","note":"arxivid:1112.1613"},{"id":"arxiv:2201.10566","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Fault-tolerant cluster states form the basis for scalable measurement-based quantum computation. Recently, new stabilizer codes for scalable circuit-based quantum computation have been introduced that have very high thresholds under biased noise where the qubit predominantly suffers from one type of error, e.g. dephasing. However, extending these advances in stabilizer codes to generate high-threshold cluster states for biased noise has been a challenge, as the standard method for foliating stabilizer codes to generate fault-tolerant cluster states does not preserve the noise bias. In this work, we overcome this barrier by introducing a generalization of the cluster state that allows us to foliate stabilizer codes in a bias-preserving way. As an example of our approach, we construct a foliated version of the XZZX code which we call the XZZX cluster state. We demonstrate that under a circuit-level-noise model, our XZZX cluster state has a threshold more than double the usual cluster state when dephasing errors are more likely than errors that cause bit flips by a factor of order ~100 or more.</jats:p>","DOI":"10.1038/s41534-023-00677-w","source":"Crossref","title":"Tailored cluster states with high threshold under biased noise","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0002-6711-4585","authenticated-orcid":false,"given":"Jahan","family":"Claes","sequence":"first","affiliation":[]},{"given":"J. Eli","family":"Bourassa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5194-0591","authenticated-orcid":false,"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2023,1,31]]},"URL":"http://dx.doi.org/10.1038/s41534-023-00677-w","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"9","note":"alternative-id:677\narxivid:2201.10566"},{"id":"arxiv:2303.16122","type":"article-journal","author":[{"given":"Hector","family":"Bombín"},{"given":"Chris","family":"Dawson"},{"given":"Naomi","family":"Nickerson"},{"given":"Mihir","family":"Pant"},{"given":"Jordan","family":"Sullivan"}],"title":"Increasing error tolerance in quantum computers with dynamic bias arrangement","issued":{"date-parts":[[2023,3,28]]},"note":"arxivid:2303.16122\narxiv_version_number:1"},{"id":"arxiv:1308.4776","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.88.060301","source":"Crossref","title":"High-threshold topological quantum error correction against biased noise","volume":"88","author":[{"given":"Ashley M.","family":"Stephens","sequence":"first","affiliation":[]},{"given":"William J.","family":"Munro","sequence":"additional","affiliation":[]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,12,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.88.060301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"060301","note":"arxivid:1308.4776"},{"id":"arxiv:2105.01144","type":"article-journal","author":[{"given":"Clarice Dias","family":"de Albuquerque"},{"given":"Giuliano Gadioli","family":"La Guardia"},{"given":"Reginaldo","family":"Palazzo"},{"given":"Cátia Regina de Oliveira Quilles","family":"Queiroz"},{"given":"Vandenberg Lopes","family":"Vieira"}],"title":"Euclidean and Hyperbolic Asymmetric Topological Quantum Codes","issued":{"date-parts":[[2021,5,3]]},"note":"arxivid:2105.01144\narxiv_version_number:1"},{"id":"arxiv:1102.3605","type":"article-journal","author":[{"given":"Martianu Frederic","family":"Ezerman"},{"given":"Radoslav","family":"Kirov"}],"title":"Nonbinary Quantum Codes from Two-Point Divisors on Hermitian Curves","issued":{"date-parts":[[2011,2,17]]},"note":"arxivid:1102.3605\narxiv_version_number:1"},{"id":"manual:-G.-G.-La-Guardia-R.-Palazzo-","type":"article-journal","author":[{"family":"Guardia","given":"G.G.","particle":"La"},{"family":"Palazzo","given":"R."},{"family":"Lavor","given":"C."}],"title":"Nonbinary quantum Reed-Solomon codes","volume":"65","container-title":"Int. J. Pure Applied Math","issue":"1","issued":"2010","page":"55–63"},{"id":"doi:10.1007/s11128-011-0269-3","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s11128-011-0269-3","page":"591-604","source":"Crossref","title":"Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes","volume":"11","author":[{"given":"Giuliano G.","family":"La Guardia","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2011,8,3]]},"URL":"http://dx.doi.org/10.1007/s11128-011-0269-3","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:269"},{"id":"arxiv:1209.0794","type":"article-journal","author":[{"given":"John","family":"Napp"},{"given":"John","family":"Preskill"}],"title":"Optimal Bacon-Shor codes","issued":{"date-parts":[[2012,9,4]]},"note":"arxivid:1209.0794\narxiv_version_number:1"},{"id":"arxiv:1211.1400","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.87.032310","source":"Crossref","title":"Fault-tolerant quantum computation with asymmetric Bacon-Shor codes","volume":"87","author":[{"given":"Peter","family":"Brooks","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,3,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.032310","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032310","note":"arxivid:1211.1400"},{"id":"arxiv:1104.5004","type":"article-journal","author":[{"given":"Yuichiro","family":"Fujiwara"},{"given":"Min-Hsiu","family":"Hsieh"}],"title":"Adaptively correcting quantum errors with entanglement","issued":{"date-parts":[[2011,4,26]]},"note":"arxivid:1104.5004\narxiv_version_number:1"},{"id":"arxiv:2411.04974","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41534-025-01074-1","source":"Crossref","title":"Tailoring dynamical codes for biased noise: the X3Z3 Floquet code","volume":"11","author":[{"given":"F.","family":"Setiawan","sequence":"first","affiliation":[]},{"given":"Campbell","family":"McLauchlan","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,22]]},"URL":"http://dx.doi.org/10.1038/s41534-025-01074-1","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"149","note":"alternative-id:1074\narxivid:2411.04974"},{"id":"arxiv:2501.16827","type":"article-journal","author":[{"given":"Zhipeng","family":"Liang"},{"given":"Fusheng","family":"Yang"},{"given":"Zhengzhong","family":"Yi"},{"given":"Xuan","family":"Wang"}],"title":"Quantum XYZ cyclic codes for biased noise","issued":{"date-parts":[[2025,1,28]]},"note":"arxivid:2501.16827\narxiv_version_number:1"},{"id":"arxiv:2305.06378","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevapplied.23.034048","source":"Crossref","title":"Discovery of optimal quantum codes via reinforcement learning","volume":"23","author":[{"ORCID":"https://orcid.org/0000-0001-7196-4040","authenticated-orcid":true,"given":"Vincent Paul","family":"Su","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California Berkeley"}]},{"ORCID":"https://orcid.org/0000-0002-5761-5474","authenticated-orcid":true,"given":"ChunJun","family":"Cao","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/02smfhw86","id-type":"ROR","asserted-by":"publisher"}],"name":"Virginia Tech"}]},{"ORCID":"https://orcid.org/0000-0001-5841-831X","authenticated-orcid":true,"given":"Hong-Ye","family":"Hu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0168r3w48","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California San Diego"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-7214-4156","authenticated-orcid":true,"given":"Yariv","family":"Yanay","sequence":"additional","affiliation":[{"name":"Laboratory for Physical Sciences"}]},{"given":"Charles","family":"Tahan","sequence":"additional","affiliation":[{"name":"Laboratory for Physical Sciences"}]},{"given":"Brian","family":"Swingle","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05abbep66","id-type":"ROR","asserted-by":"publisher"}],"name":"Brandeis University"}]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.23.034048","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"034048","note":"arxivid:2305.06378"},{"id":"arxiv:2305.11470","type":"article-journal","author":[{"given":"Caroline","family":"Mauron"},{"given":"Terry","family":"Farrelly"},{"given":"Thomas M.","family":"Stace"}],"title":"Optimization of Tensor Network Codes with Reinforcement Learning","issued":{"date-parts":[[2023,5,19]]},"note":"arxivid:2305.11470\narxiv_version_number:1"},{"id":"arxiv:2506.15905","type":"article-journal","author":[{"given":"Heather","family":"Leitch"},{"given":"Alastair","family":"Kay"}],"title":"Transversal Gates for Highly Asymmetric qLDPC Codes","issued":{"date-parts":[[2025,6,18]]},"note":"arxivid:2506.15905\narxiv_version_number:1"},{"id":"arxiv:1809.01193","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.9.021041","source":"Crossref","title":"2D Compass Codes","volume":"9","author":[{"given":"Muyuan","family":"Li","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Miller","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"given":"Yukai","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2019,5,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.021041","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021041","note":"arxivid:1809.01193"},{"id":"arxiv:2412.03808","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We can design efficient quantum error-correcting (QEC) codes by tailoring them to our choice of quantum architecture. Useful tools for constructing such codes include Clifford deformations and appropriate gauge fixings of compass codes. In this work, we find Clifford deformations that can be applied to elongated compass codes resulting in QEC codes with improved performance under noise models with errors biased towards dephasing commonly seen in quantum computing architectures. These Clifford deformations enhance decoder performance by introducing symmetries, while the stabilizers of compass codes can be selected to obtain more information on high-rate errors. As a result, the codes exhibit thresholds that increase with bias and lower logical error rates under both code capacity and phenomenological noise models. One of the Clifford deformations we explore yields QEC codes with better thresholds and logical error rates than those of the XZZX surface code at moderate biases under code capacity noise.</jats:p>","DOI":"10.22331/q-2026-04-16-2073","page":"2073","source":"Crossref","title":"Clifford-Deformed Compass Codes","volume":"10","author":[{"given":"Julie A.","family":"Campos","sequence":"first","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Physics, Duke University, Durham, NC 27708, USA"}]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Physics, Duke University, Durham, NC 27708, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA"},{"name":"Department of Chemistry, Duke University, Durham, NC 27708, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,16]]},"URL":"http://dx.doi.org/10.22331/q-2026-04-16-2073","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2412.03808"},{"id":"doi:10.1109/ICCES.2008.4772987","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icces.2008.4772987","page":"157-162","source":"Crossref","title":"Asymmetric quantum BCH codes","author":[{"given":"Salah A.","family":"Aly","sequence":"first","affiliation":[]}],"event":"Systems (ICCES)","container-title":"2008 International Conference on Computer Engineering &amp; Systems","original-title":[],"issued":{"date-parts":[[2008,11]]},"URL":"http://dx.doi.org/10.1109/ICCES.2008.4772987"},{"id":"doi:10.26421/QIC11.3-4-4","type":"journal-article","publisher":"Rinton Press","issue":"3&4","abstract":"<jats:p>Several families of nonbinary asymmetric quantum Bose-Chaudhuri-Hocquenghem (BCH) codes are presented in this paper. These quantum codes have parameters better than the ones available in the literature. Additionally, such codes can be applied in quantum systems where the asymmetry between qudit-flip and phase-shift errors is large.</jats:p>","DOI":"10.26421/qic11.3-4-4","page":"239-252","source":"Crossref","title":"New families of asymmetric quantum BCH codes","volume":"11","author":[{"given":"Giuliano G.","family":"La Guardia","sequence":"first","affiliation":[]}],"container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2011,3]]},"URL":"http://dx.doi.org/10.26421/QIC11.3-4-4","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC"},{"id":"arxiv:2202.01702","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Bias-tailoring allows quantum error correction codes to exploit qubit noise asymmetry. Recently, it was shown that a modified form of the surface code, the XZZX code, exhibits considerably improved performance under biased noise. In this work, we demonstrate that quantum low density parity check codes can be similarly bias-tailored. We introduce a bias-tailored lifted product code construction that provides the framework to expand bias-tailoring methods beyond the family of 2D topological codes. We present examples of bias-tailored lifted product codes based on classical quasi-cyclic codes and numerically assess their performance using a belief propagation plus ordered statistics decoder. Our Monte Carlo simulations, performed under asymmetric noise, show that bias-tailored codes achieve several orders of magnitude improvement in their error suppression relative to depolarising noise.</jats:p>","DOI":"10.22331/q-2023-05-15-1005","page":"1005","source":"Crossref","title":"Bias-tailored quantum LDPC codes","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0001-9202-1156","authenticated-orcid":false,"given":"Joschka","family":"Roffe","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"},{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom"}]},{"given":"Lawrence Z.","family":"Cohen","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia"}]},{"given":"Armanda O.","family":"Quintavalle","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom"},{"name":"Riverlane, Cambridge CB2 3BZ, United Kingdom"}]},{"given":"Daryus","family":"Chandra","sequence":"additional","affiliation":[{"name":"School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, United Kingdom"}]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom"},{"name":"Riverlane, Cambridge CB2 3BZ, United Kingdom"},{"name":"AWS Center for Quantum Computing, Cambridge CB1 2GA, United Kingdom"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,5,15]]},"URL":"http://dx.doi.org/10.22331/q-2023-05-15-1005","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2202.01702"},{"id":"arxiv:2001.08405","type":"article-journal","author":[{"given":"Manabu","family":"Hagiwara"},{"given":"Ayumu","family":"Nakayama"}],"title":"A Four-Qubits Code that is a Quantum Deletion Error-Correcting Code with the Optimal Length","issued":{"date-parts":[[2020,1,23]]},"note":"arxivid:2001.08405\narxiv_version_number:1"},{"id":"arxiv:2004.00814","type":"article-journal","author":[{"given":"Ayumu","family":"Nakayama"},{"given":"Manabu","family":"Hagiwara"}],"title":"Single Quantum Deletion Error-Correcting Codes","issued":{"date-parts":[[2020,4,2]]},"note":"arxivid:2004.00814\narxiv_version_number:1"},{"id":"arxiv:2102.02494","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit45174.2021.9518078","page":"1499-1503","source":"Crossref","title":"Permutation-invariant quantum coding for quantum deletion channels","author":[{"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]}],"event":"2021 IEEE International Symposium on Information Theory (ISIT)","container-title":"2021 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2021,7,12]]},"URL":"http://dx.doi.org/10.1109/ISIT45174.2021.9518078","note":"arxivid:2102.02494"},{"id":"arxiv:2102.03015","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit45174.2021.9517870","page":"1493-1498","source":"Crossref","title":"Permutation-Invariant Quantum Codes for Deletion Errors","author":[{"given":"Taro","family":"Shibayama","sequence":"first","affiliation":[]},{"given":"Manabu","family":"Hagiwara","sequence":"additional","affiliation":[]}],"event":"2021 IEEE International Symposium on Information Theory (ISIT)","container-title":"2021 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2021,7,12]]},"URL":"http://dx.doi.org/10.1109/ISIT45174.2021.9517870","note":"arxivid:2102.03015"},{"id":"arxiv:2105.07214","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw48936.2021.9611450","page":"1-6","source":"Crossref","title":"The equivalence between correctability of deletions and insertions of separable states in quantum codes","author":[{"given":"Taro","family":"Shibayama","sequence":"first","affiliation":[]},{"given":"Yingkai","family":"Ouyang","sequence":"additional","affiliation":[]}],"event":"2021 IEEE Information Theory Workshop (ITW)","container-title":"2021 IEEE Information Theory Workshop (ITW)","original-title":[],"issued":{"date-parts":[[2021,10,17]]},"URL":"http://dx.doi.org/10.1109/ITW48936.2021.9611450","note":"arxivid:2105.07214"},{"id":"arxiv:2501.07027","type":"article-journal","author":[{"given":"Taro","family":"Shibayama"}],"title":"Necessary and sufficient condition for constructing a single qudit insertion/deletion code and its decoding algorithm","issued":{"date-parts":[[2025,1,13]]},"note":"arxivid:2501.07027\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0002020","type":"journal-article","publisher":"Physical Society of Japan","issue":"11","DOI":"10.1143/jpsj.69.3540","page":"3540-3543","source":"Crossref","title":"Quantum Interleaver: Quantum Error Correction for Burst Error","volume":"69","author":[{"given":"Shiro","family":"Kawabata","sequence":"first","affiliation":[{"name":"Physical Science Division, Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305-8568"}]}],"container-title":"Journal of the Physical Society of Japan","original-title":[],"language":"en","issued":{"date-parts":[[2000,11,15]]},"URL":"http://dx.doi.org/10.1143/JPSJ.69.3540","ISSN":["0031-9015","1347-4073"],"container-title-short":"J. Phys. Soc. Jpn.","note":"alternative-id:10.1143/JPSJ.69.3540\narxivid:quant-ph/0002020"},{"id":"doi:10.1109/18.771250","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/18.771250","page":"1703-1708","source":"Crossref","title":"Spatially correlated qubit errors and burst-correcting quantum codes","volume":"45","author":[{"given":"F.","family":"Vatan","sequence":"first","affiliation":[]},{"given":"V.P.","family":"Roychowdhury","sequence":"additional","affiliation":[]},{"given":"M.P.","family":"Anantram","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[1999,7]]},"URL":"http://dx.doi.org/10.1109/18.771250","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:quant-ph/9709049","type":"article-journal","author":[{"given":"Alexei","family":"Ashikhmin"},{"given":"Simon","family":"Litsyn"}],"title":"Upper Bounds on the Size of Quantum Codes","issued":{"date-parts":[[1997,9,29]]},"note":"arxivid:quant-ph/9709049\narxiv_version_number:2"},{"id":"doi:10.1109/TIT.2005.862086","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2005.862086","page":"986-991","source":"Crossref","title":"Asymptotic bounds on quantum codes from algebraic geometry codes","volume":"52","author":[{"family":"Keqin Feng","sequence":"first","affiliation":[]},{"family":"San Ling","sequence":"additional","affiliation":[]},{"family":"Chaoping Xing","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2006,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.862086","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:1007.3655","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.83.052305","source":"Crossref","title":"Quantum error correction with degenerate codes for correlated noise","volume":"83","author":[{"given":"Giulio","family":"Chiribella","sequence":"first","affiliation":[]},{"given":"Michele","family":"Dall’Arno","sequence":"additional","affiliation":[]},{"given":"Giacomo Mauro","family":"D’Ariano","sequence":"additional","affiliation":[]},{"given":"Chiara","family":"Macchiavello","sequence":"additional","affiliation":[]},{"given":"Paolo","family":"Perinotti","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,5,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.052305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052305","note":"arxivid:1007.3655"},{"id":"doi:10.1109/TIT.2004.838088","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tit.2004.838088","page":"3323-3325","source":"Crossref","title":"A Finite Gilbert–Varshamov Bound for Pure Stabilizer Quantum Codes","volume":"50","author":[{"given":"K.","family":"Feng","sequence":"first","affiliation":[]},{"given":"Z.","family":"Ma","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2004,12]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.838088","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:quant-ph/9602022","type":"article-journal","author":[{"given":"A.","family":"Ekert"},{"given":"C.","family":"Macchiavello"}],"title":"Error Correction in Quantum Communication","issued":{"date-parts":[[1996,2,29]]},"note":"arxivid:quant-ph/9602022\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9906131","type":"article-journal","author":[{"given":"A.","family":"Ashikhmin"},{"given":"A.","family":"Barg"},{"given":"E.","family":"Knill"},{"given":"S.","family":"Litsyn"}],"title":"Quantum Error Detection II: Bounds","issued":{"date-parts":[[1999,6,30]]},"note":"arxivid:quant-ph/9906131\narxiv_version_number:1"},{"id":"doi:10.7907/m0xg-zs21","type":"thesis","categories":["quantum information theory","quantum error correcting codes","Physics","degenerate codes","quantum key distribution"],"language":"en","author":[{"family":"Smith","given":"Graeme Stewart Baird"}],"issued":{"date-parts":[[2006]]},"abstract":"This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution.  The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively.  The next four chapters present achievable rates for quantum codes in various scenarios.  The final chapter is dedicated to an upper bound on the quantum channel capacity.\n\nChapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists.  These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is impossible.\n\nChapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes.  By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes.\n\nChapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called \"twisted states.\"\n\nIn Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible.\n\nFinally, Chapter 7 presents a new upper bound on the quantum channel capacity that is both additive and convex, and which can be interpreted as the capacity of the channel for communication given access to side channels from a class of zero capacity \"cloning\" channels.  This \"clone assisted capacity\" is equal to the unassisted capacity for channels that are degradable, which we use to find new upper bounds on the capacity of a depolarizing channel.","DOI":"10.7907/M0XG-ZS21","publisher":"California Institute of Technology","title":"Upper and Lower Bounds on Quantum Codes","URL":"https://resolver.caltech.edu/CaltechETD:etd-05222006-153339","version":"Final","note":"copyright:No commercial reproduction, distribution, display or performance rights in this work are provided."},{"id":"doi:10.1016/j.jmaa.2007.08.023","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.jmaa.2007.08.023","page":"550-557","source":"Crossref","title":"The asymptotic probability distribution of the relative distance of additive quantum codes","volume":"340","author":[{"given":"Yuefei","family":"Ma","sequence":"first","affiliation":[]}],"container-title":"Journal of Mathematical Analysis and Applications","original-title":[],"language":"en","issued":{"date-parts":[[2008,4]]},"URL":"http://dx.doi.org/10.1016/j.jmaa.2007.08.023","ISSN":["0022-247X"],"container-title-short":"Journal of Mathematical Analysis and Applications","note":"alternative-id:S0022247X07010451"},{"id":"arxiv:0811.4262","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.102.110502","source":"Crossref","title":"Restrictions on Transversal Encoded Quantum Gate Sets","volume":"102","author":[{"given":"Bryan","family":"Eastin","sequence":"first","affiliation":[]},{"given":"Emanuel","family":"Knill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.102.110502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110502","note":"arxivid:0811.4262"},{"id":"doi:10.1007/978-3-540-37634-7_13","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-540-37634-7_13","page":"287-313","source":"Crossref","title":"Searching for linear codes with large minimum distance","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]}],"container-title":"Algorithms and Computation in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[null]]},"ISBN":["9783540376323"],"URL":"http://dx.doi.org/10.1007/978-3-540-37634-7_13","note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:2405.15057","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2024.3503420","page":"499-517","source":"Crossref","title":"Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes","volume":"71","author":[{"ORCID":"https://orcid.org/0000-0002-5851-2717","authenticated-orcid":false,"given":"Martianus","family":"Frederic Ezerman","sequence":"first","affiliation":[{"name":"School of Physical and Mathematical Sciences, Nanyang Technological University, Jurong West, Singapore"}]},{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gdansk, Gda&#x0144;sk, Poland"}]},{"ORCID":"https://orcid.org/0000-0002-1978-3557","authenticated-orcid":false,"given":"San","family":"Ling","sequence":"additional","affiliation":[{"name":"School of Physical and Mathematical Sciences, Nanyang Technological University, Jurong West, Singapore"}]},{"ORCID":"https://orcid.org/0000-0002-1694-9283","authenticated-orcid":false,"given":"Ferruh","family":"Özbudak","sequence":"additional","affiliation":[{"name":"Faculty of Engineering and Natural Sciences, Sabanc&#x0131; University, &#x0130;stanbul, T&#x00FC;rkiye"}]},{"ORCID":"https://orcid.org/0000-0003-2658-5441","authenticated-orcid":false,"given":"Buket","family":"Özkaya","sequence":"additional","affiliation":[{"name":"Institute of Applied Mathematics, Middle East Technical University, Ankara, T&#x00FC;rkiye"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2025,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3503420","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2405.15057"},{"id":"arxiv:2207.13707","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/prxquantum.4.040336","source":"Crossref","title":"Time-Energy Uncertainty Relation for Noisy Quantum Metrology","volume":"4","author":[{"given":"Philippe","family":"Faist","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0547-4569","authenticated-orcid":true,"given":"Mischa P.","family":"Woods","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2302-8025","authenticated-orcid":true,"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2421-4762","authenticated-orcid":true,"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,5]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.040336","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040336","note":"arxivid:2207.13707"},{"id":"arxiv:2111.06360","type":"article-journal","author":[{"given":"Zi-Wen","family":"Liu"},{"given":"Sisi","family":"Zhou"}],"title":"Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies","issued":{"date-parts":[[2023,12,8]]},"note":"arxivid:2111.06360\narxiv_version_number:4"},{"id":"arxiv:2111.06355","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin–Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed–Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics.</jats:p>","DOI":"10.1038/s41534-023-00788-4","source":"Crossref","title":"Approximate symmetries and quantum error correction","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0002-3402-9763","authenticated-orcid":false,"given":"Zi-Wen","family":"Liu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4618-8590","authenticated-orcid":false,"given":"Sisi","family":"Zhou","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,21]]},"URL":"http://dx.doi.org/10.1038/s41534-023-00788-4","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"119","note":"alternative-id:788\narxivid:2111.06355"},{"id":"arxiv:2005.11918","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Covariant codes are quantum codes such that a symmetry transformation on the logical system could be realized by a symmetry transformation on the physical system, usually with limited capability of performing quantum error correction (an important case being the Eastin–Knill theorem). The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. We also present a type of covariant codes which nearly saturates these lower bounds.</jats:p>","DOI":"10.22331/q-2021-08-09-521","page":"521","source":"Crossref","title":"New perspectives on covariant quantum error correction","volume":"5","author":[{"given":"Sisi","family":"Zhou","sequence":"first","affiliation":[{"name":"Department of Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA"}]},{"given":"Zi-Wen","family":"Liu","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada"}]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Illinois 60637, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,9]]},"URL":"http://dx.doi.org/10.22331/q-2021-08-09-521","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2005.11918"},{"id":"arxiv:2004.11893","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevlett.126.150503","source":"Crossref","title":"Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem","volume":"126","author":[{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":true,"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5550-4431","authenticated-orcid":true,"given":"Rafał","family":"Demkowicz-Dobrzański","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2021,4,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.126.150503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"150503","note":"arxivid:2004.11893"},{"id":"arxiv:2410.07045","type":"article-journal","author":[{"given":"Pragati","family":"Gupta"},{"given":"Andrea","family":"Morello"},{"given":"Barry C.","family":"Sanders"}],"title":"Universal transversal gates","issued":{"date-parts":[[2024,10,9]]},"note":"arxivid:2410.07045\narxiv_version_number:1"},{"id":"arxiv:2505.00427","type":"article-journal","author":[{"given":"Rhea","family":"Alexander"}],"title":"A new approximate Eastin-Knill theorem","issued":{"date-parts":[[2025,5,29]]},"note":"arxivid:2505.00427\narxiv_version_number:2"},{"id":"arxiv:2102.11835","type":"article-journal","author":[{"given":"Linghang","family":"Kong"},{"given":"Zi-Wen","family":"Liu"}],"title":"Charge-conserving unitaries typically generate optimal covariant quantum error-correcting codes","issued":{"date-parts":[[2021,2,23]]},"note":"arxivid:2102.11835\narxiv_version_number:1"},{"id":"arxiv:2112.01498","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.3.020314","source":"Crossref","title":"Near-Optimal Covariant Quantum Error-Correcting Codes from Random Unitaries with Symmetries","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-5854-5340","authenticated-orcid":true,"given":"Linghang","family":"Kong","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3402-9763","authenticated-orcid":true,"given":"Zi-Wen","family":"Liu","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,20]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.020314","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020314","note":"arxivid:2112.01498"},{"id":"arxiv:1612.04504","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/aa7235","page":"063029","source":"Crossref","title":"Invariant perfect tensors","volume":"19","author":[{"given":"Youning","family":"Li","sequence":"first","affiliation":[]},{"given":"Muxin","family":"Han","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2017,6,26]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aa7235","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1612.04504"},{"id":"arxiv:quant-ph/0005013","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/s0375-9601(00)00480-1","page":"213-217","source":"Crossref","title":"How entangled can two couples get?","volume":"273","author":[{"given":"A.","family":"Higuchi","sequence":"first","affiliation":[]},{"given":"A.","family":"Sudbery","sequence":"additional","affiliation":[]}],"container-title":"Physics Letters A","original-title":[],"language":"en","issued":{"date-parts":[[2000,8]]},"URL":"http://dx.doi.org/10.1016/S0375-9601(00)00480-1","ISSN":["0375-9601"],"container-title-short":"Physics Letters A","note":"alternative-id:S0375960100004801\narxivid:quant-ph/0005013"},{"id":"arxiv:2210.02483","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>Perfect tensors describe highly entangled quantum states that have attracted particular attention in the fields of quantum information theory and quantum gravity. In loop quantum gravity, the natural question arises whether SU(2) invariant tensors, which are fundamental ingredients of the basis states of spacetime, can also be perfect. In this work, we present a number of general constraints for the layout of such invariant perfect tensors (IPTs) and further describe a systematic and constructive approach to check the existence of an IPT of given valence. We apply our algorithm to show that no qubit encoding of valence 6 can be described by an IPT and close a gap to prove a no-go theorem for invariant perfect qubit encodings. We also provide two alternative proofs for the non-existence of 4-valent qubit IPTs which has been shown in [1, 2].</jats:p>","DOI":"10.1007/jhep02(2023)062","source":"Crossref","title":"No invariant perfect qubit codes","volume":"2023","author":[{"ORCID":"https://orcid.org/0000-0001-7352-513X","authenticated-orcid":false,"given":"Refik","family":"Mansuroglu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8083-7139","authenticated-orcid":false,"given":"Hanno","family":"Sahlmann","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,2,7]]},"URL":"http://dx.doi.org/10.1007/JHEP02(2023)062","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"62","note":"alternative-id:20155\narxivid:2210.02483"},{"id":"arxiv:2003.10559","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.2.010343","source":"Crossref","title":"Asymptotic Theory of Quantum Channel Estimation","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0003-4618-8590","authenticated-orcid":true,"given":"Sisi","family":"Zhou","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,3,16]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.010343","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010343","note":"arxivid:2003.10559"},{"id":"arxiv:1706.02445","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed that suppresses the noise without obscuring the signal; the optimal code, achieving the best possible precision, can be found by solving a semidefinite program.</jats:p>","DOI":"10.1038/s41467-017-02510-3","source":"Crossref","title":"Achieving the Heisenberg limit in quantum metrology using quantum error correction","volume":"9","author":[{"given":"Sisi","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Mengzhen","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":false,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2018,1,8]]},"URL":"http://dx.doi.org/10.1038/s41467-017-02510-3","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"78","note":"alternative-id:2510\narxivid:1706.02445"},{"id":"arxiv:2402.18765","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.133.170801","source":"Crossref","title":"Limits of Noisy Quantum Metrology with Restricted Quantum Controls","volume":"133","author":[{"ORCID":"https://orcid.org/0000-0003-4618-8590","authenticated-orcid":true,"given":"Sisi","family":"Zhou","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Waterloo"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.170801","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"170801","note":"arxivid:2402.18765"},{"id":"arxiv:1901.00896","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.</jats:p>","DOI":"10.22331/q-2020-07-02-288","page":"288","source":"Crossref","title":"Optimal probes and error-correction schemes in multi-parameter quantum metrology","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-9912-9186","authenticated-orcid":false,"given":"Wojciech","family":"Górecki","sequence":"first","affiliation":[{"name":"Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland"}]},{"ORCID":"https://orcid.org/0000-0003-4618-8590","authenticated-orcid":false,"given":"Sisi","family":"Zhou","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":false,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA"},{"name":"Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA"}]},{"ORCID":"https://orcid.org/0000-0001-5550-4431","authenticated-orcid":false,"given":"Rafał","family":"Demkowicz-Dobrzański","sequence":"additional","affiliation":[{"name":"Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,2]]},"URL":"http://dx.doi.org/10.22331/q-2020-07-02-288","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1901.00896"},{"id":"arxiv:1910.08472","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevresearch.2.013235","source":"Crossref","title":"Optimal approximate quantum error correction for quantum metrology","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0003-4618-8590","authenticated-orcid":true,"given":"Sisi","family":"Zhou","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.013235","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013235","note":"arxivid:1910.08472"},{"id":"arxiv:2303.00881","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.109.042406","source":"Crossref","title":"Achieving metrological limits using ancilla-free quantum error-correcting codes","volume":"109","author":[{"ORCID":"https://orcid.org/0000-0003-4618-8590","authenticated-orcid":true,"given":"Sisi","family":"Zhou","sequence":"first","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5"}]},{"ORCID":"https://orcid.org/0009-0007-5566-7223","authenticated-orcid":true,"given":"Argyris Giannisis","family":"Manes","sequence":"additional","affiliation":[{"name":"Department of Physics, Yale University, New Haven, Connecticut 06511, USA"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.109.042406","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042406","note":"arxivid:2303.00881"},{"id":"arxiv:1704.06280","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.7.041009","source":"Crossref","title":"Adaptive Quantum Metrology under General Markovian Noise","volume":"7","author":[{"given":"Rafał","family":"Demkowicz-Dobrzański","sequence":"first","affiliation":[]},{"given":"Jan","family":"Czajkowski","sequence":"additional","affiliation":[]},{"given":"Pavel","family":"Sekatski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.7.041009","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041009","note":"arxivid:1704.06280"},{"id":"arxiv:2504.13168","type":"article-journal","author":[{"given":"Hyukgun","family":"Kwon"},{"given":"Uwe R.","family":"Fischer"},{"given":"Seung-Woo","family":"Lee"},{"given":"Liang","family":"Jiang"}],"title":"Restoring Heisenberg scaling in time via autonomous quantum error correction","issued":{"date-parts":[[2026,2,28]]},"note":"arxivid:2504.13168\narxiv_version_number:2"},{"id":"arxiv:1602.07144","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.116.230502","source":"Crossref","title":"Quantum Metrology Enhanced by Repetitive Quantum Error Correction","volume":"116","author":[{"given":"Thomas","family":"Unden","sequence":"first","affiliation":[]},{"given":"Priya","family":"Balasubramanian","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Louzon","sequence":"additional","affiliation":[]},{"given":"Yuval","family":"Vinkler","sequence":"additional","affiliation":[]},{"given":"Martin B.","family":"Plenio","sequence":"additional","affiliation":[]},{"given":"Matthew","family":"Markham","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Twitchen","sequence":"additional","affiliation":[]},{"given":"Alastair","family":"Stacey","sequence":"additional","affiliation":[]},{"given":"Igor","family":"Lovchinsky","sequence":"additional","affiliation":[]},{"given":"Alexander O.","family":"Sushkov","sequence":"additional","affiliation":[]},{"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Retzker","sequence":"additional","affiliation":[]},{"given":"Boris","family":"Naydenov","sequence":"additional","affiliation":[]},{"given":"Liam P.","family":"McGuinness","sequence":"additional","affiliation":[]},{"given":"Fedor","family":"Jelezko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,6,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.116.230502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230502","note":"arxivid:1602.07144"},{"id":"arxiv:2503.15743","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/tqe.2026.3664680","page":"1-13","source":"Crossref","title":"Explaining Robust Quantum Metrology by Counting Codewords","author":[{"ORCID":"https://orcid.org/0009-0000-8935-657X","authenticated-orcid":false,"given":"Oskar","family":"Novak","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, USA"}]},{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, USA"}]}],"container-title":"IEEE Transactions on Quantum Engineering","original-title":[],"issued":{"date-parts":[[2026]]},"URL":"http://dx.doi.org/10.1109/TQE.2026.3664680","ISSN":["2689-1808"],"container-title-short":"IEEE Trans. Quantum Eng.","note":"arxivid:2503.15743"},{"id":"arxiv:2212.06285","type":"article-journal","author":[{"given":"Yingkai","family":"Ouyang"},{"given":"Gavin K.","family":"Brennen"}],"title":"Finite-round quantum error correction on symmetric quantum sensors","issued":{"date-parts":[[2026,2,17]]},"note":"arxivid:2212.06285\narxiv_version_number:5"},{"id":"arxiv:1902.07725","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Following the introduction of the task of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">reference frame error</mml:mtext></mml:mrow></mml:math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">correction</mml:mtext></mml:mrow></mml:math> \\cite{hayden2017error}, we show how, by using reference frame alignment with clocks, one can add a continuous Abelian group of transversal logical gates to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>y</mml:mi></mml:math> error-correcting code. With this we further explore a way of circumventing the no-go theorem of Eastin and Knill, which states that if local errors are correctable, the group of transversal gates must be of finite order. We are able to do this by introducing a small error on the decoding procedure that decreases with the dimension of the frames used. Furthermore, we show that there is a direct relationship between how small this error can be and how accurate quantum clocks can be: the more accurate the clock, the smaller the error; and the no-go theorem would be violated if time could be measured perfectly in quantum mechanics. The asymptotic scaling of the error is studied under a number of scenarios of reference frames and error models. The scheme is also extended to errors at unknown locations, and we show how to achieve this by simple majority voting related error correction schemes on the reference frames. In the Outlook, we discuss our results in relation to the AdS/CFT correspondence and the Page-Wooters mechanism.</jats:p>","DOI":"10.22331/q-2020-03-23-245","page":"245","source":"Crossref","title":"Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames","volume":"4","author":[{"given":"Mischa P.","family":"Woods","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, ETH Zurich, Switzerland"}]},{"ORCID":"https://orcid.org/0000-0002-5889-4022","authenticated-orcid":false,"given":"Álvaro M.","family":"Alhambra","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,23]]},"URL":"http://dx.doi.org/10.22331/q-2020-03-23-245","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1902.07725"},{"id":"arxiv:2007.09154","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevresearch.4.023107","source":"Crossref","title":"Optimal universal quantum error correction via bounded reference frames","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-0531-8929","authenticated-orcid":true,"given":"Yuxiang","family":"Yang","sequence":"first","affiliation":[]},{"given":"Yin","family":"Mo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2302-8025","authenticated-orcid":true,"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[]},{"given":"Giulio","family":"Chiribella","sequence":"additional","affiliation":[]},{"given":"Mischa P.","family":"Woods","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.4.023107","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023107","note":"arxivid:2007.09154"},{"id":"arxiv:quant-ph/0005115","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.62.062314","source":"Crossref","title":"Three qubits can be entangled in two inequivalent ways","volume":"62","author":[{"given":"W.","family":"Dür","sequence":"first","affiliation":[]},{"given":"G.","family":"Vidal","sequence":"additional","affiliation":[]},{"given":"J. I.","family":"Cirac","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,11,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.62.062314","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062314","note":"arxivid:quant-ph/0005115"},{"id":"arxiv:0708.4025","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"09","DOI":"10.1088/1126-6708/2007/09/120","page":"120-120","source":"Crossref","title":"Black holes as mirrors: quantum information in random subsystems","volume":"2007","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"issued":{"date-parts":[[2007,9,26]]},"URL":"http://dx.doi.org/10.1088/1126-6708/2007/09/120","ISSN":["1029-8479"],"container-title-short":"J. High Energy Phys.","note":"arxivid:0708.4025"},{"id":"arxiv:1903.05124","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.125.030505","source":"Crossref","title":"Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition","volume":"125","author":[{"ORCID":"https://orcid.org/0000-0002-1247-062X","authenticated-orcid":true,"given":"Soonwon","family":"Choi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5357-8821","authenticated-orcid":true,"given":"Yimu","family":"Bao","sequence":"additional","affiliation":[]},{"given":"Xiao-Liang","family":"Qi","sequence":"additional","affiliation":[]},{"given":"Ehud","family":"Altman","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.030505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030505","note":"arxivid:1903.05124"},{"id":"arxiv:1905.05195","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.10.041020","source":"Crossref","title":"Dynamical Purification Phase Transition Induced by Quantum Measurements","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1008-5178","authenticated-orcid":true,"given":"David A.","family":"Huse","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2020,10,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.10.041020","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041020","note":"arxivid:1905.05195"},{"id":"arxiv:1911.00008","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevb.101.060301","source":"Crossref","title":"Critical properties of the measurement-induced transition in random quantum circuits","volume":"101","author":[{"ORCID":"https://orcid.org/0000-0002-7575-7361","authenticated-orcid":true,"given":"Aidan","family":"Zabalo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6903-0417","authenticated-orcid":true,"given":"Justin H.","family":"Wilson","sequence":"additional","affiliation":[]},{"given":"Sarang","family":"Gopalakrishnan","sequence":"additional","affiliation":[]},{"given":"David A.","family":"Huse","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3109-640X","authenticated-orcid":true,"given":"J. H.","family":"Pixley","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2020,2,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.101.060301","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"060301","note":"arxivid:1911.00008"},{"id":"doi:10.1103/PhysRevX.9.031009","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.9.031009","source":"Crossref","title":"Measurement-Induced Phase Transitions in the Dynamics of Entanglement","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0003-0774-3563","authenticated-orcid":true,"given":"Brian","family":"Skinner","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Ruhman","sequence":"additional","affiliation":[]},{"given":"Adam","family":"Nahum","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.031009","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031009"},{"id":"doi:10.1103/PhysRevB.98.205136","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevb.98.205136","source":"Crossref","title":"Quantum Zeno effect and the many-body entanglement transition","volume":"98","author":[{"given":"Yaodong","family":"Li","sequence":"first","affiliation":[]},{"given":"Xiao","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Matthew P. A.","family":"Fisher","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2018,11,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.98.205136","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"205136"},{"id":"arxiv:1808.05949","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevb.99.224307","source":"Crossref","title":"Unitary-projective entanglement dynamics","volume":"99","author":[{"given":"Amos","family":"Chan","sequence":"first","affiliation":[]},{"given":"Rahul M.","family":"Nandkishore","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5013-0186","authenticated-orcid":true,"given":"Michael","family":"Pretko","sequence":"additional","affiliation":[]},{"given":"Graeme","family":"Smith","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,6,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.99.224307","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"224307","note":"arxivid:1808.05949"},{"id":"arxiv:2106.05881","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7","DOI":"10.1038/s41567-022-01619-7","page":"760-764","source":"Crossref","title":"Measurement-induced quantum phases realized in a trapped-ion quantum computer","volume":"18","author":[{"ORCID":"https://orcid.org/0000-0002-2977-2747","authenticated-orcid":false,"given":"Crystal","family":"Noel","sequence":"first","affiliation":[]},{"given":"Pradeep","family":"Niroula","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0019-256X","authenticated-orcid":false,"given":"Daiwei","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Andrew","family":"Risinger","sequence":"additional","affiliation":[]},{"given":"Laird","family":"Egan","sequence":"additional","affiliation":[]},{"given":"Debopriyo","family":"Biswas","sequence":"additional","affiliation":[]},{"given":"Marko","family":"Cetina","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0509-3421","authenticated-orcid":false,"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[]},{"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[]},{"given":"David A.","family":"Huse","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Monroe","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,6,2]]},"URL":"http://dx.doi.org/10.1038/s41567-022-01619-7","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:1619\narxivid:2106.05881"},{"id":"doi:10.1103/PhysRevD.100.086001","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevd.100.086001","source":"Crossref","title":"Soft mode and interior operator in the Hayden-Preskill thought experiment","volume":"100","author":[{"given":"Beni","family":"Yoshida","sequence":"first","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2019,10,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.100.086001","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"086001"},{"id":"arxiv:2007.03822","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevb.103.104306","source":"Crossref","title":"Statistical mechanics of quantum error correcting codes","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0003-3742-1944","authenticated-orcid":true,"given":"Yaodong","family":"Li","sequence":"first","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California 93106, USA"}]},{"given":"Matthew P. A.","family":"Fisher","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California 93106, USA"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2021,3,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.103.104306","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"104306","note":"arxivid:2007.03822"},{"id":"arxiv:2011.06595","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.127.235701","source":"Crossref","title":"Topological Order and Criticality in \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mi mathvariant=\"normal\">D</mml:mi></mml:mrow></mml:math>\n Monitored Random Quantum Circuits","volume":"127","author":[{"ORCID":"https://orcid.org/0000-0001-7613-3780","authenticated-orcid":true,"given":"Ali","family":"Lavasani","sequence":"first","affiliation":[{"name":"Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Yahya","family":"Alavirad","sequence":"additional","affiliation":[{"name":"Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA"},{"name":"Department of Physics, University of California at San Diego, La Jolla, California 92093, USA"}]},{"given":"Maissam","family":"Barkeshli","sequence":"additional","affiliation":[{"name":"Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.127.235701","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"235701","note":"arxivid:2011.06595"},{"id":"arxiv:2107.02194","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present a quantum error correcting code with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">dynamically generated logical qubits</mml:mtext></mml:mrow></mml:math>. When viewed as a subsystem code, the code has no logical qubits. Nevertheless, our measurement patterns generate logical qubits, allowing the code to act as a fault-tolerant quantum memory. Our particular code gives a model very similar to the two-dimensional toric code, but each measurement is a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mi>w</mml:mi><mml:mi>o</mml:mi></mml:math>-qubit Pauli measurement.</jats:p>","DOI":"10.22331/q-2021-10-19-564","page":"564","source":"Crossref","title":"Dynamically Generated Logical Qubits","volume":"5","author":[{"given":"Matthew B.","family":"Hastings","sequence":"first","affiliation":[{"name":"Station Q, Microsoft Quantum, Santa Barbara, CA 93106-6105, USA"},{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,10,19]]},"URL":"http://dx.doi.org/10.22331/q-2021-10-19-564","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2107.02194"},{"id":"arxiv:1312.7646","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2013.6620245","source":"Crossref","title":"Short random circuits define good quantum error correcting codes","author":[{"given":"Winton","family":"Brown","sequence":"first","affiliation":[]},{"given":"Omar","family":"Fawzi","sequence":"additional","affiliation":[]}],"event":"2013 IEEE International Symposium on Information Theory (ISIT)","container-title":"2013 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2013,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2013.6620245","note":"arxivid:1312.7646"},{"id":"arxiv:quant-ph/9910081","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.62.062311","source":"Crossref","title":"Quantum to classical phase transition in noisy quantum computers","volume":"62","author":[{"given":"Dorit","family":"Aharonov","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,11,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.62.062311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062311","note":"arxivid:quant-ph/9910081"},{"id":"arxiv:2111.08018","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-031-03998-0_9","page":"211-249","source":"Crossref","title":"Entanglement Dynamics in Hybrid Quantum Circuits","author":[{"given":"Andrew C.","family":"Potter","sequence":"first","affiliation":[]},{"given":"Romain","family":"Vasseur","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"language":"en","issued":{"date-parts":[[2022]]},"ISBN":["9783031039973","9783031039980"],"URL":"http://dx.doi.org/10.1007/978-3-031-03998-0_9","ISSN":["2364-9054","2364-9062"],"note":"publisher-location:Cham\narxivid:2111.08018"},{"id":"arxiv:2207.14280","type":"journal-article","publisher":"Annual Reviews","issue":"1","abstract":"<jats:p> Quantum circuits—built from local unitary gates and local measurements—are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far from equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as dynamical phase transitions in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics. </jats:p>","DOI":"10.1146/annurev-conmatphys-031720-030658","page":"335-379","source":"Crossref","title":"Random Quantum Circuits","volume":"14","author":[{"given":"Matthew P.A.","family":"Fisher","sequence":"first","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California, USA;,"}]},{"given":"Vedika","family":"Khemani","sequence":"additional","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, California, USA;"}]},{"given":"Adam","family":"Nahum","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique, École Normale Supérieure, CNRS, Université PSL, Sorbonne Université, Université de Paris, Paris, France;"}]},{"given":"Sagar","family":"Vijay","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California, USA;,"}]}],"container-title":"Annual Review of Condensed Matter Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,10]]},"URL":"http://dx.doi.org/10.1146/annurev-conmatphys-031720-030658","ISSN":["1947-5454","1947-5462"],"container-title-short":"Annu. Rev. Condens. Matter Phys.","note":"alternative-id:10.1146/annurev-conmatphys-031720-030658\narxivid:2207.14280"},{"id":"arxiv:2102.02619","type":"journal-article","publisher":"IOP Publishing","issue":"3","DOI":"10.1088/2058-9565/ac0293","page":"033002","source":"Crossref","title":"Holographic tensor network models and quantum error correction: a topical review","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-7142-0059","authenticated-orcid":false,"given":"Alexander","family":"Jahn","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3033-1292","authenticated-orcid":false,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2021,6,23]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ac0293","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2102.02619"},{"id":"arxiv:2110.14669","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","abstract":"<jats:title>Abstract</jats:title><jats:p>This article reviews the progress in our understanding of the reconstruction of the bulk spacetime in the holographic correspondence from the dual field theory including an account of how these developments have led to the reproduction of the Page curve of the Hawking radiation from black holes. We review quantum error correction and relevant recovery maps with toy examples based on tensor networks, and discuss how it provides the desired framework for bulk reconstruction in which apparent inconsistencies with properties of the operator algebra in the dual field theory are naturally resolved. The importance of understanding the modular flow in the dual field theory has been emphasized. We discuss how the state-dependence of reconstruction of black hole microstates can be formulated in the framework of quantum error correction with inputs from extremal surfaces along with a quantification of the complexity of encoding of bulk operators. Finally, we motivate and discuss a class of tractable microstate models of black holes which can illuminate how the black hole complementarity principle can emerge operationally without encountering information paradoxes, and provide new insights into generation of desirable features of encoding into the Hawking radiation.</jats:p>","DOI":"10.1140/epjc/s10052-022-10382-1","source":"Crossref","title":"Holographic spacetime, black holes and quantum error correcting codes: a review","volume":"82","author":[{"given":"Tanay","family":"Kibe","sequence":"first","affiliation":[]},{"given":"Prabha","family":"Mandayam","sequence":"additional","affiliation":[]},{"given":"Ayan","family":"Mukhopadhyay","sequence":"additional","affiliation":[]}],"container-title":"The European Physical Journal C","original-title":[],"language":"en","issued":{"date-parts":[[2022,5]]},"URL":"http://dx.doi.org/10.1140/epjc/s10052-022-10382-1","ISSN":["1434-6052"],"container-title-short":"Eur. Phys. J. C","page":"463","note":"alternative-id:10382\narxivid:2110.14669"},{"id":"arxiv:1611.01516","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevd.95.105007","source":"Crossref","title":"Entanglement from topology in Chern-Simons theory","volume":"95","author":[{"given":"Grant","family":"Salton","sequence":"first","affiliation":[]},{"given":"Brian","family":"Swingle","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Walter","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2017,5,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.95.105007","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"105007","note":"arxivid:1611.01516"},{"id":"arxiv:2510.15067","type":"article-journal","author":[{"given":"William","family":"Munizzi"},{"given":"Howard J.","family":"Schnitzer"}],"title":"Topological Preparation of Non-Stabilizer States and Clifford Evolution in $SU(2)_1$ Chern-Simons Theory","issued":{"date-parts":[[2026,2,12]]},"note":"arxivid:2510.15067\narxiv_version_number:2"},{"id":"arxiv:2501.10354","type":"article-journal","author":[{"given":"Kaifeng","family":"Bu"},{"given":"Weichen","family":"Gu"},{"given":"Xiang","family":"Li"}],"title":"Quantum locally recoverable code with intersecting recovery sets","issued":{"date-parts":[[2025,1,17]]},"note":"arxivid:2501.10354\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9604034","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.84.2525","page":"2525-2528","source":"Crossref","title":"Theory of Quantum Error Correction for General Noise","volume":"84","author":[{"given":"Emanuel","family":"Knill","sequence":"first","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]},{"given":"Lorenza","family":"Viola","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2000,3,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.84.2525","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9604034"},{"id":"arxiv:quant-ph/9702031","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.56.1721","page":"1721-1732","source":"Crossref","title":"Information-theoretic interpretation of quantum error-correcting codes","volume":"56","author":[{"given":"Nicolas J.","family":"Cerf","sequence":"first","affiliation":[]},{"given":"Richard","family":"Cleve","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.56.1721","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9702031"},{"id":"arxiv:quant-ph/9703048","type":"article-journal","author":[{"given":"Eric M.","family":"Rains"}],"title":"Nonbinary quantum codes","issued":{"date-parts":[[1997,3,25]]},"note":"arxivid:quant-ph/9703048\narxiv_version_number:1"},{"id":"manual:-A.-Winter-private-communicat","type":"document","author":[{"family":"Winter","given":"A."}],"genre":"private communication","issued":"2019"},{"id":"arxiv:1907.07733","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> of all QMDS codes with local dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>D</mml:mi></mml:math> and distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:math> is bounded by <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math>. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>r</mml:mi></mml:math>-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.</jats:p>","DOI":"10.22331/q-2020-06-18-284","page":"284","source":"Crossref","title":"Quantum Codes of Maximal Distance and Highly Entangled Subspaces","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-3856-4018","authenticated-orcid":false,"given":"Felix","family":"Huber","sequence":"first","affiliation":[{"name":"ICFO - The Institute of Photonic Sciences, 08860 Castelldefels (Barcelona), Spain"},{"name":"Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany"},{"name":"Naturwissenschaftlich-Technische Fakultät, Universität Siegen, 57068 Siegen, Germany"}]},{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdańsk, Poland"},{"name":"Max Planck Institute for the Science of Light, 91058 Erlangen, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,18]]},"URL":"http://dx.doi.org/10.22331/q-2020-06-18-284","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1907.07733"},{"id":"doi:10.1017/CBO9781139034807.014","type":"book-chapter","publisher":"Cambridge University Press","DOI":"10.1017/cbo9781139034807.014","page":"307-326","source":"Crossref","title":"Algebraic quantum coding theory","author":[{"given":"Andreas","family":"Klappenecker","sequence":"first","affiliation":[]}],"container-title":"Quantum Error Correction","original-title":[],"issued":{"date-parts":[[2013,9,12]]},"URL":"http://dx.doi.org/10.1017/CBO9781139034807.014","note":"edition-number:1"},{"id":"doi:10.1016/j.disc.2020.112189","type":"journal-article","publisher":"Elsevier BV","issue":"2","DOI":"10.1016/j.disc.2020.112189","page":"112189","source":"Crossref","title":"A class of skew cyclic codes and application in quantum codes construction","volume":"344","author":[{"given":"Hai Q.","family":"Dinh","sequence":"first","affiliation":[]},{"given":"Tushar","family":"Bag","sequence":"additional","affiliation":[]},{"given":"Ashish K.","family":"Upadhyay","sequence":"additional","affiliation":[]},{"given":"Ramakrishna","family":"Bandi","sequence":"additional","affiliation":[]},{"given":"Roengchai","family":"Tansuchat","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2021,2]]},"URL":"http://dx.doi.org/10.1016/j.disc.2020.112189","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X20303757"},{"id":"arxiv:1907.11253","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.103.022402","source":"Crossref","title":"Absolutely maximally entangled states, quantum-maximum-distance-separable codes, and quantum repeaters","volume":"103","author":[{"given":"Daniel","family":"Alsina","sequence":"first","affiliation":[]},{"given":"Mohsen","family":"Razavi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.022402","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022402","note":"arxivid:1907.11253"},{"id":"arxiv:1907.01393","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/jsac.2020.2968997","page":"449-462","source":"Crossref","title":"Quantum Data-Syndrome Codes","volume":"38","author":[{"ORCID":"https://orcid.org/0000-0002-7016-1614","authenticated-orcid":false,"given":"Alexei","family":"Ashikhmin","sequence":"first","affiliation":[]},{"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[2020,3]]},"URL":"http://dx.doi.org/10.1109/JSAC.2020.2968997","ISSN":["0733-8716","1558-0008"],"container-title-short":"IEEE J. Select. Areas Commun.","note":"arxivid:1907.01393"},{"id":"arxiv:1311.3009","type":"article-journal","author":[{"given":"Lingfei","family":"Jin"},{"given":"Chaoping","family":"Xing"}],"title":"A Construction of New Quantum MDS Codes","issued":{"date-parts":[[2020,7,11]]},"note":"arxivid:1311.3009\narxiv_version_number:3"},{"id":"arxiv:0712.4321","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"},{"given":"Andreas","family":"Klappenecker"}],"title":"Subsystem Code Constructions","issued":{"date-parts":[[2008,1,9]]},"note":"arxivid:0712.4321\narxiv_version_number:3"},{"id":"arxiv:0711.4603","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"}],"title":"A Note on Quantum Hamming Bound","issued":{"date-parts":[[2007,11,29]]},"note":"arxivid:0711.4603\narxiv_version_number:1"},{"id":"arxiv:0811.1621","type":"article-journal","author":[{"given":"David W.","family":"Kribs"},{"given":"Aron","family":"Pasieka"},{"given":"Karol","family":"Zyczkowski"}],"title":"Entropy of a quantum error correction code","issued":{"date-parts":[[2008,11,11]]},"note":"arxivid:0811.1621\narxiv_version_number:1"},{"id":"arxiv:2208.11800","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/mbits.2023.3262219","page":"33-38","source":"Crossref","title":"No $((n,K,d&lt; 127))$ Code Can Violate the Quantum Hamming Bound","volume":"2","author":[{"given":"Emanuel","family":"Dallas","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, CA, USA"}]},{"ORCID":"https://orcid.org/0000-0002-0372-2346","authenticated-orcid":false,"given":"Faidon","family":"Andreadakis","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, CA, USA"}]},{"ORCID":"https://orcid.org/0000-0002-1671-1515","authenticated-orcid":false,"given":"Daniel","family":"Lidar","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, and Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, CA, USA"}]}],"container-title":"IEEE BITS the Information Theory Magazine","original-title":[],"issued":{"date-parts":[[2022,12,1]]},"URL":"http://dx.doi.org/10.1109/MBITS.2023.3262219","ISSN":["2692-4110","2692-4080"],"container-title-short":"IEEE BITS Inform. Theory Mag.","note":"arxivid:2208.11800"},{"id":"arxiv:2306.00048","type":"article-journal","author":[{"given":"Andrew","family":"Nemec"},{"given":"Theerapat","family":"Tansuwannont"}],"title":"A Hamming-Like Bound for Degenerate Stabilizer Codes","issued":{"date-parts":[[2023,5,31]]},"note":"arxivid:2306.00048\narxiv_version_number:1"},{"id":"arxiv:0907.0049","type":"article-journal","author":[{"given":"Zhuo","family":"Li"},{"given":"Lijuan","family":"Xing"}],"title":"No More Perfect Codes: Classification of Perfect Quantum Codes","issued":{"date-parts":[[2009,7,1]]},"note":"arxivid:0907.0049\narxiv_version_number:1"},{"id":"doi:10.1002/(SICI)1520-6610(2000)8:3","type":"journal-article","publisher":"Wiley","issue":"3","DOI":"10.1002/(sici)1520-6610(2000)8:3<174::aid-jcd3>3.0.co;2-t","page":"174-188","source":"Crossref","title":"Quantum twisted codes","volume":"8","author":[{"given":"J�rgen","family":"Bierbrauer","sequence":"first","affiliation":[]},{"given":"Yves","family":"Edel","sequence":"additional","affiliation":[]}],"container-title":"Journal of Combinatorial Designs","original-title":[],"language":"en","issued":{"date-parts":[[2000]]},"URL":"http://dx.doi.org/10.1002/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T","ISSN":["1063-8539","1520-6610"],"container-title-short":"J. Combin. Designs"},{"id":"arxiv:2510.07158","type":"article-journal","author":[{"given":"Fermi","family":"Ma"},{"given":"Xinyu","family":"Tan"},{"given":"John","family":"Wright"}],"title":"Haar random codes attain the quantum Hamming bound, approximately","issued":{"date-parts":[[2025,10,8]]},"note":"arxivid:2510.07158\narxiv_version_number:1"},{"id":"arxiv:1705.05365","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Reliable qubits are difficult to engineer, but standard fault-tolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with fault-tolerance schemes with multiple encoded qubits. Here, we study the 15-qubit Hamming code, which protects seven encoded qubits to distance three. We give fault-tolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, 17 in total. In particular, individual encoded qubits within the code block can be targeted. Fault-tolerant universal computation is possible with four extra qubits, 19 in total. The procedures could enable testing more sophisticated protected circuits in small-scale quantum devices. Our main technique is to use gadgets to protect gates against correlated faults. We also take advantage of special code symmetries, and use pieceable fault tolerance.</jats:p>","DOI":"10.1038/s41534-018-0085-z","source":"Crossref","title":"Fault-tolerant quantum computation with few qubits","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-2570-2426","authenticated-orcid":false,"given":"Rui","family":"Chao","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2018,9,12]]},"URL":"http://dx.doi.org/10.1038/s41534-018-0085-z","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"42","note":"alternative-id:85\narxivid:1705.05365"},{"id":"arxiv:0901.1319","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.79.050306","source":"Crossref","title":"Generalized concatenated quantum codes","volume":"79","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Peter","family":"Shor","sequence":"additional","affiliation":[]},{"given":"Graeme","family":"Smith","sequence":"additional","affiliation":[]},{"given":"John","family":"Smolin","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,5,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.050306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"050306","note":"arxivid:0901.1319"},{"id":"arxiv:0705.1099","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.77.032309","source":"Crossref","title":"Minimal qudit code for a qubit in the phase-damping channel","volume":"77","author":[{"given":"Stefano","family":"Pirandola","sequence":"first","affiliation":[]},{"given":"Stefano","family":"Mancini","sequence":"additional","affiliation":[]},{"given":"Samuel L.","family":"Braunstein","sequence":"additional","affiliation":[]},{"given":"David","family":"Vitali","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,3,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.032309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032309","note":"arxivid:0705.1099"},{"id":"arxiv:2002.05180","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2021.3120685","page":"287-301","source":"Crossref","title":"Beyond Single-Shot Fault-Tolerant Quantum Error Correction","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0002-3949-981X","authenticated-orcid":false,"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]},{"given":"Krysta M.","family":"Svore","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3120685","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2002.05180"},{"id":"arxiv:1805.09271","type":"journal-article","publisher":"IOP Publishing","issue":"2","DOI":"10.1088/2058-9565/aafc8f","page":"025006","source":"Crossref","title":"A theory of single-shot error correction for adversarial noise","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-3903-2734","authenticated-orcid":false,"given":"Earl T","family":"Campbell","sequence":"first","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2019,2,5]]},"URL":"http://dx.doi.org/10.1088/2058-9565/aafc8f","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1805.09271"},{"id":"arxiv:2009.11790","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.2.020340","source":"Crossref","title":"Single-Shot Error Correction of Three-Dimensional Homological Product Codes","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0001-5101-5673","authenticated-orcid":true,"given":"Armanda O.","family":"Quintavalle","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":true,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9202-1156","authenticated-orcid":true,"given":"Joschka","family":"Roffe","sequence":"additional","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,6,14]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.020340","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020340","note":"arxivid:2009.11790"},{"id":"arxiv:1404.5504","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.5.031043","source":"Crossref","title":"Single-Shot Fault-Tolerant Quantum Error Correction","volume":"5","author":[{"given":"Héctor","family":"Bombín","sequence":"first","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2015,9,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.5.031043","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031043","note":"arxivid:1404.5504"},{"id":"arxiv:1808.03821","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/focs.2018.00076","page":"743-754","source":"Crossref","title":"Constant Overhead Quantum Fault-Tolerance with Quantum Expander Codes","author":[{"given":"Omar","family":"Fawzi","sequence":"first","affiliation":[]},{"given":"Antoine","family":"Grospellier","sequence":"additional","affiliation":[]},{"given":"Anthony","family":"Leverrier","sequence":"additional","affiliation":[]}],"event":"2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)","container-title":"2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)","original-title":[],"issued":{"date-parts":[[2018,10]]},"URL":"http://dx.doi.org/10.1109/FOCS.2018.00076","note":"arxivid:1808.03821"},{"id":"arxiv:2408.08865","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.110.062413","source":"Crossref","title":"Experiments with the four-dimensional surface code on a quantum charge-coupled device quantum computer","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0002-7586-2786","authenticated-orcid":true,"given":"Noah","family":"Berthusen","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0000-0001-9226-203X","authenticated-orcid":true,"given":"Joan","family":"Dreiling","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0000-0001-8933-2172","authenticated-orcid":true,"given":"Cameron","family":"Foltz","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0000-0002-2616-7410","authenticated-orcid":true,"given":"John P.","family":"Gaebler","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0009-0004-6338-4221","authenticated-orcid":true,"given":"Thomas M.","family":"Gatterman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"given":"Dan","family":"Gresh","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0000-0002-1021-0795","authenticated-orcid":true,"given":"Nathan","family":"Hewitt","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0000-0003-2560-4129","authenticated-orcid":true,"given":"Michael","family":"Mills","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"given":"Steven A.","family":"Moses","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"given":"Brian","family":"Neyenhuis","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0000-0002-0145-2899","authenticated-orcid":true,"given":"Peter","family":"Siegfried","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0000-0001-8546-9075","authenticated-orcid":true,"given":"David","family":"Hayes","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.062413","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062413","note":"arxivid:2408.08865"},{"id":"arxiv:2106.02621","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges. Here, we introduce a new topological quantum error-correcting code, the three-dimensional subsystem toric code (3D STC). The 3D STC can be realized with geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that one round of parity-check measurements suffices to perform reliable QEC with the 3D STC even in the presence of measurement errors. We also propose an efficient single-shot QEC decoding strategy for the 3D STC and numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to be<jats:italic>p</jats:italic><jats:sub>STC</jats:sub> ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.</jats:p>","DOI":"10.1038/s41467-022-33923-4","source":"Crossref","title":"Single-shot quantum error correction with the three-dimensional subsystem toric code","volume":"13","author":[{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":false,"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":false,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2022,10,21]]},"URL":"http://dx.doi.org/10.1038/s41467-022-33923-4","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"6272","note":"alternative-id:33923\narxivid:2106.02621"},{"id":"arxiv:2305.06365","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Topological subsystem codes in three spatial dimensions allow for quantum error correction with no time overhead, even in the presence of measurement noise. The physical origins of this single-shot property remain elusive, in part due to the scarcity of known models. To address this challenge, we provide a systematic construction of a class of topological subsystem codes in three dimensions built from Abelian quantum double models in one fewer dimension. Our construction not only generalizes the recently introduced subsystem toric code [Kubica and Vasmer, Nat. Commun. , 6272 (2022)] but also provides a new perspective on several aspects of the original model, including the origin of the Gauss law for gauge flux, and boundary conditions for the code family. We then numerically study the performance of the first few codes in this class against phenomenological noise to verify their single-shot property. Lastly, we discuss Hamiltonians naturally associated with these codes, and argue that they may be gapless.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020310","source":"Crossref","title":"Lifting Topological Codes: Three-Dimensional Subsystem Codes from Two-Dimensional Anyon Models","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-5638-6681","authenticated-orcid":true,"given":"Jacob C.","family":"Bridgeman","sequence":"first","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada"},{"name":"Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, Ghent B-9000, Belgium"}]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, California 91125, USA"},{"name":"California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,12]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020310","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020310","note":"arxivid:2305.06365"},{"id":"arxiv:2307.08118","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.110.075143","source":"Crossref","title":"Single-shot quantum error correction in intertwined toric codes","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0002-9809-5575","authenticated-orcid":true,"given":"Charles","family":"Stahl","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.110.075143","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"075143","note":"arxivid:2307.08118"},{"id":"arxiv:2306.12470","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum Tanner codes constitute a family of quantum low-density parity-check codes with good parameters, i.e., constant encoding rate and relative distance. In this article, we prove that quantum Tanner codes also facilitate single-shot quantum error correction (QEC) of adversarial noise, where one measurement round (consisting of constant-weight parity checks) suffices to perform reliable QEC even in the presence of measurement errors. We establish this result for both the sequential and parallel decoding algorithms introduced by Leverrier and Zémor. Furthermore, we show that in order to suppress errors over multiple repeated rounds of QEC, it suffices to run the parallel decoding algorithm for constant time in each round. Combined with good code parameters, the resulting constant-time overhead of QEC and robustness to (possibly time-correlated) adversarial noise make quantum Tanner codes alluring from the perspective of quantum fault-tolerant protocols.</jats:p>","DOI":"10.1007/s00220-024-04951-6","source":"Crossref","title":"Single-Shot Decoding of Good Quantum LDPC Codes","volume":"405","author":[{"ORCID":"https://orcid.org/0000-0003-2560-4209","authenticated-orcid":false,"given":"Shouzhen","family":"Gu","sequence":"first","affiliation":[]},{"given":"Eugene","family":"Tang","sequence":"additional","affiliation":[]},{"given":"Libor","family":"Caha","sequence":"additional","affiliation":[]},{"given":"Shin Ho","family":"Choe","sequence":"additional","affiliation":[]},{"given":"Zhiyang","family":"He","sequence":"additional","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,3]]},"URL":"http://dx.doi.org/10.1007/s00220-024-04951-6","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","page":"85","note":"alternative-id:4951\narxivid:2306.12470"},{"id":"arxiv:1312.2546","type":"article-journal","author":[{"given":"M. B.","family":"Hastings"}],"title":"Decoding in Hyperbolic Spaces: LDPC Codes With Linear Rate and Efficient Error Correction","issued":{"date-parts":[[2013,12,9]]},"note":"arxivid:1312.2546\narxiv_version_number:1"},{"id":"arxiv:0801.0831","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.78.012306","source":"Crossref","title":"Graphical nonbinary quantum error-correcting codes","volume":"78","author":[{"given":"Dan","family":"Hu","sequence":"first","affiliation":[]},{"given":"Weidong","family":"Tang","sequence":"additional","affiliation":[]},{"given":"Meisheng","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Qing","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Sixia","family":"Yu","sequence":"additional","affiliation":[]},{"given":"C. H.","family":"Oh","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,7,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.012306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012306","note":"arxivid:0801.0831"},{"id":"arxiv:0901.1353","type":"article-journal","author":[{"given":"Wen-Tai","family":"Yen"},{"given":"Li-Yi","family":"Hsu"}],"title":"Optimal Nonadditive Quantum Error-Detecting Code","issued":{"date-parts":[[2009,1,10]]},"note":"arxivid:0901.1353\narxiv_version_number:1"},{"id":"arxiv:2405.01332","type":"journal-article","publisher":"American Physical Society (APS)","issue":"21","DOI":"10.1103/physrevlett.134.210602","source":"Crossref","title":"How Much Entanglement Is Needed for Quantum Error Correction?","volume":"134","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM T. J. Watson Research Center"}]},{"given":"Dongjin","family":"Lee","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Waterloo"}]},{"given":"Zhi","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/04mte1k06","id-type":"ROR","asserted-by":"publisher"}],"name":"National Research Council Canada"}]},{"ORCID":"https://orcid.org/0000-0003-3213-1550","authenticated-orcid":true,"given":"Beni","family":"Yoshida","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.210602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"210602","note":"arxivid:2405.01332"},{"id":"arxiv:0901.1935","type":"article-journal","author":[{"given":"Sixia","family":"Yu"},{"given":"Qing","family":"Chen"},{"given":"C. H.","family":"Oh"}],"title":"Two infinite families of nonadditive quantum error-correcting codes","issued":{"date-parts":[[2009,1,14]]},"note":"arxivid:0901.1935\narxiv_version_number:1"},{"id":"arxiv:2403.08999","type":"article-journal","author":[{"given":"Eric","family":"Kubischta"},{"given":"Ian","family":"Teixeira"}],"title":"Quantum Codes and Irreducible Products of Characters","issued":{"date-parts":[[2024,4,6]]},"note":"arxivid:2403.08999\narxiv_version_number:2"},{"id":"arxiv:2507.14866","type":"article-journal","author":[{"given":"Manuel","family":"Calixto"},{"given":"Julio","family":"Guerrero"}],"title":"Wigner quasi-probability distribution for symmetric multi-quDit systems and their generalized heat kernel","issued":{"date-parts":[[2025,7,20]]},"note":"arxivid:2507.14866\narxiv_version_number:1"},{"id":"arxiv:2510.21476","type":"article-journal","author":[{"given":"Liubov A.","family":"Markovich"},{"given":"Vladimir A.","family":"Orlov"},{"given":"Alexey N.","family":"Rubtsov"},{"given":"Vladimir I.","family":"Man'ko"}],"title":"From Discrete to Continuous-Variable Systems via Jordan-Schwinger Tomographic Transformation","issued":{"date-parts":[[2026,3,16]]},"note":"arxivid:2510.21476\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0304153","type":"article-journal","author":[{"given":"Harriet","family":"Pollatsek"},{"given":"Mary Beth","family":"Ruskai"}],"title":"Permutationally Invariant Codes for Quantum Error Correction","issued":{"date-parts":[[2004,7,20]]},"note":"arxivid:quant-ph/0304153\narxiv_version_number:3"},{"id":"arxiv:1302.3247","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.90.062317","source":"Crossref","title":"Permutation-invariant quantum codes","volume":"90","author":[{"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.062317","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062317","note":"arxivid:1302.3247"},{"id":"arxiv:1512.02469","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.93.042340","source":"Crossref","title":"Permutation-invariant codes encoding more than one qubit","volume":"93","author":[{"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]},{"given":"Joseph","family":"Fitzsimons","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,4,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.042340","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042340","note":"arxivid:1512.02469"},{"id":"arxiv:2301.04989","type":"journal-article","publisher":"Rinton Press","issue":"1&2","abstract":"<jats:p>Qudit Dicke states are higher-dimensional analogues of an important class of highly-entangled completely symmetric quantum states known as (qubit) Dicke states.  A circuit for preparing arbitrary qudit Dicke states deterministically is formulated.  An explicit decomposition of the circuit in terms of elementary gates is presented, and is implemented in cirq for the qubit and qutrit cases.</jats:p>","DOI":"10.26421/qic24.1-2-2","page":"37-56","source":"Crossref","title":"Qudit Dicke state preparation","volume":"24","author":[{"given":"R. I.","family":"Nepomechie","sequence":"first","affiliation":[]},{"given":"D.","family":"Raveh","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information &amp; Computation","original-title":[],"issued":{"date-parts":[[2024,2]]},"URL":"http://dx.doi.org/10.26421/QIC24.1-2-2","ISSN":["1533-7146"],"container-title-short":"QIC","note":"arxivid:2301.04989"},{"id":"arxiv:1604.07925","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.laa.2017.06.031","page":"43-59","source":"Crossref","title":"Permutation-invariant qudit codes from polynomials","volume":"532","author":[{"ORCID":"https://orcid.org/0000-0003-1115-0074","authenticated-orcid":false,"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[]}],"container-title":"Linear Algebra and its Applications","original-title":[],"language":"en","issued":{"date-parts":[[2017,11]]},"URL":"http://dx.doi.org/10.1016/j.laa.2017.06.031","ISSN":["0024-3795"],"container-title-short":"Linear Algebra and its Applications","note":"special_numbering:C\nalternative-id:S0024379517303956\narxivid:1604.07925"},{"id":"arxiv:2105.13233","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum error correcting codes (QECCs) are the means of choice whenever quantum systems suffer errors, e.g., due to imperfect devices, environments, or faulty channels. By now, a plethora of families of codes is known, but there is no universal approach to finding new or optimal codes for a certain task and subject to specific experimental constraints. In particular, once found, a QECC is typically used in very diverse contexts, while its resilience against errors is captured in a single figure of merit, the distance of the code. This does not necessarily give rise to the most efficient protection possible given a certain known error or a particular application for which the code is employed.In this paper, we investigate the loss channel, which plays a key role in quantum communication, and in particular in quantum key distribution over long distances. We develop a numerical set of tools that allows to optimize an encoding specifically for recovering lost particles both deterministically and probabilistically, where some knowledge about <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>w</mml:mi><mml:mi>h</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi></mml:math> was lost is available, and demonstrate its capabilities. This allows us to arrive at new codes ideal for the distribution of entangled states in this particular setting, and also to investigate if encoding in qudits or allowing for non-deterministic correction proves advantageous compared to known QECCs. While we here focus on the case of losses, our methodology is applicable whenever the errors in a system can be characterized by a known linear map.</jats:p>","DOI":"10.22331/q-2022-03-11-667","page":"667","source":"Crossref","title":"Optimizing quantum codes with an application to the loss channel with partial erasure information","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-2083-7820","authenticated-orcid":false,"given":"Benjamin","family":"Desef","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics & IQST, University of Ulm"}]},{"ORCID":"https://orcid.org/0000-0003-4238-8843","authenticated-orcid":false,"given":"Martin B.","family":"Plenio","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics & IQST, University of Ulm"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,3,11]]},"URL":"http://dx.doi.org/10.22331/q-2022-03-11-667","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2105.13233"},{"id":"arxiv:2507.13308","type":"journal-article","publisher":"Walter de Gruyter GmbH","issue":"6","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    Dicke states are permutation-invariant superpositions of qubit computational basis states, which play a prominent role in quantum information science. We consider here two higher-dimensional generalizations of these states:\n                    <jats:italic>SU</jats:italic>\n                    (\n                    <jats:italic>d</jats:italic>\n                    ) spin-\n                    <jats:italic>s</jats:italic>\n                    Dicke states and\n                    <jats:italic>SU</jats:italic>\n                    (\n                    <jats:italic>2</jats:italic>\n                    ) Dicke states. We present various ways of preparing both types of qudit Dicke states on a qudit quantum computer, using two main approaches: a deterministic approach, based on exact canonical matrix product state representations; and a probabilistic approach, based on quantum phase estimation. The quantum circuits are explicit and straightforward, and are arguably simpler than those previously reported.\n                  </jats:p>","DOI":"10.2478/qic-2025-0036","page":"668-686","source":"Crossref","title":"Simple Ways of Preparing Qudit Dicke States","volume":"25","author":[{"given":"Noah B.","family":"Kerzner","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy , 153 Olin Science Bldg. , Bucknell University , Lewisburg , PA , USA"},{"name":"Department of Physics , P.O. Box 248046 , University of Miami , Coral Gables , FL , USA"}]},{"given":"Federico","family":"Galeazzi","sequence":"additional","affiliation":[{"name":"Department of Physics , P.O. Box 248046 , University of Miami , Coral Gables , FL , USA"},{"name":"Coral Gables High School , 450 Bird Rd , Coral Gables , FL , USA"}]},{"given":"Rafael I.","family":"Nepomechie","sequence":"additional","affiliation":[{"name":"Department of Physics , P.O. Box 248046 , University of Miami , Coral Gables , FL , USA"}]}],"container-title":"Quantum Information &amp; Computation","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,1]]},"URL":"http://dx.doi.org/10.2478/qic-2025-0036","ISSN":["1533-7146"],"note":"alternative-id:10.2478/qic-2025-0036\narxivid:2507.13308"},{"id":"arxiv:1007.1697","type":"article-journal","author":[{"given":"Sagarmoy","family":"Dutta"},{"given":"Piyush P","family":"Kurur"}],"title":"Quantum Cyclic Code","issued":{"date-parts":[[2010,7,10]]},"note":"arxivid:1007.1697\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9910061","type":"journal-article","publisher":"The Royal Society","issue":"2003","DOI":"10.1098/rspa.2000.0633","page":"2689-2706","source":"Crossref","title":"Cyclic quantum error–correcting codes and quantum shift registers","volume":"456","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[{"name":"Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, 76128 Karlsruhe, Germany"}]},{"given":"Thomas","family":"Beth","sequence":"additional","affiliation":[{"name":"Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, 76128 Karlsruhe, Germany"}]}],"container-title":"Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2000,11,8]]},"URL":"http://dx.doi.org/10.1098/rspa.2000.0633","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. Lond. A","note":"alternative-id:10.1098/rspa.2000.0633\narxivid:quant-ph/9910061"},{"id":"arxiv:1108.5490","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.84.062319","source":"Crossref","title":"Design of additive quantum codes via the code-word-stabilized framework","volume":"84","author":[{"given":"Alexey A.","family":"Kovalev","sequence":"first","affiliation":[]},{"given":"Ilya","family":"Dumer","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,12,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.062319","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062319","note":"arxivid:1108.5490"},{"id":"arxiv:2009.10329","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.127.040507","source":"Crossref","title":"Tensor-Network Codes","volume":"127","author":[{"ORCID":"https://orcid.org/0000-0002-7662-2303","authenticated-orcid":true,"given":"Terry","family":"Farrelly","sequence":"first","affiliation":[]},{"given":"Robert J.","family":"Harris","sequence":"additional","affiliation":[]},{"given":"Nathan A.","family":"McMahon","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2300-1394","authenticated-orcid":true,"given":"Thomas M.","family":"Stace","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2021,7,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.127.040507","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"040507","note":"arxivid:2009.10329"},{"id":"arxiv:2207.06536","type":"article-journal","author":[{"given":"Chris","family":"Akers"},{"given":"Netta","family":"Engelhardt"},{"given":"Daniel","family":"Harlow"},{"given":"Geoff","family":"Penington"},{"given":"Shreya","family":"Vardhan"}],"title":"The black hole interior from non-isometric codes and complexity","issued":{"date-parts":[[2022,10,4]]},"note":"arxivid:2207.06536\narxiv_version_number:2"},{"id":"arxiv:2201.11658","type":"article-journal","author":[{"given":"Jordan","family":"Cotler"},{"given":"Andrew","family":"Strominger"}],"title":"The Universe as a Quantum Encoder","issued":{"date-parts":[[2022,2,7]]},"note":"arxivid:2201.11658\narxiv_version_number:2"},{"id":"arxiv:0912.0963","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.81.032313","source":"Crossref","title":"Quantum information encoding, protection, and correction from trace-norm isometries","volume":"81","author":[{"given":"Francesco","family":"Ticozzi","sequence":"first","affiliation":[]},{"given":"Lorenza","family":"Viola","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,3,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.032313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032313","note":"arxivid:0912.0963"},{"id":"arxiv:2103.13404","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.2.030337","source":"Crossref","title":"Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0003-2283-3903","authenticated-orcid":true,"given":"Sam","family":"Cree","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0143-1622","authenticated-orcid":true,"given":"Kfir","family":"Dolev","sequence":"additional","affiliation":[]},{"given":"Vladimir","family":"Calvera","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":true,"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,31]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.030337","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030337","note":"arxivid:2103.13404"},{"id":"arxiv:2202.04710","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevd.106.046009","source":"Crossref","title":"Deconfinement and error thresholds in holography","volume":"106","author":[{"given":"Ning","family":"Bao","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5761-5474","authenticated-orcid":true,"given":"ChunJun","family":"Cao","sequence":"additional","affiliation":[]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.106.046009","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"046009","note":"arxivid:2202.04710"},{"id":"arxiv:1912.05649","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n</jats:title><jats:p>In AdS/CFT, there can exist local 2-to-2 bulk scattering processes even when local scattering is not possible on the boundary; these have previously been studied in con- nection with boundary correlation functions. We show that boundary regions associated with these scattering configurations must have <jats:italic>O</jats:italic>(1<jats:italic>/G</jats:italic><jats:sub><jats:italic>N</jats:italic></jats:sub>) mutual information, and hence a connected entanglement wedge. One of us previously argued for this statement from the boundary theory using operational tools in quantum information theory. We improve that argument to make it robust to small errors and provide a proof in the bulk using focusing arguments in general relativity. We also provide a direct link to entanglement wedge reconstruction by showing that the bulk scattering region must lie inside the con- nected entanglement wedge. Our construction implies the existence of nonlocal quantum computation protocols that are exponentially more efficient than the optimal protocols currently known.</jats:p>","DOI":"10.1007/jhep08(2020)132","source":"Crossref","title":"Holographic scattering requires a connected entanglement wedge","volume":"2020","author":[{"ORCID":"https://orcid.org/0000-0002-4030-5410","authenticated-orcid":false,"given":"Alex","family":"May","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8627-5237","authenticated-orcid":false,"given":"Geoff","family":"Penington","sequence":"additional","affiliation":[]},{"given":"Jonathan","family":"Sorce","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,8]]},"URL":"http://dx.doi.org/10.1007/JHEP08(2020)132","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"132","note":"alternative-id:13663\narxivid:1912.05649"},{"id":"arxiv:2401.09058","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>We investigate the link between quantum position-verification (QPV) and holography established in [1] using holographic quantum error correcting codes as toy models. By inserting the “temporal” scaling of the AdS metric by hand via the bulk Hamiltonian interaction strength, we recover a toy model with consistent causality structure. This leads to an interesting implication between two topics in quantum information: if position-based verification is secure against attacks with small entanglement then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another.</jats:p>","DOI":"10.1007/jhep08(2024)152","source":"Crossref","title":"Security of quantum position-verification limits Hamiltonian simulation via holography","volume":"2024","author":[{"ORCID":"https://orcid.org/0000-0001-5715-9838","authenticated-orcid":false,"given":"Harriet","family":"Apel","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02jx3x895","id-type":"ROR","asserted-by":"publisher"}],"name":"University College London"}]},{"ORCID":"https://orcid.org/0000-0002-5087-9346","authenticated-orcid":false,"given":"Toby","family":"Cubitt","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02jx3x895","id-type":"ROR","asserted-by":"publisher"}],"name":"University College London"}]},{"ORCID":"https://orcid.org/0000-0002-3964-5602","authenticated-orcid":false,"given":"Patrick","family":"Hayden","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f54p054","id-type":"ROR","asserted-by":"publisher"}],"name":"Stanford University"}]},{"ORCID":"https://orcid.org/0000-0002-7136-6267","authenticated-orcid":false,"given":"Tamara","family":"Kohler","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05e9bn444","id-type":"ROR","asserted-by":"publisher"}],"name":"Instituto de Ciencias Matemáticas"},{"id":[{"id":"https://ror.org/00f54p054","id-type":"ROR","asserted-by":"publisher"}],"name":"Stanford University"}]},{"ORCID":"https://orcid.org/0000-0003-2990-791X","authenticated-orcid":false,"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05e9bn444","id-type":"ROR","asserted-by":"publisher"}],"name":"Instituto de Ciencias Matemáticas"}]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,20]]},"URL":"http://dx.doi.org/10.1007/JHEP08(2024)152","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"152","note":"alternative-id:24286\narxivid:2401.09058"},{"id":"arxiv:1810.08992","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality. HQECC give a mapping of states and observables. However, they do not map local bulk Hamiltonians to local Hamiltonians on the boundary. In this work, we combine HQECC with Hamiltonian simulation theory to construct a bulk-boundary mapping between local Hamiltonians, whilst retaining all the features of the HQECC duality. This allows us to construct a duality between <jats:italic>models</jats:italic>, encompassing the relationship between bulk and boundary energy scales and time dynamics.</jats:p>\n          <jats:p>It also allows us to construct a map in the reverse direction: from local boundary Hamiltonians to the corresponding local Hamiltonian in the bulk. Under this boundary-to-bulk mapping, the bulk geometry emerges as an approximate, low-energy, effective theory living in the code-space of an (approximate) HQECC on the boundary. At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC. Moreover, the duality on the level of dynamics shows how these toy-model black holes can form dynamically.</jats:p>","DOI":"10.1007/jhep08(2019)017","source":"Crossref","title":"Toy models of holographic duality between local Hamiltonians","volume":"2019","author":[{"given":"Tamara","family":"Kohler","sequence":"first","affiliation":[]},{"given":"Toby","family":"Cubitt","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,8]]},"URL":"http://dx.doi.org/10.1007/JHEP08(2019)017","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"17","note":"alternative-id:11021\narxivid:1810.08992"},{"id":"arxiv:1801.09623","type":"article-journal","author":[{"given":"Matilde","family":"Marcolli"}],"title":"Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces","issued":{"date-parts":[[2018,1,29]]},"note":"arxivid:1801.09623\narxiv_version_number:1"},{"id":"arxiv:1812.04057","type":"article-journal","author":[{"given":"Matthew","family":"Heydeman"},{"given":"Matilde","family":"Marcolli"},{"given":"Sarthak","family":"Parikh"},{"given":"Ingmar","family":"Saberi"}],"title":"Nonarchimedean Holographic Entropy from Networks of Perfect Tensors","issued":{"date-parts":[[2018,12,10]]},"note":"arxivid:1812.04057\narxiv_version_number:1"},{"id":"arxiv:1607.03901","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-017-2904-z","page":"865-912","source":"Crossref","title":"The Ryu–Takayanagi Formula from Quantum Error Correction","volume":"354","author":[{"ORCID":"https://orcid.org/0000-0002-1005-4745","authenticated-orcid":false,"given":"Daniel","family":"Harlow","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,5,26]]},"URL":"http://dx.doi.org/10.1007/s00220-017-2904-z","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2904\narxivid:1607.03901"},{"id":"arxiv:2109.08158","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.3.020332","source":"Crossref","title":"Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-5761-5474","authenticated-orcid":true,"given":"ChunJun","family":"Cao","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3823-8757","authenticated-orcid":true,"given":"Brad","family":"Lackey","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,11]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.020332","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020332","note":"arxivid:2109.08158"},{"id":"arxiv:2109.11996","type":"journal-article","publisher":"IOP Publishing","issue":"4","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Tensor-network codes enable the construction of large stabilizer codes out of tensors describing smaller stabilizer codes. An application of tensor-network codes was an efficient and exact decoder for holographic codes. Here, we show how to write some topological codes, including the surface code and colour code, as simple tensor-network codes. We also show how to calculate distances of stabilizer codes by contracting a tensor network. The algorithm actually gives more information, including a histogram of all logical coset weights. We prove that this method is efficient in the case of stabilizer codes encoded via local log-depth circuits in one dimension and holographic codes. Using our tensor-network distance calculator, we find a modification of the rotated surface code that has the same distance but fewer minimum-weight logical operators by ‘doping’ the tensor network, i.e., we break the homogeneity of the tensor network by locally replacing tensors. For this example, this corresponds to an improvement in successful error correction of almost 2% against depolarizing noise (in the perfect-measurement setting), but comes at the cost of introducing three higher-weight stabilizers. Our general construction lets us pick a network geometry (e.g., a Euclidean lattice in the case of the surface code), and, using only a small set of seed codes (constituent tensors), build extensive codes with the potential for optimisation.</jats:p>","DOI":"10.1088/1367-2630/ac5e87","page":"043015","source":"Crossref","title":"Local tensor-network codes","volume":"24","author":[{"given":"Terry","family":"Farrelly","sequence":"first","affiliation":[]},{"given":"David K","family":"Tuckett","sequence":"additional","affiliation":[]},{"given":"Thomas M","family":"Stace","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2022,4,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ac5e87","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:2109.11996"},{"id":"arxiv:2308.05152","type":"article-journal","author":[{"given":"ChunJun","family":"Cao"},{"given":"Michael J.","family":"Gullans"},{"given":"Brad","family":"Lackey"},{"given":"Zitao","family":"Wang"}],"title":"Quantum Lego Expansion Pack: Enumerators from Tensor Networks","issued":{"date-parts":[[2024,3,2]]},"note":"arxivid:2308.05152\narxiv_version_number:2"},{"id":"arxiv:1312.4578","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.113.030501","source":"Crossref","title":"Tensor Networks and Quantum Error Correction","volume":"113","author":[{"given":"Andrew J.","family":"Ferris","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,7,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.113.030501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030501","note":"arxivid:1312.4578"},{"id":"arxiv:2510.08210","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Calculating the quantum weight enumerator polynomial (WEP) is a valuable tool for characterizing quantum error-correcting (QEC) codes, but it is computationally hard for large or complex codes. The Quantum LEGO (QL) framework provides a tensor network approach for WEP calculation, in some cases offering superpolynomial speedups over brute-force methods, provided the code exhibits area law entanglement, that a good QL layout is used, and an efficient tensor network contraction schedule is found. We analyze the performance of a hyper-optimized contraction schedule framework across QL layouts for diverse stabilizer code families. We find that the intermediate tensors in the QL networks for stabilizer WEPs are often highly sparse, invalidating the dense-tensor assumption of standard cost functions. To address this, we introduce an exact, polynomial-time Sparse Stabilizer Tensor (SST) cost function based on the rank of the parity check matrices for intermediate tensors. The SST cost function correlates perfectly with the true contraction cost, providing a significant advantage over the default cost function, which exhibits large uncertainty. Optimizing contraction schedules using the SST cost function yields substantial performance gains, achieving up to orders of magnitude improvement in actual contraction cost compared to using the dense tensor cost function. Furthermore, the precise cost estimation from the SST function offers an efficient metric to decide whether the QL-based WEP calculation is computationally superior to brute force for a given QL layout. These results, enabled by PlanqTN, a new open-source QL implementation, validate hyper-optimized contraction as a crucial technique for leveraging the QL framework to explore the QEC code design space.</jats:p>","DOI":"10.22331/q-2026-05-05-2092","page":"2092","source":"Crossref","title":"Hyper-optimized Quantum Lego Contraction Schedules","volume":"10","author":[{"given":"Balint","family":"Pato","sequence":"first","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA"}]},{"given":"June","family":"Vanlerberghe","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA"}]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA"},{"name":"Department of Physics, Duke University, Durham, NC 27708, USA"},{"name":"Department of Chemistry, Duke University, Durham, NC 27708, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,5,5]]},"URL":"http://dx.doi.org/10.22331/q-2026-05-05-2092","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2510.08210"},{"id":"arxiv:1106.5756","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.84.062306","source":"Crossref","title":"Multipartite entanglement detection from correlation tensors","volume":"84","author":[{"given":"Julio I.","family":"de Vicente","sequence":"first","affiliation":[]},{"given":"Marcus","family":"Huber","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,12,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.062306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062306","note":"arxivid:1106.5756"},{"id":"arxiv:1110.4108","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.84.062305","source":"Crossref","title":"Correlation-tensor criteria for genuine multiqubit entanglement","volume":"84","author":[{"given":"Wiesław","family":"Laskowski","sequence":"first","affiliation":[]},{"given":"Marcin","family":"Markiewicz","sequence":"additional","affiliation":[]},{"given":"Tomasz","family":"Paterek","sequence":"additional","affiliation":[]},{"given":"Marek","family":"Żukowski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,12,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.062305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062305","note":"arxivid:1110.4108"},{"id":"arxiv:2012.07317","type":"article-journal","author":[{"given":"Terry","family":"Farrelly"},{"given":"Robert J.","family":"Harris"},{"given":"Nathan A.","family":"McMahon"},{"given":"Thomas M.","family":"Stace"}],"title":"Parallel decoding of multiple logical qubits in tensor-network codes","issued":{"date-parts":[[2020,12,14]]},"note":"arxivid:2012.07317\narxiv_version_number:1"},{"id":"arxiv:2310.19538","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We apply the recent graphical framework of \"Quantum Lego\" to XP stabilizer codes where the stabilizer group is generally non-Abelian. We show that the idea of operator matching continues to hold for such codes and is sufficient for generating all their XP symmetries provided the resulting code is XP. We provide an efficient classical algorithm for tracking these symmetries under tensor contraction or conjoining. This constitutes a partial extension of the algorithm implied by the Gottesman-Knill theorem beyond Pauli stabilizer states and Clifford operations. Because conjoining transformations generate quantum operations that are universal, the XP symmetries obtained from these algorithms do not uniquely identify the resulting tensors in general. Using this extended framework, we provide examples of novel XP stabilizer codes with a higher distance than existing non-trivial XP regular codes and a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>8</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> Pauli stabilizer code with a fault-tolerant <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math> gate. For XP regular codes, we also construct a tensor-network-based maximum likelihood decoder for any independently and identically distributed single qubit error channel using weight enumerators.</jats:p>","DOI":"10.22331/q-2025-10-06-1876","page":"1876","source":"Crossref","title":"Quantum Lego and XP Stabilizer Codes","volume":"9","author":[{"given":"Ruohan","family":"Shen","sequence":"first","affiliation":[{"name":"Department of Physics, Tsinghua University, Beijing 100084, China"},{"name":"Institute for Quantum Information and Matter California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA."}]},{"given":"Yixu","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute for Advanced Study, Tsinghua University, Beijing 100084, China"}]},{"given":"ChunJun","family":"Cao","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information and Matter California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA."},{"name":"Department of Physics, Virginia Tech, Blacksburg, VA, 24061, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,6]]},"URL":"http://dx.doi.org/10.22331/q-2025-10-06-1876","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2310.19538"},{"id":"arxiv:2410.22861","type":"article-journal","author":[{"given":"Junyu","family":"Fan"},{"given":"Matthew","family":"Steinberg"},{"given":"Alexander","family":"Jahn"},{"given":"Chunjun","family":"Cao"},{"given":"Aritra","family":"Sarkar"},{"given":"Sebastian","family":"Feld"}],"title":"LEGO_HQEC: Automating the Analysis, Construction, and Decoding of Holographic Quantum Codes","issued":{"date-parts":[[2025,12,31]]},"note":"arxivid:2410.22861\narxiv_version_number:3"},{"id":"arxiv:2305.03694","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.132.110201","source":"Crossref","title":"Solvable Model of Quantum-Darwinism-Encoding Transitions","volume":"132","author":[{"ORCID":"https://orcid.org/0009-0005-1569-1603","authenticated-orcid":true,"given":"Benoît","family":"Ferté","sequence":"first","affiliation":[{"name":"Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France"}]},{"ORCID":"https://orcid.org/0000-0003-3406-6692","authenticated-orcid":true,"given":"Xiangyu","family":"Cao","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, F-75005 Paris, France"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,3,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.132.110201","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110201","note":"arxivid:2305.03694"},{"id":"arxiv:2306.14294","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41534-025-00961-x","source":"Crossref","title":"Noisy quantum trees: infinite protection without correction","volume":"11","author":[{"given":"Shiv Akshar","family":"Yadavalli","sequence":"first","affiliation":[]},{"given":"Iman","family":"Marvian","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,24]]},"URL":"http://dx.doi.org/10.1038/s41534-025-00961-x","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"151","note":"alternative-id:961\narxivid:2306.14294"},{"id":"arxiv:quant-ph/0005031","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.62.030301","source":"Crossref","title":"Entangling power of quantum evolutions","volume":"62","author":[{"given":"Paolo","family":"Zanardi","sequence":"first","affiliation":[]},{"given":"Christof","family":"Zalka","sequence":"additional","affiliation":[]},{"given":"Lara","family":"Faoro","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,8,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.62.030301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"030301","note":"arxivid:quant-ph/0005031"},{"id":"arxiv:0809.3081","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1140/epjd/e2010-10374-9","page":"261-265","source":"Crossref","title":"Undetermined states: how to find them and their applications","volume":"61","author":[{"given":"M. H.","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"W. T.","family":"Yen","sequence":"additional","affiliation":[]},{"given":"L. Y.","family":"Hsu","sequence":"additional","affiliation":[]}],"container-title":"The European Physical Journal D","original-title":[],"language":"en","issued":{"date-parts":[[2010,12,3]]},"URL":"http://dx.doi.org/10.1140/epjd/e2010-10374-9","ISSN":["1434-6060","1434-6079"],"container-title-short":"Eur. Phys. J. D","note":"alternative-id:9778\narxivid:0809.3081"},{"id":"arxiv:1211.4118","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.87.012319","source":"Crossref","title":"Exploring pure quantum states with maximally mixed reductions","volume":"87","author":[{"given":"Ludovic","family":"Arnaud","sequence":"first","affiliation":[]},{"given":"Nicolas J.","family":"Cerf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,1,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.012319","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012319","note":"arxivid:1211.4118"},{"id":"arxiv:0704.0251","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.76.042309","source":"Crossref","title":"Entanglement of subspaces and error-correcting codes","volume":"76","author":[{"given":"Gilad","family":"Gour","sequence":"first","affiliation":[]},{"given":"Nolan R.","family":"Wallach","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,10,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.042309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042309","note":"arxivid:0704.0251"},{"id":"arxiv:1002.2592","type":"journal-article","publisher":"IOP Publishing","issue":"22","DOI":"10.1088/1751-8113/43/22/225303","page":"225303","source":"Crossref","title":"Classical statistical mechanics approach to multipartite entanglement","volume":"43","author":[{"given":"P","family":"Facchi","sequence":"first","affiliation":[]},{"given":"G","family":"Florio","sequence":"additional","affiliation":[]},{"given":"U","family":"Marzolino","sequence":"additional","affiliation":[]},{"given":"G","family":"Parisi","sequence":"additional","affiliation":[]},{"given":"S","family":"Pascazio","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2010,5,12]]},"URL":"http://dx.doi.org/10.1088/1751-8113/43/22/225303","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(10)47934-2\narxivid:1002.2592"},{"id":"arxiv:quant-ph/0508070","type":"article-journal","author":[{"given":"Avanti","family":"Ketkar"},{"given":"Andreas","family":"Klappenecker"},{"given":"Santosh","family":"Kumar"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Nonbinary stabilizer codes over finite fields","issued":{"date-parts":[[2005,8,17]]},"note":"arxivid:quant-ph/0508070\narxiv_version_number:2"},{"id":"arxiv:2003.13639","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.102.022413","source":"Crossref","title":"Stochastic local operations with classical communication of absolutely maximally entangled states","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0003-0418-257X","authenticated-orcid":true,"given":"Adam","family":"Burchardt","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9168-8212","authenticated-orcid":true,"given":"Zahra","family":"Raissi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.022413","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022413","note":"arxivid:2003.13639"},{"id":"arxiv:2503.14506","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/cl23-c3jg","source":"Crossref","title":"Scalable and fault-tolerant preparation of encoded \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>\n-uniform states","volume":"112","author":[{"ORCID":"https://orcid.org/0000-0003-0411-7380","authenticated-orcid":true,"given":"Shayan","family":"Majidy","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-4766-7967","authenticated-orcid":true,"given":"Dominik","family":"Hangleiter","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California, Berkeley"}]},{"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland and NIST, College Park"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,3]]},"URL":"http://dx.doi.org/10.1103/cl23-c3jg","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042409","note":"arxivid:2503.14506"},{"id":"arxiv:2504.05394","type":"journal-article","publisher":"IOP Publishing","issue":"2","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>Absolutely maximally entangled (AME) states of multipartite quantum systems exhibit maximal entanglement across all possible bipartitions. These states lead to teleportation protocols that surpass standard teleportation schemes, determine quantum error correction codes and can be used to test performance of current term quantum processors. Several AME states can be constructed from graph states using minimal quantum resources. However, there exist other constructions that depart from the stabilizer formalism. In this work, we present explicit quantum circuits to generate exemplary non-stabilizer AME states of four subsystems with four, six, and eight levels each and analyze their capabilities to perform quantum information tasks.</jats:p>","DOI":"10.1088/2058-9565/ae46d6","page":"025011","source":"Crossref","title":"Quantum circuits for high-dimensional absolutely maximally entangled states","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0002-9595-8523","authenticated-orcid":true,"given":"Berta","family":"Casas","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2652-9482","authenticated-orcid":true,"given":"Grzegorz","family":"Rajchel-Mieldzioć","sequence":"additional","affiliation":[]},{"given":"Suhail","family":"Ahmad Rather","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0835-1644","authenticated-orcid":true,"given":"Marcin","family":"Płodzień","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9743-7927","authenticated-orcid":true,"given":"Wojciech","family":"Bruzda","sequence":"additional","affiliation":[]},{"given":"Alba","family":"Cervera-Lierta","sequence":"additional","affiliation":[]},{"given":"Karol","family":"Życzkowski","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2026,2,26]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ae46d6","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2504.05394"},{"id":"arxiv:1708.06298","type":"journal-article","publisher":"IOP Publishing","issue":"17","DOI":"10.1088/1751-8121/aaade5","page":"175301","source":"Crossref","title":"Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity","volume":"51","author":[{"ORCID":"https://orcid.org/0000-0002-3856-4018","authenticated-orcid":false,"given":"Felix","family":"Huber","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9428-5593","authenticated-orcid":false,"given":"Christopher","family":"Eltschka","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9410-5043","authenticated-orcid":false,"given":"Jens","family":"Siewert","sequence":"additional","affiliation":[]},{"given":"Otfried","family":"Gühne","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2018,3,28]]},"URL":"http://dx.doi.org/10.1088/1751-8121/aaade5","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1708.06298"},{"id":"arxiv:2009.12497","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.104.032601","source":"Crossref","title":"<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>\n-uniform quantum information masking","volume":"104","author":[{"given":"Fei","family":"Shi","sequence":"first","affiliation":[]},{"given":"Mao-Sheng","family":"Li","sequence":"additional","affiliation":[]},{"given":"Lin","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4234-3610","authenticated-orcid":true,"given":"Xiande","family":"Zhang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.104.032601","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032601","note":"arxivid:2009.12497"},{"id":"arxiv:2406.02379","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","DOI":"10.1038/s41567-025-02945-2","page":"1338-1345","source":"Crossref","title":"Entanglement accelerates quantum simulation","volume":"21","author":[{"ORCID":"https://orcid.org/0000-0002-8091-0682","authenticated-orcid":false,"given":"Qi","family":"Zhao","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0886-077X","authenticated-orcid":false,"given":"You","family":"Zhou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9903-837X","authenticated-orcid":false,"given":"Andrew M.","family":"Childs","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,14]]},"URL":"http://dx.doi.org/10.1038/s41567-025-02945-2","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2945\narxivid:2406.02379"},{"id":"arxiv:2508.04777","type":"article-journal","author":[{"given":"Grzegorz","family":"Rajchel-Mieldzioć"},{"given":"Rafał","family":"Bistroń"},{"given":"Albert","family":"Rico"},{"given":"Arul","family":"Lakshminarayan"},{"given":"Karol","family":"Życzkowski"}],"title":"Absolutely maximally entangled pure states of multipartite quantum systems","issued":{"date-parts":[[2025,12,30]]},"note":"arxivid:2508.04777\narxiv_version_number:2"},{"id":"arxiv:1204.2289","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.86.052335","source":"Crossref","title":"Absolute maximal entanglement and quantum secret sharing","volume":"86","author":[{"given":"Wolfram","family":"Helwig","sequence":"first","affiliation":[]},{"given":"Wei","family":"Cui","sequence":"additional","affiliation":[]},{"given":"José Ignacio","family":"Latorre","sequence":"additional","affiliation":[]},{"given":"Arnau","family":"Riera","sequence":"additional","affiliation":[]},{"given":"Hoi-Kwong","family":"Lo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,11,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.052335","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052335","note":"arxivid:1204.2289"},{"id":"doi:10.1007/s11128-022-03723-2","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"11","DOI":"10.1007/s11128-022-03723-2","source":"Crossref","title":"Quantum secret sharing based on quantum information masking","volume":"21","author":[{"ORCID":"https://orcid.org/0000-0001-6255-8796","authenticated-orcid":false,"given":"Chen-Ming","family":"Bai","sequence":"first","affiliation":[]},{"given":"Sujuan","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Lu","family":"Liu","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2022,11,11]]},"URL":"http://dx.doi.org/10.1007/s11128-022-03723-2","ISSN":["1573-1332"],"container-title-short":"Quantum Inf Process","page":"377","note":"alternative-id:3723"},{"id":"arxiv:1806.08781","type":"journal-article","publisher":"IOP Publishing","issue":"2","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>We examine the usefulness of applying neural networks as a variational state ansatz for many-body quantum systems in the context of quantum information-processing tasks. In the neural network state ansatz, the complex amplitude function of a quantum state is computed by a neural network. The resulting multipartite entanglement structure captured by this ansatz has proven rich enough to describe the ground states and unitary dynamics of various physical systems of interest. In the present paper, we initiate the study of neural network states in quantum information-processing tasks. We demonstrate that neural network states are capable of efficiently representing quantum codes for quantum information transmission and quantum error correction, supplying further evidence for the usefulness of neural network states to describe multipartite entanglement. In particular, we show the following main results: (a) neural network states yield quantum codes with a high coherent information for two important quantum channels, the generalized amplitude damping channel and the dephrasure channel. These codes outperform all other known codes for these channels, and cannot be found using a direct parametrization of the quantum state. (b) For the depolarizing channel, the neural network state ansatz reliably finds the best known codes given by repetition codes. (c) Neural network states can be used to represent absolutely maximally entangled states, a special type of quantum error-correcting codes. In all three cases, the neural network state ansatz provides an efficient and versatile means as a variational parametrization of these highly entangled states.</jats:p>","DOI":"10.1088/1367-2630/ab6cdd","page":"023005","source":"Crossref","title":"Quantum codes from neural networks","volume":"22","author":[{"ORCID":"https://orcid.org/0000-0003-3189-9162","authenticated-orcid":false,"given":"Johannes","family":"Bausch","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1073-9795","authenticated-orcid":false,"given":"Felix","family":"Leditzky","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2020,2,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ab6cdd","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1806.08781"},{"id":"arxiv:2210.09980","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Photons are the physical system of choice for performing experimental tests of the foundations of quantum mechanics. Furthermore, photonic quantum technology is a main player in the second quantum revolution, promising the development of better sensors, secure communications, and quantum-enhanced computation. These endeavors require generating specific quantum states or efficiently performing quantum tasks. The design of the corresponding optical experiments was historically powered by human creativity but is recently being automated with advanced computer algorithms and artificial intelligence. While several computer-designed experiments have been experimentally realized, this approach has not yet been widely adopted by the broader photonic quantum optics community. The main roadblocks consist of most systems being closed-source, inefficient, or targeted to very specific use-cases that are difficult to generalize. Here, we overcome these problems with a highly-efficient, open-source digital discovery framework PyTheus, which can employ a wide range of experimental devices from modern quantum labs to solve various tasks. This includes the discovery of highly entangled quantum states, quantum measurement schemes, quantum communication protocols, multi-particle quantum gates, as well as the optimization of continuous and discrete properties of quantum experiments or quantum states. PyTheus produces interpretable designs for complex experimental problems which human researchers can often readily conceptualize. PyTheus is an example of a powerful framework that can lead to scientific discoveries – one of the core goals of artificial intelligence in science. We hope it will help accelerate the development of quantum optics and provide new ideas in quantum hardware and technology.</jats:p>","DOI":"10.22331/q-2023-12-12-1204","page":"1204","source":"Crossref","title":"Digital Discovery of 100 diverse Quantum Experiments with PyTheus","volume":"7","author":[{"given":"Carlos","family":"Ruiz-Gonzalez","sequence":"first","affiliation":[{"name":"Max Planck Institute for the Science of Light, Erlangen, Germany."}]},{"given":"Sören","family":"Arlt","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Science of Light, Erlangen, Germany."}]},{"given":"Jan","family":"Petermann","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Science of Light, Erlangen, Germany."}]},{"given":"Sharareh","family":"Sayyad","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Science of Light, Erlangen, Germany."}]},{"given":"Tareq","family":"Jaouni","sequence":"additional","affiliation":[{"name":"Nexus for Quantum Technologies, University of Ottawa, K1N 6N5, ON, Ottawa, Canada."}]},{"given":"Ebrahim","family":"Karimi","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Science of Light, Erlangen, Germany."},{"name":"Nexus for Quantum Technologies, University of Ottawa, K1N 6N5, ON, Ottawa, Canada."}]},{"given":"Nora","family":"Tischler","sequence":"additional","affiliation":[{"name":"Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, Australia."}]},{"given":"Xuemei","family":"Gu","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Science of Light, Erlangen, Germany."}]},{"given":"Mario","family":"Krenn","sequence":"additional","affiliation":[{"name":"Max Planck Institute for the Science of Light, Erlangen, Germany."}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,12]]},"URL":"http://dx.doi.org/10.22331/q-2023-12-12-1204","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2210.09980"},{"id":"arxiv:2405.06142","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/jsait.2025.3602744","page":"1-1","source":"Crossref","title":"Codeword Stabilized Codes from m-Uniform Graph States","author":[{"ORCID":"https://orcid.org/0000-0001-5072-0098","authenticated-orcid":false,"given":"Sowrabh","family":"Sudevan","sequence":"first","affiliation":[{"name":"Department of Physical Sciences, Indian Institute of Science Education and Research, Kolkata, India"}]},{"given":"Sourin","family":"Das","sequence":"additional","affiliation":[{"name":"Department of Physical Sciences, Indian Institute of Science Education and Research, Kolkata, India"}]},{"given":"Thamadathil","family":"Aswanth","sequence":"additional","affiliation":[{"name":"Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru, India"}]},{"given":"Nupur","family":"Patanker","sequence":"additional","affiliation":[{"name":"Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru, India"}]},{"ORCID":"https://orcid.org/0000-0001-8834-0082","authenticated-orcid":false,"given":"Navin","family":"Kashyap","sequence":"additional","affiliation":[{"name":"Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru, India"}]}],"container-title":"IEEE Journal on Selected Areas in Information Theory","original-title":[],"issued":{"date-parts":[[2025]]},"URL":"http://dx.doi.org/10.1109/JSAIT.2025.3602744","ISSN":["2641-8770"],"container-title-short":"IEEE J. Sel. Areas Inf. Theory","note":"arxivid:2405.06142"},{"id":"arxiv:2104.05122","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.128.080507","source":"Crossref","title":"Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem","volume":"128","author":[{"given":"Suhail Ahmad","family":"Rather","sequence":"first","affiliation":[]},{"given":"Adam","family":"Burchardt","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9743-7927","authenticated-orcid":true,"given":"Wojciech","family":"Bruzda","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2652-9482","authenticated-orcid":true,"given":"Grzegorz","family":"Rajchel-Mieldzioć","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5891-6017","authenticated-orcid":true,"given":"Arul","family":"Lakshminarayan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0653-3639","authenticated-orcid":true,"given":"Karol","family":"Życzkowski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.128.080507","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080507","note":"arxivid:2104.05122"},{"id":"arxiv:2005.01426","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/access.2020.3043401","page":"222439-222448","source":"Crossref","title":"Modifying Method of Constructing Quantum Codes From Highly Entangled States","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-9168-8212","authenticated-orcid":false,"given":"Zahra","family":"Raissi","sequence":"first","affiliation":[{"name":"ICFO&#x2014;The Institute of Photonic Sciences, The Barcelona Institute of Science and Technology, Barcelona, Spain"}]}],"container-title":"IEEE Access","original-title":[],"issued":{"date-parts":[[2020]]},"URL":"http://dx.doi.org/10.1109/ACCESS.2020.3043401","ISSN":["2169-3536"],"container-title-short":"IEEE Access","note":"arxivid:2005.01426"},{"id":"arxiv:2004.00906","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.102.012427","source":"Crossref","title":"Planar maximally entangled states","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0002-7124-303X","authenticated-orcid":true,"given":"Mehregan","family":"Doroudiani","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5496-4647","authenticated-orcid":true,"given":"Vahid","family":"Karimipour","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.012427","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012427","note":"arxivid:2004.00906"},{"id":"arxiv:1806.06472","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.98.052301","source":"Crossref","title":"Calderbank-Shor-Steane holographic quantum error-correcting codes","volume":"98","author":[{"given":"Robert J.","family":"Harris","sequence":"first","affiliation":[]},{"given":"Nathan A.","family":"McMahon","sequence":"additional","affiliation":[]},{"given":"Gavin K.","family":"Brennen","sequence":"additional","affiliation":[]},{"given":"Thomas M.","family":"Stace","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,11,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.98.052301","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052301","note":"arxivid:1806.06472"},{"id":"arxiv:0903.1069","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.80.155131","source":"Crossref","title":"Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order","volume":"80","author":[{"given":"Zheng-Cheng","family":"Gu","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,10,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.80.155131","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"155131","note":"arxivid:0903.1069"},{"id":"arxiv:0909.4059","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.85.075125","source":"Crossref","title":"Symmetry protection of topological phases in one-dimensional quantum spin systems","volume":"85","author":[{"given":"Frank","family":"Pollmann","sequence":"first","affiliation":[]},{"given":"Erez","family":"Berg","sequence":"additional","affiliation":[]},{"given":"Ari M.","family":"Turner","sequence":"additional","affiliation":[]},{"given":"Masaki","family":"Oshikawa","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2012,2,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.85.075125","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"075125","note":"arxivid:0909.4059"},{"id":"arxiv:2502.17572","type":"article-journal","author":[{"given":"Yu-Jie","family":"Liu"},{"given":"Wen-Tao","family":"Xu"},{"given":"Frank","family":"Pollmann"},{"given":"Michael","family":"Knap"}],"title":"Information-theoretic principle of emergent 1-form symmetries","issued":{"date-parts":[[2025,11,21]]},"note":"arxivid:2502.17572\narxiv_version_number:3"},{"id":"arxiv:2407.07951","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/kw68-mkkd","source":"Crossref","title":"A quantum cellular automaton for every symmetry protected topological phase","volume":"112","author":[{"ORCID":"https://orcid.org/0000-0002-9313-7799","authenticated-orcid":true,"given":"Lukasz","family":"Fidkowski","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00cvxb145","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Washington"}]},{"ORCID":"https://orcid.org/0000-0002-1087-6853","authenticated-orcid":true,"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00d0nc645","id-type":"ROR","asserted-by":"publisher"}],"name":"Microsoft Quantum"}]},{"ORCID":"https://orcid.org/0000-0002-6182-5087","authenticated-orcid":true,"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00d0nc645","id-type":"ROR","asserted-by":"publisher"}],"name":"Microsoft Quantum"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,8]]},"URL":"http://dx.doi.org/10.1103/kw68-mkkd","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"035123","note":"arxivid:2407.07951"},{"id":"arxiv:2204.03045","type":"journal-article","publisher":"Annual Reviews","issue":"1","abstract":"<jats:p> Recent advances in our understanding of symmetry in quantum many-body systems offer the possibility of a generalized Landau paradigm that encompasses all equilibrium phases of matter. This is a brief and elementary review of some of these developments. </jats:p>","DOI":"10.1146/annurev-conmatphys-040721-021029","page":"57-82","source":"Crossref","title":"Generalized Symmetries in Condensed Matter","volume":"14","author":[{"given":"John","family":"McGreevy","sequence":"first","affiliation":[{"name":"Department of Physics, University of California at San Diego, La Jolla, California, USA;"}]}],"container-title":"Annual Review of Condensed Matter Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,10]]},"URL":"http://dx.doi.org/10.1146/annurev-conmatphys-040721-021029","ISSN":["1947-5454","1947-5462"],"container-title-short":"Annu. Rev. Condens. Matter Phys.","note":"alternative-id:10.1146/annurev-conmatphys-040721-021029\narxivid:2204.03045"},{"id":"arxiv:1805.01474","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.10.031041","source":"Crossref","title":"Symmetry-Protected Self-Correcting Quantum Memories","volume":"10","author":[{"given":"Sam","family":"Roberts","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.10.031041","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031041","note":"arxivid:1805.01474"},{"id":"arxiv:2408.14970","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We investigate the emergence of stable subspaces in the low-temperature quantum thermal dynamics of finite spin chains. Our analysis reveals the existence of effective decoherence-free qudit subspaces, persisting for timescales exponential in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi></mml:math>. Surprisingly, the appearance of metastable subspaces is not directly related to the entanglement structure of the ground state(s). Rather, they arise from symmetry relations in low-lying excited states. Despite their stability within a 'phase', practical realization of stable qubits is hindered by susceptibility to symmetry-breaking perturbations. This work highlights that there can be non-trivial quantum behavior in the thermal dynamics of noncommuting many body models, and opens the door to more extensive studies of self-correction in such systems.</jats:p>","DOI":"10.22331/q-2025-08-04-1820","page":"1820","source":"Crossref","title":"A little bit of self-correction","volume":"9","author":[{"given":"Michael J.","family":"Kastoryano","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Copenhagen, Denmark"},{"name":"AWS Center for Quantum Computing, Pasadena, CA"}]},{"ORCID":"https://orcid.org/0000-0002-3939-8170","authenticated-orcid":false,"given":"Lasse B.","family":"Kristensen","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Copenhagen, Denmark"}]},{"given":"Chi-Fang","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA"}]},{"given":"Andras","family":"Gilyén","sequence":"additional","affiliation":[{"name":"HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,4]]},"URL":"http://dx.doi.org/10.22331/q-2025-08-04-1820","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2408.14970"},{"id":"arxiv:2310.16982","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive a generic formula for a transversal\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>T</a:mi>\n                    </a:math>\n                    gate on color codes defined on general 3-manifolds, which acts as collective non-Clifford logical  gates on any triplet of logical qubits with their logical-\n                    <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <c:mi>X</c:mi>\n                    </c:math>\n                    membranes having a\n                    <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <e:msub>\n                        <e:mrow>\n                          <e:mi mathvariant=\"double-struck\">Z</e:mi>\n                        </e:mrow>\n                        <e:mn>2</e:mn>\n                      </e:msub>\n                    </e:math>\n                    triple intersection at a single point. The triple-intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory (TQFT): the\n                    <h:math xmlns:h=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <h:msubsup>\n                        <h:mrow>\n                          <h:mi mathvariant=\"double-struck\">Z</h:mi>\n                        </h:mrow>\n                        <h:mn>2</h:mn>\n                        <h:mn>3</h:mn>\n                      </h:msubsup>\n                    </h:math>\n                    gauge theory. Moreover, the transversal\n                    <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <k:mi>S</k:mi>\n                    </k:math>\n                    gate of the color code corresponds to a higher-form symmetry in TQFT supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical  gates. A construction of constant-depth circuits of the above logical gates via cup-product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple-intersection invariants for 3-manifolds, with the structure encoded into an interaction hypergraph which determines the logical gate property and also corresponds to the hypergraph magic state that can be injected into the code without distillation (“”). We also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate\n                    <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <m:mi>k</m:mi>\n                      <m:mo>/</m:mo>\n                      <m:mi>n</m:mi>\n                      <m:mo>=</m:mo>\n                      <m:mi>O</m:mi>\n                      <m:mo stretchy=\"false\">(</m:mo>\n                      <m:mn>1</m:mn>\n                      <m:mo>/</m:mo>\n                      <m:mi>log</m:mi>\n                      <m:mo></m:mo>\n                      <m:mo stretchy=\"false\">(</m:mo>\n                      <m:mi>n</m:mi>\n                      <m:mo stretchy=\"false\">)</m:mo>\n                      <m:mo stretchy=\"false\">)</m:mo>\n                    </m:math>\n                    and\n                    <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <s:mi>O</s:mi>\n                      <s:mo stretchy=\"false\">(</s:mo>\n                      <s:mi>log</s:mi>\n                      <s:mo></s:mo>\n                      <s:mo stretchy=\"false\">(</s:mo>\n                      <s:mi>n</s:mi>\n                      <s:mo stretchy=\"false\">)</s:mo>\n                      <s:mo stretchy=\"false\">)</s:mo>\n                    </s:math>\n                    distance; (2) A homological fiber-bundle code from twisting the product by an isometry of the surface based on the construction by Freedman-Meyer-Luo, with\n                    <y:math xmlns:y=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <y:mi>O</y:mi>\n                      <y:mo stretchy=\"false\">(</y:mo>\n                      <y:mn>1</y:mn>\n                      <y:mo>/</y:mo>\n                      <y:msup>\n                        <y:mi>log</y:mi>\n                        <y:mrow>\n                          <y:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n                            <y:mfrac>\n                              <y:mn>1</y:mn>\n                              <y:mn>2</y:mn>\n                            </y:mfrac>\n                          </y:mstyle>\n                        </y:mrow>\n                      </y:msup>\n                      <y:mo></y:mo>\n                      <y:mo stretchy=\"false\">(</y:mo>\n                      <y:mi>n</y:mi>\n                      <y:mo stretchy=\"false\">)</y:mo>\n                      <y:mo stretchy=\"false\">)</y:mo>\n                    </y:math>\n                    rate and\n                    <gb:math xmlns:gb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <gb:mi>O</gb:mi>\n                      <gb:mo stretchy=\"false\">(</gb:mo>\n                      <gb:msup>\n                        <gb:mi>log</gb:mi>\n                        <gb:mrow>\n                          <gb:mstyle displaystyle=\"false\" scriptlevel=\"0\">\n                            <gb:mfrac>\n                              <gb:mn>1</gb:mn>\n                              <gb:mn>2</gb:mn>\n                            </gb:mfrac>\n                          </gb:mstyle>\n                        </gb:mrow>\n                      </gb:msup>\n                      <gb:mo></gb:mo>\n                      <gb:mo stretchy=\"false\">(</gb:mo>\n                      <gb:mi>n</gb:mi>\n                      <gb:mo stretchy=\"false\">)</gb:mo>\n                      <gb:mo stretchy=\"false\">)</gb:mo>\n                    </gb:math>\n                    distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping-torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical-\n                    <ob:math xmlns:ob=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <ob:mi>X</ob:mi>\n                    </ob:math>\n                    measurements.\n                  </jats:p>","DOI":"10.1103/wcxs-w69t","source":"Crossref","title":"Non-Clifford and Parallelizable Fault-Tolerant Logical Gates on Constant and Almost-Constant Rate Homological Quantum Low-Density Parity-Check Codes via Higher Symmetries","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM Quantum, T.J. Watson Research Center"}]},{"ORCID":"https://orcid.org/0000-0001-7167-2128","authenticated-orcid":true,"given":"Shehryar","family":"Sikander","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05vt9qd57","id-type":"ROR","asserted-by":"publisher"}],"name":"Rutgers University"}]},{"given":"Elia","family":"Portnoy","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0001-9786-8196","authenticated-orcid":true,"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM Quantum, T.J. Watson Research Center"}]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM Quantum, T.J. Watson Research Center"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,11]]},"URL":"http://dx.doi.org/10.1103/wcxs-w69t","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040361","note":"arxivid:2310.16982"},{"id":"arxiv:1902.02115","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"9","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>Motivated by the close relationship between quantum error-correction, topological order, the holographic AdS/CFT duality, and tensor networks, we initiate the study of approximate quantum error-detecting codes in matrix product states (MPS). We first show that using open-boundary MPS to define boundary to bulk encoding maps yields at most constant distance error-detecting codes. These are degenerate ground spaces of gapped local Hamiltonians. To get around this no-go result, we consider excited states, i.e., we use the excitation ansatz to construct encoding maps: these yield error-detecting codes with distance Ω(<jats:italic>n</jats:italic>\n            <jats:sup>1−ν</jats:sup> ) for any <jats:italic>ν</jats:italic> ∈ (0<jats:italic>,</jats:italic> 1) and Ω(log <jats:italic>n</jats:italic>) encoded qubits. This shows that gapped systems contain — within isolated energy bands — error-detecting codes spanned by momentum eigenstates. We also consider the gapless Heisenberg-XXX model, whose energy eigenstates can be described via Bethe ansatz tensor networks. We show that it contains — within its low-energy eigenspace — an error-detecting code with the same parameter scaling. All these codes detect arbitrary <jats:italic>d</jats:italic>-local (not necessarily geometrically local) errors even though they are not permutation-invariant. This suggests that a wide range of naturally occurring many-body systems possess intrinsic error-detecting features.</jats:p>","DOI":"10.1007/jhep09(2019)021","source":"Crossref","title":"Quantum error-detection at low energies","volume":"2019","author":[{"given":"Martina","family":"Gschwendtner","sequence":"first","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]},{"given":"Burak","family":"Şahinoğlu","sequence":"additional","affiliation":[]},{"given":"Eugene","family":"Tang","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,9]]},"URL":"http://dx.doi.org/10.1007/JHEP09(2019)021","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"21","note":"alternative-id:11204\narxivid:1902.02115"},{"id":"arxiv:1008.3745","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevb.83.035107","source":"Crossref","title":"Classification of gapped symmetric phases in one-dimensional spin systems","volume":"83","author":[{"given":"Xie","family":"Chen","sequence":"first","affiliation":[]},{"given":"Zheng-Cheng","family":"Gu","sequence":"additional","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2011,1,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.83.035107","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"035107","note":"arxivid:1008.3745"},{"id":"arxiv:1010.3732","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevb.84.165139","source":"Crossref","title":"Classifying quantum phases using matrix product states and projected entangled pair states","volume":"84","author":[{"given":"Norbert","family":"Schuch","sequence":"first","affiliation":[]},{"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[]},{"given":"Ignacio","family":"Cirac","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2011,10,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.84.165139","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"165139","note":"arxivid:1010.3732"},{"id":"arxiv:1103.3323","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevb.84.235128","source":"Crossref","title":"Complete classification of one-dimensional gapped quantum phases in interacting spin systems","volume":"84","author":[{"given":"Xie","family":"Chen","sequence":"first","affiliation":[]},{"given":"Zheng-Cheng","family":"Gu","sequence":"additional","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2011,12,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.84.235128","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"235128","note":"arxivid:1103.3323"},{"id":"arxiv:1106.4772","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.87.155114","source":"Crossref","title":"Symmetry protected topological orders and the group cohomology of their symmetry group","volume":"87","author":[{"given":"Xie","family":"Chen","sequence":"first","affiliation":[]},{"given":"Zheng-Cheng","family":"Gu","sequence":"additional","affiliation":[]},{"given":"Zheng-Xin","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2013,4,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.87.155114","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"155114","note":"arxivid:1106.4772"},{"id":"arxiv:1910.00038","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.2.033116","source":"Crossref","title":"Quasi-exact quantum computation","volume":"2","author":[{"given":"Dong-Sheng","family":"Wang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Cihan","family":"Okay","sequence":"additional","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033116","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033116","note":"arxivid:1910.00038"},{"id":"doi:10.1007/BF02727953","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf02727953","page":"1-23","source":"Crossref","title":"On the theory of identical particles","volume":"37","author":[{"given":"J. M.","family":"Leinaas","sequence":"first","affiliation":[]},{"given":"J.","family":"Myrheim","sequence":"additional","affiliation":[]}],"container-title":"Il Nuovo Cimento B Series 11","original-title":[],"language":"en","issued":{"date-parts":[[1977,1]]},"URL":"http://dx.doi.org/10.1007/BF02727953","ISSN":["1826-9877"],"container-title-short":"Nuovo Cim B","note":"alternative-id:BF02727953"},{"id":"doi:10.1063/1.524510","type":"journal-article","publisher":"AIP Publishing","issue":"4","abstract":"<jats:p>Representations of the nonrelativistic current group 𝒮-𝒦 are studied in the Gel’fand–Vilenkin formalism, where 𝒮 is Schwartz’ space of rapidly decreasing functions, and 𝒦 is a group of diffeomorphisms of Rs. For the case of N identical particles, information about particle statistics is contained in a representation of 𝒦F (the stability group of a point F∈𝒮′) which factors through the permutation group SN. Starting from a quasi-invariant measure μ concentrated on a 𝒦 orbit Δ in 𝒮′, together with a suitable representation of 𝒦F for F∈Δ, sufficient conditions are developed for inducing a representation of 𝒮-𝒦. The Hilbert space for the induced representation consists of square-integrable functions on a covering space of Δ, which transform in accordance with a representation of 𝒦F. The Bose and Fermi N-particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representation of 𝒮-𝒦 determines a well-defined representation of 𝒦F; (2) equivalent representations of 𝒮-𝒦 determine equivalent representations of 𝒦F; (3) a representation of 𝒦F induces a representation of 𝒮-𝒦; and (4) equivalent representations of 𝒦F determine equivalent induced representations.</jats:p>","DOI":"10.1063/1.524510","page":"650-664","source":"Crossref","title":"Particle statistics from induced representations of a local current group","volume":"21","author":[{"given":"G. A.","family":"Goldin","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115 and Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545"}]},{"given":"R.","family":"Menikoff","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545"}]},{"given":"D. H.","family":"Sharp","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1980,4,1]]},"URL":"http://dx.doi.org/10.1063/1.524510","ISSN":["0022-2488","1089-7658"]},{"id":"doi:10.1103/PhysRevLett.49.957","type":"journal-article","publisher":"American Physical Society (APS)","issue":"14","DOI":"10.1103/physrevlett.49.957","page":"957-959","source":"Crossref","title":"Quantum Mechanics of Fractional-Spin Particles","volume":"49","author":[{"given":"Frank","family":"Wilczek","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1982,10,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.49.957","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"doi:10.1063/1.2810672","type":"journal-article","publisher":"AIP Publishing","issue":"8","DOI":"10.1063/1.2810672","page":"90-91","source":"Crossref","title":"The Ancestry of the ‘Anyon’","volume":"43","author":[{"given":"Larry","family":"Biedenharn","sequence":"first","affiliation":[{"name":"Duke University, Durham, North Carolina"}]},{"given":"Elliott","family":"Lieb","sequence":"additional","affiliation":[{"name":"Princeton University, Princeton, New Jersey"}]},{"given":"Barry","family":"Simon","sequence":"additional","affiliation":[{"name":"California Institute of Technology, Pasadena, California"}]},{"given":"Frank","family":"Wilczek","sequence":"additional","affiliation":[{"name":"Institute for Advanced Study, Princeton, New Jersey"}]}],"container-title":"Physics Today","original-title":[],"language":"en","issued":{"date-parts":[[1990,8,1]]},"URL":"http://dx.doi.org/10.1063/1.2810672","ISSN":["0031-9228","1945-0699"]},{"id":"doi:10.1007/BF01877742","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf01877742","page":"199-230","source":"Crossref","title":"Local observables and particle statistics I","volume":"23","author":[{"given":"Sergio","family":"Doplicher","sequence":"first","affiliation":[]},{"given":"Rudolf","family":"Haag","sequence":"additional","affiliation":[]},{"given":"John E.","family":"Roberts","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1971,9]]},"URL":"http://dx.doi.org/10.1007/BF01877742","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF01877742"},{"id":"doi:10.1007/BF01646454","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01646454","page":"49-85","source":"Crossref","title":"Local observables and particle statistics II","volume":"35","author":[{"given":"Sergio","family":"Doplicher","sequence":"first","affiliation":[]},{"given":"Rudolf","family":"Haag","sequence":"additional","affiliation":[]},{"given":"John E.","family":"Roberts","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1974,3]]},"URL":"http://dx.doi.org/10.1007/BF01646454","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF01646454"},{"id":"arxiv:1804.03203","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s00220-019-03630-1","page":"219-264","source":"Crossref","title":"On the Stability of Charges in Infinite Quantum Spin Systems","volume":"373","author":[{"ORCID":"https://orcid.org/0000-0002-3658-2909","authenticated-orcid":false,"given":"Matthew","family":"Cha","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5670-243X","authenticated-orcid":false,"given":"Pieter","family":"Naaijkens","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7835-3776","authenticated-orcid":false,"given":"Bruno","family":"Nachtergaele","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,17]]},"URL":"http://dx.doi.org/10.1007/s00220-019-03630-1","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:3630\narxivid:1804.03203"},{"id":"manual:-J.-Fuchs.-Affine-Lie-algebra","type":"document","author":[{"family":"Fuchs","given":"J."}],"title":"Affine Lie algebras and quantum groups: An Introduction, with applications in conformal field theory","note":"Cambridge university press, 1995"},{"id":"arxiv:1009.2117","type":"article-journal","author":[{"given":"Alexei","family":"Davydov"},{"given":"Michael","family":"Mueger"},{"given":"Dmitri","family":"Nikshych"},{"given":"Victor","family":"Ostrik"}],"title":"The Witt group of non-degenerate braided fusion categories","issued":{"date-parts":[[2011,9,6]]},"note":"arxivid:1009.2117\narxiv_version_number:2"},{"id":"arxiv:1203.4568","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-013-1723-0","page":"543-575","source":"Crossref","title":"Bicategories for Boundary Conditions and for Surface Defects in 3-d TFT","volume":"321","author":[{"given":"Jürgen","family":"Fuchs","sequence":"first","affiliation":[]},{"given":"Christoph","family":"Schweigert","sequence":"additional","affiliation":[]},{"given":"Alessandro","family":"Valentino","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2013,5,10]]},"URL":"http://dx.doi.org/10.1007/s00220-013-1723-0","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1723\narxivid:1203.4568"},{"id":"arxiv:hep-th/0510092","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.96.110404","source":"Crossref","title":"Topological Entanglement Entropy","volume":"96","author":[{"given":"Alexei","family":"Kitaev","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,3,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.96.110404","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110404","note":"arxivid:hep-th/0510092"},{"id":"arxiv:cond-mat/0510613","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.96.110405","source":"Crossref","title":"Detecting Topological Order in a Ground State Wave Function","volume":"96","author":[{"given":"Michael","family":"Levin","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,3,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.96.110405","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110405","note":"arxivid:cond-mat/0510613"},{"id":"arxiv:2509.12355","type":"article-journal","author":[{"given":"Sergi","family":"Mas-Mendoza"},{"given":"Richard D. P.","family":"East"},{"given":"Michele","family":"Filippone"},{"given":"Adolfo G.","family":"Grushin"}],"title":"A graphical diagnostic of topological order using ZX calculus","issued":{"date-parts":[[2025,9,15]]},"note":"arxivid:2509.12355\narxiv_version_number:1"},{"id":"arxiv:1111.2342","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevb.85.235151","source":"Crossref","title":"Quasiparticle statistics and braiding from ground-state entanglement","volume":"85","author":[{"given":"Yi","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Tarun","family":"Grover","sequence":"additional","affiliation":[]},{"given":"Ari","family":"Turner","sequence":"additional","affiliation":[]},{"given":"Masaki","family":"Oshikawa","sequence":"additional","affiliation":[]},{"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2012,6,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.85.235151","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"235151","note":"arxivid:1111.2342"},{"id":"arxiv:1407.2926","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-016-2594-y","page":"771-801","source":"Crossref","title":"An Invariant of Topologically Ordered States Under Local Unitary Transformations","volume":"342","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2016,2,22]]},"URL":"http://dx.doi.org/10.1007/s00220-016-2594-y","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2594\narxivid:1407.2926"},{"id":"arxiv:2110.06932","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.128.176402","source":"Crossref","title":"Chiral Central Charge from a Single Bulk Wave Function","volume":"128","author":[{"given":"Isaac H.","family":"Kim","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0689-9964","authenticated-orcid":true,"given":"Bowen","family":"Shi","sequence":"additional","affiliation":[]},{"given":"Kohtaro","family":"Kato","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.128.176402","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"176402","note":"arxivid:2110.06932"},{"id":"arxiv:2106.12627","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6613","abstract":"<jats:p>Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground-state properties of gapped Hamiltonians after learning from other Hamiltonians in the same quantum phase of matter. By contrast, under a widely accepted conjecture, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.</jats:p>","DOI":"10.1126/science.abk3333","source":"Crossref","title":"Provably efficient machine learning for quantum many-body problems","volume":"377","author":[{"ORCID":"https://orcid.org/0000-0001-5317-2613","authenticated-orcid":true,"given":"Hsin-Yuan","family":"Huang","sequence":"first","affiliation":[{"name":"Institute for Quantum Information and Matter and Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8291-648X","authenticated-orcid":true,"given":"Richard","family":"Kueng","sequence":"additional","affiliation":[{"name":"Institute for Integrated Circuits, Johannes Kepler University, Linz, Austria."}]},{"ORCID":"https://orcid.org/0000-0001-8478-4436","authenticated-orcid":true,"given":"Giacomo","family":"Torlai","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing, Pasadena, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, National Institute of Standards and Technology and University of Maryland, College Park, MD, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2421-4762","authenticated-orcid":true,"given":"John","family":"Preskill","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information and Matter and Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA."},{"name":"AWS Center for Quantum Computing, Pasadena, CA, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,23]]},"URL":"http://dx.doi.org/10.1126/science.abk3333","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","page":"eabk3333","note":"alternative-id:10.1126/science.abk3333\narxivid:2106.12627"},{"id":"arxiv:1106.6026","type":"journal-article","publisher":"American Physical Society (APS)","issue":"21","DOI":"10.1103/physrevlett.107.210501","source":"Crossref","title":"Topological Order at Nonzero Temperature","volume":"107","author":[{"given":"Matthew B.","family":"Hastings","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,11,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.107.210501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"210501","note":"arxivid:1106.6026"},{"id":"arxiv:0904.4492","type":"article-journal","author":[{"given":"Mehdi","family":"Kargarian"}],"title":"Finite temperature topological order in 2D topological color codes","issued":{"date-parts":[[2009,7,19]]},"note":"arxivid:0904.4492\narxiv_version_number:2"},{"id":"arxiv:2403.16850","type":"article-journal","author":[{"given":"Ainesh","family":"Bakshi"},{"given":"Allen","family":"Liu"},{"given":"Ankur","family":"Moitra"},{"given":"Ewin","family":"Tang"}],"title":"High-Temperature Gibbs States are Unentangled and Efficiently Preparable","issued":{"date-parts":[[2025,2,24]]},"note":"arxivid:2403.16850\narxiv_version_number:2"},{"id":"arxiv:1704.04221","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.8.021074","source":"Crossref","title":"Classification of \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo mathvariant=\"bold\" stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo mathvariant=\"bold\" stretchy=\"false\">)</mml:mo><mml:mi mathvariant=\"normal\">D</mml:mi></mml:mrow></mml:math>\n Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons","volume":"8","author":[{"given":"Tian","family":"Lan","sequence":"first","affiliation":[]},{"given":"Liang","family":"Kong","sequence":"additional","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2018,6,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.8.021074","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021074","note":"arxivid:1704.04221"},{"id":"arxiv:1808.09394","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.100.045105","source":"Crossref","title":"Topological nonlinear \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>σ</mml:mi></mml:math>\n-model, higher gauge theory, and a systematic construction of \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo></mml:mrow><mml:mn>1</mml:mn><mml:mtext>D</mml:mtext></mml:math>\n topological orders for boson systems","volume":"100","author":[{"given":"Chenchang","family":"Zhu","sequence":"first","affiliation":[]},{"given":"Tian","family":"Lan","sequence":"additional","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.100.045105","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"045105","note":"arxivid:1808.09394"},{"id":"arxiv:1403.1467","type":"article-journal","author":[{"given":"Anton","family":"Kapustin"}],"title":"Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology","issued":{"date-parts":[[2014,4,14]]},"note":"arxivid:1403.1467\narxiv_version_number:3"},{"id":"arxiv:1801.08530","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.9.021005","source":"Crossref","title":"Classification of \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mi mathvariant=\"normal\">D</mml:mi></mml:mrow></mml:math>\n Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions","volume":"9","author":[{"given":"Tian","family":"Lan","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2019,4,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.021005","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021005","note":"update-to:{\"updated\":{\"date-parts\":[[2019,4,12]],\"date-time\":\"2019-04-12T00:00:00Z\",\"timestamp\":1555027200000},\"DOI\":\"10.1103/physrevx.9.021005\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2019,4,12]],\"date-time\":\"2019-04-12T00:00:00Z\",\"timestamp\":1555027200000},\"DOI\":\"10.1103/physrevx.9.021005\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:1801.08530"},{"id":"arxiv:2104.04534","type":"journal-article","publisher":"Stichting SciPost","issue":"3","abstract":"<jats:p>We investigate the Morita equivalences of (4+1)-dimensional topological orders. We show that any (4+1)-dimensional super (fermionic) topological order admits a gapped boundary condition --- in other words, all (4+1)-dimensional super topological orders are Morita trivial.  As a result, there are no inherently gapless super (3+1)-dimensional theories. On the other hand, we show that there are infinitely many algebraically Morita-inequivalent bosonic (4+1)-dimensional topological orders.</jats:p>","DOI":"10.21468/scipostphys.13.3.068","source":"Crossref","title":"Topological Orders in (4+1)-Dimensions","volume":"13","author":[{"given":"Theo","family":"Johnson-Freyd","sequence":"first","affiliation":[{"name":"Dalhousie University"},{"name":"Perimeter Institute"}]},{"given":"Matthew","family":"Yu","sequence":"additional","affiliation":[{"name":"Perimeter Institute"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2022,9,27]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.13.3.068","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"068","note":"arxivid:2104.04534"},{"id":"arxiv:1405.5858","type":"article-journal","author":[{"given":"Liang","family":"Kong"},{"given":"Xiao-Gang","family":"Wen"}],"title":"Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions","issued":{"date-parts":[[2014,5,22]]},"note":"arxivid:1405.5858\narxiv_version_number:1"},{"id":"arxiv:1702.00673","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.nuclphysb.2017.06.023","page":"62-76","source":"Crossref","title":"Boundary-bulk relation in topological orders","volume":"922","author":[{"given":"Liang","family":"Kong","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3208-9615","authenticated-orcid":false,"given":"Hao","family":"Zheng","sequence":"additional","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[2017,9]]},"URL":"http://dx.doi.org/10.1016/j.nuclphysb.2017.06.023","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"special_numbering:C\nalternative-id:S0550321317302183\narxivid:1702.00673"},{"id":"arxiv:2003.06663","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-022-04380-3","page":"989-1033","source":"Crossref","title":"On the Classification of Topological Orders","volume":"393","author":[{"ORCID":"https://orcid.org/0000-0003-3617-155X","authenticated-orcid":false,"given":"Theo","family":"Johnson-Freyd","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,16]]},"URL":"http://dx.doi.org/10.1007/s00220-022-04380-3","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:4380\narxivid:2003.06663"},{"id":"arxiv:1109.1588","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-013-1762-6","page":"277-302","source":"Crossref","title":"Stability of Frustration-Free Hamiltonians","volume":"322","author":[{"given":"Spyridon","family":"Michalakis","sequence":"first","affiliation":[]},{"given":"Justyna P.","family":"Zwolak","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2013,7,3]]},"URL":"http://dx.doi.org/10.1007/s00220-013-1762-6","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1762\narxivid:1109.1588"},{"id":"arxiv:2110.11194","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>A gapped ground state of a quantum spin system has a natural length scale set by the gap. This length scale governs the decay of correlations. A common intuition is that this length scale also controls the spatial relaxation towards the ground state away from impurities or boundaries. The aim of this article is to take a step towards a proof of this intuition. We assume that the ground state is frustration-free and invertible, i.e. it has no long-range entanglement. Moreover, we assume the property that we are aiming to prove for one specific kind of boundary condition; namely open boundary conditions. This assumption is also known as the \"local topological quantum order\" (LTQO) condition. With these assumptions we can prove stretched exponential decay away from boundaries or impurities, for any of the ground states of the perturbed system. In contrast to most earlier results, we do not assume that the perturbations at the boundary or the impurity are small. In particular, the perturbed system itself can have long-range entanglement.</jats:p>","DOI":"10.22331/q-2022-09-08-793","page":"793","source":"Crossref","title":"Stability of invertible, frustration-free ground states against large perturbations","volume":"6","author":[{"given":"Sven","family":"Bachmann","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada"}]},{"given":"Wojciech","family":"De Roeck","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics, K.U. Leuven, 3001 Leuven, Belgium"}]},{"given":"Brecht","family":"Donvil","sequence":"additional","affiliation":[{"name":"Institute for Complex Quantum Systems and Center for IQST, Ulm University, 89069 Ulm, Germany"},{"name":"Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland"}]},{"given":"Martin","family":"Fraas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, Davis, Davis, CA, 95616, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,8]]},"URL":"http://dx.doi.org/10.22331/q-2022-09-08-793","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2110.11194"},{"id":"arxiv:1406.5090","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevb.91.125121","source":"Crossref","title":"Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity","volume":"91","author":[{"given":"Bei","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2015,3,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.91.125121","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"125121","note":"arxivid:1406.5090"},{"id":"arxiv:1407.8203","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.93.045127","source":"Crossref","title":"Renormalization group constructions of topological quantum liquids and beyond","volume":"93","author":[{"given":"Brian","family":"Swingle","sequence":"first","affiliation":[]},{"given":"John","family":"McGreevy","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2016,1,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.93.045127","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"045127","note":"arxivid:1407.8203"},{"id":"doi:10.1007/bf01217730","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/bf01217730","page":"351-399","source":"Crossref","title":"Quantum field theory and the Jones polynomial","volume":"121","author":[{"given":"Edward","family":"Witten","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1989,9]]},"URL":"http://dx.doi.org/10.1007/bf01217730","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF01217730"},{"id":"doi:10.1007/BF01238857","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/bf01238857","page":"177-254","source":"Crossref","title":"Classical and quantum conformal field theory","volume":"123","author":[{"given":"Gregory","family":"Moore","sequence":"first","affiliation":[]},{"given":"Nathan","family":"Seiberg","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1989,6]]},"URL":"http://dx.doi.org/10.1007/BF01238857","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF01238857"},{"id":"arxiv:1612.07792","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-019-03475-8","page":"1021-1124","source":"Crossref","title":"Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter: Path Integrals, Non-orientable Manifolds, and Anomalies","volume":"374","author":[{"ORCID":"https://orcid.org/0000-0002-4322-9433","authenticated-orcid":false,"given":"Maissam","family":"Barkeshli","sequence":"first","affiliation":[]},{"given":"Parsa","family":"Bonderson","sequence":"additional","affiliation":[]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[]},{"given":"Chao-Ming","family":"Jian","sequence":"additional","affiliation":[]},{"given":"Kevin","family":"Walker","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,6,26]]},"URL":"http://dx.doi.org/10.1007/s00220-019-03475-8","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:3475\narxivid:1612.07792"},{"id":"arxiv:quant-ph/0603121","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.97.050401","source":"Crossref","title":"Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order","volume":"97","author":[{"given":"S.","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"M. B.","family":"Hastings","sequence":"additional","affiliation":[]},{"given":"F.","family":"Verstraete","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,7,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.97.050401","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050401","note":"arxivid:quant-ph/0603121"},{"id":"arxiv:quant-ph/0603114","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevlett.97.150404","source":"Crossref","title":"General Entanglement Scaling Laws from Time Evolution","volume":"97","author":[{"given":"Jens","family":"Eisert","sequence":"first","affiliation":[]},{"given":"Tobias J.","family":"Osborne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,10,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.97.150404","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"150404","note":"arxivid:quant-ph/0603114"},{"id":"arxiv:2209.03964","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevlett.131.060405","source":"Crossref","title":"Shortest Route to Non-Abelian Topological Order on a Quantum Processor","volume":"131","author":[{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":true,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[]},{"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.131.060405","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"060405","note":"arxivid:2209.03964"},{"id":"arxiv:2410.23544","type":"article-journal","author":[{"given":"Hyun-Soo","family":"Kim"},{"given":"Isaac H.","family":"Kim"},{"given":"Daniel","family":"Ranard"}],"title":"Learning State Preparation Circuits for Quantum Phases of Matter","issued":{"date-parts":[[2024,11,4]]},"note":"arxivid:2410.23544\narxiv_version_number:2"},{"id":"arxiv:1311.0019","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.4.031058","source":"Crossref","title":"Thermalization, Error Correction, and Memory Lifetime for Ising Anyon Systems","volume":"4","author":[{"given":"Courtney G.","family":"Brell","sequence":"first","affiliation":[]},{"given":"Simon","family":"Burton","sequence":"additional","affiliation":[]},{"given":"Guillaume","family":"Dauphinais","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.4.031058","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031058","note":"arxivid:1311.0019"},{"id":"arxiv:hep-th/0110205","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevd.65.065022","source":"Crossref","title":"Measurability of Wilson loop operators","volume":"65","author":[{"given":"David","family":"Beckman","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2002,3,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.65.065022","ISSN":["0556-2821","1089-4918"],"container-title-short":"Phys. Rev. D","page":"065022","note":"arxivid:hep-th/0110205"},{"id":"arxiv:cond-mat/0506438","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.aop.2005.10.005","page":"2-111","source":"Crossref","title":"Anyons in an exactly solved model and beyond","volume":"321","author":[{"given":"Alexei","family":"Kitaev","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2006,1]]},"URL":"http://dx.doi.org/10.1016/j.aop.2005.10.005","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491605002381\narxivid:cond-mat/0506438"},{"id":"doi:10.7907/5NDZ-W890","type":"thesis","categories":["Physics","anyon interferometry"],"language":"en","author":[{"family":"Bonderson","given":"Parsa Hassan"}],"issued":{"date-parts":[[2007]]},"abstract":"This thesis is primarily a study of the measurement theory of non-Abelian anyons through interference experiments. We give an introduction to the theory of anyon models, providing all the formalism necessary to apply standard quantum measurement theory to such systems. This formalism is then applied to give a detailed analysis of a Mach-Zehnder interferometer for arbitrary anyon models. In this treatment, we find that the collapse behavior exhibited by a target anyon in a superposition of states is determined by the monodromy of the probe anyons with the target. Such measurements may also be used to gain knowledge that would help to properly identify the anyon model describing an unknown system. The techniques used and results obtained from this model interferometer have general applicability, and we use them to also describe the interferometry measurements in a two point-contact interferometer proposed for non-Abelian fractional quantum Hall states. Additionally, we give the complete description of a number of important examples of anyon models, as well as their corresponding quantities that are relevant for interferometry. Finally, we give a partial classification of anyon models with small numbers of particle types.","DOI":"10.7907/5NDZ-W890","publisher":"California Institute of Technology","title":"Non-Abelian Anyons and Interferometry","URL":"https://resolver.caltech.edu/CaltechETD:etd-06042007-101617","version":"Final","note":"copyright:No commercial reproduction, distribution, display or performance rights in this work are provided."},{"id":"arxiv:0707.1889","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/revmodphys.80.1083","page":"1083-1159","source":"Crossref","title":"Non-Abelian anyons and topological quantum computation","volume":"80","author":[{"given":"Chetan","family":"Nayak","sequence":"first","affiliation":[]},{"given":"Steven H.","family":"Simon","sequence":"additional","affiliation":[]},{"given":"Ady","family":"Stern","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Freedman","sequence":"additional","affiliation":[]},{"given":"Sankar","family":"Das Sarma","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2008,9,12]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.80.1083","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","note":"arxivid:0707.1889"},{"id":"arxiv:1508.02595","type":"article-journal","author":[{"given":"Bei","family":"Zeng"},{"given":"Xie","family":"Chen"},{"given":"Duan-Lu","family":"Zhou"},{"given":"Xiao-Gang","family":"Wen"}],"title":"Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems","issued":{"date-parts":[[2018,2,23]]},"note":"arxivid:1508.02595\narxiv_version_number:4"},{"id":"arxiv:1610.03911","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/revmodphys.89.041004","source":"Crossref","title":"<i>Colloquium</i>\n: Zoo of quantum-topological phases of matter","volume":"89","author":[{"given":"Xiao-Gang","family":"Wen","sequence":"first","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,12,4]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.89.041004","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","page":"041004","note":"arxivid:1610.03911"},{"id":"doi:10.1017/9781316226308","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Topological Phases of Matter are an exceptionally dynamic field of research: several of the most exciting recent experimental discoveries and conceptual advances in modern physics have originated in this field. These have generated new, topological, notions of order, interactions and excitations. This text provides an accessible, unified and comprehensive introduction to the phenomena surrounding topological matter, with detailed expositions of the underlying theoretical tools and conceptual framework, alongside accounts of the central experimental breakthroughs. Among the systems covered are topological insulators, magnets, semimetals, and superconductors. The emergence of new particles with remarkable properties such as fractional charge and statistics is discussed alongside possible applications such as fault-tolerant topological quantum computing. Suitable as a textbook for graduate or advanced undergraduate students, or as a reference for more experienced researchers, the book assumes little prior background, providing self-contained introductions to topics as varied as phase transitions, superconductivity, and localisation.</jats:p>","DOI":"10.1017/9781316226308","source":"Crossref","title":"Topological Phases of Matter","author":[{"given":"Roderich","family":"Moessner","sequence":"first","affiliation":[]},{"given":"Joel E.","family":"Moore","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2021,4,9]]},"ISBN":["9781316226308","9781107105539"],"URL":"http://dx.doi.org/10.1017/9781316226308","note":"edition-number:1"},{"id":"arxiv:2205.05565","type":"article-journal","author":[{"given":"Liang","family":"Kong"},{"given":"Zhi-Hao","family":"Zhang"}],"title":"An invitation to topological orders and category theory","issued":{"date-parts":[[2022,5,31]]},"note":"arxivid:2205.05565\narxiv_version_number:2"},{"id":"arxiv:2112.02502","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.105.042418","source":"Crossref","title":"Topological graph states and quantum error-correction codes","volume":"105","author":[{"ORCID":"https://orcid.org/0000-0002-3215-8887","authenticated-orcid":true,"given":"Pengcheng","family":"Liao","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8326-8912","authenticated-orcid":true,"given":"Barry C.","family":"Sanders","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1446-7029","authenticated-orcid":true,"given":"David L.","family":"Feder","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.042418","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042418","note":"arxivid:2112.02502"},{"id":"arxiv:quant-ph/0111035","type":"journal-article","publisher":"Elsevier BV","issue":"3-4","DOI":"10.1016/s0375-9601(02)00068-3","page":"153-157","source":"Crossref","title":"Almost any quantum spin system with short-range interactions can support toric codes","volume":"294","author":[{"given":"Maxim","family":"Raginsky","sequence":"first","affiliation":[]}],"container-title":"Physics Letters A","original-title":[],"language":"en","issued":{"date-parts":[[2002,2]]},"URL":"http://dx.doi.org/10.1016/S0375-9601(02)00068-3","ISSN":["0375-9601"],"container-title-short":"Physics Letters A","note":"alternative-id:S0375960102000683\narxivid:quant-ph/0111035"},{"id":"arxiv:2404.05867","type":"article-journal","author":[{"given":"Isaac H.","family":"Kim"},{"given":"Ting-Chun","family":"Lin"},{"given":"Daniel","family":"Ranard"},{"given":"Bowen","family":"Shi"}],"title":"Strict area law implies commuting parent Hamiltonian","issued":{"date-parts":[[2024,4,8]]},"note":"arxivid:2404.05867\narxiv_version_number:1"},{"id":"arxiv:2112.12160","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.4.020303","source":"Crossref","title":"Logical Blocks for Fault-Tolerant Topological Quantum Computation","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-6728-5002","authenticated-orcid":true,"given":"Héctor","family":"Bombín","sequence":"first","affiliation":[]},{"given":"Chris","family":"Dawson","sequence":"additional","affiliation":[]},{"given":"Ryan V.","family":"Mishmash","sequence":"additional","affiliation":[]},{"given":"Naomi","family":"Nickerson","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3104-7392","authenticated-orcid":true,"given":"Fernando","family":"Pastawski","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4652-389X","authenticated-orcid":true,"given":"Sam","family":"Roberts","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,4,7]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.020303","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020303","note":"arxivid:2112.12160"},{"id":"arxiv:2308.07844","type":"article-journal","author":[{"given":"Hector","family":"Bombin"},{"given":"Chris","family":"Dawson"},{"given":"Terry","family":"Farrelly"},{"given":"Yehua","family":"Liu"},{"given":"Naomi","family":"Nickerson"},{"given":"Mihir","family":"Pant"},{"given":"Fernando","family":"Pastawski"},{"given":"Sam","family":"Roberts"}],"title":"Fault-tolerant complexes","issued":{"date-parts":[[2023,8,15]]},"note":"arxivid:2308.07844\narxiv_version_number:1"},{"id":"arxiv:2412.13248","type":"article-journal","author":[{"given":"Benedikt","family":"Placke"},{"given":"Tibor","family":"Rakovszky"},{"given":"Nikolas P.","family":"Breuckmann"},{"given":"Vedika","family":"Khemani"}],"title":"Topological Quantum Spin Glass Order and its realization in qLDPC codes","issued":{"date-parts":[[2024,12,17]]},"note":"arxivid:2412.13248\narxiv_version_number:1"},{"id":"arxiv:2002.05166","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevresearch.2.043165","source":"Crossref","title":"Topological defect networks for fractons of all types","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-6552-488X","authenticated-orcid":true,"given":"David","family":"Aasen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8978-4531","authenticated-orcid":true,"given":"Daniel","family":"Bulmash","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4438-7107","authenticated-orcid":true,"given":"Abhinav","family":"Prem","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8036-3447","authenticated-orcid":true,"given":"Kevin","family":"Slagle","sequence":"additional","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,10,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.043165","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"043165","note":"arxivid:2002.05166"},{"id":"arxiv:2112.14717","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.4.010304","source":"Crossref","title":"Topological Defect Network Representations of Fracton Stabilizer Codes","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0003-1044-4927","authenticated-orcid":true,"given":"Zijian","family":"Song","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[]},{"given":"Wilbur","family":"Shirley","sequence":"additional","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,1,11]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.010304","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010304","note":"arxivid:2112.14717"},{"id":"arxiv:2004.12048","type":"journal-article","publisher":"IOP Publishing","issue":"50","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a\n                    <jats:italic>K</jats:italic>\n                    -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.\n                  </jats:p>","DOI":"10.1088/1751-8121/abc6c0","page":"505203","source":"Crossref","title":"In and around abelian anyon models\n                    <sup>*</sup>","volume":"53","author":[{"given":"Liang","family":"Wang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5253-6400","authenticated-orcid":false,"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2020,11,18]]},"URL":"http://dx.doi.org/10.1088/1751-8121/abc6c0","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:2004.12048"},{"id":"arxiv:2011.11165","type":"article-journal","author":[{"given":"Theo","family":"Johnson-Freyd"}],"title":"(3+1)D topological orders with only a $\\mathbb{Z}_2$-charged particle","issued":{"date-parts":[[2020,11,23]]},"note":"arxivid:2011.11165\narxiv_version_number:1"},{"id":"arxiv:2110.14654","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevb.106.165135","source":"Crossref","title":"Gravitational anomaly of \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>\n-dimensional \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi mathvariant=\"double-struck\">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>\n toric code with fermionic charges and fermionic loop self-statistics","volume":"106","author":[{"given":"Lukasz","family":"Fidkowski","sequence":"first","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2022,10,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.106.165135","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"165135","note":"arxivid:2110.14654"},{"id":"arxiv:2108.00018","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.3.030338","source":"Crossref","title":"Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"first","affiliation":[]},{"given":"Tomas","family":"Jochym-O’Connor","sequence":"additional","affiliation":[]},{"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,15]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.030338","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030338","note":"arxivid:2108.00018"},{"id":"arxiv:1305.7203","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevb.88.235103","source":"Crossref","title":"Theory of defects in Abelian topological states","volume":"88","author":[{"given":"Maissam","family":"Barkeshli","sequence":"first","affiliation":[]},{"given":"Chao-Ming","family":"Jian","sequence":"additional","affiliation":[]},{"given":"Xiao-Liang","family":"Qi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2013,12,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.88.235103","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"235103","note":"arxivid:1305.7203"},{"id":"arxiv:2210.09282","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2023.169286","page":"169286","source":"Crossref","title":"Graph gauge theory of mobile non-Abelian anyons in a qubit stabilizer code","volume":"452","author":[{"ORCID":"https://orcid.org/0000-0002-8271-516X","authenticated-orcid":false,"given":"Yuri D.","family":"Lensky","sequence":"first","affiliation":[]},{"given":"Kostyantyn","family":"Kechedzhi","sequence":"additional","affiliation":[]},{"given":"Igor","family":"Aleiner","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9554-4443","authenticated-orcid":false,"given":"Eun-Ah","family":"Kim","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,5]]},"URL":"http://dx.doi.org/10.1016/j.aop.2023.169286","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491623000714\narxivid:2210.09282"},{"id":"arxiv:1004.1838","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.105.030403","source":"Crossref","title":"Topological Order with a Twist: Ising Anyons from an Abelian Model","volume":"105","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,7,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.105.030403","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030403","note":"arxivid:1004.1838"},{"id":"arxiv:1104.5047","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-012-1500-5","page":"351-373","source":"Crossref","title":"Models for Gapped Boundaries and Domain Walls","volume":"313","author":[{"given":"Alexei","family":"Kitaev","sequence":"first","affiliation":[]},{"given":"Liang","family":"Kong","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2012,6,7]]},"URL":"http://dx.doi.org/10.1007/s00220-012-1500-5","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1500\narxivid:1104.5047"},{"id":"arxiv:1208.0928","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.86.032324","source":"Crossref","title":"Surface codes: Towards practical large-scale quantum computation","volume":"86","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Matteo","family":"Mariantoni","sequence":"additional","affiliation":[]},{"given":"John M.","family":"Martinis","sequence":"additional","affiliation":[]},{"given":"Andrew N.","family":"Cleland","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,9,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.032324","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032324","note":"arxivid:1208.0928"},{"id":"arxiv:1508.04166","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.92.245139","source":"Crossref","title":"Demonstrating non-Abelian statistics of Majorana fermions using twist defects","volume":"92","author":[{"given":"Huaixiu","family":"Zheng","sequence":"first","affiliation":[]},{"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2015,12,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.92.245139","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"245139","note":"arxivid:1508.04166"},{"id":"arxiv:1609.04673","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.7.021029","source":"Crossref","title":"Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code","volume":"7","author":[{"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[]},{"given":"Katharina","family":"Laubscher","sequence":"additional","affiliation":[]},{"given":"Markus S.","family":"Kesselring","sequence":"additional","affiliation":[]},{"given":"James R.","family":"Wootton","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2017,5,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.7.021029","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021029","note":"arxivid:1609.04673"},{"id":"arxiv:2103.08381","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.105.042417","source":"Crossref","title":"Non-Abelian statistics with mixed-boundary punctures on the toric code","volume":"105","author":[{"ORCID":"https://orcid.org/0000-0003-3874-493X","authenticated-orcid":true,"given":"Asmae","family":"Benhemou","sequence":"first","affiliation":[]},{"given":"Jiannis K.","family":"Pachos","sequence":"additional","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.042417","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042417","note":"arxivid:2103.08381"},{"id":"arxiv:2212.00042","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>The manipulation of topologically ordered phases of matter to encode and process quantum information forms the cornerstone of many approaches to fault-tolerant quantum computing. Here we demonstrate that fault-tolerant logical operations in these approaches can be interpreted as instances of anyon condensation. We present a constructive theory for anyon condensation and, in tandem, illustrate our theory explicitly using the color-code model. We show that different condensation processes are associated with a general class of domain walls, which can exist in both spacelike and timelike directions. This class includes semitransparent domain walls that condense certain subsets of anyons. We use our theory to classify topological objects and design novel fault-tolerant logic gates for the color code. As a final example, we also argue that dynamical “Floquet codes” can be viewed as a series of condensation operations. We propose a general construction for realizing planar dynamically driven codes based on condensation operations on the color code. We use our construction to introduce a new Calderbank-Shor-Steane–type Floquet code that we call the Floquet color code.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.010342","source":"Crossref","title":"Anyon Condensation and the Color Code","volume":"5","author":[{"given":"Markus S.","family":"Kesselring","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany"}]},{"given":"Julio C.","family":"Magdalena de la Fuente","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany"}]},{"ORCID":"https://orcid.org/0000-0001-7422-1517","authenticated-orcid":true,"given":"Felix","family":"Thomsen","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin 14195, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Berlin 14109, Germany"}]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia"}]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,3,11]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.010342","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010342","note":"arxivid:2212.00042"},{"id":"arxiv:2303.04846","type":"article-journal","author":[{"given":"Héctor","family":"Bombín"},{"given":"Chris","family":"Dawson"},{"given":"Ye-Hua","family":"Liu"},{"given":"Naomi","family":"Nickerson"},{"given":"Fernando","family":"Pastawski"},{"given":"Sam","family":"Roberts"}],"title":"Modular decoding: parallelizable real-time decoding for quantum computers","issued":{"date-parts":[[2023,3,8]]},"note":"arxivid:2303.04846\narxiv_version_number:1"},{"id":"arxiv:2503.02928","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/n9sq-8cxw","source":"Crossref","title":"Finite-Temperature Quantum Topological Order in Three Dimensions","volume":"135","author":[{"ORCID":"https://orcid.org/0009-0006-5406-9689","authenticated-orcid":true,"given":"Shu-Tong","family":"Zhou","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01rxvg760","id-type":"ROR","asserted-by":"publisher"}],"name":"Nanjing University"}]},{"ORCID":"https://orcid.org/0000-0001-7600-1809","authenticated-orcid":true,"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"given":"Tibor","family":"Rakovszky","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f54p054","id-type":"ROR","asserted-by":"publisher"}],"name":"Stanford University"},{"id":[{"id":"https://ror.org/02w42ss30","id-type":"ROR","asserted-by":"publisher"}],"name":"Budapest University of Technology and Economics"},{"id":[{"id":"https://ror.org/02w42ss30","id-type":"ROR","asserted-by":"publisher"}],"name":"Budapest University of Technology and Economics"}]},{"ORCID":"https://orcid.org/0000-0001-6235-6430","authenticated-orcid":true,"given":"Curt","family":"von Keyserlingk","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0220mzb33","id-type":"ROR","asserted-by":"publisher"}],"name":"King’s College London"}]},{"ORCID":"https://orcid.org/0000-0002-1740-6889","authenticated-orcid":true,"given":"Tyler D.","family":"Ellison","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,23]]},"URL":"http://dx.doi.org/10.1103/n9sq-8cxw","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"040402","note":"update-to:{\"DOI\":\"10.1103/n9sq-8cxw\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2026,1,13]],\"date-time\":\"2026-01-13T00:00:00Z\",\"timestamp\":1768262400000}}\nupdated-by:{\"DOI\":\"10.1103/n9sq-8cxw\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2026,1,13]],\"date-time\":\"2026-01-13T00:00:00Z\",\"timestamp\":1768262400000}}\narxivid:2503.02928"},{"id":"arxiv:2112.02137","type":"journal-article","publisher":"Stichting SciPost","issue":"1","abstract":"<jats:p>2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist? We study the 4+1d gauge theory with 2-form <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> gauge field (the loop-only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self “exchange” statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The SL(2,<jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula>) symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the “fractional Maxwell theory” and show how it can be Higgsed into gapped boundary conditions. We also discuss the generalization of these results from the <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> gauge group to <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_N</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula>.</jats:p>","DOI":"10.21468/scipostphys.15.1.001","source":"Crossref","title":"Loops in 4+1d topological phases","volume":"15","author":[{"given":"Xie","family":"Chen","sequence":"first","affiliation":[{"name":"California Institute of Technology"},{"name":"Walter Burke Institute for Theoretical Physics"}]},{"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[{"name":"California Institute of Technology"},{"name":"Walter Burke Institute for Theoretical Physics"}]},{"given":"Po-Shen","family":"Hsin","sequence":"additional","affiliation":[{"name":"California Institute of Technology"},{"name":"Mani L. Bhaumik Institute for Theoretical Physics"},{"name":"Walter Burke Institute for Theoretical Physics"}]},{"given":"Chao-Ming","family":"Jian","sequence":"additional","affiliation":[{"name":"Cornell University"}]},{"given":"Wilbur","family":"Shirley","sequence":"additional","affiliation":[{"name":"California Institute of Technology"},{"name":"Institute for Advanced Study"}]},{"given":"Cenke","family":"Xu","sequence":"additional","affiliation":[{"name":"University of California, Santa Barbara"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2023,7,6]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.15.1.001","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"001","note":"arxivid:2112.02137"},{"id":"arxiv:0801.0229","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevb.79.024426","source":"Crossref","title":"Exactly solvable Kitaev model in three dimensions","volume":"79","author":[{"given":"Saptarshi","family":"Mandal","sequence":"first","affiliation":[]},{"given":"Naveen","family":"Surendran","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,1,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.79.024426","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"024426","note":"arxivid:0801.0229"},{"id":"arxiv:2307.10353","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We propose a new model of quantum computation comprised of low-weight measurement sequences that simultaneously encode logical information, enable error correction, and apply logical gates. These measurement sequences constitute a new class of quantum error-correcting codes generalizing Floquet codes, which we call dynamic automorphism (DA) codes. We construct an explicit example, the DA color code, which is assembled from short measurement sequences that can realize all 72 automorphisms of the 2D color code. On a stack of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math> triangular patches, the DA color code encodes <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math> logical qubits and can implement the full logical Clifford group by a sequence of two- and, more rarely, three-qubit Pauli measurements. We also make the first step towards universal quantum computation with DA codes by introducing a 3D DA color code and showing that a non-Clifford logical gate can be realized by adaptive two-qubit measurements.</jats:p>","DOI":"10.22331/q-2024-08-27-1448","page":"1448","source":"Crossref","title":"Quantum computation from dynamic automorphism codes","volume":"8","author":[{"given":"Margarita","family":"Davydova","sequence":"first","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"},{"name":"Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA"}]},{"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA"},{"name":"Microsoft Quantum, Station Q, Santa Barbara, California, USA"}]},{"given":"Shankar","family":"Balasubramanian","sequence":"additional","affiliation":[{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"}]},{"given":"David","family":"Aasen","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Station Q, Santa Barbara, California, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,27]]},"URL":"http://dx.doi.org/10.22331/q-2024-08-27-1448","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2307.10353"},{"id":"arxiv:1912.05565","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.101.155124","source":"Crossref","title":"Exactly solvable model for a \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mi mathvariant=\"normal\">D</mml:mi></mml:math>\n beyond-cohomology symmetry-protected topological phase","volume":"101","author":[{"given":"Lukasz","family":"Fidkowski","sequence":"first","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2020,4,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.101.155124","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155124","note":"update-to:{\"updated\":{\"date-parts\":[[2022,2,22]],\"date-time\":\"2022-02-22T00:00:00Z\",\"timestamp\":1645488000000},\"DOI\":\"10.1103/physrevb.101.155124\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2022,2,22]],\"date-time\":\"2022-02-22T00:00:00Z\",\"timestamp\":1645488000000},\"DOI\":\"10.1103/physrevb.101.155124\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:1912.05565"},{"id":"arxiv:quant-ph/9608049","type":"article-journal","author":[{"given":"E.","family":"Knill"}],"title":"Group Representations, Error Bases and Quantum Codes","issued":{"date-parts":[[1996,8,29]]},"note":"arxivid:quant-ph/9608049\narxiv_version_number:1"},{"id":"arxiv:2304.08611","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.108.022424","source":"Crossref","title":"Multispin Clifford codes for angular momentum errors in spin systems","volume":"108","author":[{"ORCID":"https://orcid.org/0000-0002-6229-7087","authenticated-orcid":true,"given":"Sivaprasad","family":"Omanakuttan","sequence":"first","affiliation":[{"name":"Center for Quantum Information and Control, Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA"}]},{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":true,"given":"Jonathan A.","family":"Gross","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Venice, California 90291, USA"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.108.022424","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022424","note":"arxivid:2304.08611"},{"id":"arxiv:2312.00162","type":"article-journal","author":[{"given":"Xzavier","family":"Herbert"},{"given":"Jonathan","family":"Gross"},{"given":"Michael","family":"Newman"}],"title":"Qutrit codes within representations of SU(3)","issued":{"date-parts":[[2023,11,30]]},"note":"arxivid:2312.00162\narxiv_version_number:1"},{"id":"arxiv:2506.01843","type":"article-journal","author":[{"given":"Jonas","family":"Eidesen"}],"title":"Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes","issued":{"date-parts":[[2026,2,25]]},"note":"arxivid:2506.01843\narxiv_version_number:3"},{"id":"arxiv:quant-ph/9608048","type":"article-journal","author":[{"given":"E.","family":"Knill"}],"title":"Non-binary Unitary Error Bases and Quantum Codes","issued":{"date-parts":[[1996,10,8]]},"note":"arxivid:quant-ph/9608048\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0010082","type":"article-journal","author":[{"given":"Andreas","family":"Klappenecker"},{"given":"Martin","family":"Roetteler"}],"title":"Beyond Stabilizer Codes I: Nice Error Bases","issued":{"date-parts":[[2001,11,7]]},"note":"arxivid:quant-ph/0010082\narxiv_version_number:2"},{"id":"doi:10.1201/9781420035377-11","type":"book-chapter","publisher":"Chapman and Hall/CRC","DOI":"10.1201/9781420035377-11","page":"269-292","source":"Crossref","title":"Clifford codes","container-title":"Mathematics of Quantum Computation","original-title":[],"language":"en","issued":{"date-parts":[[2002,2,14]]},"ISBN":["9780429122798"],"URL":"http://dx.doi.org/10.1201/9781420035377-11","note":"edition-number:0"},{"id":"arxiv:0902.0383","type":"article-journal","author":[{"given":"Yong","family":"Zhang"}],"title":"Abstract Error Groups Via Jones Unitary Braid Group Representations at q=i","issued":{"date-parts":[[2009,2,2]]},"note":"arxivid:0902.0383\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0010076","type":"article-journal","author":[{"given":"Andreas","family":"Klappenecker"},{"given":"Martin","family":"Roetteler"}],"title":"Beyond Stabilizer Codes II: Clifford Codes","issued":{"date-parts":[[2001,11,7]]},"note":"arxivid:quant-ph/0010076\narxiv_version_number:2"},{"id":"manual:-A.-Klappenecker-and-M.-R-tte","type":"article-journal","author":[{"family":"Klappenecker","given":"A."},{"family":"Rötteler","given":"M."}],"title":"On the structure of nonstabilizer Clifford codes","volume":"4","container-title":"Quantum Information & Computation","issue":"2","issued":"2004","page":"152–160"},{"id":"arxiv:quant-ph/0402060","type":"article-journal","author":[{"given":"Hagiwara","family":"Manabu"},{"given":"Hideki","family":"Imai"}],"title":"Non stabilizer Clifford codes with qupits","issued":{"date-parts":[[2004,2,26]]},"note":"arxivid:quant-ph/0402060\narxiv_version_number:2"},{"id":"arxiv:1410.5487","type":"journal-article","publisher":"American Physical Society (APS)","issue":"26","DOI":"10.1103/physrevlett.113.260504","source":"Crossref","title":"Quantum Error Suppression with Commuting Hamiltonians: Two Local is Too Local","volume":"113","author":[{"given":"Iman","family":"Marvian","sequence":"first","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.113.260504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"260504","note":"arxivid:1410.5487"},{"id":"arxiv:2412.07764","type":"article-journal","author":[{"given":"Yingkang","family":"Cao"},{"given":"Suying","family":"Liu"},{"given":"Haowei","family":"Deng"},{"given":"Zihan","family":"Xia"},{"given":"Xiaodi","family":"Wu"},{"given":"Yu-Xin","family":"Wang"}],"title":"Robust analog quantum simulators by quantum error-detecting codes","issued":{"date-parts":[[2024,12,10]]},"note":"arxivid:2412.07764\narxiv_version_number:1"},{"id":"arxiv:1511.01997","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/s11128-017-1527-9","source":"Crossref","title":"Non-commuting two-local Hamiltonians for quantum error suppression","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0003-0435-655X","authenticated-orcid":false,"given":"Zhang","family":"Jiang","sequence":"first","affiliation":[]},{"given":"Eleanor G.","family":"Rieffel","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2017,2,18]]},"URL":"http://dx.doi.org/10.1007/s11128-017-1527-9","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","page":"89","note":"alternative-id:1527\narxivid:1511.01997"},{"id":"arxiv:1606.03795","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.118.030504","source":"Crossref","title":"Error Suppression for Hamiltonian-Based Quantum Computation Using Subsystem Codes","volume":"118","author":[{"given":"Milad","family":"Marvian","sequence":"first","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,1,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.118.030504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030504","note":"arxivid:1606.03795"},{"id":"arxiv:2504.19966","type":"article-journal","author":[{"given":"Natalie","family":"Parham"}],"title":"Quantum circuit lower bounds in the magic hierarchy","issued":{"date-parts":[[2025,8,28]]},"note":"arxivid:2504.19966\narxiv_version_number:3"},{"id":"doi:10.1007/978-3-662-06390-3_18","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-662-06390-3_18","page":"249-252","source":"Crossref","title":"Rigorous Results on Valence-Bond Ground States in Antiferromagnets","author":[{"given":"Ian","family":"Affleck","sequence":"first","affiliation":[]},{"given":"Tom","family":"Kennedy","sequence":"additional","affiliation":[]},{"given":"Elliott H.","family":"Lieb","sequence":"additional","affiliation":[]},{"given":"Hal","family":"Tasaki","sequence":"additional","affiliation":[]}],"container-title":"Condensed Matter Physics and Exactly Soluble Models","original-title":[],"language":"en","issued":{"date-parts":[[2004]]},"ISBN":["9783642060939","9783662063903"],"URL":"http://dx.doi.org/10.1007/978-3-662-06390-3_18","note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:quant-ph/9603022","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.55.67","page":"67-71","source":"Crossref","title":"Quantum error correction in the presence of spontaneous emission","volume":"55","author":[{"given":"M. B.","family":"Plenio","sequence":"first","affiliation":[]},{"given":"V.","family":"Vedral","sequence":"additional","affiliation":[]},{"given":"P. L.","family":"Knight","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,1,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.55.67","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9603022"},{"id":"arxiv:quant-ph/9705044","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.79.3306","page":"3306-3309","source":"Crossref","title":"Noiseless Quantum Codes","volume":"79","author":[{"given":"P.","family":"Zanardi","sequence":"first","affiliation":[]},{"given":"M.","family":"Rasetti","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1997,10,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.79.3306","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9705044"},{"id":"arxiv:quant-ph/9809081","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevlett.82.4556","page":"4556-4559","source":"Crossref","title":"Concatenating Decoherence-Free Subspaces with Quantum Error Correcting Codes","volume":"82","author":[{"given":"D. A.","family":"Lidar","sequence":"first","affiliation":[]},{"given":"D.","family":"Bacon","sequence":"additional","affiliation":[]},{"given":"K. B.","family":"Whaley","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1999,5,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.82.4556","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9809081"},{"id":"doi:10.1109/ISIT45174.2021.9518206","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit45174.2021.9518206","page":"1481-1486","source":"Crossref","title":"CSS Codes that are Oblivious to Coherent Noise","author":[{"given":"Jingzhen","family":"Hu","sequence":"first","affiliation":[{"name":"Duke University,Department of Mathematics,Durham,NC,USA,27708"}]},{"given":"Qingzhong","family":"Liang","sequence":"additional","affiliation":[{"name":"Duke University,Department of Mathematics,Durham,NC,USA,27708"}]},{"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"University of Arizona,Department of Electrical and Computer Engineering,Tucson,AZ,USA,85719"}]},{"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[{"name":"Duke University,Department of Mathematics,Durham,NC,USA,27708"}]}],"event":"2021 IEEE International Symposium on Information Theory (ISIT)","container-title":"2021 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2021,7,12]]},"URL":"http://dx.doi.org/10.1109/ISIT45174.2021.9518206"},{"id":"arxiv:2203.13924","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The field of quantum communications promises the faithful distribution of quantum information, quantum entanglement, and absolutely secret keys, however, the highest rates of these tasks are fundamentally limited by the transmission distance between quantum repeaters. The ultimate end-to-end rates of quantum communication networks are known to be achievable by an optimal entanglement distillation protocol followed by teleportation. In this work, we give a practical design for this achievability. Our ultimate design is an iterative approach, where each purification step operates on shared entangled states and detects loss errors at the highest rates allowed by physics. As a simpler design, we show that the first round of iterations can purify completely at high rates. We propose an experimental implementation using linear optics and photon-number measurements which is robust to inefficient operations and measurements, showcasing its near-term potential for real-world practical applications.</jats:p>","DOI":"10.1038/s41534-022-00641-0","source":"Crossref","title":"Achieving the ultimate end-to-end rates of lossy quantum communication networks","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0003-3457-4451","authenticated-orcid":false,"given":"Matthew S.","family":"Winnel","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9990-6341","authenticated-orcid":false,"given":"Joshua J.","family":"Guanzon","sequence":"additional","affiliation":[]},{"given":"Nedasadat","family":"Hosseinidehaj","sequence":"additional","affiliation":[]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2022,11,8]]},"URL":"http://dx.doi.org/10.1038/s41534-022-00641-0","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"129","note":"alternative-id:641\narxivid:2203.13924"},{"id":"arxiv:quant-ph/9907096","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.61.052307","source":"Crossref","title":"Protecting quantum information encoded in decoherence-free states against exchange errors","volume":"61","author":[{"given":"Daniel A.","family":"Lidar","sequence":"first","affiliation":[]},{"given":"David","family":"Bacon","sequence":"additional","affiliation":[]},{"given":"Julia","family":"Kempe","sequence":"additional","affiliation":[]},{"given":"K.","family":"Birgitta Whaley","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,4,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.61.052307","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052307","note":"arxivid:quant-ph/9907096"},{"id":"arxiv:2411.08955","type":"article-journal","author":[{"given":"Alexander","family":"Schuckert"},{"given":"Eleanor","family":"Crane"},{"given":"Alexey V.","family":"Gorshkov"},{"given":"Mohammad","family":"Hafezi"},{"given":"Michael J.","family":"Gullans"}],"title":"Fault-tolerant fermionic quantum computing","issued":{"date-parts":[[2025,7,16]]},"note":"arxivid:2411.08955\narxiv_version_number:2"},{"id":"arxiv:0809.0613","type":"article-journal","author":[{"given":"Francesco","family":"Ticozzi"},{"given":"Lorenza","family":"Viola"}],"title":"Analysis and synthesis of attractive quantum Markovian dynamics","issued":{"date-parts":[[2008,9,3]]},"note":"arxivid:0809.0613\narxiv_version_number:1"},{"id":"arxiv:1112.4860","type":"journal-article","publisher":"The Royal Society","issue":"1979","abstract":"<jats:p>\n            We provide a solution to the problem of determining whether a target pure state can be asymptotically prepared using dissipative Markovian dynamics under\n            <jats:italic>fixed</jats:italic>\n            locality constraints. Besides recovering existing results for a large class of physically relevant entangled states, our approach has the advantage of providing an explicit stabilization test solely based on the input state and constraints of the problem. Connections with the formalism of frustration-free parent Hamiltonians are discussed, as well as control implementations in terms of a switching output-feedback law.\n          </jats:p>","DOI":"10.1098/rsta.2011.0485","page":"5259-5269","source":"Crossref","title":"Stabilizing entangled states with quasi-local quantum dynamical semigroups","volume":"370","author":[{"given":"Francesco","family":"Ticozzi","sequence":"first","affiliation":[{"name":"Dipartimento di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6/B, 35131 Padova, Italy"},{"name":"Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA"}]},{"given":"Lorenza","family":"Viola","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA"}]}],"container-title":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2012,11,28]]},"URL":"http://dx.doi.org/10.1098/rsta.2011.0485","ISSN":["1364-503X","1471-2962"],"container-title-short":"Phil. Trans. R. Soc. A.","note":"alternative-id:10.1098/rsta.2011.0485\narxivid:1112.4860"},{"id":"arxiv:0803.1447","type":"article-journal","author":[{"given":"Frank","family":"Verstraete"},{"given":"Michael M.","family":"Wolf"},{"given":"J. Ignacio","family":"Cirac"}],"title":"Quantum computation, quantum state engineering, and quantum phase transitions driven by dissipation","issued":{"date-parts":[[2008,3,12]]},"note":"arxivid:0803.1447\narxiv_version_number:2"},{"id":"arxiv:1802.00010","type":"article-journal","author":[{"given":"Victor V.","family":"Albert"}],"title":"Lindbladians with multiple steady states: theory and applications","issued":{"date-parts":[[2018,1,31]]},"note":"arxivid:1802.00010\narxiv_version_number:1"},{"id":"arxiv:1904.01458","type":"journal-article","publisher":"American Physical Society (APS)","issue":"14","DOI":"10.1103/physrevb.103.144417","source":"Crossref","title":"Quantum storage in quantum ferromagnets","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0003-1115-0074","authenticated-orcid":true,"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield, United Kingdom; Singapore University of Technology and Design, 8 Somapah Road, Singapore; and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2021,4,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.103.144417","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"144417","note":"arxivid:1904.01458"},{"id":"doi:10.1063/PT.3.1641","type":"journal-article","publisher":"AIP Publishing","issue":"7","abstract":"<jats:p>Phases of matter distinguished by topological properties sometimes possess quasiparticles with bizarre statistics. Those that do may see applications in quantum information processing.</jats:p>","DOI":"10.1063/pt.3.1641","page":"38-43","source":"Crossref","title":"Topological phases and quasiparticle braiding","volume":"65","author":[{"given":"Nicholas","family":"Read","sequence":"first","affiliation":[]}],"container-title":"Physics Today","original-title":[],"language":"en","issued":{"date-parts":[[2012,7,1]]},"URL":"http://dx.doi.org/10.1063/PT.3.1641","ISSN":["0031-9228","1945-0699"]},{"id":"arxiv:cond-mat/0503554","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.72.045141","source":"Crossref","title":"Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance","volume":"72","author":[{"given":"M. B.","family":"Hastings","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2005,7,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.72.045141","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"045141","note":"arxivid:cond-mat/0503554"},{"id":"arxiv:quant-ph/0601019","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.75.032321","source":"Crossref","title":"Simulating adiabatic evolution of gapped spin systems","volume":"75","author":[{"given":"Tobias J.","family":"Osborne","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,3,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.032321","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032321","note":"arxivid:quant-ph/0601019"},{"id":"arxiv:2205.10460","type":"article-journal","author":[{"given":"Bruno","family":"Nachtergaele"}],"title":"From Lieb-Robinson Bounds to Automorphic Equivalence","issued":{"date-parts":[[2022,5,20]]},"note":"arxivid:2205.10460\narxiv_version_number:1"},{"id":"arxiv:1004.3835","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.82.155138","source":"Crossref","title":"Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order","volume":"82","author":[{"given":"Xie","family":"Chen","sequence":"first","affiliation":[]},{"given":"Zheng-Cheng","family":"Gu","sequence":"additional","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2010,10,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.82.155138","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"155138","note":"arxivid:1004.3835"},{"id":"doi:10.1126/science.273.5278.1073","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"5278","abstract":"<jats:p>Feynman's 1982 conjecture, that quantum computers can be programmed to simulate any local quantum system, is shown to be correct.</jats:p>","DOI":"10.1126/science.273.5278.1073","page":"1073-1078","source":"Crossref","title":"Universal Quantum Simulators","volume":"273","author":[{"given":"Seth","family":"Lloyd","sequence":"first","affiliation":[{"name":"The author is at the D'Arbeloff Laboratory for Information Systems and Technology, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[1996,8,23]]},"URL":"http://dx.doi.org/10.1126/science.273.5278.1073","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.273.5278.1073"},{"id":"arxiv:2211.02086","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-023-04806-6","page":"661-698","source":"Crossref","title":"Invertible Subalgebras","volume":"403","author":[{"ORCID":"https://orcid.org/0000-0002-1087-6853","authenticated-orcid":false,"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,29]]},"URL":"http://dx.doi.org/10.1007/s00220-023-04806-6","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:4806\narxivid:2211.02086"},{"id":"arxiv:2510.08503","type":"article-journal","author":[{"given":"Thomas","family":"Schuster"},{"given":"Dominik","family":"Kufel"},{"given":"Norman Y.","family":"Yao"},{"given":"Hsin-Yuan","family":"Huang"}],"title":"Hardness of recognizing phases of matter","issued":{"date-parts":[[2026,3,17]]},"note":"arxivid:2510.08503\narxiv_version_number:2"},{"id":"manual:-A.-Kitaev-Toward-Topological","type":"document","author":[{"family":"Kitaev","given":"A."}],"title":"Toward Topological Classification of Phases with Short-range Entanglement, talk at KITP","issued":"2011"},{"id":"manual:-A.-Kitaev-On-the-Classificat","type":"document","author":[{"family":"Kitaev","given":"A."}],"title":"On the Classification of Short-Range Entangled States, talk at Simons Center","issued":"2013"},{"id":"arxiv:1406.7278","type":"article-journal","author":[{"given":"Daniel S.","family":"Freed"}],"title":"Short-range entanglement and invertible field theories","issued":{"date-parts":[[2014,8,10]]},"note":"arxivid:1406.7278\narxiv_version_number:2"},{"id":"doi:10.1007/978-3-662-02520-8","type":"book","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-662-02520-8","source":"Crossref","title":"Operator Algebras and Quantum Statistical Mechanics 1","author":[{"given":"Ola","family":"Bratteli","sequence":"first","affiliation":[]},{"given":"Derek W.","family":"Robinson","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[1987]]},"ISBN":["9783642057366","9783662025208"],"URL":"http://dx.doi.org/10.1007/978-3-662-02520-8","note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:1311.2717","type":"book","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-51458-1","source":"Crossref","title":"Quantum Spin Systems on Infinite Lattices","author":[{"given":"Pieter","family":"Naaijkens","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017]]},"ISBN":["9783319514567","9783319514581"],"URL":"http://dx.doi.org/10.1007/978-3-319-51458-1","ISSN":["0075-8450","1616-6361"],"note":"publisher-location:Cham\narxivid:1311.2717"},{"id":"arxiv:1203.4565","type":"article-journal","author":[{"given":"Subir","family":"Sachdev"}],"title":"The quantum phases of matter","issued":{"date-parts":[[2012,5,22]]},"note":"arxivid:1203.4565\narxiv_version_number:4"},{"id":"doi:10.1017/9781009212717","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>This modern text describes the remarkable developments in quantum condensed matter physics following the experimental discoveries of quantum Hall effects and high temperature superconductivity in the 1980s. After a review of the phases of matter amenable to an independent particle description, entangled phases of matter are described in an accessible and unified manner. The concepts of fractionalization and emergent gauge fields are introduced using the simplest resonating valence bond insulator with an energy gap, the Z2 spin liquid. Concepts in band topology and the parton method are then combined to obtain a large variety of experimentally relevant gapped states. Correlated metallic states are described, beginning with a discussion of the Kondo effect on magnetic impurities in metals. Metals without quasiparticle excitations are introduced using the Sachdev-Ye-Kitaev model, followed by a discussion of critical Fermi surfaces and strange metals. Numerous end-of-chapter problems expand readers' comprehension and reinforce key concepts.</jats:p>","DOI":"10.1017/9781009212717","source":"Crossref","title":"Quantum Phases of Matter","author":[{"given":"Subir","family":"Sachdev","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2023,3,24]]},"ISBN":["9781009212717","9781009212694"],"URL":"http://dx.doi.org/10.1017/9781009212717","note":"edition-number:1"},{"id":"doi:10.1007/978-3-030-41265-4","type":"book","publisher":"Springer International Publishing","DOI":"10.1007/978-3-030-41265-4","source":"Crossref","title":"Physics and Mathematics of Quantum Many-Body Systems","author":[{"given":"Hal","family":"Tasaki","sequence":"first","affiliation":[]}],"container-title":"Graduate Texts in Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020]]},"ISBN":["9783030412647","9783030412654"],"URL":"http://dx.doi.org/10.1007/978-3-030-41265-4","ISSN":["1868-4513","1868-4521"],"note":"publisher-location:Cham"},{"id":"doi:10.1017/CBO9781139020916","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>A modern, graduate-level introduction to many-body physics in condensed matter, this textbook explains the tools and concepts needed for a research-level understanding of the correlated behavior of quantum fluids. Starting with an operator-based introduction to the quantum field theory of many-body physics, this textbook presents the Feynman diagram approach, Green's functions and finite-temperature many-body physics before developing the path integral approach to interacting systems. Special chapters are devoted to the concepts of Fermi liquid theory, broken symmetry, conduction in disordered systems, superconductivity and the physics of local-moment metals. A strong emphasis on concepts and numerous exercises make this an invaluable course book for graduate students in condensed matter physics. It will also interest students in nuclear, atomic and particle physics.</jats:p>","DOI":"10.1017/cbo9781139020916","source":"Crossref","title":"Introduction to Many-Body Physics","author":[{"given":"Piers","family":"Coleman","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2015,11,26]]},"ISBN":["9780521864886","9781139020916"],"URL":"http://dx.doi.org/10.1017/CBO9781139020916","note":"edition-number:1"},{"id":"doi:10.1017/9781316480649","type":"monograph","publisher":"Cambridge University Press","DOI":"10.1017/9781316480649","source":"Crossref","title":"Modern Condensed Matter Physics","author":[{"given":"Steven M.","family":"Girvin","sequence":"first","affiliation":[]},{"given":"Kun","family":"Yang","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2019,2,26]]},"ISBN":["9781316480649","9781107137394"],"URL":"http://dx.doi.org/10.1017/9781316480649","note":"edition-number:1"},{"id":"doi:10.1017/CBO9780511819681","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. A unique account of the subject, this book provides a complete, self-contained description of research on the stability of matter problem. It introduces the necessary quantum mechanics to mathematicians, and aspects of functional analysis to physicists. The topics covered include electrodynamics of classical and quantized fields, Lieb-Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, gravitational stability of stars, basics of equilibrium statistical mechanics, and the existence of the thermodynamic limit. The book is an up-to-date account for researchers, and its pedagogical style makes it suitable for advanced undergraduate and graduate courses in mathematical physics.</jats:p>","DOI":"10.1017/cbo9780511819681","source":"Crossref","title":"The Stability of Matter in Quantum Mechanics","author":[{"given":"Elliott H.","family":"Lieb","sequence":"first","affiliation":[]},{"given":"Robert","family":"Seiringer","sequence":"additional","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2009,11,5]]},"ISBN":["9780521191180","9780511819681"],"URL":"http://dx.doi.org/10.1017/CBO9780511819681","note":"edition-number:1"},{"id":"arxiv:2003.05465","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Exactly solvable models are essential in physics. For many-body spin-<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo mathvariant=\"sans-serif\">/</mml:mo></mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn mathvariant=\"sans-serif\">2</mml:mn></mml:mrow></mml:mrow></mml:math>systems, an important class of such models consists of those that can be mapped to free fermions hopping on a graph. We provide a complete characterization of models which can be solved this way. Specifically, we reduce the problem of recognizing such spin models to the graph-theoretic problem of recognizing line graphs, which has been solved optimally. A corollary of our result is a complete set of constant-sized commutation structures that constitute the obstructions to a free-fermion solution. We find that symmetries are tightly constrained in these models. Pauli symmetries correspond to either: (i) cycles on the fermion hopping graph, (ii) the fermion parity operator, or (iii) logically encoded qubits. Clifford symmetries within one of these symmetry sectors, with three exceptions, must be symmetries of the free-fermion model itself. We demonstrate how several exact free-fermion solutions from the literature fit into our formalism and give an explicit example of a new model previously unknown to be solvable by free fermions.</jats:p>","DOI":"10.22331/q-2020-06-04-278","page":"278","source":"Crossref","title":"Characterization of solvable spin models via graph invariants","volume":"4","author":[{"given":"Adrian","family":"Chapman","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, Australia"}]},{"ORCID":"https://orcid.org/0000-0002-3975-0226","authenticated-orcid":false,"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, Australia"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,4]]},"URL":"http://dx.doi.org/10.22331/q-2020-06-04-278","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2003.05465"},{"id":"arxiv:2012.07857","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-021-04220-w","page":"969-1003","source":"Crossref","title":"Free Fermions Behind the Disguise","volume":"388","author":[{"ORCID":"https://orcid.org/0000-0003-2140-4787","authenticated-orcid":false,"given":"Samuel J.","family":"Elman","sequence":"first","affiliation":[]},{"given":"Adrian","family":"Chapman","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,10,29]]},"URL":"http://dx.doi.org/10.1007/s00220-021-04220-w","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:4220\narxivid:2012.07857"},{"id":"arxiv:2505.02683","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/6nn5-vhj7","source":"Crossref","title":"Partons from stabilizer codes","volume":"112","author":[{"ORCID":"https://orcid.org/0000-0001-5494-7732","authenticated-orcid":true,"given":"Rafael A.","family":"Macêdo","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04wn09761","id-type":"ROR","asserted-by":"publisher"}],"name":"Universidade Federal do Rio Grande do Norte"},{"id":[{"id":"https://ror.org/04wn09761","id-type":"ROR","asserted-by":"publisher"}],"name":"Universidade Federal do Rio Grande do Norte"}]},{"given":"Carlo C.","family":"Bellinati","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/036rp1748","id-type":"ROR","asserted-by":"publisher"}],"name":"Instituto de Física, Universidade de São Paulo"}]},{"given":"Weslei B.","family":"Fontana","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00zdnkx70","id-type":"ROR","asserted-by":"publisher"}],"name":"National Tsing Hua University"}]},{"ORCID":"https://orcid.org/0000-0003-2016-8305","authenticated-orcid":true,"given":"Eric C.","family":"Andrade","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/036rp1748","id-type":"ROR","asserted-by":"publisher"}],"name":"Instituto de Física, Universidade de São Paulo"}]},{"ORCID":"https://orcid.org/0000-0003-2767-8535","authenticated-orcid":true,"given":"Rodrigo G.","family":"Pereira","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04wn09761","id-type":"ROR","asserted-by":"publisher"}],"name":"Universidade Federal do Rio Grande do Norte"},{"id":[{"id":"https://ror.org/04wn09761","id-type":"ROR","asserted-by":"publisher"}],"name":"Universidade Federal do Rio Grande do Norte"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,2]]},"URL":"http://dx.doi.org/10.1103/6nn5-vhj7","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"125101","note":"arxivid:2505.02683"},{"id":"arxiv:0806.2160","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.78.042324","source":"Crossref","title":"Stability of quantum concatenated-code Hamiltonians","volume":"78","author":[{"given":"Dave","family":"Bacon","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,10,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.042324","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042324","note":"arxivid:0806.2160"},{"id":"arxiv:0906.4127","type":"journal-article","publisher":"IOP Publishing","issue":"2","DOI":"10.1088/1367-2630/12/2/025018","page":"025018","source":"Crossref","title":"Topological color codes and two-body quantum lattice Hamiltonians","volume":"12","author":[{"given":"M","family":"Kargarian","sequence":"first","affiliation":[]},{"given":"H","family":"Bombin","sequence":"additional","affiliation":[]},{"given":"M A","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2010,2,26]]},"URL":"http://dx.doi.org/10.1088/1367-2630/12/2/025018","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:0906.4127"},{"id":"arxiv:1111.5817","type":"journal-article","publisher":"American Physical Society (APS)","issue":"26","DOI":"10.1103/physrevlett.109.260401","source":"Crossref","title":"Gapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism","volume":"109","author":[{"given":"Carlos","family":"Fernández-González","sequence":"first","affiliation":[]},{"given":"Norbert","family":"Schuch","sequence":"additional","affiliation":[]},{"given":"Michael M.","family":"Wolf","sequence":"additional","affiliation":[]},{"given":"J. Ignacio","family":"Cirac","sequence":"additional","affiliation":[]},{"given":"David","family":"Pérez-García","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,12,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.109.260401","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"260401","note":"arxivid:1111.5817"},{"id":"arxiv:2012.15346","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevb.104.235151","source":"Crossref","title":"On the stability of topological order in tensor network states","volume":"104","author":[{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":true,"given":"Dominic J.","family":"Williamson","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2337-5772","authenticated-orcid":true,"given":"Clement","family":"Delcamp","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Verstraete","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6494-8616","authenticated-orcid":true,"given":"Norbert","family":"Schuch","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.104.235151","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"235151","note":"arxivid:2012.15346"},{"id":"doi:10.1070/PU1982v025n04ABEH004537","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1070/pu1982v025n04abeh004537","page":"231-256","source":"Crossref","title":"The Jahn-Teller effect and magnetism: transition metal compounds","volume":"25","author":[{"given":"Kliment I","family":"Kugel'","sequence":"first","affiliation":[]},{"given":"D I","family":"Khomskiĭ","sequence":"additional","affiliation":[]}],"container-title":"Soviet Physics Uspekhi","original-title":[],"issued":{"date-parts":[[1982,4,30]]},"URL":"http://dx.doi.org/10.1070/PU1982v025n04ABEH004537","ISSN":["0038-5670"],"container-title-short":"Sov. Phys. Usp."},{"id":"arxiv:cond-mat/0501708","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevb.72.024448","source":"Crossref","title":"Quantum compass model on the square lattice","volume":"72","author":[{"given":"Julien","family":"Dorier","sequence":"first","affiliation":[]},{"given":"Federico","family":"Becca","sequence":"additional","affiliation":[]},{"given":"Frédéric","family":"Mila","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2005,7,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.72.024448","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"024448","note":"arxivid:cond-mat/0501708"},{"id":"arxiv:1303.5922","type":"article-journal","author":[{"given":"Zohar","family":"Nussinov"},{"given":"Jeroen van den","family":"Brink"}],"title":"Compass and Kitaev models -- Theory and Physical Motivations","issued":{"date-parts":[[2013,3,24]]},"note":"arxivid:1303.5922\narxiv_version_number:1"},{"id":"arxiv:1310.5664","type":"article-journal","author":[{"given":"Dorit","family":"Aharonov"},{"given":"Lior","family":"Eldar"}],"title":"Quantum Locally Testable Codes","issued":{"date-parts":[[2013,10,21]]},"note":"arxivid:1310.5664\narxiv_version_number:1"},{"id":"arxiv:1611.03790","type":"article-journal","author":[{"given":"M. B.","family":"Hastings"}],"title":"Weight Reduction for Quantum Codes","issued":{"date-parts":[[2016,11,11]]},"note":"arxivid:1611.03790\narxiv_version_number:1"},{"id":"arxiv:2004.07935","type":"article-journal","author":[{"given":"Shai","family":"Evra"},{"given":"Tali","family":"Kaufman"},{"given":"Gilles","family":"Zémor"}],"title":"Decodable quantum LDPC codes beyond the $\\sqrt{n}$ distance barrier using high dimensional expanders","issued":{"date-parts":[[2020,4,16]]},"note":"arxivid:2004.07935\narxiv_version_number:1"},{"id":"arxiv:2305.00689","type":"article-journal","author":[{"given":"Adam","family":"Wills"},{"given":"Ting-Chun","family":"Lin"},{"given":"Min-Hsiu","family":"Hsieh"}],"title":"General Distance Balancing for Quantum Locally Testable Codes","issued":{"date-parts":[[2023,5,1]]},"note":"arxivid:2305.00689\narxiv_version_number:1"},{"id":"arxiv:2309.05541","type":"article-journal","author":[{"given":"Adam","family":"Wills"},{"given":"Ting-Chun","family":"Lin"},{"given":"Min-Hsiu","family":"Hsieh"}],"title":"Tradeoff Constructions for Quantum Locally Testable Codes","issued":{"date-parts":[[2024,1,23]]},"note":"arxivid:2309.05541\narxiv_version_number:3"},{"id":"arxiv:2311.09503","type":"article-journal","author":[{"given":"Louis","family":"Golowich"},{"given":"Tali","family":"Kaufman"}],"title":"NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes","issued":{"date-parts":[[2023,11,16]]},"note":"arxivid:2311.09503\narxiv_version_number:1"},{"id":"arxiv:1309.7495","type":"article-journal","author":[{"given":"Dorit","family":"Aharonov"},{"given":"Itai","family":"Arad"},{"given":"Thomas","family":"Vidick"}],"title":"The Quantum PCP Conjecture","issued":{"date-parts":[[2013,9,28]]},"note":"arxivid:1309.7495\narxiv_version_number:1"},{"id":"arxiv:2402.07476","type":"article-journal","author":[{"given":"Irit","family":"Dinur"},{"given":"Ting-Chun","family":"Lin"},{"given":"Thomas","family":"Vidick"}],"title":"Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes","issued":{"date-parts":[[2025,9,4]]},"note":"arxivid:2402.07476\narxiv_version_number:3"},{"id":"arxiv:1501.04112","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.91.032303","source":"Crossref","title":"Perturbative instability of quantum memory based on effective long-range interactions","volume":"91","author":[{"given":"Olivier","family":"Landon-Cardinal","sequence":"first","affiliation":[]},{"given":"Beni","family":"Yoshida","sequence":"additional","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,3,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.032303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032303","note":"arxivid:1501.04112"},{"id":"arxiv:2601.20818","type":"article-journal","author":[{"given":"Gesa","family":"Dünnweber"},{"given":"Georgios","family":"Styliaris"},{"given":"Rahul","family":"Trivedi"}],"title":"Quantum Memory and Autonomous Computation in Two Dimensions","issued":{"date-parts":[[2026,1,28]]},"note":"arxivid:2601.20818\narxiv_version_number:1"},{"id":"arxiv:0811.0033","type":"article-journal","author":[{"given":"R.","family":"Alicki"},{"given":"M.","family":"Horodecki"},{"given":"P.","family":"Horodecki"},{"given":"R.","family":"Horodecki"}],"title":"On thermal stability of topological qubit in Kitaev's 4D model","issued":{"date-parts":[[2008,11,2]]},"note":"arxivid:0811.0033\narxiv_version_number:1"},{"id":"doi:10.1017/S0305004100019174","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"3","abstract":"<jats:p>Ising discussed the following model of a ferromagnetic body: Assume <jats:italic>N</jats:italic> elementary magnets of moment μ to be arranged in a regular lattice; each of them is supposed to have only two possible orientations, which we call positive and negative. Assume further that there is an interaction energy <jats:italic>U</jats:italic> for each pair of neighbouring magnets of opposite direction. Further, there is an external magnetic field of magnitude <jats:italic>H</jats:italic> such as to produce an additional energy of − μ<jats:italic>H</jats:italic> (+ μ<jats:italic>H</jats:italic>) for each magnet with positive (negative) direction.</jats:p>","DOI":"10.1017/s0305004100019174","page":"477-481","source":"Crossref","title":"On Ising's model of ferromagnetism","volume":"32","author":[{"given":"R.","family":"Peierls","sequence":"first","affiliation":[]}],"container-title":"Mathematical Proceedings of the Cambridge Philosophical Society","original-title":[],"language":"en","issued":{"date-parts":[[1936,10]]},"URL":"http://dx.doi.org/10.1017/S0305004100019174","ISSN":["0305-0041","1469-8064"],"container-title-short":"Math. Proc. Camb. Phil. Soc.","note":"alternative-id:S0305004100019174"},{"id":"arxiv:2510.08533","type":"article-journal","author":[{"given":"Thiago","family":"Bergamaschi"},{"given":"Chi-Fang","family":"Chen"}],"title":"Fast Mixing of Quantum Spin Chains at All Temperatures","issued":{"date-parts":[[2026,1,23]]},"note":"arxivid:2510.08533\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0702102","type":"journal-article","publisher":"IOP Publishing","issue":"24","DOI":"10.1088/1751-8113/40/24/012","page":"6451-6467","source":"Crossref","title":"A statistical mechanics view on Kitaev's proposal for quantum memories","volume":"40","author":[{"given":"R","family":"Alicki","sequence":"first","affiliation":[]},{"given":"M","family":"Fannes","sequence":"additional","affiliation":[]},{"given":"M","family":"Horodecki","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2007,5,30]]},"URL":"http://dx.doi.org/10.1088/1751-8113/40/24/012","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(07)43725-8\narxivid:quant-ph/0702102"},{"id":"arxiv:0709.2717","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevb.77.064302","source":"Crossref","title":"Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems","volume":"77","author":[{"given":"Zohar","family":"Nussinov","sequence":"first","affiliation":[]},{"given":"Gerardo","family":"Ortiz","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2008,2,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.77.064302","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"064302","note":"arxivid:0709.2717"},{"id":"arxiv:0810.4584","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1751-8113/42/6/065303","page":"065303","source":"Crossref","title":"On thermalization in Kitaev's 2D model","volume":"42","author":[{"given":"R","family":"Alicki","sequence":"first","affiliation":[]},{"given":"M","family":"Fannes","sequence":"additional","affiliation":[]},{"given":"M","family":"Horodecki","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2009,1,14]]},"URL":"http://dx.doi.org/10.1088/1751-8113/42/6/065303","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(09)98076-3\narxivid:0810.4584"},{"id":"arxiv:2410.01206","type":"article-journal","author":[{"given":"Zhiyan","family":"Ding"},{"given":"Zeph","family":"Landau"},{"given":"Bowen","family":"Li"},{"given":"Lin","family":"Lin"},{"given":"Ruizhe","family":"Zhang"}],"title":"Polynomial-Time Preparation of Low-Temperature Gibbs States for 2D Toric Code","issued":{"date-parts":[[2025,5,1]]},"note":"arxivid:2410.01206\narxiv_version_number:2"},{"id":"arxiv:0907.2807","type":"journal-article","publisher":"IOP Publishing","issue":"2","DOI":"10.1088/1367-2630/12/2/025013","page":"025013","source":"Crossref","title":"Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes","volume":"12","author":[{"given":"Stefano","family":"Chesi","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]},{"given":"Sergey","family":"Bravyi","sequence":"additional","affiliation":[]},{"given":"Barbara M","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2010,2,26]]},"URL":"http://dx.doi.org/10.1088/1367-2630/12/2/025013","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:0907.2807"},{"id":"arxiv:1105.4159","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevlett.107.150504","source":"Crossref","title":"Energy Landscape of 3D Spin Hamiltonians with Topological Order","volume":"107","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,10,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.107.150504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"150504","note":"arxivid:1105.4159"},{"id":"arxiv:0904.4861","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.103.080501","source":"Crossref","title":"How Long Can a Quantum Memory Withstand Depolarizing Noise?","volume":"103","author":[{"given":"Fernando","family":"Pastawski","sequence":"first","affiliation":[]},{"given":"Alastair","family":"Kay","sequence":"additional","affiliation":[]},{"given":"Norbert","family":"Schuch","sequence":"additional","affiliation":[]},{"given":"Ignacio","family":"Cirac","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.103.080501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080501","note":"arxivid:0904.4861"},{"id":"arxiv:1210.3207","type":"journal-article","publisher":"Informa UK Limited","issue":"20","DOI":"10.1080/09500340.2012.737937","page":"1717-1738","source":"Crossref","title":"Quantum memories and error correction","volume":"59","author":[{"given":"James R.","family":"Wootton","sequence":"first","affiliation":[]}],"container-title":"Journal of Modern Optics","original-title":[],"language":"en","issued":{"date-parts":[[2012,11,20]]},"URL":"http://dx.doi.org/10.1080/09500340.2012.737937","ISSN":["0950-0340","1362-3044"],"container-title-short":"Journal of Modern Optics","note":"alternative-id:10.1080/09500340.2012.737937\narxivid:1210.3207"},{"id":"arxiv:0812.4622","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.79.245122","source":"Crossref","title":"Toric-boson model: Toward a topological quantum memory at finite temperature","volume":"79","author":[{"given":"Alioscia","family":"Hamma","sequence":"first","affiliation":[]},{"given":"Claudio","family":"Castelnovo","sequence":"additional","affiliation":[]},{"given":"Claudio","family":"Chamon","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,6,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.79.245122","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"245122","note":"arxivid:0812.4622"},{"id":"arxiv:0908.4264","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.82.022305","source":"Crossref","title":"Self-correcting quantum memory in a thermal environment","volume":"82","author":[{"given":"Stefano","family":"Chesi","sequence":"first","affiliation":[]},{"given":"Beat","family":"Röthlisberger","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,8,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.82.022305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022305","note":"arxivid:0908.4264"},{"id":"arxiv:1101.6028","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.107.030504","source":"Crossref","title":"Localization of Toric Code Defects","volume":"107","author":[{"given":"Cyril","family":"Stark","sequence":"first","affiliation":[]},{"given":"Lode","family":"Pollet","sequence":"additional","affiliation":[]},{"given":"Ataç","family":"Imamoğlu","sequence":"additional","affiliation":[]},{"given":"Renato","family":"Renner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,7,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.107.030504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030504","note":"arxivid:1101.6028"},{"id":"manual:-D.-Poulin-(2011-September-14","type":"document","author":[{"family":"Poulin","given":"D."}],"title":"Decoding problem for topological quantum codes","genre":"[Conference talk].","publisher":"Simons Center for Geometry and Physics, Stony Brook University","note":"Available at \\url{http://scgp.stonybrook.edu/archives/1087}","issued":"2011-09-14","URL":"http://scgp.stonybrook.edu/archives/1087"},{"id":"arxiv:1406.2338","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/npjqi.2015.10","source":"Crossref","title":"Cellular-automaton decoders for topological quantum memories","volume":"1","author":[{"given":"Michael","family":"Herold","sequence":"first","affiliation":[]},{"given":"Earl T","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]},{"given":"Michael J","family":"Kastoryano","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2015,10,27]]},"URL":"http://dx.doi.org/10.1038/npjqi.2015.10","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"15010","note":"alternative-id:BFnpjqi201510\narxivid:1406.2338"},{"id":"arxiv:1511.05579","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/aa7099","page":"063012","source":"Crossref","title":"Cellular automaton decoders of topological quantum memories in the fault tolerant setting","volume":"19","author":[{"given":"Michael","family":"Herold","sequence":"first","affiliation":[]},{"given":"Michael J","family":"Kastoryano","sequence":"additional","affiliation":[]},{"given":"Earl T","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2017,6,6]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aa7099","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1511.05579"},{"id":"arxiv:1512.04528","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevb.94.094303","source":"Crossref","title":"Exponential lifetime improvement in topological quantum memories","volume":"94","author":[{"given":"Charles-Edouard","family":"Bardyn","sequence":"first","affiliation":[]},{"given":"Torsten","family":"Karzig","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2016,9,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.94.094303","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"094303","note":"arxivid:1512.04528"},{"id":"arxiv:1406.4227","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.113.130501","source":"Crossref","title":"3D Topological Quantum Memory with a Power-Law Energy Barrier","volume":"113","author":[{"given":"Kamil P.","family":"Michnicki","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.113.130501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130501","note":"arxivid:1406.4227"},{"id":"arxiv:2305.06389","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Subsystem quantum error-correcting codes typically involve measuring a sequence of noncommuting parity check operators. They can sometimes exhibit greater fault tolerance than conventional  codes, which use commuting checks. However, unlike subspace codes, it is unclear if subsystem codes—in particular their advantages—can be understood in terms of ground-state properties of a physical Hamiltonian. In this paper, we address this question for the three-dimensional subsystem toric code (3D STC), as recently constructed by Kubica and Vasmer [], which exhibits single-shot error correction. Motivated by a conjectured relation between single-shot properties and thermal stability, we study the zero- and finite-temperature phases of an associated noncommuting Hamiltonian. By mapping the Hamiltonian model to a pair of 3D <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\"><a:msub><a:mi mathvariant=\"double-struck\">Z</a:mi><a:mn>2</a:mn></a:msub></a:math> gauge theories coupled by a kinetic constraint, we find various phases at zero temperature, all separated by first-order transitions: There are 3D toric code-like phases with deconfined point-like excitations in the bulk, and there are phases with a confined bulk supporting a 2D toric code on the surface when appropriate boundary conditions are chosen. The latter is similar to the surface topological order present in 3D STC. However, the similarities between the single-shot correction in 3D STC and the confined phases are only partial: they share the same sets of degrees of freedom, but they are governed by different dynamical rules. Instead, we argue that the process of single-shot error correction can more suitably be associated with a path (rather than a point) in the zero-temperature phase diagram, a perspective, which inspires alternative measurement sequences enabling single-shot error correction. Moreover, since none of the above-mentioned phases survives at nonzero temperature, the single-shot error-correction property of the code does not imply thermal stability of the associated Hamiltonian phase.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevresearch.6.043007","source":"Crossref","title":"Phase diagram of the three-dimensional subsystem toric code","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-3742-1944","authenticated-orcid":true,"given":"Yaodong","family":"Li","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00f54p054","id-type":"ROR","asserted-by":"publisher"}],"name":"Stanford University"},{"id":[{"id":"https://ror.org/02t274463","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California, Santa Barbara"}]},{"given":"C. W.","family":"von Keyserlingk","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0220mzb33","id-type":"ROR","asserted-by":"publisher"}],"name":"King's College London"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM T.J. Watson Research Center"}]},{"given":"Tomas","family":"Jochym-O'Connor","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM T.J. Watson Research Center"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.6.043007","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"043007","note":"arxivid:2305.06389"},{"id":"arxiv:0907.5228","type":"article-journal","author":[{"given":"H.","family":"Bombin"},{"given":"R. W.","family":"Chhajlany"},{"given":"M.","family":"Horodecki"},{"given":"M. A.","family":"Martin-Delgado"}],"title":"Self-Correcting Quantum Computers","issued":{"date-parts":[[2012,9,26]]},"note":"arxivid:0907.5228\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0506023","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.73.012340","source":"Crossref","title":"Operator quantum error-correcting subsystems for self-correcting quantum memories","volume":"73","author":[{"given":"Dave","family":"Bacon","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,1,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.012340","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012340","note":"arxivid:quant-ph/0506023"},{"id":"arxiv:2407.20526","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.134.180601","source":"Crossref","title":"Energy Barrier of Hypergraph Product Codes","volume":"134","author":[{"given":"Guangqi","family":"Zhao","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05rrcem69","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California, Davis"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.180601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180601","note":"arxivid:2407.20526"},{"id":"arxiv:2309.16503","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41467-024-53881-3","source":"Crossref","title":"Layer codes","volume":"15","author":[{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":false,"given":"Dominic J.","family":"Williamson","sequence":"first","affiliation":[]},{"given":"Nouédyn","family":"Baspin","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,4]]},"URL":"http://dx.doi.org/10.1038/s41467-024-53881-3","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"9528","note":"alternative-id:53881\narxivid:2309.16503"},{"id":"arxiv:2510.06659","type":"article-journal","author":[{"given":"Shouzhen","family":"Gu"},{"given":"Libor","family":"Caha"},{"given":"Shin Ho","family":"Choe"},{"given":"Zhiyang","family":"He"},{"given":"Aleksander","family":"Kubica"},{"given":"Eugene","family":"Tang"}],"title":"Layer codes as partially self-correcting quantum memories","issued":{"date-parts":[[2025,10,8]]},"note":"arxivid:2510.06659\narxiv_version_number:1"},{"id":"arxiv:2510.09218","type":"article-journal","author":[{"given":"Dominic J.","family":"Williamson"}],"title":"Partial Self-Correction in Layer Codes","issued":{"date-parts":[[2025,10,10]]},"note":"arxivid:2510.09218\narxiv_version_number:1"},{"id":"arxiv:cond-mat/0212497","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/0305-4470/36/6/305","page":"1593-1604","source":"Crossref","title":"Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result","volume":"36","author":[{"given":"A","family":"Vezzani","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[2003,1,29]]},"URL":"http://dx.doi.org/10.1088/0305-4470/36/6/305","ISSN":["0305-4470"],"container-title-short":"J. Phys. A: Math. Gen.","note":"alternative-id:S0305-4470(03)56197-2\narxivid:cond-mat/0212497"},{"id":"arxiv:1002.1227","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreve.81.021108","source":"Crossref","title":"Generalization of the Peierls-Griffiths theorem for the Ising model on graphs","volume":"81","author":[{"given":"Riccardo","family":"Campari","sequence":"first","affiliation":[]},{"given":"Davide","family":"Cassi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review E","original-title":[],"language":"en","issued":{"date-parts":[[2010,2,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevE.81.021108","ISSN":["1539-3755","1550-2376"],"container-title-short":"Phys. Rev. E","page":"021108","note":"arxivid:1002.1227"},{"id":"doi:10.1239/jap/1019737983","type":"journal-article","publisher":"Cambridge University Press (CUP)","issue":"1","abstract":"<jats:p>We study Bernoulli bond percolation on Sierpiński carpet lattices, which is a class of graphs corresponding to generalized Sierpiński carpets. In this paper we give a sufficient condition for the existence of a phase transition on the lattices. The proof is suitable for graphs which have self-similarity. We also discuss the relation between the existence of a phase transition and the isoperimetric dimension.</jats:p>","DOI":"10.1239/jap/1019737983","page":"1-10","source":"Crossref","title":"Existence of phase transition of percolation on Sierpiński carpet lattices","volume":"39","author":[{"given":"Masato","family":"Shinoda","sequence":"first","affiliation":[]}],"container-title":"Journal of Applied Probability","original-title":[],"language":"en","issued":{"date-parts":[[2002,3]]},"URL":"http://dx.doi.org/10.1239/jap/1019737983","ISSN":["0021-9002","1475-6072"],"container-title-short":"Journal of Applied Probability","note":"alternative-id:S002190020002146X"},{"id":"arxiv:2206.05294","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.4.030341","source":"Crossref","title":"Self-Correction from Higher-Form Symmetry Protection on a Boundary","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-9809-5575","authenticated-orcid":true,"given":"Charles","family":"Stahl","sequence":"first","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,9,29]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.030341","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030341","note":"arxivid:2206.05294"},{"id":"arxiv:quant-ph/9911074","type":"article-journal","author":[{"given":"Alexander Yu.","family":"Vlasov"}],"title":"Error Correction with Euclidean Qubits","issued":{"date-parts":[[1999,11,23]]},"note":"arxivid:quant-ph/9911074\narxiv_version_number:2"},{"id":"arxiv:2504.19833","type":"article-journal","author":[{"given":"Valentine","family":"Nyirahafashimana"},{"given":"Nurisya Mohd","family":"Shah"},{"given":"Umair Abdul","family":"Halim"},{"given":"Mohamed","family":"Othman"},{"given":"Sharifah Kartini Said","family":"Husain"}],"title":"Quantum Error Correction in Quaternionic Hilbert Spaces","issued":{"date-parts":[[2025,4,28]]},"note":"arxivid:2504.19833\narxiv_version_number:1"},{"id":"arxiv:0811.0421","type":"journal-article","publisher":"AIP Publishing","issue":"6","abstract":"<jats:p>We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in finite-dimensional Hilbert spaces carries straightforwardly over to infinite dimensions. We also find that, at least in principle, there exist qualitatively new classes of quantum error correcting codes that have no finite-dimensional counterparts. We begin with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. The new classes of codes that arise in infinite dimensions are shown to be characterized by von Neumann algebras of types II and III, for which we give in-principle physical examples.</jats:p>","DOI":"10.1063/1.3155783","source":"Crossref","title":"Quantum error correction on infinite-dimensional Hilbert spaces","volume":"50","author":[{"given":"Cédric","family":"Bény","sequence":"first","affiliation":[{"name":"National University of Singapore 1 Centre for Quantum Technologies, , 3 Science Drive 2, Singapore 117543, Singapore"},{"name":"University of Waterloo 2 Department of Applied Mathematics and Department of Physics, , Ontario N2L 3G1, Canada"}]},{"given":"Achim","family":"Kempf","sequence":"additional","affiliation":[{"name":"University of Waterloo 2 Department of Applied Mathematics and Department of Physics, , Ontario N2L 3G1, Canada"},{"name":"University of Queensland 3 Department of Physics, , St. Lucia, Brisbane, Queensland 4072, Australia"},{"name":"University of Waterloo 4 Institute for Quantum Computing, , Ontario N2L 3G1, Canada"}]},{"given":"David W.","family":"Kribs","sequence":"additional","affiliation":[{"name":"University of Waterloo 4 Institute for Quantum Computing, , Ontario N2L 3G1, Canada"},{"name":"University of Guelph 5 Department of Mathematics and Statistics, , Guelph, Ontario N1G 2W1, Canada"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2009,6,1]]},"URL":"http://dx.doi.org/10.1063/1.3155783","ISSN":["0022-2488","1089-7658"],"page":"062108","note":"arxivid:0811.0421"},{"id":"arxiv:quant-ph/9604015","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.55.1613","page":"1613-1622","source":"Crossref","title":"Capacity of the noisy quantum channel","volume":"55","author":[{"given":"Seth","family":"Lloyd","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,3,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.55.1613","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9604015"},{"id":"preset:ShorMSRI","type":"webpage","author":[{"family":"Shor","given":"P.W."}],"title":"\\href{https://github.com/errorcorrectionzoo/eczoo_data/files/7808154/shor.pdf}{The quantum channel capacity and coherent information}, 2002","note":"obtained from the MSRI Workshop on Quantum Computation website"},{"id":"arxiv:quant-ph/0304127","type":"article-journal","author":[{"given":"I.","family":"Devetak"}],"title":"The private classical capacity and quantum capacity of a quantum channel","issued":{"date-parts":[[2004,10,21]]},"note":"arxivid:quant-ph/0304127\narxiv_version_number:6"},{"id":"doi:10.1017/9781316848142","type":"monograph","publisher":"Cambridge University Press","DOI":"10.1017/9781316848142","source":"Crossref","title":"The Theory of Quantum Information","author":[{"given":"John","family":"Watrous","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2018,4,26]]},"ISBN":["9781316848142","9781107180567"],"URL":"http://dx.doi.org/10.1017/9781316848142","note":"edition-number:1"},{"id":"arxiv:1106.1445","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Developing many of the major, exciting, pre- and post-millennium developments from the ground up, this book is an ideal entry point for graduate students into quantum information theory. Significant attention is given to quantum mechanics for quantum information theory, and careful studies of the important protocols of teleportation, superdense coding, and entanglement distribution are presented. In this new edition, readers can expect to find over 100 pages of new material, including detailed discussions of Bell's theorem, the CHSH game, Tsirelson's theorem, the axiomatic approach to quantum channels, the definition of the diamond norm and its interpretation, and a proof of the Choi–Kraus theorem. Discussion of the importance of the quantum dynamic capacity formula has been completely revised, and many new exercises and references have been added. This new edition will be welcomed by the upcoming generation of quantum information theorists and the already established community of classical information theorists.</jats:p>","DOI":"10.1017/9781316809976","source":"Crossref","title":"Quantum Information Theory","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2016,11,2]]},"ISBN":["9781107176164","9781316809976"],"URL":"http://dx.doi.org/10.1017/9781316809976","note":"edition-number:2\narxivid:1106.1445"},{"id":"arxiv:quant-ph/9712048","type":"article-journal","author":[{"given":"John","family":"Preskill"}],"title":"Fault-tolerant quantum computation","issued":{"date-parts":[[1997,12,19]]},"note":"arxivid:quant-ph/9712048\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0004072","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"An Introduction to Quantum Error Correction","issued":{"date-parts":[[2000,4,18]]},"note":"arxivid:quant-ph/0004072\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0507174","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"Quantum Error Correction and Fault-Tolerance","issued":{"date-parts":[[2005,7,18]]},"note":"arxivid:quant-ph/0507174\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0612185","type":"article-journal","author":[{"given":"Julia","family":"Kempe"}],"title":"Approaches to Quantum Error Correction","issued":{"date-parts":[[2006,12,21]]},"note":"arxivid:quant-ph/0612185\narxiv_version_number:1"},{"id":"arxiv:0904.2557","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation","issued":{"date-parts":[[2009,4,16]]},"note":"arxivid:0904.2557\narxiv_version_number:1"},{"id":"arxiv:0905.2794","type":"journal-article","publisher":"IOP Publishing","issue":"7","DOI":"10.1088/0034-4885/76/7/076001","page":"076001","source":"Crossref","title":"Quantum error correction for beginners","volume":"76","author":[{"given":"Simon J","family":"Devitt","sequence":"first","affiliation":[]},{"given":"William J","family":"Munro","sequence":"additional","affiliation":[]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]}],"container-title":"Reports on Progress in Physics","original-title":[],"issued":{"date-parts":[[2013,6,20]]},"URL":"http://dx.doi.org/10.1088/0034-4885/76/7/076001","ISSN":["0034-4885","1361-6633"],"container-title-short":"Rep. Prog. Phys.","note":"arxivid:0905.2794"},{"id":"doi:10.1103/RevModPhys.88.041001","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/revmodphys.88.041001","source":"Crossref","title":"<i>Colloquium</i>: Protecting quantum information against environmental noise","volume":"88","author":[{"given":"Dieter","family":"Suter","sequence":"first","affiliation":[]},{"given":"Gonzalo A.","family":"Álvarez","sequence":"additional","affiliation":[]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2016,10,10]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.88.041001","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","page":"041001"},{"id":"doi:10.1002/9783527805785.ch1","type":"other","publisher":"Wiley","DOI":"10.1002/9783527805785.ch1","page":"1-17","source":"Crossref","title":"Classical Information Theory and Classical Error Correction","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]}],"container-title":"Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2016,4,18]]},"ISBN":["9783527413539","9783527805785"],"URL":"http://dx.doi.org/10.1002/9783527805785.ch1","note":"edition-number:1\nalternative-id:10.1002/9783527805785.ch1\n10.1002/9783527805785"},{"id":"arxiv:1508.03695","type":"article-journal","author":[{"given":"Alexandru","family":"Paler"},{"given":"Simon J.","family":"Devitt"}],"title":"An introduction to Fault-tolerant Quantum Computing","issued":{"date-parts":[[2015,8,15]]},"note":"arxivid:1508.03695\narxiv_version_number:1"},{"id":"arxiv:1907.11157","type":"journal-article","publisher":"Informa UK Limited","issue":"3","DOI":"10.1080/00107514.2019.1667078","page":"226-245","source":"Crossref","title":"Quantum error correction: an introductory guide","volume":"60","author":[{"given":"Joschka","family":"Roffe","sequence":"first","affiliation":[{"name":"Department of Physics & Astronomy, University of Sheffield, Sheffield, UK"}]}],"container-title":"Contemporary Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,3]]},"URL":"http://dx.doi.org/10.1080/00107514.2019.1667078","ISSN":["0010-7514","1366-5812"],"container-title-short":"Contemporary Physics","note":"alternative-id:10.1080/00107514.2019.1667078\narxivid:1907.11157"},{"id":"arxiv:1910.03672","type":"reference-entry","publisher":"Oxford University Press","abstract":"<p>Quantum error correction is a set of methods to protect quantum information—that is, quantum states—from unwanted environmental interactions (<italic>decoherence</italic>) and other forms of noise. The information is stored in a <italic>quantum error-correcting code</italic>, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an <italic>error space</italic> orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific <italic>error set</italic>, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state.</p>\n               <p>Most work on quantum error correction has focused on systems of quantum bits, or <italic>qubits</italic>, which are two-level quantum systems. These can be physically realized by the states of a spin-1/2 particle, the polarization of a single photon, two distinguished levels of a trapped atom or ion, the current states of a microscopic superconducting loop, or many other physical systems. The most widely used codes are the <italic>stabilizer codes</italic>, which are closely related to classical <italic>linear</italic> codes. The code space is the joint +1 eigenspace of a set of commuting Pauli operators on <italic>n</italic> qubits, called <italic>stabilizer generators</italic>; the <italic>error syndrome</italic> is determined by measuring these operators, which allows errors to be diagnosed and corrected. A stabilizer code is characterized by three parameters [[n,k,d]], where n is the number of <italic>physical</italic> qubits, k is the number of encoded <italic>logical</italic> qubits, and d is the minimum distance of the code (the smallest number of simultaneous qubit errors that can transform one valid codeword into another). Every useful code has n&gt;k; this physical redundancy is necessary to detect and correct errors without disturbing the logical state.</p>\n               <p>Quantum error correction is used to protect information in <italic>quantum communication</italic> (where quantum states pass through noisy channels) and <italic>quantum computation</italic> (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of <italic>fault-tolerant</italic> design. Other approaches to error mitigation in quantum systems include <italic>decoherence-free subspaces</italic>, <italic>noiseless subsystems</italic>, and <italic>dynamical decoupling</italic>.</p>","DOI":"10.1093/acrefore/9780190871994.013.35","source":"Crossref","title":"Quantum Error Correction","author":[{"given":"Todd A.","family":"Brun","sequence":"first","affiliation":[]}],"container-title":"Oxford Research Encyclopedia of Physics","original-title":["Quantum Error Correction"],"language":"en","issued":{"date-parts":[[2020,2,28]]},"ISBN":["9780190871994"],"URL":"http://dx.doi.org/10.1093/acrefore/9780190871994.013.35","note":"arxivid:1910.03672"},{"id":"doi:10.1002/9781119790327.ch10","type":"other","publisher":"Wiley","DOI":"10.1002/9781119790327.ch10","page":"451-498","source":"Crossref","title":"Quantum Error Correction","container-title":"Artificial Intelligence and Quantum Computing for Advanced Wireless Networks","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,14]]},"ISBN":["9781119790297","9781119790327"],"URL":"http://dx.doi.org/10.1002/9781119790327.ch10","note":"edition-number:1\nalternative-id:10.1002/9781119790327.ch10\n10.1002/9781119790327"},{"id":"arxiv:2407.12737","type":"article-journal","author":[{"given":"Priya J.","family":"Nadkarni"},{"given":"Narayanan","family":"Rengaswamy"},{"given":"Bane","family":"Vasić"}],"title":"Tutorial on Quantum Error Correction for 2024 Quantum Information Knowledge (QuIK) Workshop","issued":{"date-parts":[[2024,7,17]]},"note":"arxivid:2407.12737\narxiv_version_number:1"},{"id":"doi:10.1017/CBO9781139034807","type":"book","publisher":"Cambridge University Press","abstract":"<jats:p>Quantum computation and information is one of the most exciting developments in science and technology of the last twenty years. To achieve large scale quantum computers and communication networks it is essential not only to overcome noise in stored quantum information, but also in general faulty quantum operations. Scalable quantum computers require a far-reaching theory of fault-tolerant quantum computation. This comprehensive text, written by leading experts in the field, focuses on quantum error correction and thoroughly covers the theory as well as experimental and practical issues. The book is not limited to a single approach, but reviews many different methods to control quantum errors, including topological codes, dynamical decoupling and decoherence-free subspaces. Basic subjects as well as advanced theory and a survey of topics from cutting-edge research make this book invaluable both as a pedagogical introduction at the graduate level and as a reference for experts in quantum information science.</jats:p>","DOI":"10.1017/cbo9781139034807","source":"Crossref","title":"Quantum Error Correction","container-title":[],"original-title":[],"editor":[{"given":"Daniel A.","family":"Lidar","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2013,9,5]]},"ISBN":["9780521897877","9781139034807"],"URL":"http://dx.doi.org/10.1017/CBO9781139034807","note":"edition-number:1"},{"id":"doi:10.1201/b15868","type":"book","publisher":"CRC Press","DOI":"10.1201/b15868","source":"Crossref","title":"Quantum Error Correction and Fault Tolerant Quantum Computing","author":[{"given":"Frank","family":"Gaitan","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2018,10,3]]},"ISBN":["9781315221885"],"URL":"http://dx.doi.org/10.1201/b15868","note":"edition-number:0"},{"id":"arxiv:hep-th/9406058","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.51.992","page":"992-997","source":"Crossref","title":"Maintaining coherence in quantum computers","volume":"51","author":[{"given":"W. G.","family":"Unruh","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1995,2,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.51.992","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:hep-th/9406058"},{"id":"doi:10.1098/rsta.1995.0106","type":"journal-article","publisher":"The Royal Society","issue":"1703","abstract":"<jats:p>\n            Technologies differ in their explicit utilization of quantum mechanical behaviour. A transistor, despite its roots in energy band structure, does not invoke quantum mechanically coherent transmission between terminals. The impressive progress in the past decade in mesoscopic physics, when combined with studies that have analysed a totally quantum mechanical computational process, suggest that we may be ready to move toward more quantum mechanical procedures for information processing. This paper is a warning signal; this possibility is beset by problems. The case will be made via two separate but complementary arguments. First, by summarizing this author's published comments on computation via totally quantum mechanical coherent Hamiltonians. The computation is likely to suffer from\n            <jats:italic>localization</jats:italic>\n            , i.e. from reflection of the computational trajectory, causing the computation to turn around. Additionally, small errors will accumulate and cause the computation to go off track. This is supplemented by analysis of specific proposals that suggest more detailed machinery than invoked in the general literature on quantum mechanical Hamiltonian computation.\n          </jats:p>","DOI":"10.1098/rsta.1995.0106","page":"367-376","source":"Crossref","title":"Is quantum mechanics useful?","volume":"353","container-title":"Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1995,12,15]]},"URL":"http://dx.doi.org/10.1098/rsta.1995.0106","ISSN":["0962-8428","2054-0299"],"container-title-short":"Phil. Trans. R. Soc. Lond. A","note":"alternative-id:10.1098/rsta.1995.0106"},{"id":"arxiv:1310.8457","type":"article-journal","author":[{"given":"Robert","family":"Alicki"}],"title":"Critique of Fault-Tolerant Quantum Information Processing","issued":{"date-parts":[[2013,10,31]]},"note":"arxivid:1310.8457\narxiv_version_number:1"},{"id":"arxiv:1105.4464","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/ncomms2076","source":"Crossref","title":"Quantum correlations with no causal order","volume":"3","author":[{"given":"Ognyan","family":"Oreshkov","sequence":"first","affiliation":[]},{"given":"Fabio","family":"Costa","sequence":"additional","affiliation":[]},{"given":"Časlav","family":"Brukner","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2012,10,2]]},"URL":"http://dx.doi.org/10.1038/ncomms2076","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"1092","note":"alternative-id:BFncomms2076\narxivid:1105.4464"},{"id":"arxiv:1210.4722","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2014.2353614","page":"7317-7329","source":"Crossref","title":"Finite Blocklength Converse Bounds for Quantum Channels","volume":"60","author":[{"given":"William","family":"Matthews","sequence":"first","affiliation":[]},{"given":"Stephanie","family":"Wehner","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2014.2353614","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1210.4722"},{"id":"arxiv:1406.7142","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2015.2439953","page":"4486-4499","source":"Crossref","title":"On the Power of PPT-Preserving and Non-Signalling Codes","volume":"61","author":[{"given":"Debbie","family":"Leung","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3560-9177","authenticated-orcid":false,"given":"William","family":"Matthews","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2015,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2015.2439953","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1406.7142"},{"id":"arxiv:2405.17567","type":"article-journal","author":[{"given":"Andrew","family":"Tanggara"},{"given":"Mile","family":"Gu"},{"given":"Kishor","family":"Bharti"}],"title":"Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction","issued":{"date-parts":[[2024,5,27]]},"note":"arxivid:2405.17567\narxiv_version_number:1"},{"id":"arxiv:2409.09416","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.110.032413","source":"Crossref","title":"Quantum resource theory of coding for error correction","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0002-7041-4298","authenticated-orcid":true,"given":"Dong-Sheng","family":"Wang","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02xtbq882","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute of Theoretical Physics"},{"id":[{"id":"https://ror.org/034t30j35","id-type":"ROR","asserted-by":"publisher"}],"name":"Chinese Academy of Sciences"},{"name":"University of Chinese Academy of Sciences"}]},{"given":"Yuan-Dong","family":"Liu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02xtbq882","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute of Theoretical Physics"},{"id":[{"id":"https://ror.org/034t30j35","id-type":"ROR","asserted-by":"publisher"}],"name":"Chinese Academy of Sciences"},{"name":"University of Chinese Academy of Sciences"}]},{"given":"Yun-Jiang","family":"Wang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05s92vm98","id-type":"ROR","asserted-by":"publisher"}],"name":"Xidian University"},{"name":"Xidian University"},{"name":"Xidian University"}]},{"given":"Shunlong","family":"Luo","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02jkmyk67","id-type":"ROR","asserted-by":"publisher"}],"name":"Academy of Mathematics and Systems Science"},{"id":[{"id":"https://ror.org/034t30j35","id-type":"ROR","asserted-by":"publisher"}],"name":"Chinese Academy of Sciences"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.032413","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032413","note":"arxivid:2409.09416"},{"id":"arxiv:2506.16614","type":"article-journal","author":[{"given":"Vincent","family":"Mutolo"},{"given":"Devon","family":"Campbell"},{"given":"Quinn","family":"Manning"},{"given":"Henri Witold","family":"Dubourg"},{"given":"Ruibin","family":"Lyu"},{"given":"Simha","family":"Sethumadhavan"},{"given":"Daniel","family":"Rubenstein"},{"given":"Salvatore","family":"Stolfo"}],"title":"Quantum Computer Fingerprinting using Error Syndromes","issued":{"date-parts":[[2025,9,25]]},"note":"arxivid:2506.16614\narxiv_version_number:2"},{"id":"doi:10.1103/PhysRevA.55.900","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.55.900","page":"900-911","source":"Crossref","title":"Theory of quantum error-correcting codes","volume":"55","author":[{"given":"Emanuel","family":"Knill","sequence":"first","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,2,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.55.900","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:quant-ph/9604024","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.54.3824","page":"3824-3851","source":"Crossref","title":"Mixed-state entanglement and quantum error correction","volume":"54","author":[{"given":"Charles H.","family":"Bennett","sequence":"first","affiliation":[]},{"given":"David P.","family":"DiVincenzo","sequence":"additional","affiliation":[]},{"given":"John A.","family":"Smolin","sequence":"additional","affiliation":[]},{"given":"William K.","family":"Wootters","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1996,11,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.54.3824","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9604024"},{"id":"doi:10.1112/jlms/s1-41.1.123","type":"journal-article","publisher":"Wiley","issue":"1","DOI":"10.1112/jlms/s1-41.1.123","page":"123-128","source":"Crossref","title":"A Generalization of Radon's Theorem","volume":"s1-41","author":[{"given":"H.","family":"Tverberg","sequence":"first","affiliation":[]}],"container-title":"Journal of the London Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[1966]]},"URL":"http://dx.doi.org/10.1112/jlms/s1-41.1.123","ISSN":["0024-6107"],"container-title-short":"Journal of the London Mathematical Society"},{"id":"doi:10.1007/978-1-4613-0039-7","type":"book","publisher":"Springer New York","DOI":"10.1007/978-1-4613-0039-7","source":"Crossref","title":"Lectures on Discrete Geometry","container-title":"Graduate Texts in Mathematics","original-title":[],"editor":[{"given":"Jiří","family":"Matoušek","sequence":"first","affiliation":[]}],"issued":{"date-parts":[[2002]]},"ISBN":["9780387953748","9781461300397"],"URL":"http://dx.doi.org/10.1007/978-1-4613-0039-7","ISSN":["0072-5285"],"note":"publisher-location:New York, NY"},{"id":"arxiv:1712.06119","type":"article-journal","author":[{"given":"Imre","family":"Bárány"},{"given":"Pablo","family":"Soberón"}],"title":"Tverberg's theorem is 50 years old: a survey","issued":{"date-parts":[[2018,5,31]]},"note":"arxivid:1712.06119\narxiv_version_number:2"},{"id":"doi:10.1007/BF01464231","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1-2","DOI":"10.1007/bf01464231","page":"113-115","source":"Crossref","title":"Mengen konvexer K�rper, die einen gemeinsamen Punkt enthalten","volume":"83","author":[{"given":"Johann","family":"Radon","sequence":"first","affiliation":[]}],"container-title":"Mathematische Annalen","original-title":[],"language":"de","issued":{"date-parts":[[1921,3]]},"URL":"http://dx.doi.org/10.1007/BF01464231","ISSN":["0025-5831","1432-1807"],"container-title-short":"Math. Ann.","note":"alternative-id:BF01464231"},{"id":"arxiv:quant-ph/9604022","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.54.2629","page":"2629-2635","source":"Crossref","title":"Quantum data processing and error correction","volume":"54","author":[{"given":"Benjamin","family":"Schumacher","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Nielsen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1996,10,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.54.2629","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9604022"},{"id":"arxiv:quant-ph/9707023","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.57.3330","page":"3330-3347","source":"Crossref","title":"Entropic bounds on coding for noisy quantum channels","volume":"57","author":[{"given":"Nicolas J.","family":"Cerf","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1998,5,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.57.3330","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9707023"},{"id":"arxiv:quant-ph/0304007","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-004-1049-z","page":"359-374","source":"Crossref","title":"Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality","volume":"246","author":[{"given":"Patrick","family":"Hayden","sequence":"first","affiliation":[]},{"given":"Richard","family":"Jozsa","sequence":"additional","affiliation":[]},{"given":"D�nes","family":"Petz","sequence":"additional","affiliation":[]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"issued":{"date-parts":[[2004,4,1]]},"URL":"http://dx.doi.org/10.1007/s00220-004-1049-z","ISSN":["0010-3616","1432-0916"],"container-title-short":"Communications in Mathematical Physics","note":"alternative-id:RNN6DKW5GBU6XYQ4\narxivid:quant-ph/0304007"},{"id":"arxiv:quant-ph/0202054","type":"article-journal","author":[{"given":"Vladimir","family":"Korepin"},{"given":"John","family":"Terilla"}],"title":"Thermodynamic interpretation of quantum error correcting criterion","issued":{"date-parts":[[2004,3,1]]},"note":"arxivid:quant-ph/0202054\narxiv_version_number:6"},{"id":"arxiv:quant-ph/0511101","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0034-4877(06)80041-8","page":"77-91","source":"Crossref","title":"Quantum error correcting codes from the compression formalism","volume":"58","author":[{"given":"Man-Duen","family":"Choi","sequence":"first","affiliation":[]},{"given":"David W.","family":"Kribs","sequence":"additional","affiliation":[]},{"given":"Karol","family":"Życzkowski","sequence":"additional","affiliation":[]}],"container-title":"Reports on Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2006,8]]},"URL":"http://dx.doi.org/10.1016/S0034-4877(06)80041-8","ISSN":["0034-4877"],"container-title-short":"Reports on Mathematical Physics","note":"alternative-id:S0034487706800418\narxivid:quant-ph/0511101"},{"id":"arxiv:math/0511278","type":"article-journal","author":[{"given":"Man-Duen","family":"Choi"},{"given":"David W.","family":"Kribs"},{"given":"Karol","family":"Zyczkowski"}],"title":"Higher-Rank Numerical Ranges and Compression Problems","issued":{"date-parts":[[2006,4,2]]},"note":"arxivid:math/0511278\narxiv_version_number:2"},{"id":"arxiv:1407.1350","type":"journal-article","publisher":"Informa UK Limited","issue":"5","DOI":"10.1080/03081087.2013.801963","page":"639-647","source":"Crossref","title":"Private quantum codes: introduction and connection with higher rank numerical ranges","volume":"62","author":[{"given":"David W.","family":"Kribs","sequence":"first","affiliation":[]},{"given":"Sarah","family":"Plosker","sequence":"additional","affiliation":[]}],"container-title":"Linear and Multilinear Algebra","original-title":[],"language":"en","issued":{"date-parts":[[2013,6,3]]},"URL":"http://dx.doi.org/10.1080/03081087.2013.801963","ISSN":["0308-1087","1563-5139"],"container-title-short":"Linear and Multilinear Algebra","note":"alternative-id:10.1080/03081087.2013.801963\narxivid:1407.1350"},{"id":"doi:10.1017/CBO9781139525343","type":"monograph","publisher":"Cambridge University Press","abstract":"<jats:p>Finally, here is a modern, self-contained text on quantum information theory suitable for graduate-level courses. Developing the subject 'from the ground up' it covers classical results as well as major advances of the past decade. Beginning with an extensive overview of classical information theory suitable for the non-expert, the author then turns his attention to quantum mechanics for quantum information theory, and the important protocols of teleportation, super-dense coding and entanglement distribution. He develops all of the tools necessary for understanding important results in quantum information theory, including capacity theorems for classical, entanglement-assisted, private and quantum communication. The book also covers important recent developments such as superadditivity of private, coherent and Holevo information, and the superactivation of quantum capacity. This book will be warmly welcomed by the upcoming generation of quantum information theorists and the already established community of classical information theorists.</jats:p>","DOI":"10.1017/cbo9781139525343","source":"Crossref","title":"Quantum Information Theory","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"issued":{"date-parts":[[2013,4,18]]},"ISBN":["9781107034259","9781139525343"],"URL":"http://dx.doi.org/10.1017/CBO9781139525343","note":"edition-number:1"},{"id":"arxiv:1507.07072","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.91.052326","source":"Crossref","title":"Solution to the mean king's problem using quantum error-correcting codes","volume":"91","author":[{"given":"Masakazu","family":"Yoshida","sequence":"first","affiliation":[]},{"given":"Gen","family":"Kimura","sequence":"additional","affiliation":[]},{"given":"Takayuki","family":"Miyadera","sequence":"additional","affiliation":[]},{"given":"Hideki","family":"Imai","sequence":"additional","affiliation":[]},{"given":"Jun","family":"Cheng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,5,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.052326","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052326","note":"arxivid:1507.07072"},{"id":"doi:10.1103/PhysRevLett.58.1385","type":"journal-article","publisher":"American Physical Society (APS)","issue":"14","DOI":"10.1103/physrevlett.58.1385","page":"1385-1387","source":"Crossref","title":"How to ascertain the values of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">sigma</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">x</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>,<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>, and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">z</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>of a spin-<i>1/2</i>particle","volume":"58","author":[{"given":"Lev","family":"Vaidman","sequence":"first","affiliation":[]},{"given":"Yakir","family":"Aharonov","sequence":"additional","affiliation":[]},{"given":"David Z.","family":"Albert","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1987,4,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.58.1385","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"doi:10.1098/rspa.1998.0165","type":"journal-article","publisher":"The Royal Society","issue":"1969","DOI":"10.1098/rspa.1998.0165","page":"355-364","source":"Crossref","title":"Continuous error correction","volume":"454","author":[{"given":"Juan Pablo","family":"Paz","sequence":"first","affiliation":[{"name":"Departamento de Física J.J. Giambiagi, FCEN, UBA, Pabellón 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina"}]},{"given":"Wojciech Hubert","family":"Zurek","sequence":"additional","affiliation":[{"name":"Theoretical Astrophysics, MSB288, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]}],"container-title":"Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1998,1,8]]},"URL":"http://dx.doi.org/10.1098/rspa.1998.0165","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. Lond. A","note":"alternative-id:10.1098/rspa.1998.0165"},{"id":"arxiv:quant-ph/9912104","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.85.856","page":"856-859","source":"Crossref","title":"Automatic Quantum Error Correction","volume":"85","author":[{"given":"Jeff P.","family":"Barnes","sequence":"first","affiliation":[]},{"given":"Warren S.","family":"Warren","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2000,7,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.85.856","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9912104"},{"id":"arxiv:quant-ph/0110111","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.65.042301","source":"Crossref","title":"Continuous quantum error correction via quantum feedback control","volume":"65","author":[{"given":"Charlene","family":"Ahn","sequence":"first","affiliation":[]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[]},{"given":"Andrew J.","family":"Landahl","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,3,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.042301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042301","note":"arxivid:quant-ph/0110111"},{"id":"arxiv:quant-ph/0501038","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.72.012306","source":"Crossref","title":"Continuous quantum error correction by cooling","volume":"72","author":[{"given":"Mohan","family":"Sarovar","sequence":"first","affiliation":[]},{"given":"G. J.","family":"Milburn","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.72.012306","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012306","note":"arxivid:quant-ph/0501038"},{"id":"arxiv:quant-ph/0511221","type":"article-journal","author":[{"given":"Ramon","family":"van Handel"},{"given":"Hideo","family":"Mabuchi"}],"title":"Optimal error tracking via quantum coding and continuous syndrome measurement","issued":{"date-parts":[[2005,11,22]]},"note":"arxivid:quant-ph/0511221\narxiv_version_number:1"},{"id":"arxiv:1311.2485","type":"article-journal","author":[{"given":"Ognyan","family":"Oreshkov"}],"title":"Continuous-time quantum error correction","issued":{"date-parts":[[2013,12,1]]},"note":"arxivid:1311.2485\narxiv_version_number:2"},{"id":"arxiv:1711.02999","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.98.012317","source":"Crossref","title":"Implementation-independent sufficient condition of the Knill-Laflamme type for the autonomous protection of logical qudits by strong engineered dissipation","volume":"98","author":[{"given":"Jae-Mo","family":"Lihm","sequence":"first","affiliation":[]},{"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]},{"given":"Uwe R.","family":"Fischer","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,7,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.98.012317","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012317","note":"arxivid:1711.02999"},{"id":"arxiv:2103.05007","type":"article-journal","author":[{"given":"José","family":"Lebreuilly"},{"given":"Kyungjoo","family":"Noh"},{"given":"Chiao-Hsuan","family":"Wang"},{"given":"Steven M.","family":"Girvin"},{"given":"Liang","family":"Jiang"}],"title":"Autonomous quantum error correction and quantum computation","issued":{"date-parts":[[2021,3,8]]},"note":"arxivid:2103.05007\narxiv_version_number:1"},{"id":"arxiv:0705.2342","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.76.022318","source":"Crossref","title":"Continuous quantum error correction for non-Markovian decoherence","volume":"76","author":[{"given":"Ognyan","family":"Oreshkov","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,8,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.022318","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022318","note":"arxivid:0705.2342"},{"id":"arxiv:quant-ph/0409187","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.68.064302","source":"Crossref","title":"Information-theoretical approach to control of quantum-mechanical systems","volume":"68","author":[{"given":"Shiro","family":"Kawabata","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,12,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.064302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"064302","note":"arxivid:quant-ph/0409187"},{"id":"arxiv:2506.21707","type":"article-journal","author":[{"given":"Anirudh","family":"Lanka"},{"given":"Shashank","family":"Hegde"},{"given":"Todd A.","family":"Brun"}],"title":"Optimizing continuous-time quantum error correction for arbitrary noise","issued":{"date-parts":[[2026,1,27]]},"note":"arxivid:2506.21707\narxiv_version_number:2"},{"id":"arxiv:2312.06664","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Quantum error correcting (QEC) codes protect quantum information from decoherence as long as error rates fall below critical error thresholds. In general, obtaining thresholds implies simulating the QEC procedure using, in general, suboptimal decoding strategies. In a few cases and for sufficiently simple noise models, optimal decoding of QEC codes can be framed as a phase transition in disordered classical spin models. In both situations, accurate estimation of thresholds demands intensive computational resources. Here we use the coherent information of the mixed state of noisy QEC codes to accurately estimate the associated optimal QEC thresholds already from small-distance codes at moderate computational cost. We show the effectiveness and versatility of our method by applying it first to the topological surface and color code under bit-flip and depolarizing noise. We then extend the coherent information based methodology to phenomenological and quantum circuit level noise settings. For all examples considered, we obtain highly accurate estimates of optimal error thresholds from small, low-distance instances of the codes, in close agreement with threshold values reported in the literature. Our findings establish the coherent information as a reliable competitive  tool for the calculation of optimal thresholds of state-of-the-art QEC codes under realistic noise models.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevresearch.6.l042014","source":"Crossref","title":"Accurate optimal quantum error correction thresholds from coherent information","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-5946-7591","authenticated-orcid":true,"given":"Luis","family":"Colmenarez","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0002-2369-6245","authenticated-orcid":true,"given":"Ze-Min","family":"Huang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/00rcxh774","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Cologne"}]},{"given":"Sebastian","family":"Diehl","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00rcxh774","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Cologne"}]},{"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.6.L042014","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"L042014","note":"arxivid:2312.06664"},{"id":"arxiv:0909.5103","type":"article-journal","author":[{"given":"Zhuo","family":"Wang"},{"given":"Kai","family":"Sun"},{"given":"Hen","family":"Fan"},{"given":"Vlatko","family":"Vedral"}],"title":"Nested Quantum Error Correction Codes","issued":{"date-parts":[[2009,9,28]]},"note":"arxivid:0909.5103\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9608012","type":"article-journal","author":[{"given":"Emanuel","family":"Knill"},{"given":"Raymond","family":"Laflamme"}],"title":"Concatenated Quantum Codes","issued":{"date-parts":[[1996,8,8]]},"note":"arxivid:quant-ph/9608012\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0606126","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.74.052333","source":"Crossref","title":"Optimal and efficient decoding of concatenated quantum block codes","volume":"74","author":[{"given":"David","family":"Poulin","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,11,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.74.052333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052333","note":"arxivid:quant-ph/0606126"},{"id":"arxiv:0910.4129","type":"journal-article","publisher":"AIP Publishing","issue":"2","DOI":"10.1063/1.3534799","page":"022201","source":"Crossref","title":"Graph concatenation for quantum codes","volume":"52","author":[{"given":"Salman","family":"Beigi","sequence":"first","affiliation":[]},{"given":"Isaac","family":"Chuang","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Shor","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,2]]},"URL":"http://dx.doi.org/10.1063/1.3534799","ISSN":["0022-2488","1089-7658"],"container-title-short":"Journal of Mathematical Physics","note":"alternative-id:10.1063/1.3534799\narxivid:0910.4129"},{"id":"arxiv:2202.08084","type":"journal-article","publisher":"Proceedings of the National Academy of Sciences","issue":"24","abstract":"<jats:p>Entanglement-assisted concatenated quantum codes (EACQCs), constructed by concatenating two quantum codes, are proposed. These EACQCs show significant advantages over standard concatenated quantum codes (CQCs). First, we prove that, unlike standard CQCs, EACQCs can beat the nondegenerate Hamming bound for entanglement-assisted quantum error-correction codes (EAQECCs). Second, we construct families of EACQCs with parameters better than the best-known standard quantum error-correction codes (QECCs) and EAQECCs. Moreover, these EACQCs require very few Einstein–Podolsky–Rosen (EPR) pairs to begin with. Finally, it is shown that EACQCs make entanglement-assisted quantum communication possible, even if the ebits are noisy. Furthermore, EACQCs can outperform CQCs in entanglement fidelity over depolarizing channels if the ebits are less noisy than the qubits. We show that the error-probability threshold of EACQCs is larger than that of CQCs when the error rate of ebits is sufficiently lower than that of qubits. Specifically, we derive a high threshold of 47% when the error probability of the preshared entanglement is 1% to that of qubits.</jats:p>","DOI":"10.1073/pnas.2202235119","source":"Crossref","title":"Entanglement-assisted concatenated quantum codes","volume":"119","author":[{"given":"Jihao","family":"Fan","sequence":"first","affiliation":[{"name":"School of Cyber Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Jun","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Yongbin","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Cyber Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[{"name":"Quantum Computing Research Center, Hon Hai Research Institute, Taipei City 114, Taiwan"}]},{"ORCID":"https://orcid.org/0000-0002-2062-131X","authenticated-orcid":false,"given":"H. Vincent","family":"Poor","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Princeton University, Princeton, NJ 08544"}]}],"container-title":"Proceedings of the National Academy of Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2022,6,10]]},"URL":"http://dx.doi.org/10.1073/pnas.2202235119","ISSN":["0027-8424","1091-6490"],"container-title-short":"Proc. Natl. Acad. Sci. U.S.A.","page":"e2202235119","note":"alternative-id:10.1073/pnas.2202235119\narxivid:2202.08084"},{"id":"arxiv:2202.00248","type":"article-journal","author":[{"given":"Tania","family":"Sidana"},{"given":"Navin","family":"Kashyap"}],"title":"Entanglement-Assisted Quantum Error-Correcting Codes over Local Frobenius Rings","issued":{"date-parts":[[2023,1,8]]},"note":"arxivid:2202.00248\narxiv_version_number:4"},{"id":"arxiv:0901.0042","type":"article-journal","author":[{"given":"Zhuo","family":"Li"},{"given":"Li-Juan","family":"Xing"},{"given":"Xin-Mei","family":"Wang"}],"title":"A family of asymptotically good quantum codes based on code concatenation","issued":{"date-parts":[[2008,12,31]]},"note":"arxivid:0901.0042\narxiv_version_number:1"},{"id":"arxiv:2210.02468","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.4.020341","source":"Crossref","title":"Floquet Codes without Parent Subsystem Codes","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-8482-5832","authenticated-orcid":true,"given":"Margarita","family":"Davydova","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[]},{"given":"Shankar","family":"Balasubramanian","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,13]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.020341","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020341","note":"arxivid:2210.02468"},{"id":"arxiv:2307.13668","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Floquet codes are a novel class of quantum error-correcting codes with dynamically generated logical qubits arising from a periodic schedule of noncommuting measurements. We utilize the interpretation of measurements in terms of condensation of topological excitations and the rewinding of measurement sequences to engineer new examples of Floquet codes. In particular, rewinding is advantageous for obtaining a desired set of instantaneous stabilizer groups on both toric and planar layouts. Our first example is a Floquet code with instantaneous stabilizer codes that have the same topological order as the three-dimensional (3D) toric code(s). This Floquet code also exhibits a splitting of the topological order of the 3D toric code under the associated sequence of measurements, i.e., an instantaneous stabilizer group of a single copy of the 3D toric code in one round transforms into an instantaneous stabilizer group of two copies of the 3D toric code up to nonlocal stabilizers in the following round. We further construct boundaries for this 3D code and argue that stacking it with two copies of the 3D subsystem toric code allows for a transversal implementation of the logical non-Clifford controlled-controlled-<a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mi>Z</a:mi></a:math> gate. We also show that the coupled-layer construction of the X-cube Floquet code can be modified by a rewinding schedule such that each of the instantaneous stabilizer codes is finite depth equivalent to the X-cube model up to toric codes; the X-cube Floquet code exhibits a splitting of the X-cube model into a copy of the X-cube model and toric codes under the measurement sequence. Our final 3D example is a generalization of the 2D Floquet toric code on the honeycomb lattice to three dimensions, which has instantaneous stabilizer codes with the same topological order as the 3D fermionic toric code.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020305","source":"Crossref","title":"Engineering 3D Floquet Codes by Rewinding","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"first","affiliation":[{"name":"Department of Physics, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Department of Physics, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA"}]},{"given":"Joseph","family":"Sullivan","sequence":"additional","affiliation":[{"name":"Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada"},{"name":"Department of Physics, Yale University, New Haven, Connecticut 06511, USA"}]},{"given":"Tyler D.","family":"Ellison","sequence":"additional","affiliation":[{"name":"Department of Physics, Yale University, New Haven, Connecticut 06511, USA"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,8]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020305","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020305","note":"arxivid:2307.13668"},{"id":"doi:10.26081/K6F65V","type":"article","author":[{"family":"Brown","given":"Ben"}],"issued":{"date-parts":[[2022]]},"DOI":"10.26081/K6F65V","publisher":"Kavli Institute for Theoretical Physics","title":"Anyon condensation and the color code","URL":"https://online.kitp.ucsb.edu/online/dynisq-c22/brown/"},{"id":"arxiv:2212.06775","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.131.120603","source":"Crossref","title":"High-Threshold Quantum Computing by Fusing One-Dimensional Cluster States","volume":"131","author":[{"ORCID":"https://orcid.org/0000-0001-5709-0906","authenticated-orcid":true,"given":"Stefano","family":"Paesani","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2023,9,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.131.120603","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120603","note":"arxivid:2212.06775"},{"id":"arxiv:2210.13154","type":"article-journal","author":[{"literal":"Wootton, James R."}],"issued":{"date-parts":[[2023]]},"container-title":"University of Basel","DOI":"10.5451/UNIBAS-EP96333","publisher":"University of Basel","title":"Quantum error correction: From the blackboard to the cloud","URL":"https://edoc.unibas.ch/96333/","note":"arxivid:2210.13154"},{"id":"arxiv:2407.08566","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Analyzing and developing new quantum error-correcting (QEC) schemes is one of the most prominent tasks in quantum computing research. In such efforts, introducing time dynamics explicitly in both analysis and design of error-correcting protocols constitutes an important cornerstone. In this work, we present a graphical formalism based on tensor networks to capture the logical action and error-correcting capabilities of any Clifford circuit with Pauli measurements. We showcase the functioning of the formalism on new Floquet codes derived from topological subsystem codes, which we call XYZ . Based on the projective symmetries of the building blocks of the tensor network we develop a framework of . Pauli flows allow for a graphical understanding of all quantities entering an error-correction analysis of a circuit, including different types of QEC experiments, such as memory and stability experiments. We lay out how to derive a well-defined decoding problem from the tensor-network representation of a protocol and its Pauli flows alone, independent of any stabilizer code or fixed circuit. Importantly, this framework applies to all Clifford protocols and encompasses both measurement-based and circuit-based approaches to fault tolerance. We apply our method to our new family of dynamical codes, which are in the same topological phase as the <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:math>-dimensional color code, making them a promising candidate for low-overhead logical gates. In contrast to its static counterpart, the dynamical protocol applies a <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><d:msub><d:mrow><d:mi mathvariant=\"double-struck\">Z</d:mi></d:mrow><d:mn>3</d:mn></d:msub></d:math> automorphism to the logical Pauli group every three time steps. We highlight some of its topological properties and comment on the anyon physics behind a planar layout. Lastly, we benchmark the performance of the <h:math xmlns:h=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><h:mi>XYZ</h:mi></h:math> ruby code on a torus by performing both memory and stability experiments and find competitive circuit-level noise thresholds of approximately equal to <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><k:mn>0.18</k:mn><k:mi mathvariant=\"normal\">%</k:mi></k:math>, comparable with other Floquet codes and <o:math xmlns:o=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><o:mn>2</o:mn><o:mo>+</o:mo><o:mn>1</o:mn></o:math>-dimensional color codes.</jats:p>","DOI":"10.1103/prxquantum.6.010360","source":"Crossref","title":"<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><mml:mi>XYZ</mml:mi></mml:math>\n Ruby Code: Making a Case for a Three-Colored Graphical Calculus for Quantum Error Correction in Spacetime","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-3779-0080","authenticated-orcid":true,"given":"Julio C.","family":"Magdalena de la Fuente","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"}]},{"ORCID":"https://orcid.org/0009-0002-8811-1401","authenticated-orcid":true,"given":"Josias","family":"Old","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"},{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"}]},{"given":"Alex","family":"Townsend-Teague","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"}]},{"ORCID":"https://orcid.org/0009-0000-8290-8247","authenticated-orcid":true,"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"},{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"}]},{"ORCID":"https://orcid.org/0000-0003-3033-1292","authenticated-orcid":true,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"},{"id":[{"id":"https://ror.org/02aj13c28","id-type":"ROR","asserted-by":"publisher"}],"name":"Helmholtz-Zentrum Berlin für Materialien und Energie"}]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":true,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"},{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,28]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.010360","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010360","note":"arxivid:2407.08566"},{"id":"arxiv:2303.17664","type":"article-journal","author":[{"given":"Joseph","family":"Sullivan"},{"given":"Rui","family":"Wen"},{"given":"Andrew C.","family":"Potter"}],"title":"Floquet codes and phases in twist-defect networks","issued":{"date-parts":[[2023,3,30]]},"note":"arxivid:2303.17664\narxiv_version_number:1"},{"id":"arxiv:2110.09545","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We introduce a simple construction of boundary conditions for the honeycomb code \\cite{honeycomb} that uses only pairwise checks and allows parallelogram geometries at the cost of modifying the bulk measurement sequence. We discuss small instances of the code.</jats:p>","DOI":"10.22331/q-2022-04-21-693","page":"693","source":"Crossref","title":"Boundaries for the Honeycomb Code","volume":"6","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"},{"name":"Station Q, Microsoft Quantum, Santa Barbara, CA 93106-6105, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,21]]},"URL":"http://dx.doi.org/10.22331/q-2022-04-21-693","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2110.09545"},{"id":"arxiv:2110.05348","type":"article-journal","author":[{"given":"Christophe","family":"Vuillot"}],"title":"Planar Floquet Codes","issued":{"date-parts":[[2021,12,14]]},"note":"arxivid:2110.05348\narxiv_version_number:2"},{"id":"arxiv:2203.11137","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevb.106.085122","source":"Crossref","title":"Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes","volume":"106","author":[{"ORCID":"https://orcid.org/0000-0002-6552-488X","authenticated-orcid":true,"given":"David","family":"Aasen","sequence":"first","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105, USA"},{"name":"Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA"}]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105, USA"},{"name":"Department of Mathematics, University of California, Santa Barbara, California 93106, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105, USA"},{"name":"Microsoft Quantum, Redmond, Washington 98052, USA"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.106.085122","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"085122","note":"arxivid:2203.11137"},{"id":"arxiv:2306.08027","type":"article-journal","author":[{"given":"Tyler D.","family":"Ellison"},{"given":"Joseph","family":"Sullivan"},{"given":"Arpit","family":"Dua"}],"title":"Floquet codes with a twist","issued":{"date-parts":[[2023,9,19]]},"note":"arxivid:2306.08027\narxiv_version_number:3"},{"id":"arxiv:1406.2690","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.92.012336","source":"Crossref","title":"Adiabatic topological quantum computing","volume":"92","author":[{"given":"Chris","family":"Cesare","sequence":"first","affiliation":[]},{"given":"Andrew J.","family":"Landahl","sequence":"additional","affiliation":[]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"Alice","family":"Neels","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,7,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.92.012336","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012336","note":"arxivid:1406.2690"},{"id":"arxiv:1411.4248","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.91.022302","source":"Crossref","title":"Fault-tolerant holonomic quantum computation in surface codes","volume":"91","author":[{"given":"Yi-Cong","family":"Zheng","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,2,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.022302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022302","note":"arxivid:1411.4248"},{"id":"arxiv:quant-ph/9904011","type":"journal-article","publisher":"Elsevier BV","issue":"2-3","DOI":"10.1016/s0375-9601(99)00803-8","page":"94-99","source":"Crossref","title":"Holonomic quantum computation","volume":"264","author":[{"given":"Paolo","family":"Zanardi","sequence":"first","affiliation":[]},{"given":"Mario","family":"Rasetti","sequence":"additional","affiliation":[]}],"container-title":"Physics Letters A","original-title":[],"language":"en","issued":{"date-parts":[[1999,12]]},"URL":"http://dx.doi.org/10.1016/S0375-9601(99)00803-8","ISSN":["0375-9601"],"container-title-short":"Physics Letters A","note":"alternative-id:S0375960199008038\narxivid:quant-ph/9904011"},{"id":"arxiv:2108.10457","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Recently, Hastings &amp; Haah introduced a quantum memory defined on the honeycomb lattice. Remarkably, this honeycomb code assembles weight-six parity checks using only two-local measurements. The sparse connectivity and two-local measurements are desirable features for certain hardware, while the weight-six parity checks enable robust performance in the circuit model.In this work, we quantify the robustness of logical qubits preserved by the honeycomb code using a correlated minimum-weight perfect-matching decoder. Using Monte Carlo sampling, we estimate the honeycomb code's threshold in different error models, and project how efficiently it can reach the \"teraquop regime\" where trillions of quantum logical operations can be executed reliably. We perform the same estimates for the rotated surface code, and find a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.2</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>0.3</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> for the honeycomb code compared to a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>0.7</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> for the surface code in a controlled-not circuit model. In a circuit model with native two-body measurements, the honeycomb code achieves a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi><mml:mo>&amp;#x003C;</mml:mo><mml:mi>p</mml:mi><mml:mo>&amp;#x003C;</mml:mo><mml:mn>2.0</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>p</mml:mi></mml:math> is the collective error rate of the two-body measurement gate - including both measurement and correlated data depolarization error processes. With such gates at a physical error rate of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>, we project that the honeycomb code can reach the teraquop regime with only <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>600</mml:mn></mml:math> physical qubits. </jats:p>","DOI":"10.22331/q-2021-12-20-605","page":"605","source":"Crossref","title":"A Fault-Tolerant Honeycomb Memory","volume":"5","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Austin","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Broughton","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,20]]},"URL":"http://dx.doi.org/10.22331/q-2021-12-20-605","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2108.10457"},{"id":"arxiv:2202.11845","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We improve the planar honeycomb code by describing boundaries that need no additional physical connectivity, and by optimizing the shape of the qubit patch. We then benchmark the code using Monte Carlo sampling to estimate logical error rates and derive metrics including thresholds, lambdas, and teraquop qubit counts. We determine that the planar honeycomb code can create a logical qubit with one-in-a-trillion logical error rates using 7000 physical qubits at a 0.1% gate-level error rate (or 900 physical qubits given native two-qubit parity measurements). Our results cement the honeycomb code as a promising candidate for two-dimensional qubit architectures with sparse connectivity.</jats:p>","DOI":"10.22331/q-2022-09-21-813","page":"813","source":"Crossref","title":"Benchmarking the Planar Honeycomb Code","volume":"6","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Matt","family":"McEwen","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"},{"name":"University of California, Santa Barbara, 93106, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,21]]},"URL":"http://dx.doi.org/10.22331/q-2022-09-21-813","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2202.11845"},{"id":"arxiv:2202.11829","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.4.010310","source":"Crossref","title":"Performance of Planar Floquet Codes with Majorana-Based Qubits","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-8337-634X","authenticated-orcid":true,"given":"Adam","family":"Paetznick","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5982-8107","authenticated-orcid":true,"given":"Christina","family":"Knapp","sequence":"additional","affiliation":[]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[]},{"given":"Bela","family":"Bauer","sequence":"additional","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[]},{"given":"Marcus P.","family":"da Silva","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,1,25]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.010310","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010310","note":"arxivid:2202.11829"},{"id":"arxiv:2007.00307","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The surface code is a prominent topological error-correcting code exhibiting high fault-tolerance accuracy thresholds. Conventional schemes for error correction with the surface code place qubits on a planar grid and assume native CNOT gates between the data qubits with nearest-neighbor ancilla qubits.Here, we present surface code error-correction schemes using <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">only</mml:mtext></mml:mrow></mml:math> Pauli measurements on single qubits and on pairs of nearest-neighbor qubits. In particular, we provide several qubit layouts that offer favorable trade-offs between qubit overhead, circuit depth and connectivity degree. We also develop minimized measurement sequences for syndrome extraction, enabling reduced logical error rates and improved fault-tolerance thresholds.Our work applies to topologically protected qubits realized with Majorana zero modes and to similar systems in which multi-qubit Pauli measurements rather than CNOT gates are the native operations.</jats:p>","DOI":"10.22331/q-2020-10-28-352","page":"352","source":"Crossref","title":"Optimization of the surface code design for Majorana-based qubits","volume":"4","author":[{"given":"Rui","family":"Chao","sequence":"first","affiliation":[{"name":"University of Southern California, Los Angeles, CA, USA"}]},{"given":"Michael E.","family":"Beverland","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,10,28]]},"URL":"http://dx.doi.org/10.22331/q-2020-10-28-352","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2007.00307"},{"id":"arxiv:2206.12780","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In this paper, I present a way to compile the surface code into two-body parity measurements (\"pair measurements\"), where the pair measurements run along the edges of a Cairo pentagonal tiling. The resulting circuit improves on prior work by Chao et al. by using fewer pair measurements per four-body stabilizer measurement (5 instead of 6) and fewer time steps per round of stabilizer measurement (6 instead of 10). Using Monte Carlo sampling, I show that these improvements increase the threshold of the surface code when compiling into pair measurements from <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.2</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.4</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>, and also that they improve the teraquop footprint at a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> physical gate error rate from <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>6000</mml:mn></mml:math> qubits to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>3000</mml:mn></mml:math> qubits. However, I also show that the teraquop footprint of Chao et al's construction improves more quickly than mine as physical error rate decreases, and is likely better below a physical gate error rate of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.03</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> (due to bidirectional hook errors in my construction). I also compare to the planar honeycomb code, showing that although this work does noticeably reduce the gap between the surface code and the honeycomb code (when compiling into pair measurements), the honeycomb code is still more efficient (threshold <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.8</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>, teraquop footprint at <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>1000</mml:mn></mml:math>).</jats:p>","DOI":"10.22331/q-2023-10-25-1156","page":"1156","source":"Crossref","title":"A Pair Measurement Surface Code on Pentagons","volume":"7","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,25]]},"URL":"http://dx.doi.org/10.22331/q-2023-10-25-1156","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2206.12780"},{"id":"arxiv:2310.12981","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We devise a new realization of the surface code on a rectangular lattice of qubits utilizing single-qubit and nearest-neighbor two-qubit Pauli measurements and three auxiliary qubits per plaquette. This realization gains substantial advantages over prior pairwise measurement-based realizations of the surface code. It has a short operation period of 4 steps and our performance analysis for a standard circuit noise model yields a high fault-tolerance threshold of approximately <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.66</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>. The syndrome extraction circuits avoid bidirectional hook errors, so we can achieve full code distance by choosing appropriate boundary conditions. We also construct variants of the syndrome extraction circuits that entirely prevent hook errors, at the cost of larger circuit depth. This achieves full distance regardless of boundary conditions, with only a modest decrease in the threshold. Furthermore, we propose an efficient strategy for dealing with dead components (qubits and measurements) in our surface code realization, which can be adopted more generally for other surface code realizations. This new surface code realization is highly optimized for Majorana-based hardware, accounting for constraints imposed by layouts and the implementation of measurements, making it competitive with the recently proposed Floquet codes.</jats:p>","DOI":"10.22331/q-2024-08-02-1429","page":"1429","source":"Crossref","title":"Improved Pairwise Measurement-Based Surface Code","volume":"8","author":[{"given":"Linnea","family":"Grans-Samuelsson","sequence":"first","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105 USA"}]},{"given":"Ryan V.","family":"Mishmash","sequence":"additional","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105 USA"}]},{"given":"David","family":"Aasen","sequence":"additional","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105 USA"}]},{"given":"Christina","family":"Knapp","sequence":"additional","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105 USA"}]},{"given":"Bela","family":"Bauer","sequence":"additional","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105 USA"}]},{"given":"Brad","family":"Lackey","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Redmond, Washington 98052, USA"}]},{"given":"Marcus P. da","family":"Silva","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Redmond, Washington 98052, USA"}]},{"given":"Parsa","family":"Bonderson","sequence":"additional","affiliation":[{"name":"Microsoft Station Q, Santa Barbara, California 93106-6105 USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,2]]},"URL":"http://dx.doi.org/10.22331/q-2024-08-02-1429","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2310.12981"},{"id":"arxiv:2410.07065","type":"article-journal","author":[{"given":"Théo","family":"Dessertaine"},{"given":"Boris","family":"Bourdoncle"},{"given":"Aurélie","family":"Denys"},{"given":"Grégoire","family":"de Gliniasty"},{"given":"Pierre Colonna","family":"d'Istria"},{"given":"Gerard","family":"Valentí-Rojas"},{"given":"Shane","family":"Mansfield"},{"given":"Paul","family":"Hilaire"}],"title":"Enhanced Fault-tolerance in Photonic Quantum Computing: Comparing the Honeycomb Floquet Code and the Surface Code in Tailored Architecture","issued":{"date-parts":[[2026,1,8]]},"note":"arxivid:2410.07065\narxiv_version_number:3"},{"id":"arxiv:1610.05289","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevb.95.235305","source":"Crossref","title":"Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes","volume":"95","author":[{"given":"Torsten","family":"Karzig","sequence":"first","affiliation":[]},{"given":"Christina","family":"Knapp","sequence":"additional","affiliation":[]},{"given":"Roman M.","family":"Lutchyn","sequence":"additional","affiliation":[]},{"given":"Parsa","family":"Bonderson","sequence":"additional","affiliation":[]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[]},{"given":"Chetan","family":"Nayak","sequence":"additional","affiliation":[]},{"given":"Jason","family":"Alicea","sequence":"additional","affiliation":[]},{"given":"Karsten","family":"Flensberg","sequence":"additional","affiliation":[]},{"given":"Stephan","family":"Plugge","sequence":"additional","affiliation":[]},{"given":"Yuval","family":"Oreg","sequence":"additional","affiliation":[]},{"given":"Charles M.","family":"Marcus","sequence":"additional","affiliation":[]},{"given":"Michael H.","family":"Freedman","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,6,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.95.235305","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"235305","note":"arxivid:1610.05289"},{"id":"arxiv:2310.06917","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>We consider the <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:msub><a:mrow><a:mi mathvariant=\"double-struck\">Z</a:mi></a:mrow><a:mn>2</a:mn></a:msub></a:math> toric code, surface code, and Floquet code defined on a nonorientable surface, which can be considered as families of codes extending Shor’s nine-qubit code. We investigate the fault-tolerant logical gates of the <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><e:msub><e:mrow><e:mi mathvariant=\"double-struck\">Z</e:mi></e:mrow><e:mn>2</e:mn></e:msub></e:math> toric code in this setup, which corresponds to <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><i:mi>e</i:mi><i:mo stretchy=\"false\">↔</i:mo><i:mi>m</i:mi></i:math> exchanging symmetry of the underlying <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><m:msub><m:mrow><m:mi mathvariant=\"double-struck\">Z</m:mi></m:mrow><m:mn>2</m:mn></m:msub></m:math> gauge theory. We find that nonorientable geometry provides a new way for the emergent symmetry to act on the code space, and discover the new realization of the fault-tolerant Hadamard gate of the two-dimensional surface code with a single cross cap connecting the vertices nonlocally along a slit, dubbed a nonorientable surface code. This Hadamard gate can be realized by a constant-depth local unitary circuit modulo nonlocality caused by a cross cap. Via folding, the nonorientable surface code can be turned into a bilayer local quantum code, where the folded cross cap is equivalent to a bilayer twist terminated on a gapped boundary and the logical Hadamard only contains local gates with intralayer couplings when being away from the cross cap, as opposed to the interlayer couplings on each site needed in the case of the folded surface code. We further obtain the complete logical Clifford gate set for a stack of nonorientable surface codes and similarly for codes defined on Klein-bottle geometries. We then construct the honeycomb Floquet code in the presence of a single cross cap, and find that the period of the sequential Pauli measurements acts as a <q:math xmlns:q=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><q:mi>H</q:mi><q:mi>Z</q:mi></q:math> logical gate on the single logical qubit, where the cross cap enriches the dynamics compared with the orientable case. We find that the dynamics of the honeycomb Floquet code is precisely described by a condensation operator of the <t:math xmlns:t=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><t:msub><t:mrow><t:mi mathvariant=\"double-struck\">Z</t:mi></t:mrow><t:mn>2</t:mn></t:msub></t:math> gauge theory, and illustrate the exotic dynamics of our code in terms of a condensation operator supported at a nonorientable surface.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020360","source":"Crossref","title":"Cross-Cap Defects and Fault-Tolerant Logical Gates in the Surface Code and the Honeycomb Floquet Code","volume":"5","author":[{"given":"Ryohei","family":"Kobayashi","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM T.J. Watson Research Center"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,13]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020360","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020360","note":"arxivid:2310.06917"},{"id":"arxiv:2211.05784","type":"article-journal","author":[{"given":"Zhehao","family":"Zhang"},{"given":"David","family":"Aasen"},{"given":"Sagar","family":"Vijay"}],"title":"The X-Cube Floquet Code","issued":{"date-parts":[[2022,11,10]]},"note":"arxivid:2211.05784\narxiv_version_number:1"},{"id":"arxiv:2403.04163","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We ask what is the general framework for a quantum error correcting code that is defined by a sequence of measurements. Recently, there has been much interest in Floquet codes and space-time codes. In this work, we define and study the distance of a dynamical code. This is a subtle concept and difficult to determine: At any given time, the system will be in a subspace which forms a quantum error-correcting code with a given distance, but the full error correction capability of that code may not be available due to the schedule of measurements associated with the code. We address this challenge by developing an algorithm that tracks information we have learned about the error syndromes through the protocol and put that together to determine the distance of a dynamical code, in a non-fault-tolerant context. We use the tools developed for the algorithm to analyze the initialization and masking properties of a generic Floquet code. Further, we look at properties of dynamical codes under the constraint of geometric locality with a view to understand whether the fundamental limitations on logical gates and code parameters imposed by geometric locality for traditional codes can be surpassed in the dynamical paradigm. We find that codes with a limited number of long range connectivity will not allow non-Clifford gates to be implemented with finite depth circuits in the 2D setting.</jats:p>","DOI":"10.22331/q-2025-10-20-1886","page":"1886","source":"Crossref","title":"Error Correction in Dynamical Codes","volume":"9","author":[{"given":"Esther Xiaozhen","family":"Fu","sequence":"first","affiliation":[{"name":"QuICS, University of Maryland, College Park, MD 20742, USA."}]},{"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[{"name":"QuICS, University of Maryland, College Park, MD 20742, USA."},{"name":"Computer Science Department, University of Maryland, College Park, MD 20742, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,20]]},"URL":"http://dx.doi.org/10.22331/q-2025-10-20-1886","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2403.04163"},{"id":"arxiv:2410.17240","type":"article-journal","author":[{"given":"Benjamin","family":"Rodatz"},{"given":"Boldizsár","family":"Poór"},{"given":"Aleks","family":"Kissinger"}],"title":"Floquetifying stabiliser codes with distance-preserving rewrites","issued":{"date-parts":[[2024,12,16]]},"note":"arxivid:2410.17240\narxiv_version_number:2"},{"id":"arxiv:2307.11136","type":"journal-article","publisher":"Open Publishing Association","DOI":"10.4204/eptcs.384.14","page":"265-303","source":"Crossref","title":"Floquetifying the Colour Code","volume":"384","author":[{"given":"Alex","family":"Townsend-Teague","sequence":"first","affiliation":[]},{"given":"Julio","family":"Magdalena de la Fuente","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Kesselring","sequence":"additional","affiliation":[]}],"container-title":"Electronic Proceedings in Theoretical Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,30]]},"URL":"http://dx.doi.org/10.4204/EPTCS.384.14","ISSN":["2075-2180"],"container-title-short":"Electron. Proc. Theor. Comput. Sci.","note":"arxivid:2307.11136"},{"id":"arxiv:2504.08918","type":"article-journal","author":[{"given":"Yichen","family":"Xu"},{"given":"Arpit","family":"Dua"}],"title":"Fault-tolerant protocols through spacetime concatenation","issued":{"date-parts":[[2026,1,19]]},"note":"arxivid:2504.08918\narxiv_version_number:4"},{"id":"arxiv:2510.05549","type":"article-journal","author":[{"given":"Keller","family":"Blackwell"},{"given":"Jeongwan","family":"Haah"}],"title":"The code distance of Floquet codes","issued":{"date-parts":[[2025,10,7]]},"note":"arxivid:2510.05549\narxiv_version_number:1"},{"id":"arxiv:2307.03715","type":"article-journal","author":[{"given":"David","family":"Aasen"},{"given":"Jeongwan","family":"Haah"},{"given":"Parsa","family":"Bonderson"},{"given":"Zhenghan","family":"Wang"},{"given":"Matthew","family":"Hastings"}],"title":"Fault-Tolerant Hastings-Haah Codes in the Presence of Dead Qubits","issued":{"date-parts":[[2023,7,26]]},"note":"arxivid:2307.03715\narxiv_version_number:2"},{"id":"arxiv:2405.15854","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Floquet codes are an intriguing generalisation of stabiliser and subsystem codes, which can provide good fault-tolerant characteristics while benefiting from reduced connectivity requirements in hardware. A recent question of interest has been how to run Floquet codes on devices which have defective – and therefore unusable – qubits. This is an under-studied issue of crucial importance for running such codes on realistic hardware. To address this challenge, we introduce a new method of accommodating defective qubits on a wide range of two-dimensional Floquet codes, which requires no additional connectivity in the underlying quantum hardware, no modifications to the original Floquet code's measurement schedule, can accommodate boundaries, and is optimal in terms of the number of qubits and stabilisers removed. We numerically demonstrate that, using this method, the planar honeycomb code is fault tolerant up to a fabrication defect probability of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>12</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>. We find the fault-tolerant performance of this code under defect noise is competitive with that of the surface code, despite its sparser connectivity. We finally propose multiple ways this approach can be adapted to the underlying hardware, through utilising any additional connectivity available, and treating defective auxiliary qubits separately to defective data qubits. Our work therefore serves as a guide for the implementation of Floquet codes in realistic quantum hardware.</jats:p>","DOI":"10.22331/q-2024-12-12-1562","page":"1562","source":"Crossref","title":"Accommodating Fabrication Defects on Floquet Codes with Minimal Hardware Requirements","volume":"8","author":[{"given":"Campbell","family":"McLauchlan","sequence":"first","affiliation":[{"name":"Riverlane, St Andrews House, 59 St Andrews Street, Cambridge, CB2 3BZ, United Kingdom"},{"name":"Department of Physics and Astronomy, University College London, London, WC1E 6BT, United Kingdom"}]},{"given":"György P.","family":"Gehér","sequence":"additional","affiliation":[{"name":"Riverlane, St Andrews House, 59 St Andrews Street, Cambridge, CB2 3BZ, United Kingdom"}]},{"given":"Alexandra E.","family":"Moylett","sequence":"additional","affiliation":[{"name":"Riverlane, St Andrews House, 59 St Andrews Street, Cambridge, CB2 3BZ, United Kingdom"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,12]]},"URL":"http://dx.doi.org/10.22331/q-2024-12-12-1562","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2405.15854"},{"id":"arxiv:2409.13681","type":"article-journal","author":[{"given":"Linnea","family":"Grans-Samuelsson"},{"given":"David","family":"Aasen"},{"given":"Parsa","family":"Bonderson"}],"title":"A fault-tolerant pairwise measurement-based code on eight qubits","issued":{"date-parts":[[2024,9,20]]},"note":"arxivid:2409.13681\narxiv_version_number:1"},{"id":"arxiv:2504.02749","type":"article-journal","author":[{"given":"M. Sohaib","family":"Alam"},{"given":"Jun","family":"Zen"},{"given":"Thomas R.","family":"Scruby"}],"title":"Bacon-Shor Board Games","issued":{"date-parts":[[2026,2,6]]},"note":"arxivid:2504.02749\narxiv_version_number:2"},{"id":"arxiv:2308.03750","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>We construct families of Floquet codes derived from color-code tilings of closed hyperbolic surfaces. These codes have weight-two check operators, a finite encoding rate and can be decoded efficiently with minimum-weight perfect matching. We also construct semi-hyperbolic Floquet codes, which have improved distance scaling, and are obtained via a fine-graining procedure. Using a circuit-based noise model that assumes direct two-qubit measurements, we show that semi-hyperbolic Floquet codes can be <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mn>48</a:mn></a:math> times more efficient than planar honeycomb codes and therefore over <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><d:mn>100</d:mn></d:math> times more efficient than alternative compilations of the surface code to two-qubit measurements, even at physical error rates of 0.3% to 1%. We further demonstrate that semi-hyperbolic Floquet codes can have a teraquop footprint of only 32 physical qubits per logical qubit at a noise strength of 0.1%. For standard circuit-level depolarizing noise at <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><g:mi>p</g:mi><g:mo>=</g:mo><g:mn>0.1</g:mn><g:mi mathvariant=\"normal\">%</g:mi></g:math>, we find a <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><k:mn>30</k:mn></k:math> times improvement over planar honeycomb codes and a <n:math xmlns:n=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><n:mn>5.6</n:mn></n:math> times improvement over surface codes. Finally, we analyze small instances that are amenable to near-term experiments, including a Floquet code derived from the Bolza surface that encodes four logical qubits into 16 physical qubits.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.040327","source":"Crossref","title":"Constructions and Performance of Hyperbolic and Semi-Hyperbolic Floquet Codes","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0001-9880-5218","authenticated-orcid":true,"given":"Oscar","family":"Higgott","sequence":"first","affiliation":[]},{"given":"Nikolas P.","family":"Breuckmann","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,22]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.040327","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040327","note":"arxivid:2308.03750"},{"id":"arxiv:2309.10033","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>A central goal in quantum error correction is to reduce the overhead of fault-tolerant quantum computing by increasing noise thresholds and reducing the number of physical qubits required to sustain a logical qubit. We introduce a potential path towards this goal based on a family of dynamically generated quantum error correcting codes that we call \"hyperbolic Floquet codes.'' These codes are defined by a specific sequence of non-commuting two-body measurements arranged periodically in time that stabilize a topological code on a hyperbolic manifold with negative curvature. We focus on a family of lattices for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> qubits that, according to our prescription that defines the code, provably achieve a finite encoding rate <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>8</mml:mn><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> while still requiring only two-body measurements. Similar to hyperbolic surface codes, the distance of the code at each time-step scales at most logarithmically in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>. The family of lattices we choose indicates that this scaling is achievable in practice. We develop and benchmark an efficient matching-based decoder that provides evidence of a threshold near 0.1% in a phenomenological noise model and 0.25% in an entangling measurements noise model. Utilizing weight-two check operators and a qubit connectivity of 3, one of our hyperbolic Floquet codes uses 400 physical qubits to encode 52 logical qubits with a code distance of 8, i.e., it is a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>400</mml:mn><mml:mo>,</mml:mo><mml:mn>52</mml:mn><mml:mo>,</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> code. At small error rates, comparable logical error suppression to this code requires 5x as many physical qubits (1924) when using the honeycomb Floquet code with the same noise model and decoder.</jats:p>","DOI":"10.22331/q-2025-09-05-1849","page":"1849","source":"Crossref","title":"Fault-tolerant hyperbolic Floquet quantum error correcting codes","volume":"9","author":[{"given":"Ali","family":"Fahimniya","sequence":"first","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Hossein","family":"Dehghani","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Kishor","family":"Bharti","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, # 16-16 Connexis, Singapore 138632, Republic of Singapore"}]},{"given":"Sheryl","family":"Mathew","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Alicia J.","family":"Kollár","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"},{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,5]]},"URL":"http://dx.doi.org/10.22331/q-2025-09-05-1849","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2309.10033"},{"id":"arxiv:2501.14029","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce65121.2025.00076","page":"649-657","source":"Crossref","title":"Distributed Quantum Error Correction Based on Hyperbolic Floquet Codes","author":[{"given":"Evan","family":"Sutcliffe","sequence":"first","affiliation":[{"name":"Nu Quantum Ltd.,Cambridge,United Kingdom"}]},{"given":"Bhargavi","family":"Jonnadula","sequence":"additional","affiliation":[{"name":"Nu Quantum Ltd.,Cambridge,United Kingdom"}]},{"given":"Claire","family":"Le Gall","sequence":"additional","affiliation":[{"name":"Nu Quantum Ltd.,Cambridge,United Kingdom"}]},{"given":"Alexandra E.","family":"Moylett","sequence":"additional","affiliation":[{"name":"Nu Quantum Ltd.,Cambridge,United Kingdom"}]},{"given":"Coral M.","family":"Westoby","sequence":"additional","affiliation":[{"name":"Nu Quantum Ltd.,Cambridge,United Kingdom"}]}],"event":"2025 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2025 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2025,8,30]]},"URL":"http://dx.doi.org/10.1109/QCE65121.2025.00076","note":"arxivid:2501.14029"},{"id":"doi:10.26081/K6WH2V","type":"article","author":[{"literal":"Arpit Dua"}],"issued":{"date-parts":[[2022]]},"DOI":"10.26081/K6WH2V","publisher":"Kavli Institute for Theoretical Physics","title":"Subsystem Codes","URL":"https://online.kitp.ucsb.edu/online/dynisq22/dua/"},{"id":"preset:albertBoulder21","type":"document","author":[{"family":"Albert","given":"V.V."}],"title":"\\href{https://boulderschool.yale.edu/sites/default/files/files/BSS_notes_-_VVA.pdf}{Boulder School 2023 Lecture notes}"},{"id":"arxiv:2311.17985","type":"article-journal","author":[{"given":"Jon","family":"Nelson"},{"given":"Gregory","family":"Bentsen"},{"given":"Steven T.","family":"Flammia"},{"given":"Michael J.","family":"Gullans"}],"title":"Fault-Tolerant Quantum Memory using Low-Depth Random Circuit Codes","issued":{"date-parts":[[2023,11,29]]},"note":"arxivid:2311.17985\narxiv_version_number:1"},{"id":"arxiv:2210.10808","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.4.030313","source":"Crossref","title":"Crystalline Quantum Circuits","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0003-4229-8282","authenticated-orcid":true,"given":"Grace M.","family":"Sommers","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1008-5178","authenticated-orcid":true,"given":"David A.","family":"Huse","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,31]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.030313","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030313","note":"update-to:{\"updated\":{\"date-parts\":[[2023,11,15]],\"date-time\":\"2023-11-15T00:00:00Z\",\"timestamp\":1700006400000},\"DOI\":\"10.1103/prxquantum.4.030313\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2023,11,15]],\"date-time\":\"2023-11-15T00:00:00Z\",\"timestamp\":1700006400000},\"DOI\":\"10.1103/prxquantum.4.030313\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2210.10808"},{"id":"arxiv:1703.01517","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.2.033042","source":"Crossref","title":"Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel","volume":"2","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6041-9554","authenticated-orcid":true,"given":"Gilles","family":"Zémor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033042","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033042","note":"arxivid:1703.01517"},{"id":"arxiv:2010.09775","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.11.031066","source":"Crossref","title":"Quantum Coding with Low-Depth Random Circuits","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"first","affiliation":[]},{"given":"Stefan","family":"Krastanov","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1008-5178","authenticated-orcid":true,"given":"David A.","family":"Huse","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2021,9,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.11.031066","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031066","note":"arxivid:2010.09775"},{"id":"arxiv:quant-ph/0512247","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s00220-006-0118-x","page":"107-136","source":"Crossref","title":"Quantum State Merging and Negative Information","volume":"269","author":[{"given":"Michał","family":"Horodecki","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Oppenheim","sequence":"additional","affiliation":[]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2006,10,21]]},"URL":"http://dx.doi.org/10.1007/s00220-006-0118-x","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:118\narxivid:quant-ph/0512247"},{"id":"doi:10.1137/0717034","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","DOI":"10.1137/0717034","page":"403-409","source":"Crossref","title":"The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators","volume":"17","author":[{"given":"G. W.","family":"Stewart","sequence":"first","affiliation":[]}],"container-title":"SIAM Journal on Numerical Analysis","original-title":[],"language":"en","issued":{"date-parts":[[1980,6]]},"URL":"http://dx.doi.org/10.1137/0717034","ISSN":["0036-1429","1095-7170"],"container-title-short":"SIAM J. Numer. Anal.","note":"alternative-id:10.1137/0717034"},{"id":"arxiv:0712.2558","type":"article-journal","author":[{"given":"Rochus","family":"Klesse"}],"title":"A random-coding based proof for the quantum coding theorem","issued":{"date-parts":[[2007,12,16]]},"note":"arxivid:0712.2558\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0701102","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.75.062315","source":"Crossref","title":"Approximate quantum error correction, random codes, and quantum channel capacity","volume":"75","author":[{"given":"Rochus","family":"Klesse","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,6,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.062315","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062315","note":"arxivid:quant-ph/0701102"},{"id":"arxiv:2212.05071","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Recent work [M. J. Gullans  , ] has shown that quantum error correcting codes defined by random Clifford encoding circuits can achieve a nonzero encoding rate in correcting errors even if the random circuits on <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\"><a:mi>n</a:mi></a:math> qubits, embedded in one spatial dimension (1D), have a logarithmic depth <b:math xmlns:b=\"http://www.w3.org/1998/Math/MathML\"><b:mrow><b:mi>d</b:mi><b:mo>=</b:mo><b:mi>O</b:mi><b:mo>(</b:mo><b:mo form=\"prefix\">log</b:mo><b:mi>n</b:mi><b:mo>)</b:mo></b:mrow></b:math>. However, this was demonstrated only for a simple erasure noise model. In this work, we discover that, for the same class of codes, this desired property indeed holds for the conventional Pauli noise model. Specifically, we numerically demonstrate that the hashing bound, i.e., a rate known to be achieved with <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\"><d:mrow><d:mi>d</d:mi><d:mo>=</d:mo><d:mi>O</d:mi><d:mo>(</d:mo><d:mi>n</d:mi><d:mo>)</d:mo></d:mrow></d:math>-depth random encoding circuits, can be attained for the above codes even when the circuit depth is restricted to <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\"><e:mrow><e:mi>d</e:mi><e:mo>=</e:mo><e:mi>O</e:mi><e:mo>(</e:mo><e:mo form=\"prefix\">log</e:mo><e:mi>n</e:mi><e:mo>)</e:mo></e:mrow></e:math> in 1D for depolarizing noise of various strengths. This analysis is made possible with our development of a tensor-network maximum-likelihood decoding algorithm that works efficiently for <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\"><g:mo form=\"prefix\">log</g:mo></g:math>-depth encoding circuits in 1D.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevresearch.6.023055","source":"Crossref","title":"Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder","volume":"6","author":[{"given":"Andrew S.","family":"Darmawan","sequence":"first","affiliation":[{"name":"Yukawa Institute of Theoretical Physics (YITP), Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan"},{"name":"JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan"}]},{"ORCID":"https://orcid.org/0000-0003-1285-6968","authenticated-orcid":true,"given":"Yoshifumi","family":"Nakata","sequence":"additional","affiliation":[{"name":"Yukawa Institute of Theoretical Physics (YITP), Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan"}]},{"ORCID":"https://orcid.org/0000-0001-7171-1713","authenticated-orcid":true,"given":"Shiro","family":"Tamiya","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan"}]},{"ORCID":"https://orcid.org/0000-0003-3521-831X","authenticated-orcid":true,"given":"Hayata","family":"Yamasaki","sequence":"additional","affiliation":[{"name":"Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan"},{"name":"JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.6.023055","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023055","note":"arxivid:2212.05071"},{"id":"arxiv:quant-ph/0609067","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.99.070502","source":"Crossref","title":"Simple Proof of Equivalence between Adiabatic Quantum Computation and the Circuit Model","volume":"99","author":[{"given":"Ari","family":"Mizel","sequence":"first","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]},{"given":"Morgan","family":"Mitchell","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,8,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.99.070502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070502","note":"arxivid:quant-ph/0609067"},{"id":"arxiv:1311.6101","type":"journal-article","publisher":"IOP Publishing","issue":"19","DOI":"10.1088/1751-8113/47/19/195304","page":"195304","source":"Crossref","title":"Space-time circuit-to-Hamiltonian construction and its applications","volume":"47","author":[{"given":"Nikolas P","family":"Breuckmann","sequence":"first","affiliation":[]},{"given":"Barbara M","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2014,4,23]]},"URL":"http://dx.doi.org/10.1088/1751-8113/47/19/195304","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1311.6101"},{"id":"arxiv:2304.05943","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Adam","family":"Paetznick"}],"title":"Spacetime codes of Clifford circuits","issued":{"date-parts":[[2023,5,26]]},"note":"arxivid:2304.05943\narxiv_version_number:2"},{"id":"arxiv:2210.15844","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"Opportunities and Challenges in Fault-Tolerant Quantum Computation","issued":{"date-parts":[[2022,10,28]]},"note":"arxivid:2210.15844\narxiv_version_number:1"},{"id":"arxiv:2506.17181","type":"article-journal","author":[{"given":"Benjamin","family":"Rodatz"},{"given":"Boldizsár","family":"Poór"},{"given":"Aleks","family":"Kissinger"}],"title":"Fault Tolerance by Construction","issued":{"date-parts":[[2026,3,31]]},"note":"arxivid:2506.17181\narxiv_version_number:4"},{"id":"arxiv:2502.16408","type":"article-journal","author":[{"given":"Kai R.","family":"Ott"},{"given":"Bence","family":"Hetényi"},{"given":"Michael E.","family":"Beverland"}],"title":"Decision-tree decoders for general quantum LDPC codes","issued":{"date-parts":[[2025,2,23]]},"note":"arxivid:2502.16408\narxiv_version_number:1"},{"id":"arxiv:2103.02202","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>This paper presents “Stim\", a fast simulator for quantum stabilizer circuits. The paper explains how Stim works and compares it to existing tools. With no foreknowledge, Stim can analyze a distance 100 surface code circuit (20 thousand qubits, 8 million gates, 1 million measurements) in 15 seconds and then begin sampling full circuit shots at a rate of 1 kHz. Stim uses a stabilizer tableau representation, similar to Aaronson and Gottesman's CHP simulator, but with three main improvements. First, Stim improves the asymptotic complexity of deterministic measurement from quadratic to linear by tracking the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>v</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi><mml:mi>s</mml:mi><mml:mi>e</mml:mi></mml:math> of the circuit's stabilizer tableau. Second, Stim improves the constant factors of the algorithm by using a cache-friendly data layout and 256 bit wide SIMD instructions. Third, Stim only uses expensive stabilizer tableau simulation to create an initial reference sample. Further samples are collected in bulk by using that sample as a reference for batches of Pauli frames propagating through the circuit.</jats:p>","DOI":"10.22331/q-2021-07-06-497","page":"497","source":"Crossref","title":"Stim: a fast stabilizer circuit simulator","volume":"5","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,7,6]]},"URL":"http://dx.doi.org/10.22331/q-2021-07-06-497","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2103.02202"},{"id":"arxiv:1010.5506","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/s11128-013-0704-8","page":"957-990","source":"Crossref","title":"Dualities and identities for entanglement-assisted quantum codes","volume":"13","author":[{"given":"Ching-Yi","family":"Lai","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2013,12,17]]},"URL":"http://dx.doi.org/10.1007/s11128-013-0704-8","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:704\narxivid:1010.5506"},{"id":"arxiv:quant-ph/9904023","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevlett.83.3081","page":"3081-3084","source":"Crossref","title":"Entanglement-Assisted Classical Capacity of Noisy Quantum Channels","volume":"83","author":[{"given":"Charles H.","family":"Bennett","sequence":"first","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]},{"given":"John A.","family":"Smolin","sequence":"additional","affiliation":[]},{"given":"Ashish V.","family":"Thapliyal","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1999,10,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.83.3081","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9904023"},{"id":"arxiv:2408.17290","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>The optimal rate of reliable communication over a quantum channel can be enhanced by preshared entanglement. Whereas the enhancement may be unbounded in infinite-dimensional settings even when the input power is constrained, a long-standing conjecture asserts that the ratio between the entanglement-assisted and unassisted classical capacities is bounded in finite-dimensional settings [Bennett , ]. In this Letter, we prove this conjecture by showing that their ratio is upper bounded by <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mrow><a:mi>o</a:mi><a:mo stretchy=\"false\">(</a:mo><a:msup><a:mrow><a:mi>d</a:mi></a:mrow><a:mrow><a:mn>2</a:mn></a:mrow></a:msup><a:mo stretchy=\"false\">)</a:mo></a:mrow></a:math>, where <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>d</e:mi></e:math> is the input dimension of the channel. An application to quantum communication with noisy encoders and decoders is given.</jats:p>","DOI":"10.1103/physrevlett.134.020802","source":"Crossref","title":"Fundamental Limit on the Power of Entanglement Assistance in Quantum Communication","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0002-2562-2629","authenticated-orcid":true,"given":"Lasse H.","family":"Wolff","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/035b05819","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Copenhagen"}]},{"ORCID":"https://orcid.org/0000-0003-0834-613X","authenticated-orcid":true,"given":"Paula","family":"Belzig","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Waterloo"}]},{"ORCID":"https://orcid.org/0000-0003-2281-3355","authenticated-orcid":true,"given":"Matthias","family":"Christandl","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/035b05819","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Copenhagen"}]},{"ORCID":"https://orcid.org/0000-0002-0450-6792","authenticated-orcid":true,"given":"Bergfinnur","family":"Durhuus","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/035b05819","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Copenhagen"}]},{"ORCID":"https://orcid.org/0000-0001-5410-3329","authenticated-orcid":true,"given":"Marco","family":"Tomamichel","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01tgyzw49","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Singapore"},{"id":[{"id":"https://ror.org/01mgdzc49","id-type":"ROR","asserted-by":"publisher"}],"name":"Centre for Quantum Technologies"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.020802","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020802","note":"arxivid:2408.17290"},{"id":"arxiv:0806.4214","type":"article-journal","author":[{"given":"Mark M.","family":"Wilde"}],"title":"Quantum Coding with Entanglement","issued":{"date-parts":[[2008,6,25]]},"note":"arxivid:0806.4214\narxiv_version_number:1"},{"id":"arxiv:1109.3358","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.84.062321","source":"Crossref","title":"Entanglement-assisted codeword stabilized quantum codes","volume":"84","author":[{"given":"Jeonghwan","family":"Shin","sequence":"first","affiliation":[]},{"given":"Jun","family":"Heo","sequence":"additional","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,12,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.062321","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062321","note":"arxivid:1109.3358"},{"id":"arxiv:1210.1895","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.112.240502","source":"Crossref","title":"Class of Highly Entangled Many-Body States that can be Efficiently Simulated","volume":"112","author":[{"given":"G.","family":"Evenbly","sequence":"first","affiliation":[]},{"given":"G.","family":"Vidal","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,6,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.112.240502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240502","note":"arxivid:1210.1895"},{"id":"arxiv:0906.5532","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2011.2104590","page":"1761-1769","source":"Crossref","title":"High Performance Entanglement-Assisted Quantum LDPC Codes Need Little Entanglement","volume":"57","author":[{"given":"Min-Hsiu","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"Wen-Tai","family":"Yen","sequence":"additional","affiliation":[]},{"given":"Li-Yi","family":"Hsu","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2011,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2011.2104590","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0906.5532"},{"id":"arxiv:0803.0100","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.79.032340","source":"Crossref","title":"Entanglement-assisted quantum quasicyclic low-density parity-check codes","volume":"79","author":[{"given":"Min-Hsiu","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]},{"given":"Igor","family":"Devetak","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.032340","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032340","note":"arxivid:0803.0100"},{"id":"doi:10.1364/OL.35.001464","type":"journal-article","publisher":"Optica Publishing Group","issue":"9","DOI":"10.1364/ol.35.001464","page":"1464","source":"Crossref","title":"Photonic entanglement-assisted quantum low-density parity-check encoders and decoders","volume":"35","author":[{"given":"Ivan B.","family":"Djordjevic","sequence":"first","affiliation":[]}],"container-title":"Optics Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,4,30]]},"URL":"http://dx.doi.org/10.1364/OL.35.001464","ISSN":["0146-9592","1539-4794"],"container-title-short":"Opt. Lett."},{"id":"arxiv:0712.2223","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.81.042333","source":"Crossref","title":"Entanglement-assisted quantum convolutional coding","volume":"81","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,4,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.042333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042333","note":"arxivid:0712.2223"},{"id":"arxiv:0807.3803","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","DOI":"10.1007/s11128-010-0179-9","page":"509-540","source":"Crossref","title":"Quantum convolutional coding with shared entanglement: general structure","volume":"9","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2010,6,8]]},"URL":"http://dx.doi.org/10.1007/s11128-010-0179-9","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:179\narxivid:0807.3803"},{"id":"arxiv:0812.4449","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.79.032313","source":"Crossref","title":"Extra shared entanglement reduces memory demand in quantum convolutional coding","volume":"79","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.032313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032313","note":"arxivid:0812.4449"},{"id":"doi:10.1109/ISIT.2011.6034165","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2011.6034165","source":"Crossref","title":"Entanglement boosts quantum turbo codes","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[]}],"event":"2011 IEEE International Symposium on Information Theory - ISIT","container-title":"2011 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2011,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2011.6034165"},{"id":"arxiv:1010.1256","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2013.2292052","page":"1203-1222","source":"Crossref","title":"Entanglement-Assisted Quantum Turbo Codes","volume":"60","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[]},{"given":"Zunaira","family":"Babar","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2014,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2292052","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1010.1256"},{"id":"arxiv:1302.4150","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2013.2246274","page":"4020-4024","source":"Crossref","title":"Duality in Entanglement-Assisted Quantum Error Correction","volume":"59","author":[{"given":"Ching-Yi","family":"Lai","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2246274","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1302.4150"},{"id":"arxiv:1302.5081","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.110.170501","source":"Crossref","title":"Quantum Error Correction via Less Noisy Qubits","volume":"110","author":[{"given":"Yuichiro","family":"Fujiwara","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2013,4,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.110.170501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"170501","note":"arxivid:1302.5081"},{"id":"doi:10.1007/s11128-015-1143-5","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","DOI":"10.1007/s11128-015-1143-5","page":"4427-4447","source":"Crossref","title":"On entanglement-assisted quantum codes achieving the entanglement-assisted Griesmer bound","volume":"14","author":[{"given":"Ruihu","family":"Li","sequence":"first","affiliation":[]},{"given":"Xueliang","family":"Li","sequence":"additional","affiliation":[]},{"given":"Luobin","family":"Guo","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2015,10,19]]},"URL":"http://dx.doi.org/10.1007/s11128-015-1143-5","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:1143"},{"id":"doi:10.1103/PhysRevA.87.032309","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.87.032309","source":"Crossref","title":"Linear Plotkin bound for entanglement-assisted quantum codes","volume":"87","author":[{"given":"Luobin","family":"Guo","sequence":"first","affiliation":[]},{"given":"Ruihu","family":"Li","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,3,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.032309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032309"},{"id":"arxiv:2405.07242","type":"article-journal","author":[{"given":"Abhi Kumar","family":"Sharma"},{"given":"Shayan Srinivasa","family":"Garani"}],"title":"Fault-Tolerant Quantum LDPC Encoders","issued":{"date-parts":[[2024,5,12]]},"note":"arxivid:2405.07242\narxiv_version_number:1"},{"id":"arxiv:2007.01249","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.103.l020601","source":"Crossref","title":"Entanglement-assisted quantum communication beating the quantum Singleton bound","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":true,"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.L020601","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"L020601","note":"arxivid:2007.01249"},{"id":"arxiv:2207.05647","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/s11128-023-04211-x","source":"Crossref","title":"Constructing quantum error-correcting codes that require a variable amount of entanglement","volume":"23","author":[{"ORCID":"https://orcid.org/0000-0003-1482-4783","authenticated-orcid":false,"given":"Gaojun","family":"Luo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5851-2717","authenticated-orcid":false,"given":"Martianus Frederic","family":"Ezerman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1978-3557","authenticated-orcid":false,"given":"San","family":"Ling","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,26]]},"URL":"http://dx.doi.org/10.1007/s11128-023-04211-x","ISSN":["1573-1332"],"container-title-short":"Quantum Inf Process","page":"4","note":"alternative-id:4211\narxivid:2207.05647"},{"id":"arxiv:2205.13732","type":"article-journal","author":[{"given":"Daiki","family":"Ueno"},{"given":"Ryutaroh","family":"Matsumoto"}],"title":"Explicit method to make shortened stabilizer EAQECC from stabilizer QECC","issued":{"date-parts":[[2022,5,27]]},"note":"arxivid:2205.13732\narxiv_version_number:1"},{"id":"arxiv:2412.16082","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/qn1w-q165","source":"Crossref","title":"Bounds on concatenated entanglement-assisted quantum error-correcting codes","volume":"112","author":[{"ORCID":"https://orcid.org/0009-0009-0139-798X","authenticated-orcid":true,"given":"Nihar Ranjan","family":"Dash","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03yacj906","id-type":"ROR","asserted-by":"publisher"}],"name":"Indian Institute of Technology Jodhpur"}]},{"ORCID":"https://orcid.org/0009-0001-9181-3677","authenticated-orcid":true,"given":"Sanjoy","family":"Dutta","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00wyj1j88","id-type":"ROR","asserted-by":"publisher"}],"name":"Poornaprajna Institute of Scientific Research"}]},{"ORCID":"https://orcid.org/0000-0001-7581-2546","authenticated-orcid":true,"given":"R.","family":"Srikanth","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00wyj1j88","id-type":"ROR","asserted-by":"publisher"}],"name":"Poornaprajna Institute of Scientific Research"}]},{"given":"Subhashish","family":"Banerjee","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03yacj906","id-type":"ROR","asserted-by":"publisher"}],"name":"Indian Institute of Technology Jodhpur"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,10]]},"URL":"http://dx.doi.org/10.1103/qn1w-q165","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052419","note":"arxivid:2412.16082"},{"id":"arxiv:0803.1495","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.78.012337","source":"Crossref","title":"Encoding one logical qubit into six physical qubits","volume":"78","author":[{"given":"Bilal","family":"Shaw","sequence":"first","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]},{"given":"Ognyan","family":"Oreshkov","sequence":"additional","affiliation":[]},{"given":"Isaac","family":"Kremsky","sequence":"additional","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,7,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.012337","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012337","note":"arxivid:0803.1495"},{"id":"arxiv:quant-ph/0205128","type":"proceedings-article","publisher":"IEEE Comput. Soc","DOI":"10.1109/sfcs.2002.1181969","page":"449-458","source":"Crossref","title":"Authentication of quantum messages","author":[{"given":"H.","family":"Barnum","sequence":"first","affiliation":[]},{"given":"C.","family":"Crepeau","sequence":"additional","affiliation":[]},{"given":"D.","family":"Gottesman","sequence":"additional","affiliation":[]},{"given":"A.","family":"Smith","sequence":"additional","affiliation":[]},{"given":"A.","family":"Tapp","sequence":"additional","affiliation":[]}],"event":"43rd Annual IEEE Symposium on Foundations of Computer Science","container-title":"The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.","original-title":[],"issued":{"date-parts":[[null]]},"URL":"http://dx.doi.org/10.1109/SFCS.2002.1181969","note":"arxivid:quant-ph/0205128"},{"id":"arxiv:2303.08294","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present the construction of standard entanglement-assisted (EA) qubit Reed-Muller (RM) codes and their tensor product variants from classical RM codes. We show that the EA RM codes obtained using the CSS construction have zero coding rate and negative catalytic rate. We further show that EA codes constructed from these same classical RM codes using the tensor product code (TPC) construction have positive coding rate and provide a subclass of EA RM TPCs that have positive catalytic rate, thus establishing the coding analog of superadditivity for this family of codes, useful towards quantum communications. We also generalize this analysis to obtain conditions for EA TPCs from classical codes to have positive catalytic rate when their corresponding EA CSS codes have zero rate.</jats:p>","DOI":"10.22331/q-2024-05-02-1329","page":"1329","source":"Crossref","title":"Entanglement-assisted Quantum Reed-Muller Tensor Product Codes","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-1351-2959","authenticated-orcid":false,"given":"Priya J.","family":"Nadkarni","sequence":"first","affiliation":[{"name":"Department of Electronic Systems Engineering, Indian Institute of Science, Bengaluru, India, 560012"}]},{"ORCID":"https://orcid.org/0000-0002-1523-8260","authenticated-orcid":false,"given":"Praveen","family":"Jayakumar","sequence":"additional","affiliation":[{"name":"Department of Electronic Systems Engineering, Indian Institute of Science, Bengaluru, India, 560012"},{"name":"Department of Chemistry, University of Toronto, Toronto, Canada, M5S 3H6"}]},{"ORCID":"https://orcid.org/0009-0002-1496-4639","authenticated-orcid":false,"given":"Arpit","family":"Behera","sequence":"additional","affiliation":[{"name":"Department of Electronic Systems Engineering, Indian Institute of Science, Bengaluru, India, 560012"},{"name":"Department of Complex Systems, Weizmann Institute of Science, Rehovot, Israel, 7610001"}]},{"ORCID":"https://orcid.org/0000-0002-2459-1445","authenticated-orcid":false,"given":"Shayan Srinivasa","family":"Garani","sequence":"additional","affiliation":[{"name":"Department of Electronic Systems Engineering, Indian Institute of Science, Bengaluru, India, 560012"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,2]]},"URL":"http://dx.doi.org/10.22331/q-2024-05-02-1329","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2303.08294"},{"id":"arxiv:0803.3096","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.78.032335","source":"Crossref","title":"Physical underpinnings of privacy","volume":"78","author":[{"given":"Joseph M.","family":"Renes","sequence":"first","affiliation":[]},{"given":"Jean-Christian","family":"Boileau","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,9,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.032335","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032335","note":"arxivid:0803.3096"},{"id":"arxiv:1201.2906","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2012.6284203","source":"Crossref","title":"Quantum polar codes for arbitrary channels","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[]}],"event":"2012 IEEE International Symposium on Information Theory - ISIT","container-title":"2012 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2012,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2012.6284203","note":"arxivid:1201.2906"},{"id":"arxiv:1109.5346","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2013.2250575","page":"4718-4729","source":"Crossref","title":"Polar Codes for Degradable Quantum Channels","volume":"59","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2013.2250575","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1109.5346"},{"id":"arxiv:1109.3195","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.109.050504","source":"Crossref","title":"Efficient Polar Coding of Quantum Information","volume":"109","author":[{"given":"Joseph M.","family":"Renes","sequence":"first","affiliation":[]},{"given":"Frédéric","family":"Dupuis","sequence":"additional","affiliation":[]},{"given":"Renato","family":"Renner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,8,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.109.050504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050504","note":"arxivid:1109.3195"},{"id":"arxiv:1307.1136","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"11","DOI":"10.1109/tit.2015.2468084","page":"6395-6414","source":"Crossref","title":"Efficient Quantum Polar Codes Requiring No Preshared Entanglement","volume":"61","author":[{"given":"Joseph M.","family":"Renes","sequence":"first","affiliation":[]},{"given":"David","family":"Sutter","sequence":"additional","affiliation":[]},{"given":"Frederic","family":"Dupuis","sequence":"additional","affiliation":[]},{"given":"Renato","family":"Renner","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2015,11]]},"URL":"http://dx.doi.org/10.1109/TIT.2015.2468084","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1307.1136"},{"id":"arxiv:2304.04743","type":"article-journal","author":[{"given":"Anqi","family":"Gong"},{"given":"Joseph M.","family":"Renes"}],"title":"Improved Logical Error Rate via List Decoding of Quantum Polar Codes","issued":{"date-parts":[[2023,4,10]]},"note":"arxivid:2304.04743\narxiv_version_number:1"},{"id":"arxiv:2209.06673","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.108.042605","source":"Crossref","title":"Fault-tolerant preparation of quantum polar codes encoding one logical qubit","volume":"108","author":[{"ORCID":"https://orcid.org/0000-0002-3227-9216","authenticated-orcid":true,"given":"Ashutosh","family":"Goswami","sequence":"first","affiliation":[]},{"given":"Mehdi","family":"Mhalla","sequence":"additional","affiliation":[]},{"given":"Valentin","family":"Savin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.108.042605","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042605","note":"update-to:{\"DOI\":\"10.1103/physreva.108.042605\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2023,10,25]],\"date-time\":\"2023-10-25T00:00:00Z\",\"timestamp\":1698192000000}}\nupdated-by:{\"DOI\":\"10.1103/physreva.108.042605\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2023,10,25]],\"date-time\":\"2023-10-25T00:00:00Z\",\"timestamp\":1698192000000}}\narxivid:2209.06673"},{"id":"manual:-K.-Noh-href-https://github.c","type":"document","author":[{"family":"Noh","given":"K."}],"title":"\\href{https://github.com/errorcorrectionzoo/eczoo_data/files/7652763/Leung_code_as_quantum_polar_code.pdf}{Leung code as quantum polar code}","issued":"2017"},{"id":"arxiv:2505.22142","type":"article-journal","author":[{"given":"Keita","family":"Hidaka"},{"given":"Dina","family":"Abdelhadi"},{"given":"Ruediger","family":"Urbanke"}],"title":"Interpolation of Quantum Polar Codes and Quantum Reed-Muller Codes","issued":{"date-parts":[[2025,6,6]]},"note":"arxivid:2505.22142\narxiv_version_number:2"},{"id":"arxiv:1611.07528","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.119.020501","source":"Crossref","title":"Towards Holography via Quantum Source-Channel Codes","volume":"119","author":[{"given":"Fernando","family":"Pastawski","sequence":"first","affiliation":[]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]},{"given":"Henrik","family":"Wilming","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,7,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.119.020501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020501","note":"arxivid:1611.07528"},{"id":"arxiv:2406.09555","type":"article-journal","author":[{"given":"Shengqi","family":"Sang"},{"given":"Timothy H.","family":"Hsieh"},{"given":"Yijian","family":"Zou"}],"title":"Approximate quantum error correcting codes from conformal field theory","issued":{"date-parts":[[2024,11,9]]},"note":"arxivid:2406.09555\narxiv_version_number:3"},{"id":"arxiv:2504.08724","type":"article-journal","author":[{"given":"Anatoly","family":"Dymarsky"},{"given":"Johan","family":"Henriksson"},{"given":"Brian","family":"McPeak"}],"title":"Holographic duality from Howe duality: Chern-Simons gravity as an ensemble of code CFTs","issued":{"date-parts":[[2025,12,10]]},"note":"arxivid:2504.08724\narxiv_version_number:2"},{"id":"arxiv:2003.05451","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>We reconsider the black hole firewall puzzle, emphasizing that quantum error- correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation.</jats:p>","DOI":"10.1007/jhep06(2020)031","source":"Crossref","title":"The ghost in the radiation: robust encodings of the black hole interior","volume":"2020","author":[{"given":"Isaac","family":"Kim","sequence":"first","affiliation":[]},{"given":"Eugene","family":"Tang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2421-4762","authenticated-orcid":false,"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,3]]},"URL":"http://dx.doi.org/10.1007/JHEP06(2020)031","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"31","note":"alternative-id:13130\narxivid:2003.05451"},{"id":"arxiv:2203.01961","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>We study the quantum error correction properties of the black hole interior in a toy model for an evaporating black hole: Jackiw-Teitelboim gravity entangled with a non-gravitational bath. After the Page time, the black hole interior degrees of freedom in this system are encoded in the bath Hilbert space. We use the gravitational path integral to show that the interior density matrix is correctable against the action of quantum operations on the bath which (i) do not have prior access to details of the black hole microstates, and (ii) do not have a large, negative coherent information with respect to the maximally mixed state on the bath, with the lower bound controlled by the black hole entropy and code subspace dimension. Thus, the encoding of the black hole interior in the radiation is robust against generic, low-rank quantum operations. For erasure errors, gravity comes within an <jats:italic>O</jats:italic>(1) distance of saturating the Singleton bound on the tolerance of error correcting codes. For typical errors in the bath to corrupt the interior, they must have a rank that is a large multiple of the bath Hilbert space dimension, with the precise coefficient set by the black hole entropy and code subspace dimension.</jats:p>","DOI":"10.1007/jhep07(2023)189","source":"Crossref","title":"Quantum error correction in the black hole interior","volume":"2023","author":[{"given":"Vijay","family":"Balasubramanian","sequence":"first","affiliation":[]},{"given":"Arjun","family":"Kar","sequence":"additional","affiliation":[]},{"given":"Cathy","family":"Li","sequence":"additional","affiliation":[]},{"given":"Onkar","family":"Parrikar","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,25]]},"URL":"http://dx.doi.org/10.1007/JHEP07(2023)189","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"189","note":"alternative-id:21401\narxivid:2203.01961"},{"id":"arxiv:cond-mat/9212030","type":"journal-article","publisher":"American Physical Society (APS)","issue":"21","DOI":"10.1103/physrevlett.70.3339","page":"3339-3342","source":"Crossref","title":"Gapless spin-fluid ground state in a random quantum Heisenberg magnet","volume":"70","author":[{"given":"Subir","family":"Sachdev","sequence":"first","affiliation":[]},{"given":"Jinwu","family":"Ye","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1993,5,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.70.3339","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:cond-mat/9212030"},{"id":"manual:-A.-Kitaev-A-simple-model-of-","type":"chapter","author":[{"family":"Kitaev","given":"A."}],"title":"A simple model of quantum holography (part 2","container-title":"Entanglement in Strongly-Correlated Quantum Matter","issued":"2015","page":"38"},{"id":"arxiv:1910.10173","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevb.101.125112","source":"Crossref","title":"Low-rank Sachdev-Ye-Kitaev models","volume":"101","author":[{"ORCID":"https://orcid.org/0000-0002-0627-2475","authenticated-orcid":true,"given":"Jaewon","family":"Kim","sequence":"first","affiliation":[]},{"given":"Xiangyu","family":"Cao","sequence":"additional","affiliation":[]},{"given":"Ehud","family":"Altman","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.101.125112","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"125112","note":"arxivid:1910.10173"},{"id":"arxiv:2010.10545","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevb.103.l081113","source":"Crossref","title":"Dirac fast scramblers","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0002-0627-2475","authenticated-orcid":true,"given":"Jaewon","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Physics, University of California, Berkeley, California 94720, USA"}]},{"given":"Ehud","family":"Altman","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California, Berkeley, California 94720, USA"},{"name":"Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3406-6692","authenticated-orcid":true,"given":"Xiangyu","family":"Cao","sequence":"additional","affiliation":[{"name":"Department of Physics, University of California, Berkeley, California 94720, USA"},{"name":"Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 75005 Paris, France"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.103.L081113","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"L081113","note":"arxivid:2010.10545"},{"id":"arxiv:2203.05058","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>We analyze the error correcting properties of the Sachdev-Ye-Kitaev model, with errors that correspond to erasures of subsets of fermions. We study the limit where the number of fermions erased is large but small compared to the total number of fermions. We compute the <jats:italic>price</jats:italic> of the quantum error correcting code, defined as the number of physical qubits needed to reconstruct whether a given operator has been acted upon the thermal state or not. By thinking about reconstruction via quantum teleportation, we argue for a bound that relates the price to the ordinary operator size in systems that display so-called <jats:italic>detailed size winding</jats:italic> [1]. We then find that in SYK the price roughly saturates this bound. Computing the price requires computing modular flowed correlators with respect to the density matrix associated to a subset of fermions. We offer an interpretation of these correlators as probing a quantum extremal surface in the AdS dual of SYK. In the large <jats:italic>N</jats:italic> limit, the operator algebras associated to subsets of fermions in SYK satisfy half-sided modular inclusion, which is indicative of an emergent Type III1 von Neumann algebra. We discuss the relationship between the emergent algebra of half-sided modular inclusions and bulk symmetry generators.</jats:p>","DOI":"10.1007/jhep06(2022)039","source":"Crossref","title":"Quantum error correction in SYK and bulk emergence","volume":"2022","author":[{"given":"Venkatesa","family":"Chandrasekaran","sequence":"first","affiliation":[]},{"given":"Adam","family":"Levine","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,6,8]]},"URL":"http://dx.doi.org/10.1007/JHEP06(2022)039","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"39","note":"alternative-id:18519\narxivid:2203.05058"},{"id":"arxiv:2410.24225","type":"article-journal","author":[{"given":"Jaewon","family":"Kim"},{"given":"Ehud","family":"Altman"},{"given":"Jong Yeon","family":"Lee"}],"title":"Error Threshold of SYK Codes from Strong-to-Weak Parity Symmetry Breaking","issued":{"date-parts":[[2024,10,31]]},"note":"arxivid:2410.24225\narxiv_version_number:1"},{"id":"arxiv:1704.01589","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.7.031048","source":"Crossref","title":"Combining Topological Hardware and Topological Software: Color-Code Quantum Computing with Topological Superconductor Networks","volume":"7","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[]},{"given":"Markus S.","family":"Kesselring","sequence":"additional","affiliation":[]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]},{"given":"Felix","family":"von Oppen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2017,9,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.7.031048","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031048","note":"arxivid:1704.01589"},{"id":"arxiv:1708.05012","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevb.96.205413","source":"Crossref","title":"Braiding by Majorana tracking and long-range CNOT gates with color codes","volume":"96","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[]},{"given":"Felix","family":"von Oppen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,11,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.96.205413","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"205413","note":"arxivid:1708.05012"},{"id":"arxiv:1801.08143","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevb.97.205404","source":"Crossref","title":"Quantum computing with Majorana fermion codes","volume":"97","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[]},{"given":"Felix","family":"von Oppen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.97.205404","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"205404","note":"arxivid:1801.08143"},{"id":"arxiv:2211.11777","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Majorana zero modes (MZMs) are promising candidates for topologically-protected quantum computing hardware, however their large-scale use will likely require quantum error correction. Majorana surface codes (MSCs) have been proposed to achieve this. However, many MSC properties remain unexplored. We present a unified framework for MSC \"twist defects\" <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mtext>&amp;#x2013;</mml:mtext></mml:math> anyon-like objects encoding quantum information. We show that twist defects in MSCs can encode twice the amount of topologically protected information as in qubit-based codes or other MSC encoding schemes. This is due to twists encoding both logical qubits and \"logical MZMs,\" with the latter enhancing the protection microscopic MZMs can offer. We explain how to perform universal computation with logical qubits and logical MZMs while potentially using far fewer resources than in other MSC schemes. All Clifford gates can be implemented on logical qubits by braiding twist defects. We introduce lattice-surgery-based techniques for computing with logical MZMs and logical qubits, achieving the effect of Clifford gates with zero time overhead. We also show that logical MZMs may result in improved spatial overheads for sufficiently low rates of quasi-particle poisoning. Finally, we introduce a novel MSC analogue of transversal gates that achieves encoded Clifford gates in small codes by braiding microscopic MZMs. MSC twist defects thus open new paths towards fault-tolerant quantum computation.</jats:p>","DOI":"10.22331/q-2024-07-10-1400","page":"1400","source":"Crossref","title":"A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation","volume":"8","author":[{"given":"Campbell","family":"McLauchlan","sequence":"first","affiliation":[{"name":"DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK"}]},{"given":"Benjamin","family":"Béri","sequence":"additional","affiliation":[{"name":"DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK"},{"name":"T.C.M. Group, Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge, CB3 0HE, UK"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,10]]},"URL":"http://dx.doi.org/10.22331/q-2024-07-10-1400","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2211.11777"},{"id":"arxiv:1504.01724","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.5.041038","source":"Crossref","title":"Majorana Fermion Surface Code for Universal Quantum Computation","volume":"5","author":[{"given":"Sagar","family":"Vijay","sequence":"first","affiliation":[]},{"given":"Timothy H.","family":"Hsieh","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Fu","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2015,12,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.5.041038","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041038","note":"arxivid:1504.01724"},{"id":"arxiv:1509.05345","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.116.050501","source":"Crossref","title":"Towards Realistic Implementations of a Majorana Surface Code","volume":"116","author":[{"given":"L. A.","family":"Landau","sequence":"first","affiliation":[]},{"given":"S.","family":"Plugge","sequence":"additional","affiliation":[]},{"given":"E.","family":"Sela","sequence":"additional","affiliation":[]},{"given":"A.","family":"Altland","sequence":"additional","affiliation":[]},{"given":"S. M.","family":"Albrecht","sequence":"additional","affiliation":[]},{"given":"R.","family":"Egger","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,2,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.116.050501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050501","note":"arxivid:1509.05345"},{"id":"arxiv:quant-ph/0003137","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1006/aphy.2002.6254","page":"210-226","source":"Crossref","title":"Fermionic Quantum Computation","volume":"298","author":[{"given":"Sergey B.","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Alexei Yu.","family":"Kitaev","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2002,5]]},"URL":"http://dx.doi.org/10.1006/aphy.2002.6254","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491602962548\narxivid:quant-ph/0003137"},{"id":"arxiv:quant-ph/0108033","type":"article-journal","author":[{"given":"E.","family":"Knill"}],"title":"Fermionic Linear Optics and Matchgates","issued":{"date-parts":[[2001,8,8]]},"note":"arxivid:quant-ph/0108033\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0108010","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.65.032325","source":"Crossref","title":"Classical simulation of noninteracting-fermion quantum circuits","volume":"65","author":[{"given":"Barbara M.","family":"Terhal","sequence":"first","affiliation":[]},{"given":"David P.","family":"DiVincenzo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,3,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.032325","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032325","note":"arxivid:quant-ph/0108010"},{"id":"arxiv:quant-ph/0404180","type":"article-journal","author":[{"given":"Sergey","family":"Bravyi"}],"title":"Lagrangian representation for fermionic linear optics","issued":{"date-parts":[[2004,9,20]]},"note":"arxivid:quant-ph/0404180\narxiv_version_number:2"},{"id":"arxiv:1806.01275","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Majorana-based quantum computing seeks to use the non-local nature of Majorana zero modes to store and manipulate quantum information in a topologically protected way. While noise is anticipated to be significantly suppressed in such systems, finite temperature and system size result in residual errors. In this work, we connect the underlying physical error processes in Majorana-based systems to the noise models used in a fault tolerance analysis. Standard qubit-based noise models built from Pauli operators do not capture leading order noise processes arising from quasiparticle poisoning events, thus it is not obvious<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">a priori</mml:mtext></mml:mrow></mml:math>that such noise models can be usefully applied to a Majorana-based system. We develop stochastic Majorana noise models that are generalizations of the standard qubit-based models and connect the error probabilities defining these models to parameters of the physical system. Using these models, we compute pseudo-thresholds for the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math>Bacon-Shor subsystem code. Our results emphasize the importance of correlated errors induced in multi-qubit measurements. Moreover, we find that for sufficiently fast quasiparticle relaxation the errors are well described by Pauli operators. This work bridges the divide between physical errors in Majorana-based quantum computing architectures and the significance of these errors in a quantum error correcting code.</jats:p>","DOI":"10.22331/q-2018-09-03-88","page":"88","source":"Crossref","title":"Modeling noise and error correction for Majorana-based quantum computing","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-5982-8107","authenticated-orcid":false,"given":"Christina","family":"Knapp","sequence":"first","affiliation":[{"name":"Department of Physics, University of California, Santa Barbara, California 93106 USA"}]},{"ORCID":"https://orcid.org/0000-0003-0954-4509","authenticated-orcid":false,"given":"Michael","family":"Beverland","sequence":"additional","affiliation":[{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052 USA"}]},{"ORCID":"https://orcid.org/0000-0001-7315-0526","authenticated-orcid":false,"given":"Dmitry I.","family":"Pikulin","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, California 93106-6105 USA"}]},{"ORCID":"https://orcid.org/0000-0003-0834-0547","authenticated-orcid":false,"given":"Torsten","family":"Karzig","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, California 93106-6105 USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2018,9,3]]},"URL":"http://dx.doi.org/10.22331/q-2018-09-03-88","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1806.01275"},{"id":"arxiv:2412.16081","type":"article-journal","author":[{"given":"Robert","family":"Ott"},{"given":"Daniel","family":"González-Cuadra"},{"given":"Torsten V.","family":"Zache"},{"given":"Peter","family":"Zoller"},{"given":"Adam M.","family":"Kaufman"},{"given":"Hannes","family":"Pichler"}],"title":"Error-corrected fermionic quantum processors with neutral atoms","issued":{"date-parts":[[2024,12,20]]},"note":"arxivid:2412.16081\narxiv_version_number:1"},{"id":"doi:10.1145/380752.380785","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/380752.380785","page":"114-123","source":"Crossref","title":"Quantum computers that can be simulated classically in polynomial time","author":[{"given":"Leslie G.","family":"Valiant","sequence":"first","affiliation":[{"name":"Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA"}]}],"event":"STOC01: 33rd ACM Symposium on Theory of Computing","container-title":"Proceedings of the thirty-third annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2001,7,6]]},"URL":"http://dx.doi.org/10.1145/380752.380785","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/380752.380785\n10.1145/380752"},{"id":"arxiv:0804.4050","type":"journal-article","publisher":"The Royal Society","issue":"2100","abstract":"<jats:p>Let<jats:italic>G</jats:italic>(<jats:italic>A</jats:italic>, <jats:italic>B</jats:italic>) denote the two-qubit gate that acts as the one-qubit<jats:italic>SU</jats:italic>(2) gates<jats:italic>A</jats:italic>and<jats:italic>B</jats:italic>in the even and odd parity subspaces, respectively, of two qubits. Using a Clifford algebra formalism, we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allow the same gates to act only on n.n. and next n.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the Jordan–Wigner representation of a Clifford algebra, we show how one may generalize the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits.</jats:p>","DOI":"10.1098/rspa.2008.0189","page":"3089-3106","source":"Crossref","title":"Matchgates and classical simulation of quantum circuits","volume":"464","author":[{"given":"Richard","family":"Jozsa","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of BristolMerchant Venturers Building, Bristol BS8 1UB, UK"}]},{"given":"Akimasa","family":"Miyake","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics, University of InnsbruckTechnikerstrasse 25, 6020 Innsbruck, Austria"},{"name":"Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences6020 Innsbruck, Austria"}]}],"container-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2008,7,22]]},"URL":"http://dx.doi.org/10.1098/rspa.2008.0189","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. A.","note":"alternative-id:10.1098/rspa.2008.0189\narxivid:0804.4050"},{"id":"arxiv:2407.11319","type":"article-journal","author":[{"given":"Valérie","family":"Bettaque"},{"given":"Brian","family":"Swingle"}],"title":"The Structure of the Majorana Clifford Group","issued":{"date-parts":[[2025,7,8]]},"note":"arxivid:2407.11319\narxiv_version_number:3"},{"id":"arxiv:1308.1463","type":"journal-issue","publisher":"Rinton Press","issue":"11&12","DOI":"10.26421/qic14.11-12","source":"Crossref","title":[],"volume":"14","container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2014,9]]},"URL":"http://dx.doi.org/10.26421/QIC14.11-12","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC","note":"arxivid:1308.1463"},{"id":"arxiv:1602.03539","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.93.062332","source":"Crossref","title":"Efficient classical simulation of matchgate circuits with generalized inputs and measurements","volume":"93","author":[{"given":"Daniel J.","family":"Brod","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,6,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.062332","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062332","note":"arxivid:1602.03539"},{"id":"arxiv:1905.08584","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.123.080503","source":"Crossref","title":"All Pure Fermionic Non-Gaussian States Are Magic States for Matchgate Computations","volume":"123","author":[{"given":"M.","family":"Hebenstreit","sequence":"first","affiliation":[]},{"given":"R.","family":"Jozsa","sequence":"additional","affiliation":[]},{"given":"B.","family":"Kraus","sequence":"additional","affiliation":[]},{"given":"S.","family":"Strelchuk","sequence":"additional","affiliation":[]},{"given":"M.","family":"Yoganathan","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.123.080503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080503","note":"arxivid:1905.08584"},{"id":"arxiv:2501.06179","type":"article-journal","author":[{"given":"Luke","family":"Coffman"},{"given":"Graeme","family":"Smith"},{"given":"Xun","family":"Gao"}],"title":"Measuring Non-Gaussian Magic in Fermions: Convolution, Entropy, and the Violation of Wick's Theorem and the Matchgate Identity","issued":{"date-parts":[[2025,1,10]]},"note":"arxivid:2501.06179\narxiv_version_number:1"},{"id":"arxiv:1706.00023","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevx.8.011044","source":"Crossref","title":"Low-Depth Quantum Simulation of Materials","volume":"8","author":[{"given":"Ryan","family":"Babbush","sequence":"first","affiliation":[]},{"given":"Nathan","family":"Wiebe","sequence":"additional","affiliation":[]},{"given":"Jarrod","family":"McClean","sequence":"additional","affiliation":[]},{"given":"James","family":"McClain","sequence":"additional","affiliation":[]},{"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[]},{"given":"Garnet Kin-Lic","family":"Chan","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2018,3,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.8.011044","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011044","note":"arxivid:1706.00023"},{"id":"arxiv:1404.0897","type":"article-journal","author":[{"given":"Fabian","family":"Hassler"}],"title":"Majorana Qubits","issued":{"date-parts":[[2014,4,3]]},"note":"arxivid:1404.0897\narxiv_version_number:1"},{"id":"arxiv:1505.02576","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevb.92.235136","source":"Crossref","title":"A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations","volume":"92","author":[{"given":"Sagar","family":"Vijay","sequence":"first","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Fu","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2015,12,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.92.235136","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"235136","note":"arxivid:1505.02576"},{"id":"arxiv:2508.09928","type":"article-journal","author":[{"given":"Maryam","family":"Mudassar"},{"given":"Alexander","family":"Schuckert"},{"given":"Daniel","family":"Gottesman"}],"title":"Fault tolerant Operations in Majorana-based Quantum Codes: Gates, Measurements and High Rate Constructions","issued":{"date-parts":[[2025,9,16]]},"note":"arxivid:2508.09928\narxiv_version_number:2"},{"id":"arxiv:cond-mat/0010440","type":"journal-article","publisher":"Uspekhi Fizicheskikh Nauk (UFN) Journal","issue":"10S","DOI":"10.1070/1063-7869/44/10s/s29","page":"131-136","source":"Crossref","title":"Unpaired Majorana fermions in quantum wires","volume":"44","author":[{"given":"A Yu","family":"Kitaev","sequence":"first","affiliation":[]}],"container-title":"Physics-Uspekhi","original-title":[],"issued":{"date-parts":[[2001,10,1]]},"URL":"http://dx.doi.org/10.1070/1063-7869/44/10S/S29","ISSN":["1468-4780"],"container-title-short":"Phys.-Usp.","note":"arxivid:cond-mat/0010440"},{"id":"arxiv:2402.07829","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.110.032430","source":"Crossref","title":"Encoding Majorana codes","volume":"110","author":[{"given":"Maryam","family":"Mudassar","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/049s0rh22","id-type":"ROR","asserted-by":"publisher"}],"name":"Dartmouth College"}]},{"given":"Riley W.","family":"Chien","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049s0rh22","id-type":"ROR","asserted-by":"publisher"}],"name":"Dartmouth College"}]},{"ORCID":"https://orcid.org/0009-0002-2948-3395","authenticated-orcid":true,"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.032430","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032430","note":"arxivid:2402.07829"},{"id":"arxiv:1409.4724","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.90.042326","source":"Crossref","title":"Parafermion stabilizer codes","volume":"90","author":[{"given":"Utkan","family":"Güngördü","sequence":"first","affiliation":[]},{"given":"Rabindra","family":"Nepal","sequence":"additional","affiliation":[]},{"given":"Alexey A.","family":"Kovalev","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,10,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.042326","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042326","note":"arxivid:1409.4724"},{"id":"manual:-A.-Kubica-private-communicat","type":"document","author":[{"family":"Kubica","given":"A."}],"genre":"private communication,","issued":"2019"},{"id":"arxiv:1905.03268","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.1.033079","source":"Crossref","title":"Majorana dimers and holographic quantum error-correcting codes","volume":"1","author":[{"ORCID":"https://orcid.org/0000-0002-7142-0059","authenticated-orcid":true,"given":"A.","family":"Jahn","sequence":"first","affiliation":[]},{"given":"M.","family":"Gluza","sequence":"additional","affiliation":[]},{"given":"F.","family":"Pastawski","sequence":"additional","affiliation":[]},{"given":"J.","family":"Eisert","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2019,11,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.1.033079","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033079","note":"arxivid:1905.03268"},{"id":"arxiv:2311.01779","type":"article-journal","author":[{"given":"Sourav","family":"Kundu"},{"given":"Ben W.","family":"Reichardt"}],"title":"Majorana qubit codes that also correct odd-weight errors","issued":{"date-parts":[[2023,11,3]]},"note":"arxivid:2311.01779\narxiv_version_number:1"},{"id":"doi:10.1088/1367-2630/ad4737","type":"journal-article","publisher":"IOP Publishing","issue":"7","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>A potential platform for topological quantum computation is the Majorana-based tetron architecture. Its building blocks are superconducting islands called tetrons, which host four Majorana zero modes. Existing error correcting codes can correct even-weight errors on tetrons. In a previous proposal by us, we had shown that incorporating tetrons in the stabilizer group allows us to correct a combination of odd-weight errors and even-weight errors on tetrons. In this work, we show that inclusion of tetrons in the gauge group lets us create subsystem codes from conventional Pauli stabilizer codes, which can correct both kinds of errors. Compared to the previous approach, the current approach lets us construct codes with fewer stabilizer generators. This leads to shorter fault-tolerant sequence length, and improves the fault-tolerant pseudothreshold by as much as 84%.</jats:p>","DOI":"10.1088/1367-2630/ad4737","page":"073029","source":"Crossref","title":"Majorana subsystem qubit codes that also correct odd-weight errors","volume":"26","author":[{"ORCID":"https://orcid.org/0000-0002-2912-0159","authenticated-orcid":true,"given":"Sourav","family":"Kundu","sequence":"first","affiliation":[]},{"given":"Ben","family":"Reichardt","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2024,7,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ad4737","ISSN":["1367-2630"],"container-title-short":"New J. Phys."},{"id":"arxiv:1108.3845","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00220-012-1606-9","page":"641-692","source":"Crossref","title":"Disorder-Assisted Error Correction in Majorana Chains","volume":"316","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2012,10,25]]},"URL":"http://dx.doi.org/10.1007/s00220-012-1606-9","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:1606\narxivid:1108.3845"},{"id":"arxiv:1711.08196","type":"journal-article","publisher":"Stichting SciPost","issue":"1","abstract":"<jats:p>Active quantum error correction on topological codes is one of the\nmost promising routes to long-term qubit storage. In view of future\napplications, the scalability of the used decoding algorithms in\nphysical implementations is crucial. In this work, we focus on the\none-dimensional Majorana chain and construct a strictly local decoder\nbased on a self-dual cellular automaton. We study numerically and\nanalytically its performance and exploit these results to contrive a\nscalable decoder with exponentially growing decoherence times in the\npresence of noise. Our results pave the way for scalable and modular\ndesigns of actively corrected one-dimensional topological quantum\nmemories.</jats:p>","DOI":"10.21468/scipostphys.4.1.007","source":"Crossref","title":"Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0003-1477-8625","authenticated-orcid":false,"given":"Nicolai","family":"Lang","sequence":"first","affiliation":[{"name":"University of Stuttgart"}]},{"given":"Hans  Peter","family":"Büchler","sequence":"additional","affiliation":[{"name":"University of Stuttgart"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2018,1,31]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.4.1.007","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"007","note":"arxivid:1711.08196"},{"id":"arxiv:1411.7751","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The realization of Majorana zero modes is in the centre of intense theoretical and experimental investigations. Unfortunately, their exchange that can reveal their exotic statistics needs manipulations that are still beyond our experimental capabilities. Here we take an alternative approach. Through the Jordan–Wigner transformation, the Kitaev's chain supporting two Majorana zero modes is mapped to the spin-1/2 chain. We experimentally simulated the spin system and its evolution with a photonic quantum simulator. This allows us to probe the geometric phase, which corresponds to the exchange of two Majorana zero modes positioned at the ends of a three-site chain. Finally, we demonstrate the immunity of quantum information encoded in the Majorana zero modes against local errors through the simulator. Our photonic simulator opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures.</jats:p>","DOI":"10.1038/ncomms13194","source":"Crossref","title":"Simulating the exchange of Majorana zero modes with a photonic system","volume":"7","author":[{"given":"Jin-Shi","family":"Xu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3018-7219","authenticated-orcid":false,"given":"Kai","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Yong-Jian","family":"Han","sequence":"additional","affiliation":[]},{"given":"Chuan-Feng","family":"Li","sequence":"additional","affiliation":[]},{"given":"Jiannis K.","family":"Pachos","sequence":"additional","affiliation":[]},{"given":"Guang-Can","family":"Guo","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2016,10,25]]},"URL":"http://dx.doi.org/10.1038/ncomms13194","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"13194","note":"alternative-id:BFncomms13194\narxivid:1411.7751"},{"id":"arxiv:2206.00563","type":"journal-article","publisher":"IOP Publishing","issue":"2","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>The simulation of systems of interacting fermions is one of the most anticipated applications of quantum computers. The most interesting simulations will require a fault-tolerant quantum computer, and building such a device remains a long-term goal. However, the capabilities of existing noisy quantum processors have steadily improved, sparking an interest in running simulations that, while not necessarily classically intractable, may serve as device benchmarks and help elucidate the challenges to achieving practical applications on near-term devices. Systems of <jats:italic>non</jats:italic>-interacting fermions are ideally suited to serve these purposes. While they display rich physics and generate highly entangled states when simulated on a quantum processor, their classical tractability enables experimental results to be verified even at large system sizes that would typically defy classical simulation. In this work, we use a noisy superconducting quantum processor to prepare Majorana zero modes (MZMs) as eigenstates of the Kitaev chain Hamiltonian, a model of non-interacting fermions. Our work builds on previous experiments with non-interacting fermionic systems. Previous work demonstrated error mitigation techniques applicable to the special case of Slater determinants. Here, we show how to extend these techniques to the case of general fermionic Gaussian states, and demonstrate them by preparing MZMs on systems of up to seven qubits.</jats:p>","DOI":"10.1088/2058-9565/acb796","page":"025010","source":"Crossref","title":"Simulating Majorana zero modes on a noisy quantum processor","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0001-6459-6374","authenticated-orcid":true,"given":"Kevin J","family":"Sung","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4077-9744","authenticated-orcid":true,"given":"Marko J","family":"Rančić","sequence":"additional","affiliation":[]},{"given":"Olivia T","family":"Lanes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6731-6814","authenticated-orcid":true,"given":"Nicholas T","family":"Bronn","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2023,2,9]]},"URL":"http://dx.doi.org/10.1088/2058-9565/acb796","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2206.00563"},{"id":"arxiv:2203.15083","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Electrons are indivisible elementary particles, yet paradoxically a collection of them can act as a fraction of a single electron, exhibiting exotic and useful properties. One such collective excitation, known as a topological Majorana mode, is naturally stable against perturbations, such as unwanted local noise, and can thereby robustly store quantum information. As such, Majorana modes serve as the basic primitive of topological quantum computing, providing resilience to errors. However, their demonstration on quantum hardware has remained elusive. Here, we demonstrate a verifiable identification and braiding of topological Majorana modes using a superconducting quantum processor as a quantum simulator. By simulating fermions on a one-dimensional lattice subject to a periodic drive, we confirm the existence of Majorana modes localized at the edges, and distinguish them from other trivial modes. To simulate a basic logical operation of topological quantum computing known as braiding, we propose a non-adiabatic technique, whose implementation reveals correct braiding statistics in our experiments. This work could further be used to study topological models of matter using circuit-based simulations, and shows that long-sought quantum phenomena can be realized by anyone in cloud-run quantum simulations, whereby accelerating fundamental discoveries in quantum science and technology.</jats:p>","DOI":"10.1038/s41467-023-37725-0","source":"Crossref","title":"Observing and braiding topological Majorana modes on programmable quantum simulators","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0001-8415-6560","authenticated-orcid":false,"given":"Nikhil","family":"Harle","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4193-6254","authenticated-orcid":false,"given":"Oles","family":"Shtanko","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9187-8147","authenticated-orcid":false,"given":"Ramis","family":"Movassagh","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2023,4,21]]},"URL":"http://dx.doi.org/10.1038/s41467-023-37725-0","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"2286","note":"alternative-id:37725\narxivid:2203.15083"},{"id":"arxiv:2204.11372","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6621","abstract":"<jats:p>\n            Inherent symmetry of a quantum system may protect its otherwise fragile states. Leveraging such protection requires testing its robustness against uncontrolled environmental interactions. Using 47 superconducting qubits, we implement the one-dimensional kicked Ising model, which exhibits nonlocal Majorana edge modes (MEMs) with\n            <jats:inline-formula>\n              <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                <mml:mrow>\n                  <mml:msub>\n                    <mml:mi>ℤ</mml:mi>\n                    <mml:mn>2</mml:mn>\n                  </mml:msub>\n                </mml:mrow>\n              </mml:math>\n            </jats:inline-formula>\n            parity symmetry. We find that any multiqubit Pauli operator overlapping with the MEMs exhibits a uniform late-time decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This characteristic allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Furthermore, the MEMs are found to be resilient against certain symmetry-breaking noise owing to a prethermalization mechanism. Our work elucidates the complex interplay between noise and symmetry-protected edge modes in a solid-state environment.\n          </jats:p>","DOI":"10.1126/science.abq5769","page":"785-790","source":"Crossref","title":"Noise-resilient edge modes on a chain of superconducting qubits","volume":"378","author":[{"ORCID":"https://orcid.org/0000-0003-0507-0211","authenticated-orcid":true,"given":"X.","family":"Mi","sequence":"first","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9003-5914","authenticated-orcid":true,"given":"M.","family":"Sonner","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics, University of Geneva, Geneva, Switzerland."}]},{"given":"M. Y.","family":"Niu","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5396-3626","authenticated-orcid":true,"given":"K. W.","family":"Lee","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-3732-168X","authenticated-orcid":true,"given":"B.","family":"Foxen","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"R.","family":"Acharya","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"I.","family":"Aleiner","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"T. I.","family":"Andersen","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"F.","family":"Arute","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6486-7100","authenticated-orcid":true,"given":"K.","family":"Arya","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A.","family":"Asfaw","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"J.","family":"Atalaya","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6523-6730","authenticated-orcid":true,"given":"J. C.","family":"Bardin","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."},{"name":"Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5547-691X","authenticated-orcid":true,"given":"J.","family":"Basso","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5264-4025","authenticated-orcid":true,"given":"A.","family":"Bengtsson","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"G.","family":"Bortoli","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8517-7475","authenticated-orcid":true,"given":"A.","family":"Bourassa","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"L.","family":"Brill","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"M.","family":"Broughton","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5139-7931","authenticated-orcid":true,"given":"B. B.","family":"Buckley","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"D. A.","family":"Buell","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-8474-6317","authenticated-orcid":true,"given":"B.","family":"Burkett","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-7445-1654","authenticated-orcid":true,"given":"N.","family":"Bushnell","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"Z.","family":"Chen","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"B.","family":"Chiaro","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6164-0007","authenticated-orcid":true,"given":"R.","family":"Collins","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"P.","family":"Conner","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"W.","family":"Courtney","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A. L.","family":"Crook","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"D. M.","family":"Debroy","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-3727-7380","authenticated-orcid":true,"given":"S.","family":"Demura","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A.","family":"Dunsworth","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-7134-5733","authenticated-orcid":true,"given":"D.","family":"Eppens","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"C.","family":"Erickson","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"L.","family":"Faoro","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"E.","family":"Farhi","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"R.","family":"Fatemi","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"L.","family":"Flores","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"E.","family":"Forati","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9048-2896","authenticated-orcid":true,"given":"A. G.","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"W.","family":"Giang","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"C.","family":"Gidney","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"D.","family":"Gilboa","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"M.","family":"Giustina","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A. G.","family":"Dau","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":true,"given":"J. A.","family":"Gross","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-9867-2371","authenticated-orcid":true,"given":"S.","family":"Habegger","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-9412-0553","authenticated-orcid":true,"given":"M. P.","family":"Harrigan","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-7051-5558","authenticated-orcid":true,"given":"M.","family":"Hoffmann","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"S.","family":"Hong","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"T.","family":"Huang","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A.","family":"Huff","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"W. J.","family":"Huggins","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8777-6789","authenticated-orcid":true,"given":"L. B.","family":"Ioffe","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"S. V.","family":"Isakov","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"J.","family":"Iveland","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"E.","family":"Jeffrey","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-0435-655X","authenticated-orcid":true,"given":"Z.","family":"Jiang","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"C.","family":"Jones","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-9160-5176","authenticated-orcid":true,"given":"D.","family":"Kafri","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-0136-1428","authenticated-orcid":true,"given":"K.","family":"Kechedzhi","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"T.","family":"Khattar","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-3397-0444","authenticated-orcid":true,"given":"S.","family":"Kim","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A. Y.","family":"Kitaev","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."},{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA."}]},{"given":"P. V.","family":"Klimov","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A. R.","family":"Klots","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A. N.","family":"Korotkov","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."},{"name":"Department of Electrical and Computer Engineering, University of California, Riverside, CA, USA."}]},{"given":"F.","family":"Kostritsa","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-0253-4183","authenticated-orcid":true,"given":"J. M.","family":"Kreikebaum","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-9804-2185","authenticated-orcid":true,"given":"D.","family":"Landhuis","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"P.","family":"Laptev","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"K.-M.","family":"Lau","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"J.","family":"Lee","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"L.","family":"Laws","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"W.","family":"Liu","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A.","family":"Locharla","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9576-573X","authenticated-orcid":true,"given":"O.","family":"Martin","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2809-0509","authenticated-orcid":true,"given":"J. R.","family":"McClean","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9544-141X","authenticated-orcid":true,"given":"M.","family":"McEwen","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."},{"name":"Department of Physics, University of California, Santa Barbara, CA, USA."}]},{"given":"B.","family":"Meurer Costa","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-1904-5917","authenticated-orcid":true,"given":"K. C.","family":"Miao","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-4065-9400","authenticated-orcid":true,"given":"M.","family":"Mohseni","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"S.","family":"Montazeri","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A.","family":"Morvan","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"E.","family":"Mount","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8497-6363","authenticated-orcid":true,"given":"W.","family":"Mruczkiewicz","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-7760-9186","authenticated-orcid":true,"given":"O.","family":"Naaman","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-5548-0051","authenticated-orcid":true,"given":"M.","family":"Neeley","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6563-3568","authenticated-orcid":true,"given":"C.","family":"Neill","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"M.","family":"Newman","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8406-6626","authenticated-orcid":true,"given":"T. E.","family":"O’Brien","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A.","family":"Opremcak","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"A.","family":"Petukhov","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"R.","family":"Potter","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"C.","family":"Quintana","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-3963-1830","authenticated-orcid":true,"given":"N. C.","family":"Rubin","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"N.","family":"Saei","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-8188-364X","authenticated-orcid":true,"given":"D.","family":"Sank","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"K.","family":"Sankaragomathi","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5865-0813","authenticated-orcid":true,"given":"K. J.","family":"Satzinger","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9611-7275","authenticated-orcid":true,"given":"C.","family":"Schuster","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"M. J.","family":"Shearn","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"V.","family":"Shvarts","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"D.","family":"Strain","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"Y.","family":"Su","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-1310-9174","authenticated-orcid":true,"given":"M.","family":"Szalay","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"G.","family":"Vidal","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"B.","family":"Villalonga","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"C.","family":"Vollgraff-Heidweiller","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9803-7471","authenticated-orcid":true,"given":"T.","family":"White","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-1806-5454","authenticated-orcid":true,"given":"Z.","family":"Yao","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-0837-1028","authenticated-orcid":true,"given":"P.","family":"Yeh","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"J.","family":"Yoo","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2585-2424","authenticated-orcid":true,"given":"A.","family":"Zalcman","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"Y.","family":"Zhang","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-3037-2003","authenticated-orcid":true,"given":"N.","family":"Zhu","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9681-6746","authenticated-orcid":true,"given":"H.","family":"Neven","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-6266-9269","authenticated-orcid":true,"given":"D.","family":"Bacon","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"J.","family":"Hilton","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6449-2273","authenticated-orcid":true,"given":"E.","family":"Lucero","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-6979-9533","authenticated-orcid":true,"given":"R.","family":"Babbush","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-1090-7584","authenticated-orcid":true,"given":"S.","family":"Boixo","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6371-6140","authenticated-orcid":true,"given":"A.","family":"Megrant","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-7473-6745","authenticated-orcid":true,"given":"Y.","family":"Chen","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"given":"J.","family":"Kelly","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-3000-6732","authenticated-orcid":true,"given":"V.","family":"Smelyanskiy","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2461-0271","authenticated-orcid":true,"given":"D. A.","family":"Abanin","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."},{"name":"Department of Theoretical Physics, University of Geneva, Geneva, Switzerland."}]},{"ORCID":"https://orcid.org/0000-0003-1917-3879","authenticated-orcid":true,"given":"P.","family":"Roushan","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2022,11,18]]},"URL":"http://dx.doi.org/10.1126/science.abq5769","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.abq5769\narxivid:2204.11372"},{"id":"arxiv:0904.2771","type":"article-journal","author":[{"given":"Alexei","family":"Kitaev"},{"given":"Chris","family":"Laumann"}],"title":"Topological phases and quantum computation","issued":{"date-parts":[[2009,4,20]]},"note":"arxivid:0904.2771\narxiv_version_number:1"},{"id":"arxiv:2208.07367","type":"journal-article","publisher":"Stichting SciPost","issue":"4","abstract":"<jats:p>\n                    (3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>\\mathbb{Z}_2</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:msub>\n                            <mml:mi>ℤ</mml:mi>\n                            <mml:mn>2</mml:mn>\n                          </mml:msub>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    gauge theory with fermionic charges, in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>\\mathbb{Z}_2 \\times \\mathbb{Z}_2</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>ℤ</mml:mi>\n                              <mml:mn>2</mml:mn>\n                            </mml:msub>\n                            <mml:mo>×</mml:mo>\n                            <mml:msub>\n                              <mml:mi>ℤ</mml:mi>\n                              <mml:mn>2</mml:mn>\n                            </mml:msub>\n                          </mml:mrow>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral (\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>D_n</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:msub>\n                            <mml:mi>D</mml:mi>\n                            <mml:mi>n</mml:mi>\n                          </mml:msub>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    ) and alternating (\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>A_6</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:msub>\n                            <mml:mi>A</mml:mi>\n                            <mml:mn>6</mml:mn>\n                          </mml:msub>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    ) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>H^4</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:msup>\n                            <mml:mi>H</mml:mi>\n                            <mml:mn>4</mml:mn>\n                          </mml:msup>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>A_6</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:msub>\n                            <mml:mi>A</mml:mi>\n                            <mml:mn>6</mml:mn>\n                          </mml:msub>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    gauge theory.\n                  </jats:p>","DOI":"10.21468/scipostphys.14.4.065","source":"Crossref","title":"Codimension-2 defects and higher symmetries in (3+1)D topological phases","volume":"14","author":[{"given":"Maissam","family":"Barkeshli","sequence":"first","affiliation":[{"name":"Joint Quantum Institute"}]},{"given":"Yu-An","family":"Chen","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute"},{"name":"University of Maryland, College Park"}]},{"given":"Sheng-Jie","family":"Huang","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute"}]},{"given":"Ryohei","family":"Kobayashi","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute"}]},{"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"name":"Harvard University"},{"name":"Walter Burke Institute for Theoretical Physics"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"IBM Quantum"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2023,4,11]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.14.4.065","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"065","note":"arxivid:2208.07367"},{"id":"arxiv:1906.01045","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.102.022403","source":"Crossref","title":"Fault-tolerant quantum gates with defects in topological stabilizer codes","volume":"102","author":[{"given":"Paul","family":"Webster","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.022403","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022403","note":"arxivid:1906.01045"},{"id":"arxiv:1609.01697","type":"journal-article","publisher":"IOP Publishing","issue":"1","DOI":"10.1088/1367-2630/aa54e1","page":"012001","source":"Crossref","title":"Majorana box qubits","volume":"19","author":[{"given":"Stephan","family":"Plugge","sequence":"first","affiliation":[]},{"given":"Asbjørn","family":"Rasmussen","sequence":"additional","affiliation":[]},{"given":"Reinhold","family":"Egger","sequence":"additional","affiliation":[]},{"given":"Karsten","family":"Flensberg","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2017,1,11]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aa54e1","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1609.01697"},{"id":"arxiv:1006.4395","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","DOI":"10.1038/nphys1915","page":"412-417","source":"Crossref","title":"Non-Abelian statistics and topological quantum information processing in 1D wire networks","volume":"7","author":[{"given":"Jason","family":"Alicea","sequence":"first","affiliation":[]},{"given":"Yuval","family":"Oreg","sequence":"additional","affiliation":[]},{"given":"Gil","family":"Refael","sequence":"additional","affiliation":[]},{"given":"Felix","family":"von Oppen","sequence":"additional","affiliation":[]},{"given":"Matthew P. A.","family":"Fisher","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,2,13]]},"URL":"http://dx.doi.org/10.1038/nphys1915","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nature Phys","note":"alternative-id:BFnphys1915\narxivid:1006.4395"},{"id":"arxiv:1511.05153","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.6.031016","source":"Crossref","title":"Milestones Toward Majorana-Based Quantum Computing","volume":"6","author":[{"given":"David","family":"Aasen","sequence":"first","affiliation":[]},{"given":"Michael","family":"Hell","sequence":"additional","affiliation":[]},{"given":"Ryan V.","family":"Mishmash","sequence":"additional","affiliation":[]},{"given":"Andrew","family":"Higginbotham","sequence":"additional","affiliation":[]},{"given":"Jeroen","family":"Danon","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Leijnse","sequence":"additional","affiliation":[]},{"given":"Thomas S.","family":"Jespersen","sequence":"additional","affiliation":[]},{"given":"Joshua A.","family":"Folk","sequence":"additional","affiliation":[]},{"given":"Charles M.","family":"Marcus","sequence":"additional","affiliation":[]},{"given":"Karsten","family":"Flensberg","sequence":"additional","affiliation":[]},{"given":"Jason","family":"Alicea","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2016,8,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.6.031016","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031016","note":"arxivid:1511.05153"},{"id":"arxiv:2004.02124","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.2.033255","source":"Crossref","title":"Readout of Majorana qubits","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-2649-9554","authenticated-orcid":true,"given":"Jacob F.","family":"Steiner","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2537-7256","authenticated-orcid":true,"given":"Felix","family":"von Oppen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033255","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033255","note":"arxivid:2004.02124"},{"id":"arxiv:2502.12252","type":"article-journal","author":[{"given":"David","family":"Aasen"},{"given":"Morteza","family":"Aghaee"},{"given":"Zulfi","family":"Alam"},{"given":"Mariusz","family":"Andrzejczuk"},{"given":"Andrey","family":"Antipov"},{"given":"Mikhail","family":"Astafev"},{"given":"Lukas","family":"Avilovas"},{"given":"Amin","family":"Barzegar"},{"given":"Bela","family":"Bauer"},{"given":"Jonathan","family":"Becker"},{"given":"Juan M.","family":"Bello-Rivas"},{"given":"Umesh","family":"Bhaskar"},{"given":"Alex","family":"Bocharov"},{"given":"Srini","family":"Boddapati"},{"given":"David","family":"Bohn"},{"given":"Jouri","family":"Bommer"},{"given":"Parsa","family":"Bonderson"},{"given":"Jan","family":"Borovsky"},{"given":"Leo","family":"Bourdet"},{"given":"Samuel","family":"Boutin"},{"given":"Tom","family":"Brown"},{"given":"Gary","family":"Campbell"},{"given":"Lucas","family":"Casparis"},{"given":"Srivatsa","family":"Chakravarthi"},{"given":"Rui","family":"Chao"},{"given":"Benjamin J.","family":"Chapman"},{"given":"Sohail","family":"Chatoor"},{"given":"Anna Wulff","family":"Christensen"},{"given":"Patrick","family":"Codd"},{"given":"William","family":"Cole"},{"given":"Paul","family":"Cooper"},{"given":"Fabiano","family":"Corsetti"},{"given":"Ajuan","family":"Cui"},{"given":"Wim","family":"van Dam"},{"given":"Tareq El","family":"Dandachi"},{"given":"Sahar","family":"Daraeizadeh"},{"given":"Adrian","family":"Dumitrascu"},{"given":"Andreas","family":"Ekefjärd"},{"given":"Saeed","family":"Fallahi"},{"given":"Luca","family":"Galletti"},{"given":"Geoff","family":"Gardner"},{"given":"Raghu","family":"Gatta"},{"given":"Haris","family":"Gavranovic"},{"given":"Michael","family":"Goulding"},{"given":"Deshan","family":"Govender"},{"given":"Flavio","family":"Griggio"},{"given":"Ruben","family":"Grigoryan"},{"given":"Sebastian","family":"Grijalva"},{"given":"Sergei","family":"Gronin"},{"given":"Jan","family":"Gukelberger"},{"given":"Jeongwan","family":"Haah"},{"given":"Marzie","family":"Hamdast"},{"given":"Esben Bork","family":"Hansen"},{"given":"Matthew","family":"Hastings"},{"given":"Sebastian","family":"Heedt"},{"given":"Samantha","family":"Ho"},{"given":"Justin","family":"Hogaboam"},{"given":"Laurens","family":"Holgaard"},{"given":"Kevin","family":"Van Hoogdalem"},{"given":"Jinnapat","family":"Indrapiromkul"},{"given":"Henrik","family":"Ingerslev"},{"given":"Lovro","family":"Ivancevic"},{"given":"Sarah","family":"Jablonski"},{"given":"Thomas","family":"Jensen"},{"given":"Jaspreet","family":"Jhoja"},{"given":"Jeffrey","family":"Jones"},{"given":"Kostya","family":"Kalashnikov"},{"given":"Ray","family":"Kallaher"},{"given":"Rachpon","family":"Kalra"},{"given":"Farhad","family":"Karimi"},{"given":"Torsten","family":"Karzig"},{"given":"Seth","family":"Kimes"},{"given":"Vadym","family":"Kliuchnikov"},{"given":"Maren Elisabeth","family":"Kloster"},{"given":"Christina","family":"Knapp"},{"given":"Derek","family":"Knee"},{"given":"Jonne","family":"Koski"},{"given":"Pasi","family":"Kostamo"},{"given":"Jamie","family":"Kuesel"},{"given":"Brad","family":"Lackey"},{"given":"Tom","family":"Laeven"},{"given":"Jeffrey","family":"Lai"},{"given":"Gijs","family":"de Lange"},{"given":"Thorvald","family":"Larsen"},{"given":"Jason","family":"Lee"},{"given":"Kyunghoon","family":"Lee"},{"given":"Grant","family":"Leum"},{"given":"Kongyi","family":"Li"},{"given":"Tyler","family":"Lindemann"},{"given":"Marijn","family":"Lucas"},{"given":"Roman","family":"Lutchyn"},{"given":"Morten Hannibal","family":"Madsen"},{"given":"Nash","family":"Madulid"},{"given":"Michael","family":"Manfra"},{"given":"Signe Brynold","family":"Markussen"},{"given":"Esteban","family":"Martinez"},{"given":"Marco","family":"Mattila"},{"given":"Jake","family":"Mattinson"},{"given":"Robert","family":"McNeil"},{"given":"Antonio Rodolph","family":"Mei"},{"given":"Ryan V.","family":"Mishmash"},{"given":"Gopakumar","family":"Mohandas"},{"given":"Christian","family":"Mollgaard"},{"given":"Michiel","family":"de Moor"},{"given":"Trevor","family":"Morgan"},{"given":"George","family":"Moussa"},{"given":"Anirudh","family":"Narla"},{"given":"Chetan","family":"Nayak"},{"given":"Jens Hedegaard","family":"Nielsen"},{"given":"William Hvidtfelt Padkær","family":"Nielsen"},{"given":"Frédéric","family":"Nolet"},{"given":"Mike","family":"Nystrom"},{"given":"Eoin","family":"O'Farrell"},{"given":"Keita","family":"Otani"},{"given":"Adam","family":"Paetznick"},{"given":"Camille","family":"Papon"},{"given":"Andres","family":"Paz"},{"given":"Karl","family":"Petersson"},{"given":"Luca","family":"Petit"},{"given":"Dima","family":"Pikulin"},{"given":"Diego Olivier Fernandez","family":"Pons"},{"given":"Sam","family":"Quinn"},{"given":"Mohana","family":"Rajpalke"},{"given":"Alejandro Alcaraz","family":"Ramirez"},{"given":"Katrine","family":"Rasmussen"},{"given":"David","family":"Razmadze"},{"given":"Ben","family":"Reichardt"},{"given":"Yuan","family":"Ren"},{"given":"Ken","family":"Reneris"},{"given":"Roy","family":"Riccomini"},{"given":"Ivan","family":"Sadovskyy"},{"given":"Lauri","family":"Sainiemi"},{"given":"Juan Carlos Estrada","family":"Saldaña"},{"given":"Irene","family":"Sanlorenzo"},{"given":"Simon","family":"Schaal"},{"given":"Emma","family":"Schmidgall"},{"given":"Cristina","family":"Sfiligoj"},{"given":"Marcus P.","family":"da Silva"},{"given":"Shilpi","family":"Singh"},{"given":"Sarat","family":"Sinha"},{"given":"Mathias","family":"Soeken"},{"given":"Patrick","family":"Sohr"},{"given":"Tomas","family":"Stankevic"},{"given":"Lieuwe","family":"Stek"},{"given":"Patrick","family":"Strøm-Hansen"},{"given":"Eric","family":"Stuppard"},{"given":"Aarthi","family":"Sundaram"},{"given":"Henri","family":"Suominen"},{"given":"Judith","family":"Suter"},{"given":"Satoshi","family":"Suzuki"},{"given":"Krysta","family":"Svore"},{"given":"Sam","family":"Teicher"},{"given":"Nivetha","family":"Thiyagarajah"},{"given":"Raj","family":"Tholapi"},{"given":"Mason","family":"Thomas"},{"given":"Dennis","family":"Tom"},{"given":"Emily","family":"Toomey"},{"given":"Josh","family":"Tracy"},{"given":"Matthias","family":"Troyer"},{"given":"Michelle","family":"Turley"},{"given":"Matthew D.","family":"Turner"},{"given":"Shivendra","family":"Upadhyay"},{"given":"Ivan","family":"Urban"},{"given":"Alexander","family":"Vaschillo"},{"given":"Dmitrii","family":"Viazmitinov"},{"given":"Dominik","family":"Vogel"},{"given":"Zhenghan","family":"Wang"},{"given":"John","family":"Watson"},{"given":"Alex","family":"Webster"},{"given":"Joseph","family":"Weston"},{"given":"Timothy","family":"Williamson"},{"given":"Georg W.","family":"Winkler"},{"given":"David J.","family":"van Woerkom"},{"given":"Brian Paquelet","family":"Wütz"},{"given":"Chung Kai","family":"Yang"},{"given":"Richard","family":"Yu"},{"given":"Emrah","family":"Yucelen"},{"given":"Jesús Herranz","family":"Zamorano"},{"given":"Roland","family":"Zeisel"},{"given":"Guoji","family":"Zheng"},{"given":"Justin","family":"Zilke"},{"given":"Andrew","family":"Zimmerman"}],"title":"Roadmap to fault tolerant quantum computation using topological qubit arrays","issued":{"date-parts":[[2025,7,18]]},"note":"arxivid:2502.12252\narxiv_version_number:3"},{"id":"arxiv:cond-mat/9305017","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevb.49.8955","page":"8955-8982","source":"Crossref","title":"Odd-frequency pairing in the Kondo lattice","volume":"49","author":[{"given":"P.","family":"Coleman","sequence":"first","affiliation":[]},{"given":"E.","family":"Miranda","sequence":"additional","affiliation":[]},{"given":"A.","family":"Tsvelik","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[1994,4,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.49.8955","ISSN":["0163-1829","1095-3795"],"container-title-short":"Phys. Rev. B","note":"arxivid:cond-mat/9305017"},{"id":"arxiv:cond-mat/9504006","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevb.52.6611","page":"6611-6627","source":"Crossref","title":"Simple formulation of the two-channel Kondo model","volume":"52","author":[{"given":"P.","family":"Coleman","sequence":"first","affiliation":[]},{"given":"L. B.","family":"Ioffe","sequence":"additional","affiliation":[]},{"given":"A. M.","family":"Tsvelik","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[1995,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.52.6611","ISSN":["0163-1829","1095-3795"],"container-title-short":"Phys. Rev. B","note":"arxivid:cond-mat/9504006"},{"id":"arxiv:2204.11888","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevresearch.5.013169","source":"Crossref","title":"Monopole Josephson effects in a Dirac spin liquid","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0003-4305-8600","authenticated-orcid":true,"given":"Gautam","family":"Nambiar","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Bulmash","sequence":"additional","affiliation":[]},{"given":"Victor","family":"Galitski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.5.013169","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013169","note":"arxivid:2204.11888"},{"id":"arxiv:quant-ph/0205004","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.90.016803","source":"Crossref","title":"Quantum Orders in an Exact Soluble Model","volume":"90","author":[{"given":"Xiao-Gang","family":"Wen","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2003,1,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.90.016803","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"016803","note":"arxivid:quant-ph/0205004"},{"id":"arxiv:1701.05052","type":"article-journal","author":[{"given":"Ananda","family":"Roy"},{"given":"David P.","family":"DiVincenzo"}],"title":"Topological Quantum Computing","issued":{"date-parts":[[2017,1,18]]},"note":"arxivid:1701.05052\narxiv_version_number:1"},{"id":"arxiv:1711.00515","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2018.03.024","page":"234-253","source":"Crossref","title":"Exact bosonization in two spatial dimensions and a new class of lattice gauge theories","volume":"393","author":[{"given":"Yu-An","family":"Chen","sequence":"first","affiliation":[]},{"given":"Anton","family":"Kapustin","sequence":"additional","affiliation":[]},{"given":"Đorđe","family":"Radičević","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2018,6]]},"URL":"http://dx.doi.org/10.1016/j.aop.2018.03.024","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491618300873\narxivid:1711.00515"},{"id":"arxiv:2201.05153","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.4.010326","source":"Crossref","title":"Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-8810-9355","authenticated-orcid":true,"given":"Yu-An","family":"Chen","sequence":"first","affiliation":[]},{"given":"Yijia","family":"Xu","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,13]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.010326","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010326","note":"update-to:{\"DOI\":\"10.1103/prxquantum.4.010326\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2024,2,9]],\"date-time\":\"2024-02-09T00:00:00Z\",\"timestamp\":1707436800000}}\nupdated-by:{\"DOI\":\"10.1103/prxquantum.4.010326\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2024,2,9]],\"date-time\":\"2024-02-09T00:00:00Z\",\"timestamp\":1707436800000}}\narxivid:2201.05153"},{"id":"arxiv:1710.02270","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Surface codes are building blocks of quantum computing platforms based on 2D arrays of qubits responsible for detecting and correcting errors. The error suppression achieved by the surface code is usually estimated by simulating toy noise models describing random Pauli errors. However, Pauli noise models fail to capture coherent processes such as systematic unitary errors caused by imperfect control pulses. Here we report the first large-scale simulation of quantum error correction protocols based on the surface code in the presence of coherent noise. We observe that the standard Pauli approximation provides an accurate estimate of the error threshold but underestimates the logical error rate in the sub-threshold regime. We find that for large code size the logical-level noise is well approximated by random Pauli errors even though the physical-level noise is coherent. Our work demonstrates that coherent effects do not significantly change the error correcting threshold of surface codes. This gives more confidence in the viability of the fault-tolerance architecture pursued by several experimental groups.</jats:p>","DOI":"10.1038/s41534-018-0106-y","source":"Crossref","title":"Correcting coherent errors with surface codes","volume":"4","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Matthias","family":"Englbrecht","sequence":"additional","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]},{"given":"Nolan","family":"Peard","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2018,10,31]]},"URL":"http://dx.doi.org/10.1038/s41534-018-0106-y","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"55","note":"alternative-id:106\narxivid:1710.02270"},{"id":"arxiv:1912.04319","type":"journal-article","publisher":"IOP Publishing","issue":"7","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>We study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.</jats:p>","DOI":"10.1088/1367-2630/ab8e5c","page":"073066","source":"Crossref","title":"Coherence in logical quantum channels","volume":"22","author":[{"ORCID":"https://orcid.org/0000-0003-4665-8839","authenticated-orcid":false,"given":"Joseph K","family":"Iverson","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2421-4762","authenticated-orcid":false,"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2020,7,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ab8e5c","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1912.04319"},{"id":"arxiv:1104.5485","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevb.84.125125","source":"Crossref","title":"SU(2) slave fermion solution of the Kitaev honeycomb lattice model","volume":"84","author":[{"given":"F. J.","family":"Burnell","sequence":"first","affiliation":[]},{"given":"Chetan","family":"Nayak","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2011,9,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.84.125125","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"125125","note":"arxivid:1104.5485"},{"id":"arxiv:quant-ph/0511178","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.73.042313","source":"Crossref","title":"Universal quantum computation with the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>ν</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mo>∕</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>fractional quantum Hall state","volume":"73","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,4,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.042313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042313","note":"arxivid:quant-ph/0511178"},{"id":"arxiv:2205.10114","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.106.062401","source":"Crossref","title":"Scalable quantum control and non-Abelian anyon creation in the Kitaev honeycomb model","volume":"106","author":[{"ORCID":"https://orcid.org/0000-0003-3473-9236","authenticated-orcid":true,"given":"Omar","family":"Raii","sequence":"first","affiliation":[]},{"given":"Florian","family":"Mintert","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4063-1264","authenticated-orcid":true,"given":"Daniel","family":"Burgarth","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.106.062401","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062401","note":"arxivid:2205.10114"},{"id":"arxiv:1908.02613","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevb.101.035101","source":"Crossref","title":"Lattice models that realize \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi mathvariant=\"double-struck\">Z</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>\n-1 symmetry-protected topological states for even \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>","volume":"101","author":[{"ORCID":"https://orcid.org/0000-0002-8435-1224","authenticated-orcid":true,"given":"Lokman","family":"Tsui","sequence":"first","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA"}]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2020,1,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.101.035101","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"035101","note":"arxivid:1908.02613"},{"id":"arxiv:2303.15615","type":"journal-article","publisher":"IOP Publishing","issue":"10","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli <jats:italic>X</jats:italic> and <jats:italic>Z</jats:italic> gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.</jats:p>","DOI":"10.1088/1367-2630/acfc5f","page":"103018","source":"Crossref","title":"Transversal diagonal logical operators for stabiliser codes","volume":"25","author":[{"ORCID":"https://orcid.org/0000-0002-5300-1643","authenticated-orcid":false,"given":"Mark A","family":"Webster","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5101-5673","authenticated-orcid":false,"given":"Armanda O","family":"Quintavalle","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2023,10,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/acfc5f","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:2303.15615"},{"id":"arxiv:2503.15751","type":"article-journal","author":[{"given":"Margarita","family":"Davydova"},{"given":"Andreas","family":"Bauer"},{"given":"Julio C. Magdalena","family":"de la Fuente"},{"given":"Mark","family":"Webster"},{"given":"Dominic J.","family":"Williamson"},{"given":"Benjamin J.","family":"Brown"}],"title":"Universal fault tolerant quantum computation in 2D without getting tied in knots","issued":{"date-parts":[[2025,5,15]]},"note":"arxivid:2503.15751\narxiv_version_number:2"},{"id":"arxiv:2305.03766","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7999","DOI":"10.1038/s41586-023-06934-4","page":"505-511","source":"Crossref","title":"Non-Abelian topological order and anyons on a trapped-ion processor","volume":"626","author":[{"given":"Mohsin","family":"Iqbal","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":false,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":false,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[]},{"given":"Sara L.","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Joan M.","family":"Dreiling","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8460-4175","authenticated-orcid":false,"given":"Caroline","family":"Figgatt","sequence":"additional","affiliation":[]},{"given":"John P.","family":"Gaebler","sequence":"additional","affiliation":[]},{"given":"Jacob","family":"Johansen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2560-4129","authenticated-orcid":false,"given":"Michael","family":"Mills","sequence":"additional","affiliation":[]},{"given":"Steven A.","family":"Moses","sequence":"additional","affiliation":[]},{"given":"Juan M.","family":"Pino","sequence":"additional","affiliation":[]},{"given":"Anthony","family":"Ransford","sequence":"additional","affiliation":[]},{"given":"Mary","family":"Rowe","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Siegfried","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3124-2999","authenticated-orcid":false,"given":"Russell P.","family":"Stutz","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Foss-Feig","sequence":"additional","affiliation":[]},{"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1480-6406","authenticated-orcid":false,"given":"Henrik","family":"Dreyer","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2024,2,14]]},"URL":"http://dx.doi.org/10.1038/s41586-023-06934-4","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:6934\narxivid:2305.03766"},{"id":"manual:-B.-J.-Brown-private-communic","type":"article-journal","author":[{"family":"Brown","given":"B.J."}],"container-title":"private communication","issued":"2025"},{"id":"arxiv:1507.05676","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00220-016-2604-0","page":"389-419","source":"Crossref","title":"Double Semions in Arbitrary Dimension","volume":"347","author":[{"given":"Michael H.","family":"Freedman","sequence":"first","affiliation":[]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2016,3,29]]},"URL":"http://dx.doi.org/10.1007/s00220-016-2604-0","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:2604\narxivid:1507.05676"},{"id":"arxiv:2203.00103","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We propose an extension to the Pauli stabiliser formalism that includes fractional <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>&amp;#x03C0;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>N</mml:mi></mml:math> rotations around the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> axis for some integer <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>. The resulting generalised stabiliser formalism – denoted the XP stabiliser formalism – allows for a wider range of states and codespaces to be represented. We describe the states which arise in the formalism, and demonstrate an equivalence between XP stabiliser states and 'weighted hypergraph states' – a generalisation of both hypergraph and weighted graph states. Given an arbitrary set of XP operators, we present algorithms for determining the codespace and logical operators for an XP code. Finally, we consider whether measurements of XP operators on XP codes can be classically simulated.</jats:p>","DOI":"10.22331/q-2022-09-22-815","page":"815","source":"Crossref","title":"The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases","volume":"6","author":[{"given":"Mark A.","family":"Webster","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"},{"name":"Sydney Quantum Academy, Sydney, NSW, Australia"}]},{"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"},{"name":"Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, University of Copenhagen, 2100 Copenhagen, Denmark"}]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, NSW 2006, Australia"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,22]]},"URL":"http://dx.doi.org/10.22331/q-2022-09-22-815","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2203.00103"},{"id":"preset:Webster23","type":"thesis","author":[{"family":"Webster","given":"M."}],"title":"The XP Stabilizer Formalism","genre":"PhD thesis,","publisher":"University of Sydney","issued":"2023"},{"id":"arxiv:quant-ph/0407075","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.94.097203","source":"Crossref","title":"Entanglement in Spin Chains and Lattices with Long-Range Ising-Type Interactions","volume":"94","author":[{"given":"W.","family":"Dür","sequence":"first","affiliation":[]},{"given":"L.","family":"Hartmann","sequence":"additional","affiliation":[]},{"given":"M.","family":"Hein","sequence":"additional","affiliation":[]},{"given":"M.","family":"Lewenstein","sequence":"additional","affiliation":[]},{"given":"H.-J.","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,3,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.94.097203","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"097203","note":"arxivid:quant-ph/0407075"},{"id":"arxiv:quant-ph/0602096","type":"article-journal","author":[{"given":"M.","family":"Hein"},{"given":"W.","family":"Dür"},{"given":"J.","family":"Eisert"},{"given":"R.","family":"Raussendorf"},{"given":"M. Van den","family":"Nest"},{"given":"H. -J.","family":"Briegel"}],"title":"Entanglement in Graph States and its Applications","issued":{"date-parts":[[2006,2,11]]},"note":"arxivid:quant-ph/0602096\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0602230","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevlett.97.107206","source":"Crossref","title":"Ground-State Approximation for Strongly Interacting Spin Systems in Arbitrary Spatial Dimension","volume":"97","author":[{"given":"S.","family":"Anders","sequence":"first","affiliation":[]},{"given":"M. B.","family":"Plenio","sequence":"additional","affiliation":[]},{"given":"W.","family":"Dür","sequence":"additional","affiliation":[]},{"given":"F.","family":"Verstraete","sequence":"additional","affiliation":[]},{"given":"H.-J.","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,9,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.97.107206","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"107206","note":"arxivid:quant-ph/0602230"},{"id":"arxiv:1211.5554","type":"journal-article","publisher":"IOP Publishing","issue":"11","DOI":"10.1088/1367-2630/15/11/113022","page":"113022","source":"Crossref","title":"Quantum hypergraph states","volume":"15","author":[{"given":"M","family":"Rossi","sequence":"first","affiliation":[]},{"given":"M","family":"Huber","sequence":"additional","affiliation":[]},{"given":"D","family":"Bruß","sequence":"additional","affiliation":[]},{"given":"C","family":"Macchiavello","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2013,11,11]]},"URL":"http://dx.doi.org/10.1088/1367-2630/15/11/113022","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1211.5554"},{"id":"arxiv:1404.6492","type":"journal-article","publisher":"IOP Publishing","issue":"33","DOI":"10.1088/1751-8113/47/33/335303","page":"335303","source":"Crossref","title":"Entanglement and nonclassical properties of hypergraph states","volume":"47","author":[{"given":"Otfried","family":"Gühne","sequence":"first","affiliation":[]},{"given":"Martí","family":"Cuquet","sequence":"additional","affiliation":[]},{"given":"Frank E S","family":"Steinhoff","sequence":"additional","affiliation":[]},{"given":"Tobias","family":"Moroder","sequence":"additional","affiliation":[]},{"given":"Matteo","family":"Rossi","sequence":"additional","affiliation":[]},{"given":"Dagmar","family":"Bruß","sequence":"additional","affiliation":[]},{"given":"Barbara","family":"Kraus","sequence":"additional","affiliation":[]},{"given":"Chiara","family":"Macchiavello","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2014,8,5]]},"URL":"http://dx.doi.org/10.1088/1751-8113/47/33/335303","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1404.6492"},{"id":"arxiv:1410.3904","type":"journal-article","publisher":"IOP Publishing","issue":"9","DOI":"10.1088/1751-8113/48/9/095301","page":"095301","source":"Crossref","title":"Local unitary symmetries of hypergraph states","volume":"48","author":[{"given":"David W","family":"Lyons","sequence":"first","affiliation":[]},{"given":"Daniel J","family":"Upchurch","sequence":"additional","affiliation":[]},{"given":"Scott N","family":"Walck","sequence":"additional","affiliation":[]},{"given":"Chase D","family":"Yetter","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2015,2,4]]},"URL":"http://dx.doi.org/10.1088/1751-8113/48/9/095301","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1410.3904"},{"id":"arxiv:1805.12093","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.99.052304","source":"Crossref","title":"Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states","volume":"99","author":[{"given":"Mariami","family":"Gachechiladze","sequence":"first","affiliation":[]},{"given":"Otfried","family":"Gühne","sequence":"additional","affiliation":[]},{"given":"Akimasa","family":"Miyake","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,5,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.052304","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052304","note":"arxivid:1805.12093"},{"id":"doi:10.1038/s41598-019-49968-3","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Measurement-based quantum computing is one of the most promising quantum computing models. Although various universal resource states have been proposed so far, it was open whether only two Pauli bases are enough for both of universal measurement-based quantum computing and its verification. In this paper, we construct a universal hypergraph state that only requires <jats:italic>X</jats:italic> and <jats:italic>Z</jats:italic>-basis measurements for universal measurement-based quantum computing. We also show that universal measurement-based quantum computing on our hypergraph state can be verified in polynomial time using only <jats:italic>X</jats:italic> and <jats:italic>Z</jats:italic>-basis measurements. Furthermore, in order to demonstrate an advantage of our hypergraph state, we construct a verifiable blind quantum computing protocol that requires only <jats:italic>X</jats:italic> and <jats:italic>Z</jats:italic>-basis measurements for the client.</jats:p>","DOI":"10.1038/s41598-019-49968-3","source":"Crossref","title":"Quantum computational universality of hypergraph states with Pauli-X and Z basis measurements","volume":"9","author":[{"given":"Yuki","family":"Takeuchi","sequence":"first","affiliation":[]},{"given":"Tomoyuki","family":"Morimae","sequence":"additional","affiliation":[]},{"given":"Masahito","family":"Hayashi","sequence":"additional","affiliation":[]}],"container-title":"Scientific Reports","original-title":[],"language":"en","issued":{"date-parts":[[2019,9,19]]},"URL":"http://dx.doi.org/10.1038/s41598-019-49968-3","ISSN":["2045-2322"],"container-title-short":"Sci Rep","page":"13585","note":"alternative-id:49968"},{"id":"arxiv:1710.04631","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.123.110502","source":"Crossref","title":"Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains","volume":"123","author":[{"given":"Fernando G. S. L.","family":"Brandão","sequence":"first","affiliation":[]},{"given":"Elizabeth","family":"Crosson","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4488-2300","authenticated-orcid":true,"given":"M. Burak","family":"Şahinoğlu","sequence":"additional","affiliation":[]},{"given":"John","family":"Bowen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,9,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.123.110502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110502","note":"arxivid:1710.04631"},{"id":"arxiv:1906.03669","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>The eigenstate thermalization hypothesis (ETH) is a powerful conjecture for understanding how statistical mechanics emerges in a large class of many-body quantum systems. It has also been interpreted in a CFT context, and, in particular, holographic CFTs are expected to satisfy ETH. Recently, it was observed that the ETH condition corresponds to a necessary and sufficient condition for an approximate quantum error correcting code (AQECC), implying the presence of AQECCs in systems satisfying ETH. In this paper, we explore the properties of ETH as an error correcting code and show that there exists an explicit universal recovery channel for the code. Based on the analysis, we discuss a generalization that all chaotic theories contain error correcting codes. We then specialize to AdS/CFT to demonstrate the possibility of total bulk reconstruction in black holes with a well-defined macroscopic geometry. When combined with the existing AdS/CFT error correction story, this shows that black holes are enormously robust against erasure errors.</jats:p>","DOI":"10.1007/jhep08(2019)152","source":"Crossref","title":"Eigenstate thermalization hypothesis and approximate quantum error correction","volume":"2019","author":[{"given":"Ning","family":"Bao","sequence":"first","affiliation":[]},{"given":"Newton","family":"Cheng","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,8]]},"URL":"http://dx.doi.org/10.1007/JHEP08(2019)152","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"152","note":"alternative-id:11156\narxivid:1906.03669"},{"id":"arxiv:quant-ph/0506037","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1140/epjd/e2005-00252-0","page":"241-248","source":"Crossref","title":"Suppressing decoherence of quantum algorithms by jump codes","volume":"36","author":[{"given":"O.","family":"Kern","sequence":"first","affiliation":[]},{"given":"G.","family":"Alber","sequence":"additional","affiliation":[]}],"container-title":"The European Physical Journal D","original-title":[],"language":"en","issued":{"date-parts":[[2005,9,13]]},"URL":"http://dx.doi.org/10.1140/epjd/e2005-00252-0","ISSN":["1434-6060","1434-6079"],"container-title-short":"Eur. Phys. J. D","note":"alternative-id:252\narxivid:quant-ph/0506037"},{"id":"manual:-M.-Jimbo-and-K.-Shiromoto-Qu","type":"chapter","author":[{"family":"Jimbo","given":"M."},{"family":"Shiromoto","given":"K."}],"title":"Quantum jump codes and related combinatorial designs","container-title":"Information Security, Coding Theory and Related Combinatorics","publisher":"IOS Press","issued":"2011","page":"285–311"},{"id":"arxiv:1910.00471","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","DOI":"10.1137/20m1337375","page":"1410-1460","source":"Crossref","title":"Error Thresholds for Arbitrary Pauli Noise","volume":"50","author":[{"given":"Johannes","family":"Bausch","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1073-9795","authenticated-orcid":true,"given":"Felix","family":"Leditzky","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[2021,1]]},"URL":"http://dx.doi.org/10.1137/20M1337375","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/20M1337375\narxivid:1910.00471"},{"id":"arxiv:0708.1021","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2008.2008136","page":"433-438","source":"Crossref","title":"Codeword Stabilized Quantum Codes","volume":"55","author":[{"given":"Andrew","family":"Cross","sequence":"first","affiliation":[]},{"given":"Graeme","family":"Smith","sequence":"additional","affiliation":[]},{"given":"John A.","family":"Smolin","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2009,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.2008136","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:0708.1021"},{"id":"arxiv:cs/0610159","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2008.917720","page":"1700-1707","source":"Crossref","title":"Boolean Functions, Projection Operators, and Quantum Error Correcting Codes","volume":"54","author":[{"given":"Vaneet","family":"Aggarwal","sequence":"first","affiliation":[]},{"given":"A. Robert","family":"Calderbank","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.917720","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:cs/0610159"},{"id":"manual:-B.-Zeng.-Quantum-operations-","type":"thesis","author":[{"family":"Zeng","given":"B."}],"title":"Quantum operations and codes beyond the Stabilizer-Clifford framework","genre":"PhD thesis,","publisher":"Massachusetts Institute of Technology","issued":"2009"},{"id":"arxiv:0912.3245","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.81.052337","source":"Crossref","title":"Structured error recovery for code-word-stabilized quantum codes","volume":"81","author":[{"given":"Yunfan","family":"Li","sequence":"first","affiliation":[]},{"given":"Ilya","family":"Dumer","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,5,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.052337","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052337","note":"arxivid:0912.3245"},{"id":"arxiv:0712.1979","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.78.042303","source":"Crossref","title":"Quantum-error-correcting codes using qudit graph states","volume":"78","author":[{"given":"Shiang Yong","family":"Looi","sequence":"first","affiliation":[]},{"given":"Li","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Vlad","family":"Gheorghiu","sequence":"additional","affiliation":[]},{"given":"Robert B.","family":"Griffiths","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,10,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.042303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042303","note":"arxivid:0712.1979"},{"id":"arxiv:2107.11286","type":"article-journal","author":[{"given":"Upendra S.","family":"Kapshikar"}],"title":"The Diagonal Distance of CWS Codes","issued":{"date-parts":[[2021,8,3]]},"note":"arxivid:2107.11286\narxiv_version_number:2"},{"id":"arxiv:0907.2038","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevlett.104.190501","source":"Crossref","title":"Clustered Error Correction of Codeword-Stabilized Quantum Codes","volume":"104","author":[{"given":"Yunfan","family":"Li","sequence":"first","affiliation":[]},{"given":"Ilya","family":"Dumer","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,5,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.104.190501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"190501","note":"arxivid:0907.2038"},{"id":"doi:10.1109/ISIT.2010.5513671","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2010.5513671","page":"2662-2666","source":"Crossref","title":"Clustered bounded-distance decoding of codeword-stabilized quantum codes","author":[{"given":"Yunfan","family":"Li","sequence":"first","affiliation":[]},{"given":"Ilya","family":"Dumer","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"event":"2010 IEEE International Symposium on Information Theory - ISIT","container-title":"2010 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2010,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2010.5513671"},{"id":"preset:Li10","type":"thesis","author":[{"family":"Li","given":"Y."}],"title":"Codeword Stabilized Quantum Codes and Their Error Correction","genre":"PhD thesis,","publisher":"UC Riverside","issued":"2010"},{"id":"doi:10.1017/CBO9781139034807.012","type":"book-chapter","publisher":"Cambridge University Press","DOI":"10.1017/cbo9781139034807.012","page":"261-278","source":"Crossref","title":"Nonadditive quantum codes","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Martin","family":"Rötteler","sequence":"additional","affiliation":[]}],"container-title":"Quantum Error Correction","original-title":[],"issued":{"date-parts":[[2013,9,12]]},"URL":"http://dx.doi.org/10.1017/CBO9781139034807.012","note":"edition-number:1"},{"id":"preset:Cross09","type":"thesis","author":[{"family":"Cross","given":"A.W."}],"title":"Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes","genre":"PhD thesis,","publisher":"Massachusetts Institute of Technology","issued":"2008"},{"id":"arxiv:2304.08363","type":"article-journal","author":[{"given":"Zipeng","family":"Wu"},{"given":"Song","family":"Cheng"},{"given":"Bei","family":"Zeng"}],"title":"A ZX-Calculus Approach for the Construction of Graph Codes","issued":{"date-parts":[[2024,3,28]]},"note":"arxivid:2304.08363\narxiv_version_number:3"},{"id":"doi:10.1090/S0002-9947-07-04242-0","type":"journal-article","publisher":"American Mathematical Society (AMS)","issue":"4","abstract":"<p>\n                    In this paper, we present a characterization of (binary and non-binary) quantum error-correcting codes. Based on this characterization, we introduce a method to construct\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p</mml:mi>\n                          <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    -ary quantum codes using Boolean functions satisfying a system of certain quadratic relations. As a consequence of the construction, we are able to construct quantum codes of minimum distance\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n                        <mml:semantics>\n                          <mml:mn>2</mml:mn>\n                          <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    . In particular, we produce a class of binary quantum\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis left-parenthesis n comma 2 Superscript n minus 2 Baseline minus one half StartBinomialOrMatrix n minus 1 Choose left-parenthesis n minus 1 right-parenthesis slash 2 EndBinomialOrMatrix comma 2 right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(</mml:mo>\n                            <mml:mo stretchy=\"false\">(</mml:mo>\n                            <mml:mi>n</mml:mi>\n                            <mml:mo>,</mml:mo>\n                            <mml:msup>\n                              <mml:mn>2</mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n</mml:mi>\n                                <mml:mo>\n                                  −\n                                  \n                                </mml:mo>\n                                <mml:mn>2</mml:mn>\n                              </mml:mrow>\n                            </mml:msup>\n                            <mml:mo>\n                              −\n                              \n                            </mml:mo>\n                            <mml:mfrac>\n                              <mml:mn>1</mml:mn>\n                              <mml:mn>2</mml:mn>\n                            </mml:mfrac>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow>\n                                <mml:mstyle scriptlevel=\"0\">\n                                  <mml:mrow class=\"MJX-TeXAtom-OPEN\">\n                                    <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo>\n                                  </mml:mrow>\n                                </mml:mstyle>\n                                <mml:mfrac linethickness=\"0\">\n                                  <mml:mrow>\n                                    <mml:mi>n</mml:mi>\n                                    <mml:mo>\n                                      −\n                                      \n                                    </mml:mo>\n                                    <mml:mn>1</mml:mn>\n                                  </mml:mrow>\n                                  <mml:mrow>\n                                    <mml:mo stretchy=\"false\">(</mml:mo>\n                                    <mml:mi>n</mml:mi>\n                                    <mml:mo>\n                                      −\n                                      \n                                    </mml:mo>\n                                    <mml:mn>1</mml:mn>\n                                    <mml:mo stretchy=\"false\">)</mml:mo>\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mo>/</mml:mo>\n                                    </mml:mrow>\n                                    <mml:mn>2</mml:mn>\n                                  </mml:mrow>\n                                </mml:mfrac>\n                                <mml:mstyle scriptlevel=\"0\">\n                                  <mml:mrow class=\"MJX-TeXAtom-CLOSE\">\n                                    <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo>\n                                  </mml:mrow>\n                                </mml:mstyle>\n                              </mml:mrow>\n                            </mml:mrow>\n                            <mml:mo>,</mml:mo>\n                            <mml:mn>2</mml:mn>\n                            <mml:mo stretchy=\"false\">)</mml:mo>\n                            <mml:mo stretchy=\"false\">)</mml:mo>\n                          </mml:mrow>\n                          <mml:annotation encoding=\"application/x-tex\">((n,2^{n-2}-\\frac 12{n-1\\choose (n-1)/2},2))</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    -codes for odd length\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 5\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n</mml:mi>\n                            <mml:mo>\n                              ≥\n                              \n                            </mml:mo>\n                            <mml:mn>5</mml:mn>\n                          </mml:mrow>\n                          <mml:annotation encoding=\"application/x-tex\">n\\ge 5</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    . For\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 11\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n</mml:mi>\n                            <mml:mo>\n                              ≥\n                              \n                            </mml:mo>\n                            <mml:mn>11</mml:mn>\n                          </mml:mrow>\n                          <mml:annotation encoding=\"application/x-tex\">n\\ge 11</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    , this improves the result by Rains in\n                    <italic>Quantum codes of minimal distance two</italic>\n                    , 1999, showing the existence of binary quantum\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis left-parenthesis n comma 3 dot 2 Superscript n minus 4 Baseline comma 2 right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(</mml:mo>\n                            <mml:mo stretchy=\"false\">(</mml:mo>\n                            <mml:mi>n</mml:mi>\n                            <mml:mo>,</mml:mo>\n                            <mml:mn>3</mml:mn>\n                            <mml:mo>\n                              ⋅\n                              \n                            </mml:mo>\n                            <mml:msup>\n                              <mml:mn>2</mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n</mml:mi>\n                                <mml:mo>\n                                  −\n                                  \n                                </mml:mo>\n                                <mml:mn>4</mml:mn>\n                              </mml:mrow>\n                            </mml:msup>\n                            <mml:mo>,</mml:mo>\n                            <mml:mn>2</mml:mn>\n                            <mml:mo stretchy=\"false\">)</mml:mo>\n                            <mml:mo stretchy=\"false\">)</mml:mo>\n                          </mml:mrow>\n                          <mml:annotation encoding=\"application/x-tex\">((n,3\\cdot 2^{n-4},2))</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    -codes for odd\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 5\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n</mml:mi>\n                            <mml:mo>\n                              ≥\n                              \n                            </mml:mo>\n                            <mml:mn>5</mml:mn>\n                          </mml:mrow>\n                          <mml:annotation encoding=\"application/x-tex\">n\\ge 5</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    . Moreover, our binary quantum\n                    <inline-formula content-type=\"math/mathml\">\n                      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis left-parenthesis n comma 2 Superscript n minus 2 Baseline minus one half StartBinomialOrMatrix n minus 1 Choose left-parenthesis n minus 1 right-parenthesis slash 2 EndBinomialOrMatrix comma 2 right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(</mml:mo>\n                            <mml:mo stretchy=\"false\">(</mml:mo>\n                            <mml:mi>n</mml:mi>\n                            <mml:mo>,</mml:mo>\n                            <mml:msup>\n                              <mml:mn>2</mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n</mml:mi>\n                                <mml:mo>\n                                  −\n                                  \n                                </mml:mo>\n                                <mml:mn>2</mml:mn>\n                              </mml:mrow>\n                            </mml:msup>\n                            <mml:mo>\n                              −\n                              \n                            </mml:mo>\n                            <mml:mfrac>\n                              <mml:mn>1</mml:mn>\n                              <mml:mn>2</mml:mn>\n                            </mml:mfrac>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow>\n                                <mml:mstyle scriptlevel=\"0\">\n                                  <mml:mrow class=\"MJX-TeXAtom-OPEN\">\n                                    <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo>\n                                  </mml:mrow>\n                                </mml:mstyle>\n                                <mml:mfrac linethickness=\"0\">\n                                  <mml:mrow>\n                                    <mml:mi>n</mml:mi>\n                                    <mml:mo>\n                                      −\n                                      \n                                    </mml:mo>\n                                    <mml:mn>1</mml:mn>\n                                  </mml:mrow>\n                                  <mml:mrow>\n                                    <mml:mo stretchy=\"false\">(</mml:mo>\n                                    <mml:mi>n</mml:mi>\n                                    <mml:mo>\n                                      −\n                                      \n                                    </mml:mo>\n                                    <mml:mn>1</mml:mn>\n                                    <mml:mo stretchy=\"false\">)</mml:mo>\n                                    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                      <mml:mo>/</mml:mo>\n                                    </mml:mrow>\n                                    <mml:mn>2</mml:mn>\n                                  </mml:mrow>\n                                </mml:mfrac>\n                                <mml:mstyle scriptlevel=\"0\">\n                                  <mml:mrow class=\"MJX-TeXAtom-CLOSE\">\n                                    <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo>\n                                  </mml:mrow>\n                                </mml:mstyle>\n                              </mml:mrow>\n                            </mml:mrow>\n                            <mml:mo>,</mml:mo>\n                            <mml:mn>2</mml:mn>\n                            <mml:mo stretchy=\"false\">)</mml:mo>\n                            <mml:mo stretchy=\"false\">)</mml:mo>\n                          </mml:mrow>\n                          <mml:annotation encoding=\"application/x-tex\">((n,2^{n-2}-\\frac 12{n-1\\choose (n-1)/2},2))</mml:annotation>\n                        </mml:semantics>\n                      </mml:math>\n                    </inline-formula>\n                    -codes of odd length achieve the Singleton bound asymptotically. Finally, based on our characterization some propagation rules of quantum codes are proposed and the rules are similar to those in classical coding theory. It turns out that some new quantum codes are found through these propagation rules.\n                  </p>","DOI":"10.1090/s0002-9947-07-04242-0","page":"2007-2019","source":"Crossref","title":"A new construction of quantum error-correcting codes","volume":"360","author":[{"given":"Keqin","family":"Feng","sequence":"first","affiliation":[]},{"given":"Chaoping","family":"Xing","sequence":"additional","affiliation":[]}],"container-title":"Transactions of the American Mathematical Society","original-title":[],"language":"en","issued":{"date-parts":[[2007,10,23]]},"URL":"http://dx.doi.org/10.1090/S0002-9947-07-04242-0","ISSN":["1088-6850","0002-9947"],"container-title-short":"Trans. Amer. Math. Soc.","note":"special_numbering:875\nalternative-id:S0002-9947-07-04242-0"},{"id":"arxiv:1907.04537","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.100.062303","source":"Crossref","title":"Heuristic construction of codeword stabilized codes","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0001-6041-8538","authenticated-orcid":true,"given":"Alex","family":"Rigby","sequence":"first","affiliation":[]},{"given":"J. C.","family":"Olivier","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Jarvis","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.100.062303","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062303","note":"arxivid:1907.04537"},{"id":"arxiv:quant-ph/9703002","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.79.953","page":"953-954","source":"Crossref","title":"A Nonadditive Quantum Code","volume":"79","author":[{"given":"Eric M.","family":"Rains","sequence":"first","affiliation":[]},{"given":"R. H.","family":"Hardin","sequence":"additional","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]},{"given":"N. J. A.","family":"Sloane","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1997,8,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.79.953","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9703002"},{"id":"arxiv:0709.1780","type":"article-journal","author":[{"given":"Sixia","family":"Yu"},{"given":"Qing","family":"Chen"},{"given":"C. H.","family":"Oh"}],"title":"Graphical Quantum Error-Correcting Codes","issued":{"date-parts":[[2007,9,12]]},"note":"arxivid:0709.1780\narxiv_version_number:1"},{"id":"arxiv:1111.5445","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.85.062312","source":"Crossref","title":"Experimental implementation of a codeword-stabilized quantum code","volume":"85","author":[{"given":"Jingfu","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,6,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.85.062312","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062312","note":"arxivid:1111.5445"},{"id":"arxiv:quant-ph/0210097","type":"article-journal","author":[{"given":"V.","family":"Arvind"},{"given":"Piyush P","family":"Kurur"},{"given":"K. R.","family":"Parthasarathy"}],"title":"Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group","issued":{"date-parts":[[2002,10,12]]},"note":"arxivid:quant-ph/0210097\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9703016","type":"article-journal","author":[{"given":"Markus","family":"Grassl"},{"given":"Thomas","family":"Beth"}],"title":"A Note on Non-Additive Quantum Codes","issued":{"date-parts":[[1997,3,10]]},"note":"arxivid:quant-ph/9703016\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9710031","type":"article-journal","author":[{"given":"Vwani P.","family":"Roychowdhury"},{"given":"Farrokh","family":"Vatan"}],"title":"On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes","issued":{"date-parts":[[1997,10,9]]},"note":"arxivid:quant-ph/9710031\narxiv_version_number:1"},{"id":"arxiv:2311.07265","type":"article-journal","author":[{"given":"Jing-Lei","family":"Xia"}],"title":"Quotient Space Quantum Codes","issued":{"date-parts":[[2024,9,6]]},"note":"arxivid:2311.07265\narxiv_version_number:5"},{"id":"arxiv:1901.02913","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2018.8437497","page":"796-800","source":"Crossref","title":"Hybrid Codes","author":[{"given":"Andrew","family":"Nemec","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[]}],"event":"2018 IEEE International Symposium on Information Theory (ISIT)","container-title":"2018 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2018,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2018.8437497","note":"arxivid:1901.02913"},{"id":"arxiv:1806.03702","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","DOI":"10.1007/s10773-023-05439-0","source":"Crossref","title":"A Unification of the Coding Theory and OAQEC Perspectives on Hybrid Codes","volume":"62","author":[{"given":"Shayan","family":"Majidy","sequence":"first","affiliation":[]}],"container-title":"International Journal of Theoretical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,7]]},"URL":"http://dx.doi.org/10.1007/s10773-023-05439-0","ISSN":["1572-9575"],"container-title-short":"Int J Theor Phys","page":"177","note":"alternative-id:5439\narxivid:1806.03702"},{"id":"arxiv:0801.0821","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2008.4595008","page":"359-363","source":"Crossref","title":"Unified quantum convolutional coding","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"event":"2008 IEEE International Symposium on Information Theory - ISIT","container-title":"2008 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2008.4595008","note":"arxivid:0801.0821"},{"id":"arxiv:2002.11075","type":"article-journal","author":[{"given":"Andrew","family":"Nemec"},{"given":"Andreas","family":"Klappenecker"}],"title":"Nonbinary Error-Detecting Hybrid Codes","issued":{"date-parts":[[2020,2,25]]},"note":"arxivid:2002.11075\narxiv_version_number:1"},{"id":"arxiv:2304.11442","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We introduce a stabilizer formalism for the general quantum error correction framework called operator algebra quantum error correction (OAQEC), which generalizes Gottesman&amp;apos;s formulation for traditional quantum error correcting codes (QEC) and Poulin&amp;apos;s for operator quantum error correction and subsystem codes (OQEC). The construction generates hybrid classical-quantum stabilizer codes and we formulate a theorem that fully characterizes the Pauli errors that are correctable for a given code, generalizing the fundamental theorems for the QEC and OQEC stabilizer formalisms. We discover hybrid versions of the Bacon-Shor subsystem codes motivated by the formalism, and we apply the theorem to derive a result that gives the distance of such codes. We show how some recent hybrid subspace code constructions are captured by the formalism, and we also indicate how it extends to qudits.</jats:p>","DOI":"10.22331/q-2024-02-21-1261","page":"1261","source":"Crossref","title":"Stabilizer Formalism for Operator Algebra Quantum Error Correction","volume":"8","author":[{"given":"Guillaume","family":"Dauphinais","sequence":"first","affiliation":[{"name":"Xanadu, Toronto, ON M5G 2C8, Canada"}]},{"given":"David W.","family":"Kribs","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON M5G 2C8, Canada"},{"name":"Department of Mathematics & Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada"}]},{"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"name":"Xanadu, Toronto, ON M5G 2C8, Canada"},{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,2,21]]},"URL":"http://dx.doi.org/10.22331/q-2024-02-21-1261","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2304.11442"},{"id":"arxiv:2409.11312","type":"article-journal","author":[{"given":"Theerapat","family":"Tansuwannont"},{"given":"Andrew","family":"Nemec"}],"title":"Synchronizable hybrid subsystem codes","issued":{"date-parts":[[2024,9,25]]},"note":"arxivid:2409.11312\narxiv_version_number:2"},{"id":"arxiv:2603.04564","type":"article-journal","author":[{"given":"Vismay","family":"Joshi"},{"given":"Anubhab","family":"Rudra"},{"given":"Sourav","family":"Dutta"},{"given":"Siddharth","family":"Dhomkar"},{"given":"Prabha","family":"Mandayam"}],"title":"Demonstrating Noise-adapted Quantum Error Correction With Break-Even Performance","issued":{"date-parts":[[2026,4,1]]},"note":"arxivid:2603.04564\narxiv_version_number:2"},{"id":"arxiv:2310.17652","type":"article-journal","author":[{"given":"Eric","family":"Kubischta"},{"given":"Ian","family":"Teixeira"}],"title":"Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates","issued":{"date-parts":[[2024,10,4]]},"note":"arxivid:2310.17652\narxiv_version_number:3"},{"id":"arxiv:2310.05358","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We construct a new family of permutationally invariant codes that correct <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math> Pauli errors for any <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo>&amp;#x2265;</mml:mo><mml:mn>1</mml:mn></mml:math>. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math> optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math> Pauli errors from the previously known results for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math> to any number of errors. For small <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi></mml:math>, these conditions can be used to construct new examples of codes by computer.</jats:p>","DOI":"10.22331/q-2024-04-30-1321","page":"1321","source":"Crossref","title":"A family of permutationally invariant quantum codes","volume":"8","author":[{"given":"Arda","family":"Aydin","sequence":"first","affiliation":[{"name":"Department of ECE and Institute for Systems Research, University of Maryland, College Park, MD 20742"}]},{"given":"Max A.","family":"Alekseyev","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The George Washington University, Washington, DC 20052"}]},{"ORCID":"https://orcid.org/0000-0003-1985-4623","authenticated-orcid":false,"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[{"name":"Department of ECE and Institute for Systems Research, University of Maryland, College Park, MD 20742"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,30]]},"URL":"http://dx.doi.org/10.22331/q-2024-04-30-1321","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2310.05358"},{"id":"arxiv:2411.13142","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    Transversal gates on quantum error correction codes have been a promising approach for fault-tolerant quantum computing, but are limited by the Eastin-Knill no-go theorem. Existing solutions such as gate teleportation and magic state distillation are resource-intensive. We present a measurement-free code-switching protocol for universal quantum computation, switching between a stabilizer code for transversal Clifford gates and a permutation-invariant (PI) code for transversal non-Clifford gates that are logical\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>Z</a:mi>\n                    </a:math>\n                    rotations for any rational multiple of\n                    <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <c:mi>π</c:mi>\n                    </c:math>\n                    . The novel non-Clifford gates enabled by this code-switching protocol provide for a lower gate count implementation of a universal gate set relative to the Clifford + \n                    <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <e:mi>T</e:mi>\n                    </e:math>\n                    gate set. To achieve this, we present a protocol for performing controlled- gates between the codes using near-term quantum control operations that employ a catalytic bosonic mode. We also present a new class of PI codes with tunable code distance, supporting transversal non-Clifford gates, and demonstrate their reduced gate count overhead relative to a comparable stabilizer code to stabilizer code-switching scheme.\n                  </jats:p>","DOI":"10.1103/p28m-hb6x","source":"Crossref","title":"Measurement-Free Code-Switching Protocol for Low-Overhead Quantum Computation Using Permutation-Invariant Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-1115-0074","authenticated-orcid":true,"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05krs5044","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sheffield"}]},{"given":"Yumang","family":"Jing","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01sf06y89","id-type":"ROR","asserted-by":"publisher"}],"name":"Macquarie University"}]},{"ORCID":"https://orcid.org/0000-0002-6019-966X","authenticated-orcid":true,"given":"Gavin K.","family":"Brennen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01sf06y89","id-type":"ROR","asserted-by":"publisher"}],"name":"Macquarie University"},{"name":"BTQ Technologies, 16-104 555 Burrard Street, Vancouver, British Columbia, Canada V7X 1M8"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,20]]},"URL":"http://dx.doi.org/10.1103/p28m-hb6x","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040341","note":"arxivid:2411.13142"},{"id":"doi:10.22331/qv-2024-05-13-80","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","DOI":"10.22331/qv-2024-05-13-80","page":"80","source":"Crossref","title":"A new take on permutation-invariant quantum codes","volume":"8","author":[{"given":"Yingkai","family":"Ouyang","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom"}]}],"container-title":"Quantum Views","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,13]]},"URL":"http://dx.doi.org/10.22331/qv-2024-05-13-80","container-title-short":"Quantum Views"},{"id":"arxiv:2509.03413","type":"article-journal","author":[{"given":"Lewis","family":"Bulled"},{"given":"Yingkai","family":"Ouyang"}],"title":"An angular momentum approach to quantum insertion errors","issued":{"date-parts":[[2025,9,3]]},"note":"arxivid:2509.03413\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9906114","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.85.194","page":"194-197","source":"Crossref","title":"Pauli Exchange Errors in Quantum Computation","volume":"85","author":[{"given":"Mary Beth","family":"Ruskai","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2000,7,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.85.194","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9906114"},{"id":"arxiv:2602.08780","type":"article-journal","author":[{"given":"Lewis","family":"Bulled"},{"given":"Yingkai","family":"Ouyang"}],"title":"The equivalence of quantum deletion and insertion errors on permutation-invariant codes","issued":{"date-parts":[[2026,2,9]]},"note":"arxivid:2602.08780\narxiv_version_number:1"},{"id":"doi:10.1103/PhysRevA.99.012335","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.99.012335","source":"Crossref","title":"Initializing a permutation-invariant quantum error-correction code","volume":"99","author":[{"given":"Chunfeng","family":"Wu","sequence":"first","affiliation":[]},{"given":"Yimin","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Chu","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Yingkai","family":"Ouyang","sequence":"additional","affiliation":[]},{"given":"Gangcheng","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Xun-Li","family":"Feng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,1,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.012335","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012335"},{"id":"arxiv:1904.07358","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-030-25027-0_9","page":"126-139","source":"Crossref","title":"Deterministic Preparation of Dicke States","author":[{"given":"Andreas","family":"Bärtschi","sequence":"first","affiliation":[]},{"given":"Stephan","family":"Eidenbenz","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2019]]},"ISBN":["9783030250263","9783030250270"],"URL":"http://dx.doi.org/10.1007/978-3-030-25027-0_9","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Cham\narxivid:1904.07358"},{"id":"arxiv:1908.01120","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevlett.125.190403","source":"Crossref","title":"Geometric Pathway to Scalable Quantum Sensing","volume":"125","author":[{"ORCID":"https://orcid.org/0000-0003-2322-6486","authenticated-orcid":true,"given":"Mattias T.","family":"Johnsson","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9067-5500","authenticated-orcid":true,"given":"Nabomita Roy","family":"Mukty","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4063-1264","authenticated-orcid":true,"given":"Daniel","family":"Burgarth","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Volz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6019-966X","authenticated-orcid":true,"given":"Gavin K.","family":"Brennen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,11,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.190403","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"190403","note":"arxivid:1908.01120"},{"id":"arxiv:2307.14840","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Fault tolerant quantum computers repetitively apply a four-step procedure: First, perform a few one and two-qubit quantum gates. Second, perform a syndrome measurement on a subset of the qubits. Third, perform fast classical computations to establish if and where errors occurred. And, fourth, correct the errors with a correction step. The next iteration applies the same procedure with new one and two-qubit gates. Even though current error-rates prohibit this procedure to work and fault tolerant quantum computing remains a distant goal, the same procedure can already prove useful today. In this work we make use of this four-step scheme not to carry out fault-tolerant computations, but to enhance short, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>t</mml:mi></mml:math>-depth, quantum circuits that perform 1 qubit gates and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>i</mml:mi><mml:mi>g</mml:mi><mml:mi>h</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi></mml:math> 2 qubit gates.We introduce a new computational model called <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">Local Alternating Quantum Classical Computations</mml:mtext></mml:mrow></mml:math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"sans-serif\">(LAQCC)</mml:mtext></mml:mrow></mml:math>. In this model, qubits are placed in a grid and they can only interact with their direct neighbors; the quantum circuits are of constant depth with intermediate measurements; a classical controller can perform log-depth computations on these intermediate measurement outcomes and control future quantum operations based on the outcome. This model fits naturally between quantum algorithms in the NISQ era and full-fledged fault-tolerant quantum computation. We first prove that any Clifford circuit has an equivalent <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"sans-serif\">LAQCC</mml:mtext></mml:mrow></mml:math> circuit, and that any <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"sans-serif\">LAQCC</mml:mtext></mml:mrow></mml:math> circuit can be simulated by a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"sans-serif\">Q</mml:mi><mml:mi mathvariant=\"sans-serif\">N</mml:mi><mml:msup><mml:mi mathvariant=\"sans-serif\">C</mml:mi><mml:mn mathvariant=\"sans-serif\">1</mml:mn></mml:msup></mml:mrow></mml:math>circuit. Next, we conjecture the non-simulatability of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"sans-serif\">LAQCC</mml:mtext></mml:mrow></mml:math> by showing that <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"sans-serif\">LAQCC</mml:mtext></mml:mrow></mml:math> contains the class of Instantaneous Quantum Polynomial-time circuits. We also show that any <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"sans-serif\">LAQCC</mml:mtext></mml:mrow></mml:math> circuit with polynomial-sized quantum circuits and unbounded classical computations is contained in the class of quantum circuits equipped with post-selection gates with respect to the task of state preparation. We continue by presenting <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"sans-serif\">LAQCC</mml:mtext></mml:mrow></mml:math> implementations of different subroutines, including OR-gates, quantum Fourier transforms and Threshold gates. These subroutines prove vital in constructing three state preparation routines in the main part of this work. Preparing a uniform superposition uses constant-depth arithmetic gates, combined with an exact Grover implementation by Long. For the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>W</mml:mi></mml:math>-state, we employ a compress-uncompress method to switch between a binary and one-hot encoding. This method extends to the more generalized Dicke-states, the superposition of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-bit strings of Hamming weight <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>, for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, but fails for higher <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math> due to the birthday paradox. We extend this protocol to a protocol that prepares many-body scar states, highly excited states with low entanglement and longer coherence times than states with the same energy density. We present a circuit for preparing Dicke-states for larger <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math> requiring log-depth circuits that maps between the factoradic number system and the combinatorial number system.</jats:p>","DOI":"10.22331/q-2024-12-09-1552","page":"1552","source":"Crossref","title":"State preparation by shallow circuits using feed forward","volume":"8","author":[{"given":"Harry","family":"Buhrman","sequence":"first","affiliation":[{"name":"QuSoft, CWI & University of Amsterdam, Amsterdam, the Netherlands"}]},{"given":"Marten","family":"Folkertsma","sequence":"additional","affiliation":[{"name":"QuSoft, CWI & University of Amsterdam, Amsterdam, the Netherlands"}]},{"given":"Bruno","family":"Loff","sequence":"additional","affiliation":[{"name":"LASIGE & Department of Mathematics, University of Lisbon"}]},{"given":"Niels M. P.","family":"Neumann","sequence":"additional","affiliation":[{"name":"QuSoft, CWI & University of Amsterdam, Amsterdam, the Netherlands"},{"name":"The Netherlands Organisation for Applied Scientific Research (TNO), Delft, the Netherlands"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,9]]},"URL":"http://dx.doi.org/10.22331/q-2024-12-09-1552","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2307.14840"},{"id":"arxiv:2403.07604","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","abstract":"<jats:p>We introduce protocols to prepare many-body quantum states with quantum circuits assisted by local operations and classical communication. We show that by lifting the requirement of exact preparation, one can substantially save resources. In particular, the so-called <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>W</a:mi></a:math> and, more generally, Dicke states require a circuit depth and number of ancillas per site that are independent of the system size. As a by-product of our work, we introduce an efficient scheme to implement certain nonlocal, non-Clifford unitary operators. We also discuss how similar ideas may be applied in the preparation of eigenstates of well-known spin models, both free and interacting.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevlett.133.230401","source":"Crossref","title":"Approximating Many-Body Quantum States with Quantum Circuits and Measurements","volume":"133","author":[{"ORCID":"https://orcid.org/0000-0002-0107-3338","authenticated-orcid":true,"given":"Lorenzo","family":"Piroli","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01111rn36","id-type":"ROR","asserted-by":"publisher"}],"name":"Università di Bologna"},{"id":[{"id":"https://ror.org/04j0x0h93","id-type":"ROR","asserted-by":"publisher"}],"name":"INFN Sezione di Bologna"}]},{"ORCID":"https://orcid.org/0000-0002-6809-8505","authenticated-orcid":true,"given":"Georgios","family":"Styliaris","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01vekys64","id-type":"ROR","asserted-by":"publisher"}],"name":"Max-Planck-Institut für Quantenoptik"},{"id":[{"id":"https://ror.org/04xrcta15","id-type":"ROR","asserted-by":"publisher"}],"name":"Munich Center for Quantum Science and Technology (MCQST)"}]},{"ORCID":"https://orcid.org/0000-0003-3359-1743","authenticated-orcid":true,"given":"J. Ignacio","family":"Cirac","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01vekys64","id-type":"ROR","asserted-by":"publisher"}],"name":"Max-Planck-Institut für Quantenoptik"},{"id":[{"id":"https://ror.org/04xrcta15","id-type":"ROR","asserted-by":"publisher"}],"name":"Munich Center for Quantum Science and Technology (MCQST)"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.230401","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230401","note":"arxivid:2403.07604"},{"id":"arxiv:2411.03428","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/9gjk-rgql","source":"Crossref","title":"Efficient Preparation of Dicke States","volume":"136","author":[{"ORCID":"https://orcid.org/0009-0002-5616-2968","authenticated-orcid":true,"given":"Jeffery","family":"Yu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0002-5005-3763","authenticated-orcid":true,"given":"Sean R.","family":"Muleady","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0003-2848-1216","authenticated-orcid":true,"given":"Yu-Xin","family":"Wang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"given":"Nathan","family":"Schine","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0003-0509-3421","authenticated-orcid":true,"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0002-9903-837X","authenticated-orcid":true,"given":"Andrew M.","family":"Childs","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,20]]},"URL":"http://dx.doi.org/10.1103/9gjk-rgql","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030601","note":"arxivid:2411.03428"},{"id":"arxiv:quant-ph/0111017","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.65.032327","source":"Crossref","title":"Simulation of many-body interactions by conditional geometric phases","volume":"65","author":[{"given":"Xiaoguang","family":"Wang","sequence":"first","affiliation":[]},{"given":"Paolo","family":"Zanardi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,3,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.032327","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032327","note":"arxivid:quant-ph/0111017"},{"id":"arxiv:quant-ph/0409084","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tac.2006.871942","page":"448-459","source":"Crossref","title":"Generalized Performance of Concatenated Quantum Codes—A Dynamical Systems Approach","volume":"51","author":[{"given":"J.","family":"Fern","sequence":"first","affiliation":[]},{"given":"J.","family":"Kempe","sequence":"additional","affiliation":[]},{"given":"S.N.","family":"Simic","sequence":"additional","affiliation":[]},{"given":"S.","family":"Sastry","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Automatic Control","original-title":[],"language":"en","issued":{"date-parts":[[2006,3]]},"URL":"http://dx.doi.org/10.1109/TAC.2006.871942","ISSN":["0018-9286"],"container-title-short":"IEEE Trans. Automat. Contr.","note":"arxivid:quant-ph/0409084"},{"id":"arxiv:2203.16488","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.129.240502","source":"Crossref","title":"Distributed Quantum Error Correction for Chip-Level Catastrophic Errors","volume":"129","author":[{"ORCID":"https://orcid.org/0000-0002-8738-9420","authenticated-orcid":true,"given":"Qian","family":"Xu","sequence":"first","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA"}]},{"ORCID":"https://orcid.org/0000-0001-5419-5999","authenticated-orcid":true,"given":"Alireza","family":"Seif","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4797-9733","authenticated-orcid":true,"given":"Haoxiong","family":"Yan","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA"}]},{"given":"Nam","family":"Mannucci","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA"}]},{"ORCID":"https://orcid.org/0000-0002-9249-8285","authenticated-orcid":true,"given":"Bernard Ousmane","family":"Sane","sequence":"additional","affiliation":[{"name":"Graduate School of Media and Governance, Keio University, 5322 Endo, Fujisawa 252-0882, Japan"}]},{"ORCID":"https://orcid.org/0000-0002-5044-9514","authenticated-orcid":true,"given":"Rodney","family":"Van Meter","sequence":"additional","affiliation":[{"name":"Faculty of Environment and Information Studies, Keio University, 5322 Endo, Fujisawa 252-0882, Japan"}]},{"ORCID":"https://orcid.org/0000-0003-4981-4294","authenticated-orcid":true,"given":"Andrew N.","family":"Cleland","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA"},{"name":"Center for Molecular Engineering and Material Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.129.240502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240502","note":"arxivid:2203.16488"},{"id":"arxiv:2502.14835","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Device error rates on current quantum computers have improved enough to where demonstrations of error correction below break even are now possible. Still, the circuits required for quantum error correction introduce significant overhead and sometimes inject more errors than they correct. In this work, we introduce adaptive syndrome extraction as a scheme to improve code performance and reduce the quantum error-correction cycle time by measuring only the stabilizer generators that are likely to provide useful syndrome information. We provide a concrete example of the scheme through the [[4,2,2]] code concatenated with a hypergraph product code and a syndrome-extraction cycle that uses quantum error detection to modify the syndrome-extraction circuits in real time. Compared to nonconcatenated codes and nonadaptive syndrome extraction, we find that the adaptive scheme achieves over an order of magnitude lower logical error rates while requiring fewer  gates and physical qubits. Furthermore, we show how to achieve fault-tolerant universal logical computation with [[4,2,2]]-concatenated hypergraph product codes.</jats:p>","DOI":"10.1103/ps3r-wf84","source":"Crossref","title":"Adaptive Syndrome Extraction","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-7586-2786","authenticated-orcid":true,"given":"Noah","family":"Berthusen","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"}]},{"ORCID":"https://orcid.org/0000-0003-2249-3478","authenticated-orcid":true,"given":"Shi Jie Samuel","family":"Tan","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"}]},{"ORCID":"https://orcid.org/0000-0003-3843-9935","authenticated-orcid":true,"given":"Eric","family":"Huang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"}]},{"ORCID":"https://orcid.org/0009-0002-2948-3395","authenticated-orcid":true,"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,16]]},"URL":"http://dx.doi.org/10.1103/ps3r-wf84","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030307","note":"arxivid:2502.14835"},{"id":"arxiv:quant-ph/0206061","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.66.032304","source":"Crossref","title":"Exact performance of concatenated quantum codes","volume":"66","author":[{"given":"Benjamin","family":"Rahn","sequence":"first","affiliation":[]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[]},{"given":"Hideo","family":"Mabuchi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,9,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.66.032304","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032304","note":"arxivid:quant-ph/0206061"},{"id":"arxiv:2408.05260","type":"article-journal","author":[{"given":"Matthias","family":"Christandl"},{"given":"Omar","family":"Fawzi"},{"given":"Ashutosh","family":"Goswami"}],"title":"Fault-tolerant quantum input/output","issued":{"date-parts":[[2025,12,2]]},"note":"arxivid:2408.05260\narxiv_version_number:3"},{"id":"arxiv:2506.13619","type":"article-journal","author":[{"given":"Daniel","family":"Litinski"}],"title":"Blocklet concatenation: Low-overhead fault-tolerant protocols for fusion-based quantum computation","issued":{"date-parts":[[2025,6,16]]},"note":"arxivid:2506.13619\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9702058","type":"journal-article","publisher":"The Royal Society","issue":"1969","DOI":"10.1098/rspa.1998.0166","page":"365-384","source":"Crossref","title":"Resilient quantum computation: error models and thresholds","volume":"454","author":[{"given":"Emanuel","family":"Knill","sequence":"first","affiliation":[{"name":"CIC‐3, MS B265, Los Alamos National Laboratory, NM 87545, USA"}]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[{"name":"T‐6, MS B288, Los Alamos National Laboratory, NM 87545, USA"}]},{"given":"Wojciech H.","family":"Zurek","sequence":"additional","affiliation":[{"name":"T‐6, MS B288, Los Alamos National Laboratory, NM 87545, USA"}]}],"container-title":"Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1998,1,8]]},"URL":"http://dx.doi.org/10.1098/rspa.1998.0166","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. Lond. A","note":"alternative-id:10.1098/rspa.1998.0166\narxivid:quant-ph/9702058"},{"id":"arxiv:quant-ph/9705031","type":"journal-article","publisher":"The Royal Society","issue":"1969","DOI":"10.1098/rspa.1998.0167","page":"385-410","source":"Crossref","title":"Reliable quantum computers","volume":"454","author":[{"given":"John","family":"Preskill","sequence":"first","affiliation":[{"name":"Charles C. Lauritsen Laboratory of High Energy Physics, California Institute of Technology, Pasadena, CA 91125, USA"}]}],"container-title":"Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1998,1,8]]},"URL":"http://dx.doi.org/10.1098/rspa.1998.0167","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. Lond. A","note":"alternative-id:10.1098/rspa.1998.0167\narxivid:quant-ph/9705031"},{"id":"arxiv:quant-ph/9903099","type":"journal-article","publisher":"Informa UK Limited","issue":"2-3","DOI":"10.1080/09500340008244046","page":"333-345","source":"Crossref","title":"Fault-tolerant quantum computation with local gates","volume":"47","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]}],"container-title":"Journal of Modern Optics","original-title":[],"language":"en","issued":{"date-parts":[[2000,2]]},"URL":"http://dx.doi.org/10.1080/09500340008244046","ISSN":["0950-0340","1362-3044"],"container-title-short":"Journal of Modern Optics","note":"alternative-id:10.1080/09500340008244046\narxivid:quant-ph/9903099"},{"id":"arxiv:quant-ph/9906129","type":"article-journal","author":[{"given":"Dorit","family":"Aharonov"},{"given":"Michael","family":"Ben-Or"}],"title":"Fault-Tolerant Quantum Computation With Constant Error Rate","issued":{"date-parts":[[1999,6,30]]},"note":"arxivid:quant-ph/9906129\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0410047","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.72.022317","source":"Crossref","title":"Local fault-tolerant quantum computation","volume":"72","author":[{"given":"Krysta M.","family":"Svore","sequence":"first","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]},{"given":"David P.","family":"DiVincenzo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,8,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.72.022317","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022317","note":"arxivid:quant-ph/0410047"},{"id":"arxiv:quant-ph/0504218","type":"article-journal","author":[{"given":"Panos","family":"Aliferis"},{"given":"Daniel","family":"Gottesman"},{"given":"John","family":"Preskill"}],"title":"Quantum accuracy threshold for concatenated distance-3 codes","issued":{"date-parts":[[2005,10,21]]},"note":"arxivid:quant-ph/0504218\narxiv_version_number:3"},{"id":"arxiv:quant-ph/0703230","type":"article-journal","author":[{"given":"Panos","family":"Aliferis"}],"title":"Level Reduction and the Quantum Threshold Theorem","issued":{"date-parts":[[2011,7,11]]},"note":"arxivid:quant-ph/0703230\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0604090","type":"article-journal","author":[{"given":"Krysta M.","family":"Svore"},{"given":"David P.","family":"DiVincenzo"},{"given":"Barbara M.","family":"Terhal"}],"title":"Noise Threshold for a Fault-Tolerant Two-Dimensional Lattice Architecture","issued":{"date-parts":[[2006,11,3]]},"note":"arxivid:quant-ph/0604090\narxiv_version_number:3"},{"id":"arxiv:2404.02280","type":"article-journal","author":[{"given":"A.","family":"Paetznick"},{"given":"M. P.","family":"da Silva"},{"given":"C.","family":"Ryan-Anderson"},{"given":"J. M.","family":"Bello-Rivas"},{"given":"J. P.","family":"Campora"},{"given":"A.","family":"Chernoguzov"},{"given":"J. M.","family":"Dreiling"},{"given":"C.","family":"Foltz"},{"given":"F.","family":"Frachon"},{"given":"J. P.","family":"Gaebler"},{"given":"T. M.","family":"Gatterman"},{"given":"L.","family":"Grans-Samuelsson"},{"given":"D.","family":"Gresh"},{"given":"D.","family":"Hayes"},{"given":"N.","family":"Hewitt"},{"given":"C.","family":"Holliman"},{"given":"C. V.","family":"Horst"},{"given":"J.","family":"Johansen"},{"given":"D.","family":"Lucchetti"},{"given":"Y.","family":"Matsuoka"},{"given":"M.","family":"Mills"},{"given":"S. A.","family":"Moses"},{"given":"B.","family":"Neyenhuis"},{"given":"A.","family":"Paz"},{"given":"J.","family":"Pino"},{"given":"P.","family":"Siegfried"},{"given":"A.","family":"Sundaram"},{"given":"D.","family":"Tom"},{"given":"S. J.","family":"Wernli"},{"given":"M.","family":"Zanner"},{"given":"R. P.","family":"Stutz"},{"given":"K. M.","family":"Svore"}],"title":"Demonstration of logical qubits and repeated error correction with better-than-physical error rates","issued":{"date-parts":[[2024,11,17]]},"note":"arxivid:2404.02280\narxiv_version_number:3"},{"id":"arxiv:2411.03683","type":"article-journal","author":[{"given":"Shiro","family":"Tamiya"},{"given":"Masato","family":"Koashi"},{"given":"Hayata","family":"Yamasaki"}],"title":"Polylog-time- and constant-space-overhead fault-tolerant quantum computation with quantum low-density parity-check codes","issued":{"date-parts":[[2024,12,5]]},"note":"arxivid:2411.03683\narxiv_version_number:2"},{"id":"arxiv:2303.04798","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Constant-rate low-density parity-check (LDPC) codes are promising candidates for constructing efficient fault-tolerant quantum memories. However, if physical gates are subject to geometric-locality constraints, it becomes challenging to realize these codes. In this paper, we construct a new family of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>K</mml:mi><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> codes, referred to as hierarchical codes, that encode a number of logical qubits <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">&amp;#x03A9;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. The <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math> element of this code family is obtained by concatenating a constant-rate quantum LDPC code with a surface code; nearest-neighbor gates in two dimensions are sufficient to implement the corresponding syndrome-extraction circuit and achieve a threshold. Below threshold the logical failure rate vanishes superpolynomially as a function of the distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. We present a bilayer architecture for implementing the syndrome-extraction circuit, and estimate the logical failure rate for this architecture. Under conservative assumptions, we find that the hierarchical code outperforms the basic encoding where all logical qubits are encoded in the surface code.</jats:p>","DOI":"10.22331/q-2025-05-05-1728","page":"1728","source":"Crossref","title":"Hierarchical memories: Simulating quantum LDPC codes with local gates","volume":"9","author":[{"given":"Christopher A.","family":"Pattison","sequence":"first","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125"}]},{"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Stanford University, Stanford, CA, 94305"},{"name":"Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305"}]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125"},{"name":"AWS Center for Quantum Computing, Pasadena CA 91125"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,5]]},"URL":"http://dx.doi.org/10.22331/q-2025-05-05-1728","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2303.04798"},{"id":"arxiv:2312.04522","type":"article-journal","author":[{"given":"Craig","family":"Gidney"},{"given":"Michael","family":"Newman"},{"given":"Peter","family":"Brooks"},{"given":"Cody","family":"Jones"}],"title":"Yoked surface codes","issued":{"date-parts":[[2023,12,7]]},"note":"arxivid:2312.04522\narxiv_version_number:1"},{"id":"arxiv:2404.19005","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Realizing computationally complex quantum circuits in the presence of noise and imperfections is a challenging task. While fault-tolerant quantum computing provides a route to reducing noise, it requires a large overhead for generic algorithms. Here, we develop and analyze a hardware-efficient, fault-tolerant approach to realizing complex sampling circuits. We co-design the circuits with the appropriate quantum error-correcting codes for efficient implementation in a reconfigurable neutral atom-array architecture, constituting what we call a  of the sampling algorithm. Specifically, we consider a family of <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mtext>⟦</a:mtext><a:msup><a:mn>2</a:mn><a:mi>D</a:mi></a:msup><a:mo>,</a:mo><a:mi>D</a:mi><a:mo>,</a:mo><a:mn>2</a:mn><a:mtext>⟧</a:mtext></a:math> quantum error-detecting codes whose transversal and permutation gate set can realize arbitrary degree-<c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>D</c:mi></c:math> instantaneous quantum polynomial (IQP) circuits. Using native operations of the code and the atom-array hardware, we compile a fault-tolerant and fast-scrambling family of such IQP circuits in a hypercube geometry, realized recently in the experiments by Bluvstein  [Nature 626, 7997 (2024)]. We develop a theory of second-moment properties of degree-<e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>D</e:mi></e:math> IQP circuits for analyzing hardness and verification of random sampling by mapping to a statistical mechanics model. We provide strong evidence that sampling from these hypercube IQP circuits is classically hard to simulate even at relatively low depths. We analyze the linear cross-entropy benchmark (XEB) in comparison to the average fidelity and, depending on the local noise rate, find two different asymptotic regimes. To realize a fully scalable approach, we first show that Bell sampling from degree-4 IQP circuits is classically intractable and can be efficiently validated. We further devise new families of <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:mtext>⟦</g:mtext><g:mi>O</g:mi><g:mo stretchy=\"false\">(</g:mo><g:msup><g:mi>d</g:mi><g:mi>D</g:mi></g:msup><g:mo stretchy=\"false\">)</g:mo><g:mo>,</g:mo><g:mi>D</g:mi><g:mo>,</g:mo><g:mi>d</g:mi><g:mtext>⟧</g:mtext></g:math> color codes of increasing distance <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:mi>d</k:mi></k:math>, permitting exponential error suppression for transversal IQP sampling. Our results highlight fault-tolerant compiling as a powerful tool in co-designing algorithms with specific error-correcting codes and realistic hardware.</jats:p>","DOI":"10.1103/prxquantum.6.020338","source":"Crossref","title":"Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-4766-7967","authenticated-orcid":true,"given":"Dominik","family":"Hangleiter","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST/University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0003-0605-8791","authenticated-orcid":true,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"given":"Dolev","family":"Bluvstein","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-5298-3112","authenticated-orcid":true,"given":"Madelyn","family":"Cain","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0001-5775-9542","authenticated-orcid":true,"given":"Nishad","family":"Maskara","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"given":"Xun","family":"Gao","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"},{"id":[{"id":"https://ror.org/008hybe55","id-type":"ROR","asserted-by":"publisher"}],"name":"JILA"},{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"}]},{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":true,"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[{"name":"AWS Center for Quantum Computing"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"}]},{"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST/University of Maryland, College Park"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,28]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.020338","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020338","note":"arxivid:2404.19005"},{"id":"arxiv:0902.2658","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.80.022313","source":"Crossref","title":"Accuracy threshold for concatenated error detection in one dimension","volume":"80","author":[{"given":"Ashley M.","family":"Stephens","sequence":"first","affiliation":[]},{"given":"Zachary W. E.","family":"Evans","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.022313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022313","note":"arxivid:0902.2658"},{"id":"arxiv:0902.4506","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1007/s11128-011-0312-4","page":"1511-1521","source":"Crossref","title":"Optimal correction of concatenated fault-tolerant quantum codes","volume":"11","author":[{"given":"Z. W. E.","family":"Evans","sequence":"first","affiliation":[]},{"given":"A. M.","family":"Stephens","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2011,9,27]]},"URL":"http://dx.doi.org/10.1007/s11128-011-0312-4","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:312\narxivid:0902.4506"},{"id":"arxiv:quant-ph/0608018","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/focs.2006.64","page":"420-428","source":"Crossref","title":"Postselection threshold against biased noise","author":[{"given":"Ben","family":"Reichardt","sequence":"first","affiliation":[]}],"event":"2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)","container-title":"2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)","original-title":[],"issued":{"date-parts":[[2006]]},"URL":"http://dx.doi.org/10.1109/FOCS.2006.64","note":"arxivid:quant-ph/0608018"},{"id":"arxiv:quant-ph/0703264","type":"article-journal","author":[{"given":"Panos","family":"Aliferis"},{"given":"Daniel","family":"Gottesman"},{"given":"John","family":"Preskill"}],"title":"Accuracy threshold for postselected quantum computation","issued":{"date-parts":[[2007,9,17]]},"note":"arxivid:quant-ph/0703264\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0612073","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.76.042311","source":"Crossref","title":"Fault-tolerant linear optics quantum computation by error-detecting quantum state transfer","volume":"76","author":[{"given":"Jaeyoon","family":"Cho","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,10,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.042311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042311","note":"arxivid:quant-ph/0612073"},{"id":"arxiv:2402.09606","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n          <jats:p>The essential requirement for fault-tolerant quantum computation (FTQC) is the total protocol design to achieve a fair balance of all the critical factors relevant to its practical realization, such as the space overhead, the threshold, and the modularity. A major obstacle in realizing FTQC with conventional protocols, such as those based on the surface code and the concatenated Steane code, has been the space overhead, i.e., the required number of physical qubits per logical qubit. Protocols based on high-rate quantum low-density parity-check (LDPC) codes gather considerable attention as a way to reduce the space overhead, but problematically, the existing fault-tolerant protocols for such quantum LDPC codes sacrifice other factors. Here, we construct a new fault-tolerant protocol to meet these requirements simultaneously based on more recent progress on the techniques for concatenated codes rather than quantum LDPC codes, achieving a constant space overhead, a high threshold, and flexibility in modular architecture designs. In particular, under a physical error rate of 0.1%, our protocol reduces the space overhead to achieve the logical CNOT error rates 10<jats:sup>−10</jats:sup> and 10<jats:sup>−24</jats:sup> by more than 90% and 96%, respectively, compared to the protocol for the surface code. Furthermore, our protocol achieves the threshold of 2.5% under a conventional circuit-level error model, substantially outperforming that of the surface code. The use of concatenated codes also naturally introduces abstraction layers essential for the modularity of FTQC architectures. These results indicate that the code-concatenation approach opens a way to significantly save qubits in realizing FTQC while fulfilling the other essential requirements for the practical protocol design.</jats:p>","DOI":"10.1038/s41534-025-01035-8","source":"Crossref","title":"Concatenate codes, save qubits","volume":"11","author":[{"given":"Satoshi","family":"Yoshida","sequence":"first","affiliation":[]},{"given":"Shiro","family":"Tamiya","sequence":"additional","affiliation":[]},{"given":"Hayata","family":"Yamasaki","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,31]]},"URL":"http://dx.doi.org/10.1038/s41534-025-01035-8","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"88","note":"alternative-id:1035\narxivid:2402.09606"},{"id":"arxiv:1604.04062","type":"journal-article","publisher":"Rinton Press","issue":"15&16","abstract":"<jats:p>We analyze the properties of a 2D topological code derived by concatenating the J4, 2, 2K code with the toric/surface code, or alternatively by removing check operators from the 2D square-octagon or 4.8.8 color code. We show that the resulting code has a circuit-based noise threshold of ~ 0.41% (compared to ~ 0.6% for the toric code in a similar scenario), which is higher than any known 2D color code. We believe that the construction may be of interest for hardware in which one wants to use both long-range two-qubit gates as well as short-range gates between small clusters of qubits.</jats:p>","DOI":"10.26421/qic16.15-16-1","page":"1261-1281","source":"Crossref","title":"Noise thresholds for the [4,2,2]-concatenated toric code","volume":"16","author":[{"given":"Ben","family":"Criger","sequence":"first","affiliation":[]},{"given":"Barbara","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2016,11]]},"URL":"http://dx.doi.org/10.26421/QIC16.15-16-1","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC","note":"arxivid:1604.04062"},{"id":"arxiv:quant-ph/0111003","type":"article-journal","author":[{"given":"Benjamin","family":"Rahn"},{"given":"Andrew C.","family":"Doherty"},{"given":"Hideo","family":"Mabuchi"}],"title":"Exact and Approximate Performance of Concatenated Quantum Codes","issued":{"date-parts":[[2001,12,12]]},"note":"arxivid:quant-ph/0111003\narxiv_version_number:2"},{"id":"arxiv:1204.4221","type":"article-journal","author":[{"given":"Adam M.","family":"Meier"},{"given":"Bryan","family":"Eastin"},{"given":"Emanuel","family":"Knill"}],"title":"Magic-state distillation with the four-qubit code","issued":{"date-parts":[[2012,4,18]]},"note":"arxivid:1204.4221\narxiv_version_number:1"},{"id":"arxiv:2403.16054","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"36","abstract":"<jats:p>Standard approaches to quantum error correction for fault-tolerant quantum computing are based on encoding a single logical qubit into many physical ones, resulting in asymptotically zero encoding rates and therefore huge resource overheads. To overcome this issue, high-rate quantum codes, such as quantum low-density parity-check codes, have been studied over the past decade. In this case, however, it is difficult to perform logical gates in parallel while maintaining low overheads. Here, we propose concatenated high-rate small-size quantum error-detecting codes as a family of high-rate quantum codes. Their simple structure allows for a geometrical interpretation using hypercubes corresponding to logical qubits. We thus call them many-hypercube codes. They can realize both high rates, e.g., 30% (64 logical qubits are encoded into 216 physical ones), and parallelizability of logical gates. Developing dedicated decoder and encoders, we achieve high error thresholds even in a circuit-level noise model. Thus, the many-hypercube codes will pave the way to high-performance fault-tolerant quantum computing.</jats:p>","DOI":"10.1126/sciadv.adp6388","source":"Crossref","title":"High-performance fault-tolerant quantum computing with many-hypercube codes","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0003-1145-6620","authenticated-orcid":true,"given":"Hayato","family":"Goto","sequence":"first","affiliation":[{"name":"RIKEN Center for Quantum Computing (RQC), Wako, Saitama 351-0198, Japan."},{"name":"Corporate Research &amp; Development Center, Toshiba Corporation, 1 Komukai Toshiba-cho, Saiwai-ku, Kawasaki-shi 212-8582, Japan."}]}],"container-title":"Science Advances","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,6]]},"URL":"http://dx.doi.org/10.1126/sciadv.adp6388","ISSN":["2375-2548"],"container-title-short":"Sci. Adv.","page":"eadp6388","note":"alternative-id:10.1126/sciadv.adp6388\narxivid:2403.16054"},{"id":"arxiv:2209.09390","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We investigate a family of fault-tolerant quantum error correction schemes based on the concatenation of small error detection or error correction codes with the three-dimensional cluster state. We propose fault-tolerant state preparation and decoding schemes that effectively convert every circuit-level error into an erasure error, leveraging the cluster state's high threshold against such errors. We find a set of codes for which such a conversion is possible, and study their performance against the standard circuit-level depolarizing model. Our best performing scheme, which is based on a concatenation with a classical code, improves the threshold by <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>16.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> and decreases the spacetime overhead by <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>32</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> compared to the scheme without concatenation, with each scheme subject to a physical error rate of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math> and achieving a logical error rate of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:math>.</jats:p>","DOI":"10.22331/q-2023-08-22-1089","page":"1089","source":"Crossref","title":"Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction","volume":"7","author":[{"given":"Zhaoyi","family":"Li","sequence":"first","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, CA 94305, USA"}]},{"given":"Isaac","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of California, Davis, CA 95616, USA"}]},{"given":"Patrick","family":"Hayden","sequence":"additional","affiliation":[{"name":"Department of Physics, Stanford University, Stanford, CA 94305, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,22]]},"URL":"http://dx.doi.org/10.22331/q-2023-08-22-1089","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2209.09390"},{"id":"arxiv:2207.08826","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>Scaling up quantum computers to attain substantial speedups over classical computing requires fault tolerance. Conventionally, protocols for fault-tolerant quantum computation demand excessive space overheads by using many physical qubits for each logical qubit. A more recent protocol using quantum analogues of low-density parity-check codes needs only a constant space overhead that does not grow with the number of logical qubits. However, the overhead in the processing time required to implement this protocol grows polynomially with the number of computational steps. To address these problems, here we introduce an alternative approach to constant-space-overhead fault-tolerant quantum computing using a concatenation of multiple small-size quantum codes rather than a single large-size quantum low-density parity-check code. We develop techniques for concatenating different quantum Hamming codes with growing size. As a result, we construct a low-overhead protocol to achieve constant space overhead and only quasi-polylogarithmic time overhead simultaneously. Our protocol is fault tolerant even if a decoder has a non-constant runtime, unlike the existing constant-space-overhead protocol. This code concatenation approach will make possible a large class of quantum speedups with feasibly bounded space overhead yet negligibly short time overhead.</jats:p>","DOI":"10.1038/s41567-023-02325-8","page":"247-253","source":"Crossref","title":"Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0003-3521-831X","authenticated-orcid":false,"given":"Hayata","family":"Yamasaki","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4518-1461","authenticated-orcid":false,"given":"Masato","family":"Koashi","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,16]]},"URL":"http://dx.doi.org/10.1038/s41567-023-02325-8","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2325\narxivid:2207.08826"},{"id":"arxiv:2407.16176","type":"article-journal","author":[{"given":"Menglong","family":"Fang"},{"given":"Daiqin","family":"Su"}],"title":"Quantum memory based on concatenating surface codes and quantum Hamming codes","issued":{"date-parts":[[2025,4,1]]},"note":"arxivid:2407.16176\narxiv_version_number:2"},{"id":"arxiv:1007.4601","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/j.aop.2010.10.009","page":"15-95","source":"Crossref","title":"Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes","volume":"326","author":[{"given":"Beni","family":"Yoshida","sequence":"first","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,1]]},"URL":"http://dx.doi.org/10.1016/j.aop.2010.10.009","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491610001867\narxivid:1007.4601"},{"id":"arxiv:1503.02065","type":"journal-article","publisher":"IOP Publishing","issue":"8","DOI":"10.1088/1367-2630/17/8/083026","page":"083026","source":"Crossref","title":"Unfolding the color code","volume":"17","author":[{"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"given":"Beni","family":"Yoshida","sequence":"additional","affiliation":[]},{"given":"Fernando","family":"Pastawski","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2015,8,13]]},"URL":"http://dx.doi.org/10.1088/1367-2630/17/8/083026","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1503.02065"},{"id":"arxiv:1804.00866","type":"article-journal","author":[{"given":"Arun B.","family":"Aloshious"},{"given":"Arjun Nitin","family":"Bhagoji"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"On the Local Equivalence of 2D Color Codes and Surface Codes with Applications","issued":{"date-parts":[[2018,4,3]]},"note":"arxivid:1804.00866\narxiv_version_number:1"},{"id":"arxiv:1801.04255","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.100.012312","source":"Crossref","title":"Three-dimensional surface codes: Transversal gates and fault-tolerant architectures","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":true,"given":"Michael","family":"Vasmer","sequence":"first","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.100.012312","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012312","note":"arxivid:1801.04255"},{"id":"arxiv:2007.12152","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.102.062402","source":"Crossref","title":"Minimal distances for certain quantum product codes and tensor products of chain complexes","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0002-1871-5034","authenticated-orcid":true,"given":"Weilei","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.062402","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062402","note":"arxivid:2007.12152"},{"id":"arxiv:quant-ph/9610042","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.56.33","page":"33-38","source":"Crossref","title":"Codes for the quantum erasure channel","volume":"56","author":[{"given":"M.","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Th.","family":"Beth","sequence":"additional","affiliation":[]},{"given":"T.","family":"Pellizzari","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,7,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.56.33","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9610042"},{"id":"arxiv:2511.01080","type":"article-journal","author":[{"given":"Jonathan","family":"Kunjummen"},{"given":"Jacob M.","family":"Taylor"}],"title":"In situ calibration of unitary operations during quantum error correction","issued":{"date-parts":[[2025,11,2]]},"note":"arxivid:2511.01080\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0203003","type":"journal-article","publisher":"AIP Publishing","issue":"9","abstract":"<jats:p>A conjecture arising naturally in the investigation of additivity of classical information capacity of quantum channels states that the maximal purity of outputs from a quantum channel, as measured by the p-norm, should be multiplicative with respect to the tensor product of channels. We disprove this conjecture for p&amp;gt;4.79. The same example (with p=∞) also disproves a conjecture for the multiplicativity of the injective norm of Hilbert space tensor products.</jats:p>","DOI":"10.1063/1.1498491","page":"4353-4357","source":"Crossref","title":"Counterexample to an additivity conjecture for output purity of quantum channels","volume":"43","author":[{"given":"R. F.","family":"Werner","sequence":"first","affiliation":[{"name":"Institut für Mathematische Physik, TU Braunschweig, Mendelssohnstr.3, 38106 Braunschweig, Germany"}]},{"given":"A. S.","family":"Holevo","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute, Gubkina 8, 117966 Moscow, Russia"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2002,9,1]]},"URL":"http://dx.doi.org/10.1063/1.1498491","ISSN":["0022-2488","1089-7658"],"note":"arxivid:quant-ph/0203003"},{"id":"arxiv:1904.04713","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"5","DOI":"10.1109/tit.2021.3063093","page":"2857-2877","source":"Crossref","title":"Polarization of Quantum Channels Using Clifford-Based Channel Combining","volume":"67","author":[{"given":"Frederic","family":"Dupuis","sequence":"first","affiliation":[]},{"given":"Ashutosh","family":"Goswami","sequence":"additional","affiliation":[]},{"given":"Mehdi","family":"Mhalla","sequence":"additional","affiliation":[]},{"given":"Valentin","family":"Savin","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2021,5]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3063093","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1904.04713"},{"id":"arxiv:2302.03063","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.13.041013","source":"Crossref","title":"High-Threshold Codes for Neutral-Atom Qubits with Biased Erasure Errors","volume":"13","author":[{"ORCID":"https://orcid.org/0009-0002-4587-5620","authenticated-orcid":true,"given":"Kaavya","family":"Sahay","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6686-777X","authenticated-orcid":true,"given":"Junlan","family":"Jin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6711-4585","authenticated-orcid":true,"given":"Jahan","family":"Claes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8673-052X","authenticated-orcid":true,"given":"Jeff D.","family":"Thompson","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5194-0591","authenticated-orcid":true,"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.13.041013","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041013","note":"arxivid:2302.03063"},{"id":"arxiv:quant-ph/0505153","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.95.230503","source":"Crossref","title":"Quantum Error Correction in Spatially Correlated Quantum Noise","volume":"95","author":[{"given":"Rochus","family":"Klesse","sequence":"first","affiliation":[]},{"given":"Sandra","family":"Frank","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,11,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.95.230503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230503","note":"arxivid:quant-ph/0505153"},{"id":"arxiv:2012.01894","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.2.030201","source":"Crossref","title":"Quantum Stochastic Processes and Quantum non-Markovian Phenomena","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-6987-5513","authenticated-orcid":true,"given":"Simon","family":"Milz","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2054-9901","authenticated-orcid":true,"given":"Kavan","family":"Modi","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,7,14]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.2.030201","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030201","note":"arxivid:2012.01894"},{"id":"arxiv:quant-ph/9610040","type":"article-journal","author":[{"given":"Peter","family":"Shor"},{"given":"Raymond","family":"Laflamme"}],"title":"Quantum MacWilliams Identities","issued":{"date-parts":[[1996,10,25]]},"note":"arxivid:quant-ph/9610040\narxiv_version_number:1"},{"id":"arxiv:2405.19643","type":"article-journal","author":[{"given":"Alon","family":"Kukliansky"},{"given":"Brad","family":"Lackey"}],"title":"Quantum Circuit Tensors and Enumerators with Applications to Quantum Fault Tolerance","issued":{"date-parts":[[2024,5,30]]},"note":"arxivid:2405.19643\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9906126","type":"article-journal","author":[{"given":"Alexei","family":"Ashikhmin"},{"given":"Alexander","family":"Barg"},{"given":"Emanuel","family":"Knill"},{"given":"Simon","family":"Litsyn"}],"title":"Quantum Error Detection I: Statement of the Problem","issued":{"date-parts":[[1999,6,30]]},"note":"arxivid:quant-ph/9906126\narxiv_version_number:1"},{"id":"arxiv:2107.14252","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The performance of quantum error correction can be significantly improved if detailed information about the noise is available, allowing to optimize both codes and decoders. It has been proposed to estimate error rates from the syndrome measurements done anyway during quantum error correction. While these measurements preserve the encoded quantum state, it is currently not clear how much information about the noise can be extracted in this way. So far, apart from the limit of vanishing error rates, rigorous results have only been established for some specific codes.\nIn this work, we rigorously resolve the question for arbitrary stabilizer codes. The main result is that a stabilizer code can be used to estimate Pauli channels with correlations across a number of qubits given by the pure distance. This result does not rely on the limit of vanishing error rates, and applies even if high weight errors occur frequently. Moreover, it also allows for measurement errors within the framework of quantum data-syndrome codes. Our proof combines Boolean Fourier analysis, combinatorics and elementary algebraic geometry. It is our hope that this work opens up interesting applications, such as the online adaptation of a decoder to time-varying noise.</jats:p>","DOI":"10.22331/q-2022-09-19-809","page":"809","source":"Crossref","title":"Pauli channels can be estimated from syndrome measurements in quantum error correction","volume":"6","author":[{"given":"Thomas","family":"Wagner","sequence":"first","affiliation":[{"name":"Institut für Theoretische Physik, Heinrich-Heine-University Düsseldorf, Germany"}]},{"given":"Hermann","family":"Kampermann","sequence":"additional","affiliation":[{"name":"Institut für Theoretische Physik, Heinrich-Heine-University Düsseldorf, Germany"}]},{"given":"Dagmar","family":"Bruß","sequence":"additional","affiliation":[{"name":"Institut für Theoretische Physik, Heinrich-Heine-University Düsseldorf, Germany"}]},{"given":"Martin","family":"Kliesch","sequence":"additional","affiliation":[{"name":"Institut für Theoretische Physik, Heinrich-Heine-University Düsseldorf, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,19]]},"URL":"http://dx.doi.org/10.22331/q-2022-09-19-809","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2107.14252"},{"id":"arxiv:quant-ph/9612015","type":"article-journal","author":[{"given":"Eric M.","family":"Rains"}],"title":"Quantum Weight Enumerators","issued":{"date-parts":[[1996,12,2]]},"note":"arxivid:quant-ph/9612015\narxiv_version_number:1"},{"id":"arxiv:1702.06990","type":"article-journal","author":[{"given":"Patrick","family":"Rall"}],"title":"Signed quantum weight enumerators characterize qubit magic state distillation","issued":{"date-parts":[[2017,2,22]]},"note":"arxivid:1702.06990\narxiv_version_number:1"},{"id":"arxiv:0810.2574","type":"article-journal","author":[{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Quantum stabilizer codes and beyond","issued":{"date-parts":[[2008,10,14]]},"note":"arxivid:0810.2574\narxiv_version_number:1"},{"id":"doi:10.1016/j.aam.2020.102085","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aam.2020.102085","page":"102085","source":"Crossref","title":"Weight enumerators for nonbinary asymmetric quantum codes and their applications","volume":"121","author":[{"given":"Chuangqiang","family":"Hu","sequence":"first","affiliation":[]},{"given":"Shudi","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Stephen S.-T.","family":"Yau","sequence":"additional","affiliation":[]}],"container-title":"Advances in Applied Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2020,10]]},"URL":"http://dx.doi.org/10.1016/j.aam.2020.102085","ISSN":["0196-8858"],"container-title-short":"Advances in Applied Mathematics","note":"special_numbering:C\nalternative-id:S0196885820300889"},{"id":"arxiv:2211.02756","type":"article-journal","author":[{"given":"ChunJun","family":"Cao"},{"given":"Brad","family":"Lackey"}],"title":"Quantum weight enumerators and tensor networks","issued":{"date-parts":[[2023,3,30]]},"note":"arxivid:2211.02756\narxiv_version_number:2"},{"id":"arxiv:2408.10323","type":"article-journal","author":[{"given":"Gerard Anglès","family":"Munné"},{"given":"Andrew","family":"Nemec"},{"given":"Felix","family":"Huber"}],"title":"SDP bounds on quantum codes","issued":{"date-parts":[[2025,9,10]]},"note":"arxivid:2408.10323\narxiv_version_number:2"},{"id":"arxiv:2603.19901","type":"article-journal","author":[{"given":"Gerard Anglès","family":"Munné"},{"given":"Felix","family":"Huber"}],"title":"SDP bounds on quantum codes: rational certificates","issued":{"date-parts":[[2026,3,20]]},"note":"arxivid:2603.19901\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0607039","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2008.928269","page":"4208-4217","source":"Crossref","title":"The Quantum Capacity With Symmetric Side Channels","volume":"54","author":[{"given":"Graeme","family":"Smith","sequence":"first","affiliation":[]},{"given":"John A.","family":"Smolin","sequence":"additional","affiliation":[]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.928269","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0607039"},{"id":"arxiv:1801.02019","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/comst.2017.2786748","page":"1149-1205","source":"Crossref","title":"A Survey on Quantum Channel Capacities","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0002-4209-7619","authenticated-orcid":false,"given":"Laszlo","family":"Gyongyosi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2883-8919","authenticated-orcid":false,"given":"Sandor","family":"Imre","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6349-1044","authenticated-orcid":false,"given":"Hung Viet","family":"Nguyen","sequence":"additional","affiliation":[]}],"container-title":"IEEE Communications Surveys &amp; Tutorials","original-title":[],"issued":{"date-parts":[[2018]]},"URL":"http://dx.doi.org/10.1109/COMST.2017.2786748","ISSN":["1553-877X","2373-745X"],"container-title-short":"IEEE Commun. Surv. Tutorials","note":"arxivid:1801.02019"},{"id":"arxiv:2408.07067","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>Despite the central importance of quantum entanglement in quantum technologies, understanding the optimal ways to exploit it is still beyond our reach, and even measuring entanglement in an operationally meaningful way is prohibitively difficult. Here we study two fundamental tasks in the processing of entanglement: entanglement testing, which is a quantum state discrimination problem concerned with detecting entanglement in the many-copy regime, and entanglement distillation, which is concerned with purifying entanglement from noisy entangled states. We introduce a way of benchmarking the performance of distillation that focuses on the best achievable error rather than its yield in the asymptotic limit. When the underlying set of operations used for entanglement distillation is the axiomatic class of non-entangling operations, we show that the two figures of merit for entanglement testing and distillation coincide. We solve both problems by proving a generalized quantum Sanov’s theorem, which enables the exact evaluation of the asymptotic error rates of composite quantum hypothesis testing. We show in particular that the asymptotic figure of merit is given by the reverse relative entropy of entanglement, a single-letter quantity that can be evaluated using only a single copy of a quantum state—a distinct feature among measures of entanglement that quantify the optimal performance of information-theoretic tasks.</jats:p>","DOI":"10.1038/s41567-026-03182-x","page":"439-445","source":"Crossref","title":"Asymptotic quantification of entanglement with a single copy","volume":"22","author":[{"ORCID":"https://orcid.org/0000-0003-3290-3557","authenticated-orcid":false,"given":"Ludovico","family":"Lami","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0428-3429","authenticated-orcid":false,"given":"Mario","family":"Berta","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7225-071X","authenticated-orcid":false,"given":"Bartosz","family":"Regula","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,3]]},"URL":"http://dx.doi.org/10.1038/s41567-026-03182-x","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:3182\narxivid:2408.07067"},{"id":"arxiv:1612.01011","type":"article-journal","author":[{"given":"M. B.","family":"Hastings"}],"title":"Turning Gate Synthesis Errors into Incoherent Errors","issued":{"date-parts":[[2016,12,3]]},"note":"arxivid:1612.01011\narxiv_version_number:1"},{"id":"arxiv:0907.3189","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.103.170504","source":"Crossref","title":"Tight Noise Thresholds for Quantum Computation with Perfect Stabilizer Operations","volume":"103","author":[{"given":"Wim","family":"van Dam","sequence":"first","affiliation":[]},{"given":"Mark","family":"Howard","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,10,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.103.170504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"170504","note":"arxivid:0907.3189"},{"id":"arxiv:1011.2497","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.83.032310","source":"Crossref","title":"Noise thresholds for higher-dimensional systems using the discrete Wigner function","volume":"83","author":[{"given":"Wim","family":"van Dam","sequence":"first","affiliation":[]},{"given":"Mark","family":"Howard","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,3,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.032310","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032310","note":"arxivid:1011.2497"},{"id":"arxiv:2310.05958","type":"article-journal","author":[{"given":"John","family":"van de Wetering"},{"given":"Matt","family":"Amy"}],"title":"Optimising quantum circuits is generally hard","issued":{"date-parts":[[2024,8,12]]},"note":"arxivid:2310.05958\narxiv_version_number:3"},{"id":"arxiv:2503.06045","type":"article-journal","author":[{"given":"Avimita","family":"Chatterjee"},{"given":"Archisman","family":"Ghosh"},{"given":"Swaroop","family":"Ghosh"}],"title":"The Art of Optimizing T-Depth for Quantum Error Correction in Large-Scale Quantum Computing","issued":{"date-parts":[[2025,3,11]]},"note":"arxivid:2503.06045\narxiv_version_number:2"},{"id":"arxiv:1303.2042","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tcad.2014.2341953","page":"1476-1489","source":"Crossref","title":"Polynomial-Time T-Depth Optimization of Clifford+T Circuits Via Matroid Partitioning","volume":"33","author":[{"given":"Matthew","family":"Amy","sequence":"first","affiliation":[]},{"given":"Dmitri","family":"Maslov","sequence":"additional","affiliation":[]},{"given":"Michele","family":"Mosca","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems","original-title":[],"issued":{"date-parts":[[2014,10]]},"URL":"http://dx.doi.org/10.1109/TCAD.2014.2341953","ISSN":["0278-0070","1937-4151"],"container-title-short":"IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.","note":"arxivid:1303.2042"},{"id":"arxiv:1308.4134","type":"article-journal","author":[{"given":"David","family":"Gosset"},{"given":"Vadym","family":"Kliuchnikov"},{"given":"Michele","family":"Mosca"},{"given":"Vincent","family":"Russo"}],"title":"An algorithm for the T-count","issued":{"date-parts":[[2013,8,19]]},"note":"arxivid:1308.4134\narxiv_version_number:1"},{"id":"arxiv:1601.07601","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/physrevlett.116.250501","source":"Crossref","title":"Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates","volume":"116","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"David","family":"Gosset","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,6,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.116.250501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"250501","note":"arxivid:1601.07601"},{"id":"arxiv:1710.07345","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>We develop and implement automated methods for optimizing quantum circuits of the size and type expected in quantum computations that outperform classical computers. We show how to handle continuous gate parameters and report a collection of fast algorithms capable of optimizing large-scale quantum circuits. For the suite of benchmarks considered, we obtain substantial reductions in gate counts. In particular, we provide better optimization in significantly less time than previous approaches, while making minimal structural changes so as to preserve the basic layout of the underlying quantum algorithms. Our results help bridge the gap between the computations that can be run on existing hardware and those that are expected to outperform classical computers.</jats:p>","DOI":"10.1038/s41534-018-0072-4","source":"Crossref","title":"Automated optimization of large quantum circuits with continuous parameters","volume":"4","author":[{"given":"Yunseong","family":"Nam","sequence":"first","affiliation":[]},{"given":"Neil J.","family":"Ross","sequence":"additional","affiliation":[]},{"given":"Yuan","family":"Su","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9903-837X","authenticated-orcid":false,"given":"Andrew M.","family":"Childs","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7381-4556","authenticated-orcid":false,"given":"Dmitri","family":"Maslov","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,10]]},"URL":"http://dx.doi.org/10.1038/s41534-018-0072-4","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"23","note":"alternative-id:72\narxivid:1710.07345"},{"id":"arxiv:1712.01557","type":"article-journal","author":[{"given":"Luke","family":"Heyfron"},{"given":"Earl T.","family":"Campbell"}],"title":"An Efficient Quantum Compiler that reduces $T$ count","issued":{"date-parts":[[2018,6,7]]},"note":"arxivid:1712.01557\narxiv_version_number:3"},{"id":"arxiv:1808.00128","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">s</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">t</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">a</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">b</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">i</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">l</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">i</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">z</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">e</mml:mi><mml:mi class=\"MJX-tex-mathit\" mathvariant=\"italic\">r</mml:mi></mml:mrow></mml:math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">rank</mml:mtext></mml:mrow></mml:math>, which for a pure state<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>ψ</mml:mi></mml:math>is defined to be the smallest integer<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>χ</mml:mi></mml:math>such that<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>ψ</mml:mi></mml:math>is a superposition of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>χ</mml:mi></mml:math>stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>χ</mml:mi><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:math>stabilizer states and sample from the full<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-bit output distribution, improving on previous simulations which used<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:math>stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.</jats:p>","DOI":"10.22331/q-2019-09-02-181","page":"181","source":"Crossref","title":"Simulation of quantum circuits by low-rank stabilizer decompositions","volume":"3","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[{"name":"IBM T.J. Watson Research Center, Yorktown Heights NY 10598"}]},{"given":"Dan","family":"Browne","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London, UK"}]},{"given":"Padraic","family":"Calpin","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London, UK"}]},{"given":"Earl","family":"Campbell","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield, UK"}]},{"given":"David","family":"Gosset","sequence":"additional","affiliation":[{"name":"IBM T.J. Watson Research Center, Yorktown Heights NY 10598"},{"name":"Department of Combinatorics & Optimization and Institute for Quantum Computing, University of Waterloo, Waterloo, Canada"}]},{"given":"Mark","family":"Howard","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield, UK"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,9,2]]},"URL":"http://dx.doi.org/10.22331/q-2019-09-02-181","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1808.00128"},{"id":"arxiv:2101.12223","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.3.020361","source":"Crossref","title":"Fast Estimation of Outcome Probabilities for Quantum Circuits","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0003-3782-7602","authenticated-orcid":true,"given":"Hakop","family":"Pashayan","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0124-1389","authenticated-orcid":true,"given":"Oliver","family":"Reardon-Smith","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0683-5469","authenticated-orcid":true,"given":"Kamil","family":"Korzekwa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,6,23]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.020361","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020361","note":"arxivid:2101.12223"},{"id":"arxiv:2110.10292","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>We design an algorithm to determine the (minimum) T-count of any <jats:italic>n</jats:italic>-qubit (<jats:italic>n</jats:italic> ≥ 1) unitary <jats:italic>W</jats:italic> of size 2<jats:sup><jats:italic>n</jats:italic></jats:sup> × 2<jats:sup><jats:italic>n</jats:italic></jats:sup>, over the Clifford+T gate set. The space and time complexity of our algorithm are <jats:inline-formula><jats:alternatives><jats:tex-math>$$O\\left({2}^{2n}\\right)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O</mml:mi>\n                    <mml:mfenced>\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mn>2</mml:mn>\n                          </mml:mrow>\n                          <mml:mrow>\n                            <mml:mn>2</mml:mn>\n                            <mml:mi>n</mml:mi>\n                          </mml:mrow>\n                        </mml:msup>\n                      </mml:mrow>\n                    </mml:mfenced>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$O\\left({2}^{2n{{{{\\mathcal{T}}}}}_{\\epsilon }(W)+4n}\\right)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O</mml:mi>\n                    <mml:mfenced>\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mn>2</mml:mn>\n                          </mml:mrow>\n                          <mml:mrow>\n                            <mml:mn>2</mml:mn>\n                            <mml:mi>n</mml:mi>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi>T</mml:mi>\n                              </mml:mrow>\n                              <mml:mrow>\n                                <mml:mi>ϵ</mml:mi>\n                              </mml:mrow>\n                            </mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(</mml:mo>\n                              <mml:mrow>\n                                <mml:mi>W</mml:mi>\n                              </mml:mrow>\n                              <mml:mo>)</mml:mo>\n                            </mml:mrow>\n                            <mml:mo>+</mml:mo>\n                            <mml:mn>4</mml:mn>\n                            <mml:mi>n</mml:mi>\n                          </mml:mrow>\n                        </mml:msup>\n                      </mml:mrow>\n                    </mml:mfenced>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula>, respectively. <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{{\\mathcal{T}}}}}_{\\epsilon }(W)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mrow>\n                        <mml:mi>T</mml:mi>\n                      </mml:mrow>\n                      <mml:mrow>\n                        <mml:mi>ϵ</mml:mi>\n                      </mml:mrow>\n                    </mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(</mml:mo>\n                      <mml:mrow>\n                        <mml:mi>W</mml:mi>\n                      </mml:mrow>\n                      <mml:mo>)</mml:mo>\n                    </mml:mrow>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula> (<jats:italic>ϵ</jats:italic>-T-count) is the (minimum) T-count of an exactly implementable unitary <jats:italic>U</jats:italic> (<jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\mathcal{T}}}}(U)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>T</mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(</mml:mo>\n                      <mml:mrow>\n                        <mml:mi>U</mml:mi>\n                      </mml:mrow>\n                      <mml:mo>)</mml:mo>\n                    </mml:mrow>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula>), such that <jats:italic>d</jats:italic>(<jats:italic>U</jats:italic>,<jats:italic>W</jats:italic>) ≤ <jats:italic>ϵ</jats:italic> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\\mathcal{T}}}}(U)\\le {{{\\mathcal{T}}}}({U}^{{\\prime} })$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>T</mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(</mml:mo>\n                      <mml:mrow>\n                        <mml:mi>U</mml:mi>\n                      </mml:mrow>\n                      <mml:mo>)</mml:mo>\n                    </mml:mrow>\n                    <mml:mo>≤</mml:mo>\n                    <mml:mi>T</mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(</mml:mo>\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi>U</mml:mi>\n                          </mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>′</mml:mo>\n                          </mml:mrow>\n                        </mml:msup>\n                      </mml:mrow>\n                      <mml:mo>)</mml:mo>\n                    </mml:mrow>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula> where <jats:inline-formula><jats:alternatives><jats:tex-math>$${U}^{{\\prime} }$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>U</mml:mi>\n                    </mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>′</mml:mo>\n                    </mml:mrow>\n                  </mml:msup>\n                </mml:math></jats:alternatives></jats:inline-formula> is any exactly implementable unitary with <jats:inline-formula><jats:alternatives><jats:tex-math>$$d({U}^{{\\prime} },W)\\le \\epsilon$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d</mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(</mml:mo>\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi>U</mml:mi>\n                          </mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>′</mml:mo>\n                          </mml:mrow>\n                        </mml:msup>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>W</mml:mi>\n                      </mml:mrow>\n                      <mml:mo>)</mml:mo>\n                    </mml:mrow>\n                    <mml:mo>≤</mml:mo>\n                    <mml:mi>ϵ</mml:mi>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula>. <jats:italic>d</jats:italic>(. , .) is the global phase invariant distance. Our algorithm can also be used to determine the (minimum) T-depth as well as the minimum non-Clifford-gate count or depth required to implement any multi-qubit unitary with a finite universal gate set like Clifford+CS, Clifford+V, etc. For small enough <jats:italic>ϵ</jats:italic>, we can synthesize the optimal circuits.</jats:p>","DOI":"10.1038/s41534-022-00651-y","source":"Crossref","title":"T-count and T-depth of any multi-qubit unitary","volume":"8","author":[{"given":"Vlad","family":"Gheorghiu","sequence":"first","affiliation":[]},{"given":"Michele","family":"Mosca","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6463-9100","authenticated-orcid":false,"given":"Priyanka","family":"Mukhopadhyay","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2022,11,30]]},"URL":"http://dx.doi.org/10.1038/s41534-022-00651-y","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"141","note":"alternative-id:651\narxivid:2110.10292"},{"id":"arxiv:2506.15147","type":"article-journal","author":[{"given":"Isaac H.","family":"Kim"}],"title":"Catalytic $z$-rotations in constant $T$-depth","issued":{"date-parts":[[2026,2,12]]},"note":"arxivid:2506.15147\narxiv_version_number:2"},{"id":"arxiv:1903.10477","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.102.022406","source":"Crossref","title":"Reducing the number of non-Clifford gates in quantum circuits","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0002-6090-9684","authenticated-orcid":true,"given":"Aleks","family":"Kissinger","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5405-8959","authenticated-orcid":true,"given":"John","family":"van de Wetering","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.022406","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022406","note":"arxivid:1903.10477"},{"id":"arxiv:1911.09039","type":"journal-article","publisher":"Open Publishing Association","DOI":"10.4204/eptcs.318.9","page":"131-149","source":"Crossref","title":"Techniques to Reduce π/4-Parity-Phase Circuits, Motivated by the ZX Calculus","volume":"318","author":[{"given":"Niel","family":"de Beaudrap","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Oxford"}]},{"given":"Xiaoning","family":"Bian","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Dalhousie University"}]},{"given":"Quanlong","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Oxford; Cambridge Quantum Computing Ltd."}]}],"container-title":"Electronic Proceedings in Theoretical Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2020,5,1]]},"URL":"http://dx.doi.org/10.4204/EPTCS.318.9","ISSN":["2075-2180"],"container-title-short":"Electron. Proc. Theor. Comput. Sci.","note":"arxivid:1911.09039"},{"id":"arxiv:2004.05164","type":"document","categories":["T-count","Parity-phase operations","Phase gadgets","Clifford hierarchy","ZX calculus","Computer systems organization → Quantum computing"],"language":"en","author":[{"family":"de Beaudrap","given":"Niel"},{"family":"Bian","given":"Xiaoning"},{"family":"Wang","given":"Quanlong"}],"contributor":[{"family":"Flammia","given":"Steven T."}],"issued":{"date-parts":[[2020]]},"abstract":"In fault-tolerant quantum computing systems, realising (approximately) universal quantum computation is usually described in terms of realising Clifford+T operations, which is to say a circuit of CNOT, Hadamard, and π/2-phase rotations, together with T operations (π/4-phase rotations). For many error correcting codes, fault-tolerant realisations of Clifford operations are significantly less resource-intensive than those of T gates, which motivates finding ways to realise the same transformation involving T-count (the number of T gates involved) which is as low as possible. Investigations into this problem [Matthew Amy et al., 2013; Gosset et al., 2014; Matthew Amy et al., 2014; Matthew Amy et al., 2018; Earl T. Campbell and Mark Howard, 2017; Matthew Amy and Michele Mosca, 2019] has led to observations that this problem is closely related to NP-hard tensor decomposition problems [Luke E. Heyfron and Earl T. Campbell, 2018] and is tantamount to the difficult problem of decoding exponentially long Reed-Muller codes [Matthew Amy and Michele Mosca, 2019]. This problem then presents itself as one for which must be content in practise with approximate optimisation, in which one develops an array of tactics to be deployed through some pragmatic strategy. In this vein, we describe techniques to reduce the T-count, based on the effective application of \"spider nest identities\": easily recognised products of parity-phase operations which are equivalent to the identity operation. We demonstrate the effectiveness of such techniques by obtaining improvements in the T-counts of a number of circuits, in run-times which are typically less than the time required to make a fresh cup of coffee.","DOI":"10.4230/LIPICS.TQC.2020.11","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"Fast and Effective Techniques for T-Count Reduction via Spider Nest Identities","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.11","note":"copyright:Creative Commons Attribution 3.0 Unported license\narxivid:2004.05164"},{"id":"arxiv:2109.01076","type":"journal-article","publisher":"IOP Publishing","issue":"4","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the ZX-calculus. Recently it was shown that quantum circuits can be classically simulated by expressing the non-stabiliser gates in a circuit as magic state injections and decomposing them in chunks of 2–6 states at a time, obtaining sums of (efficiently-simulable) stabiliser states with many fewer terms than the naive approach. We adapt these techniques from the original setting of Clifford circuits with magic state injection to generic ZX-diagrams and show that, by interleaving this ‘chunked’ decomposition with a ZX-calculus-based simplification strategy, we can obtain stabiliser decompositions that are many orders of magnitude smaller than existing approaches. We illustrate this technique to perform exact norm calculations (and hence strong simulation) on the outputs of random 50- and 100-qubit Clifford + T circuits with up to 70 T-gates as well as a family of hidden shift circuits previously considered by Bravyi and Gosset with over 1000 T-gates.</jats:p>","DOI":"10.1088/2058-9565/ac5d20","page":"044001","source":"Crossref","title":"Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-6090-9684","authenticated-orcid":true,"given":"Aleks","family":"Kissinger","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5405-8959","authenticated-orcid":true,"given":"John","family":"van de Wetering","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2022,7,21]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ac5d20","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2109.01076"},{"id":"arxiv:2105.15048","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The general problem of quantum compiling is to approximate any unitary transformation that describes the quantum computation as a sequence of elements selected from a finite base of universal quantum gates. The Solovay-Kitaev theorem guarantees the existence of such an approximating sequence. Though, the solutions to the quantum compiling problem suffer from a tradeoff between the length of the sequences, the precompilation time, and the execution time. Traditional approaches are time-consuming, unsuitable to be employed during computation. Here, we propose a deep reinforcement learning method as an alternative strategy, which requires a single precompilation procedure to learn a general strategy to approximate single-qubit unitaries. We show that this approach reduces the overall execution time, improving the tradeoff between the length of the sequence and execution time, potentially allowing real-time operations.</jats:p>","DOI":"10.1038/s42005-021-00684-3","source":"Crossref","title":"Quantum compiling by deep reinforcement learning","volume":"4","author":[{"given":"Lorenzo","family":"Moro","sequence":"first","affiliation":[]},{"given":"Matteo G. A.","family":"Paris","sequence":"additional","affiliation":[]},{"given":"Marcello","family":"Restelli","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9839-202X","authenticated-orcid":false,"given":"Enrico","family":"Prati","sequence":"additional","affiliation":[]}],"container-title":"Communications Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,6]]},"URL":"http://dx.doi.org/10.1038/s42005-021-00684-3","ISSN":["2399-3650"],"container-title-short":"Commun Phys","page":"178","note":"alternative-id:684\narxivid:2105.15048"},{"id":"arxiv:2103.07585","type":"article-journal","author":[{"given":"Thomas","family":"Fösel"},{"given":"Murphy Yuezhen","family":"Niu"},{"given":"Florian","family":"Marquardt"},{"given":"Li","family":"Li"}],"title":"Quantum circuit optimization with deep reinforcement learning","issued":{"date-parts":[[2021,3,13]]},"note":"arxivid:2103.07585\narxiv_version_number:1"},{"id":"arxiv:2212.04508","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/dac56929.2023.10248002","page":"1-6","source":"Crossref","title":"Compiler Optimization for Quantum Computing Using Reinforcement Learning","author":[{"given":"Nils","family":"Quetschlich","sequence":"first","affiliation":[{"name":"Technical University of Munich,Chair for Design Automation,Germany"}]},{"given":"Lukas","family":"Burgholzer","sequence":"additional","affiliation":[{"name":"Johannes Kepler University,Institute for Integrated Circuits,Linz,Austria"}]},{"given":"Robert","family":"Wille","sequence":"additional","affiliation":[{"name":"Technical University of Munich,Chair for Design Automation,Germany"}]}],"event":"2023 60th ACM/IEEE Design Automation Conference (DAC)","container-title":"2023 60th ACM/IEEE Design Automation Conference (DAC)","original-title":[],"issued":{"date-parts":[[2023,7,9]]},"URL":"http://dx.doi.org/10.1109/DAC56929.2023.10248002","note":"arxivid:2212.04508"},{"id":"arxiv:2402.14396","type":"article-journal","author":[{"given":"Francisco J. R.","family":"Ruiz"},{"given":"Tuomas","family":"Laakkonen"},{"given":"Johannes","family":"Bausch"},{"given":"Matej","family":"Balog"},{"given":"Mohammadamin","family":"Barekatain"},{"given":"Francisco J. H.","family":"Heras"},{"given":"Alexander","family":"Novikov"},{"given":"Nathan","family":"Fitzpatrick"},{"given":"Bernardino","family":"Romera-Paredes"},{"given":"John","family":"van de Wetering"},{"given":"Alhussein","family":"Fawzi"},{"given":"Konstantinos","family":"Meichanetzidis"},{"given":"Pushmeet","family":"Kohli"}],"title":"Quantum Circuit Optimization with AlphaTensor","issued":{"date-parts":[[2024,3,5]]},"note":"arxivid:2402.14396\narxiv_version_number:2"},{"id":"arxiv:2404.14865","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce60285.2024.00102","page":"824-835","source":"Crossref","title":"Unitary Synthesis of Clifford+T Circuits with Reinforcement Learning","author":[{"given":"Sebastian","family":"Rietsch","sequence":"first","affiliation":[{"name":"Fraunhofer Institute for Integrated Circuits IIS,N&#x00FC;rnberg,Germany,90411"}]},{"given":"Abhishek Y.","family":"Dubey","sequence":"additional","affiliation":[{"name":"Fraunhofer Institute for Integrated Circuits IIS,N&#x00FC;rnberg,Germany,90411"}]},{"given":"Christian","family":"Ufrecht","sequence":"additional","affiliation":[{"name":"Fraunhofer Institute for Integrated Circuits IIS,N&#x00FC;rnberg,Germany,90411"}]},{"given":"Maniraman","family":"Periyasamy","sequence":"additional","affiliation":[{"name":"Fraunhofer Institute for Integrated Circuits IIS,N&#x00FC;rnberg,Germany,90411"}]},{"given":"Axel","family":"Plinge","sequence":"additional","affiliation":[{"name":"Fraunhofer Institute for Integrated Circuits IIS,N&#x00FC;rnberg,Germany,90411"}]},{"given":"Christopher","family":"Mutschler","sequence":"additional","affiliation":[{"name":"Fraunhofer Institute for Integrated Circuits IIS,N&#x00FC;rnberg,Germany,90411"}]},{"given":"Daniel D.","family":"Scherer","sequence":"additional","affiliation":[{"name":"Fraunhofer Institute for Integrated Circuits IIS,N&#x00FC;rnberg,Germany,90411"}]}],"event":"2024 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2024 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2024,9,15]]},"URL":"http://dx.doi.org/10.1109/QCE60285.2024.00102","note":"arxivid:2404.14865"},{"id":"arxiv:2511.09951","type":"article-journal","author":[{"given":"Remmy","family":"Zen"},{"given":"Maximilian","family":"Nägele"},{"given":"Florian","family":"Marquardt"}],"title":"Reusability Report: Optimizing T-count in General Quantum Circuits with AlphaTensor-Quantum","issued":{"date-parts":[[2025,11,13]]},"note":"arxivid:2511.09951\narxiv_version_number:1"},{"id":"arxiv:2504.09391","type":"article-journal","author":[{"given":"Archisman","family":"Ghosh"},{"given":"Avimita","family":"Chatterjee"},{"given":"Swaroop","family":"Ghosh"}],"title":"Survival of the Optimized: An Evolutionary Approach to T-depth Reduction","issued":{"date-parts":[[2025,7,12]]},"note":"arxivid:2504.09391\narxiv_version_number:2"},{"id":"arxiv:2405.19302","type":"article-journal","author":[{"given":"Vadym","family":"Kliuchnikov"},{"given":"Sebastian","family":"Schönnenbeck"}],"title":"Multi-qubit circuit synthesis and Hermitian lattices","issued":{"date-parts":[[2024,5,29]]},"note":"arxivid:2405.19302\narxiv_version_number:1"},{"id":"arxiv:1812.00954","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Efficient synthesis of arbitrary quantum states and unitaries from a universal fault-tolerant gate-set e.g. Clifford+T is a key subroutine in quantum computation. As large quantum algorithms feature many qubits that encode coherent quantum information but remain idle for parts of the computation, these should be used if it minimizes overall gate counts, especially that of the expensive T-gates. We present a quantum algorithm for preparing any dimension-<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math> pure quantum state specified by a list of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math> classical numbers, that realizes a trade-off between space and T-gates. Our scheme uses <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math> clean qubits and a tunable number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>&amp;#x03BB;</mml:mi><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>N</mml:mi></mml:mrow></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math> dirty qubits, to reduce the T-gate cost to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mi>N</mml:mi><mml:mi>&amp;#x03BB;</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mi>&amp;#x03BB;</mml:mi><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mi>N</mml:mi><mml:mi>&amp;#x03F5;</mml:mi></mml:mfrac></mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mrow><mml:mi>log</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>N</mml:mi></mml:mrow></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:mfrac></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math>. This trade-off is optimal up to logarithmic factors, proven through an unconditional gate counting lower bound, and is, in the best case, a quadratic improvement in T-count over prior ancillary-free approaches. We prove similar statements for unitary synthesis by reduction to state preparation. Underlying our constructions is a T-efficient circuit implementation of a quantum oracle for arbitrary classical data.</jats:p>","DOI":"10.22331/q-2024-06-17-1375","page":"1375","source":"Crossref","title":"Trading T gates for dirty qubits in state preparation and unitary synthesis","volume":"8","author":[{"given":"Guang Hao","family":"Low","sequence":"first","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Washington, Redmond, USA"},{"name":"Azure Quantum, Microsoft, Washington, Redmond, USA"}]},{"given":"Vadym","family":"Kliuchnikov","sequence":"additional","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Washington, Redmond, USA"},{"name":"Azure Quantum, Microsoft, Washington, Redmond, USA"}]},{"given":"Luke","family":"Schaeffer","sequence":"additional","affiliation":[{"name":"Quantum Architectures and Computation, Microsoft Research, Washington, Redmond, USA"},{"name":"Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA"},{"name":"Joint Center for Quantum Information and Computer Science, University of Maryland, Maryland, College Park, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,17]]},"URL":"http://dx.doi.org/10.22331/q-2024-06-17-1375","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1812.00954"},{"id":"arxiv:2411.04790","type":"article-journal","author":[{"given":"David","family":"Gosset"},{"given":"Robin","family":"Kothari"},{"given":"Kewen","family":"Wu"}],"title":"Quantum state preparation with optimal T-count","issued":{"date-parts":[[2025,10,10]]},"note":"arxivid:2411.04790\narxiv_version_number:2"},{"id":"arxiv:2203.10064","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the works \\cite{Hastings2017} and \\cite{Campbell2017}, we show that taking probabilistic mixtures of channels to solve fallback \\cite{BRS2015} and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msqrt><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"normal\">T</mml:mi></mml:mrow></mml:msqrt></mml:math>gate set we achieve an average non-Clifford gate count of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.23</mml:mn><mml:msub><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03B5;</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mn>2.13</mml:mn></mml:math>and T-count<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.56</mml:mn><mml:msub><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03B5;</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>+</mml:mo><mml:mn>5.3</mml:mn></mml:math>with mixed fallback approximations for diamond norm accuracy<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B5;</mml:mi></mml:math>.This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msqrt><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"normal\">T</mml:mi></mml:mrow></mml:msqrt></mml:math>). We also provide detailed numerical results for Clifford+T and Clifford+<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msqrt><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"normal\">T</mml:mi></mml:mrow></mml:msqrt></mml:math>gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices.</jats:p>","DOI":"10.22331/q-2023-12-18-1208","page":"1208","source":"Crossref","title":"Shorter quantum circuits via single-qubit gate approximation","volume":"7","author":[{"given":"Vadym","family":"Kliuchnikov","sequence":"first","affiliation":[{"name":"Microsoft Quantum, Redmond, WA, US"},{"name":"Microsoft Quantum, Toronto, ON, CA"}]},{"given":"Kristin","family":"Lauter","sequence":"additional","affiliation":[{"name":"Facebook AI Research, Seattle, WA, US"}]},{"given":"Romy","family":"Minko","sequence":"additional","affiliation":[{"name":"University of Oxford, Oxford, UK"},{"name":"Heilbronn Institute for Mathematical Research, University of Bristol, Bristol, UK"}]},{"given":"Adam","family":"Paetznick","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Redmond, WA, US"}]},{"given":"Christophe","family":"Petit","sequence":"additional","affiliation":[{"name":"University of Birmingham, Birmingham, UK"},{"name":"Université Libre de Bruxelles, Brussels, Belgium"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,18]]},"URL":"http://dx.doi.org/10.22331/q-2023-12-18-1208","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2203.10064"},{"id":"arxiv:quant-ph/0205115","type":"article-journal","author":[{"given":"Yaoyun","family":"Shi"}],"title":"Both Toffoli and Controlled-NOT need little help to do universal quantum computation","issued":{"date-parts":[[2002,5,26]]},"note":"arxivid:quant-ph/0205115\narxiv_version_number:2"},{"id":"arxiv:1212.5069","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.87.022328","source":"Crossref","title":"Low-overhead constructions for the fault-tolerant Toffoli gate","volume":"87","author":[{"given":"Cody","family":"Jones","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,2,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.022328","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022328","note":"arxivid:1212.5069"},{"id":"arxiv:quant-ph/0404089","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.93.130502","source":"Crossref","title":"Quantum Circuits for General Multiqubit Gates","volume":"93","author":[{"given":"Mikko","family":"Möttönen","sequence":"first","affiliation":[]},{"given":"Juha J.","family":"Vartiainen","sequence":"additional","affiliation":[]},{"given":"Ville","family":"Bergholm","sequence":"additional","affiliation":[]},{"given":"Martti M.","family":"Salomaa","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2004,9,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.93.130502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130502","note":"arxivid:quant-ph/0404089"},{"id":"manual:-P.-Sarnak-(2015-February).-L","type":"document","author":[{"family":"Sarnak","given":"P."}],"title":"Letter to Scott Aaronson and Andy Pollington on the Solovay-Kitaev Theorem and Golden Gates","issued":"2015-02"},{"id":"arxiv:1704.02106","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aim.2017.06.022","page":"869-901","source":"Crossref","title":"Super-Golden-Gates for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" altimg=\"si1.gif\" overflow=\"scroll\"><mml:mi>P</mml:mi><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy=\"true\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"true\">)</mml:mo></mml:mrow></mml:math>","volume":"327","author":[{"given":"Ori","family":"Parzanchevski","sequence":"first","affiliation":[]},{"given":"Peter","family":"Sarnak","sequence":"additional","affiliation":[]}],"container-title":"Advances in Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2018,3]]},"URL":"http://dx.doi.org/10.1016/j.aim.2017.06.022","ISSN":["0001-8708"],"container-title-short":"Advances in Mathematics","note":"special_numbering:C\nalternative-id:S0001870817301640\narxivid:1704.02106"},{"id":"arxiv:2205.03007","type":"article-journal","author":[{"given":"Terrence Richard","family":"Blackman"},{"given":"Zachary","family":"Stier"}],"title":"Fast Navigation with Icosahedral Golden Gates","issued":{"date-parts":[[2022,5,6]]},"note":"arxivid:2205.03007\narxiv_version_number:1"},{"id":"arxiv:2509.09047","type":"article-journal","author":[{"given":"Rahul","family":"Dalal"},{"given":"Shai","family":"Evra"},{"given":"Ori","family":"Parzanchevski"}],"title":"Multi-Qubit Golden Gates","issued":{"date-parts":[[2025,9,10]]},"note":"arxivid:2509.09047\narxiv_version_number:1"},{"id":"arxiv:1904.01124","type":"journal-article","publisher":"IOP Publishing","issue":"3","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    Treating stabilizer operations as free, we establish lower bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks. Our bounds apply to adaptive computations using measurements with an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational subroutines including the quantum adder and the multiply-controlled\n                    <jats:italic>Z</jats:italic>\n                    gate. To prove our resource conversion bounds we introduce two new monotones, the stabilizer nullity and the dyadic monotone, and make use of the already-known stabilizer extent. We consider conversions that borrow resource states, known as catalyst states, and return them at the end of the algorithm. We show that catalysis is necessary for many conversions and introduce new catalytic conversions, some of which are optimal. By finding a canonical form for post-selected stabilizer computations, we show that approximating a single-qubit unitary to within diamond-norm precision\n                    <jats:italic>ɛ</jats:italic>\n                    requires at least 1/7 ⋅ log\n                    <jats:sub>2</jats:sub>\n                    (1/\n                    <jats:italic>ɛ</jats:italic>\n                    ) − 4/3\n                    <jats:italic>T</jats:italic>\n                    -states on average. This is the first lower bound that applies to synthesis protocols using fall-back, mixing techniques, and where the number of ancillas used can depend on\n                    <jats:italic>ɛ</jats:italic>\n                    . Up to multiplicative factors, we optimally lower bound the number of\n                    <jats:italic>T</jats:italic>\n                    or\n                    <jats:italic>CCZ</jats:italic>\n                    states needed to implement the ubiquitous modular adder and multiply-controlled-\n                    <jats:italic>Z</jats:italic>\n                    operations. When the probability of Pauli measurement outcomes is 1/2, some of our bounds become tight to within a small additive constant.\n                  </jats:p>","DOI":"10.1088/2058-9565/ab8963","page":"035009","source":"Crossref","title":"Lower bounds on the non-Clifford resources for quantum computations","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0003-0954-4509","authenticated-orcid":false,"given":"Michael","family":"Beverland","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3903-2734","authenticated-orcid":false,"given":"Earl","family":"Campbell","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6910-185X","authenticated-orcid":false,"given":"Mark","family":"Howard","sequence":"additional","affiliation":[]},{"given":"Vadym","family":"Kliuchnikov","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2020,5,28]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ab8963","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1904.01124"},{"id":"arxiv:2510.07223","type":"article-journal","author":[{"given":"David","family":"Gosset"},{"given":"Robin","family":"Kothari"},{"given":"Chenyi","family":"Zhang"}],"title":"Multi-qubit Toffoli with exponentially fewer T gates","issued":{"date-parts":[[2025,10,8]]},"note":"arxivid:2510.07223\narxiv_version_number:1"},{"id":"arxiv:1608.06596","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.95.012329","source":"Crossref","title":"Diagonal gates in the Clifford hierarchy","volume":"95","author":[{"given":"Shawn X.","family":"Cui","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[]},{"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,1,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.95.012329","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012329","note":"arxivid:1608.06596"},{"id":"arxiv:1902.04022","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.100.022304","source":"Crossref","title":"Unifying the Clifford hierarchy via symmetric matrices over rings","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":true,"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[]},{"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.100.022304","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022304","note":"arxivid:1902.04022"},{"id":"arxiv:2212.05398","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Here we study the unitary groups that can be constructed using elements from the qubit Clifford Hierarchy. We first provide a necessary and sufficient canonical form that semi-Clifford and generalized semi-Clifford elements must satisfy to be in the Clifford Hierarchy. Then we classify the groups that can be formed from such elements. Up to Clifford conjugation, we classify all such groups that can be constructed using generalized semi-Clifford elements in the Clifford Hierarchy. We discuss a possible minor exception to this classification in the appendix. This may not be a full classification of all groups in the qubit Clifford Hierarchy as it is not currently known if all elements in the Clifford Hierarchy must be generalized semi-Clifford. In addition to the diagonal gate groups found by Cui et al., we show that many non-isomorphic (to the diagonal gate groups) generalized symmetric groups are also contained in the Clifford Hierarchy. Finally, as an application of this classification, we examine restrictions on transversal gates given by the structure of the groups enumerated herein which may be of independent interest.</jats:p>","DOI":"10.22331/q-2024-06-13-1370","page":"1370","source":"Crossref","title":"On Groups in the Qubit Clifford Hierarchy","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-3230-6712","authenticated-orcid":false,"given":"Jonas T.","family":"Anderson","sequence":"first","affiliation":[{"name":"Northrop Grumman Corporation"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,13]]},"URL":"http://dx.doi.org/10.22331/q-2024-06-13-1370","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2212.05398"},{"id":"arxiv:2410.11818","type":"article-journal","author":[{"given":"Zhiyang","family":"He"},{"given":"Luke","family":"Robitaille"},{"given":"Xinyu","family":"Tan"}],"title":"Permutation gates in the third level of the Clifford hierarchy","issued":{"date-parts":[[2024,10,15]]},"note":"arxivid:2410.11818\narxiv_version_number:1"},{"id":"arxiv:2501.07939","type":"article-journal","author":[{"given":"Nadish","family":"de Silva"},{"given":"Oscar","family":"Lautsch"}],"title":"The Clifford hierarchy for one qubit or qudit","issued":{"date-parts":[[2025,1,14]]},"note":"arxivid:2501.07939\narxiv_version_number:1"},{"id":"arxiv:2402.13863","type":"article-journal","author":[{"given":"Shin Ho","family":"Choe"},{"given":"Robert","family":"Koenig"}],"title":"How to fault-tolerantly realize any quantum circuit with local operations","issued":{"date-parts":[[2024,2,21]]},"note":"arxivid:2402.13863\narxiv_version_number:1"},{"id":"arxiv:2409.11616","type":"article-journal","author":[{"given":"Eric","family":"Kubischta"},{"given":"Ian","family":"Teixeira"}],"title":"Flexible Fault Tolerant Gate Gadgets","issued":{"date-parts":[[2024,9,18]]},"note":"arxivid:2409.11616\narxiv_version_number:1"},{"id":"arxiv:2202.00119","type":"document","categories":["Fault-tolerant quantum computation","quantum error correction","Theory of computation → Quantum computation theory","Hardware → Quantum error correction and fault tolerance"],"language":"en","author":[{"family":"Fawzi","given":"Omar"},{"family":"Müller-Hermes","given":"Alexander"},{"family":"Shayeghi","given":"Ala"}],"contributor":[{"family":"Braverman","given":"Mark"}],"editor":[{"family":"Braverman","given":"Mark"}],"issued":{"date-parts":[[2022]]},"abstract":"The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation stating that arbitrarily long quantum computations can be performed with a polylogarithmic overhead provided the noise level is below a constant level. A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance. \r\nFor any non-unitary qubit channel 𝒩 and any quantum fault tolerance schemes against i.i.d. noise modeled by 𝒩, we prove a lower bound of max{Q(𝒩)^{-1}n,α_𝒩 log T} on the number of physical qubits, for circuits of length T and width n. Here, Q(𝒩) denotes the quantum capacity of 𝒩 and α_𝒩 > 0 is a constant only depending on the channel 𝒩. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and in the case of unital noise, we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman and Hassidim (2013).","container-title":"LIPIcs, Volume 215, ITCS 2022","DOI":"10.4230/LIPICS.ITCS.2022.68","volume":"215","number":"68","page":"68:1-68:20","page-first":"68:1","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.68","note":"copyright:Creative Commons Attribution 4.0 International license\narxivid:2202.00119"},{"id":"arxiv:2302.04317","type":"article-journal","author":[{"given":"Nouédyn","family":"Baspin"},{"given":"Omar","family":"Fawzi"},{"given":"Ala","family":"Shayeghi"}],"title":"A lower bound on the overhead of quantum error correction in low dimensions","issued":{"date-parts":[[2023,2,8]]},"note":"arxivid:2302.04317\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0402104","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.71.012336","source":"Crossref","title":"Fault-tolerant quantum computation for local non-Markovian noise","volume":"71","author":[{"given":"Barbara M.","family":"Terhal","sequence":"first","affiliation":[]},{"given":"Guido","family":"Burkard","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,1,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.012336","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012336","note":"arxivid:quant-ph/0402104"},{"id":"arxiv:quant-ph/0510231","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.96.050504","source":"Crossref","title":"Fault-Tolerant Quantum Computation with Long-Range Correlated Noise","volume":"96","author":[{"given":"Dorit","family":"Aharonov","sequence":"first","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,2,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.96.050504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050504","note":"arxivid:quant-ph/0510231"},{"id":"arxiv:quant-ph/0511065","type":"article-journal","author":[{"given":"Panos","family":"Aliferis"},{"given":"Barbara M.","family":"Terhal"}],"title":"Fault-Tolerant Quantum Computation for Local Leakage Faults","issued":{"date-parts":[[2006,5,26]]},"note":"arxivid:quant-ph/0511065\narxiv_version_number:5"},{"id":"arxiv:quant-ph/0307166","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1023/b:qinp.0000004127.09741.9b","page":"249-258","source":"Crossref","title":"Scaling and Renormalization in Fault-Tolerant Quantum Computers","volume":"2","author":[{"given":"Maxim","family":"Raginsky","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2003,6]]},"URL":"http://dx.doi.org/10.1023/B:QINP.0000004127.09741.9b","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Information Processing","note":"alternative-id:476054\narxivid:quant-ph/0307166"},{"id":"arxiv:quant-ph/0607047","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.98.020501","source":"Crossref","title":"Effective Fault-Tolerant Quantum Computation with Slow Measurements","volume":"98","author":[{"given":"David P.","family":"DiVincenzo","sequence":"first","affiliation":[]},{"given":"Panos","family":"Aliferis","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,1,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.98.020501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020501","note":"arxivid:quant-ph/0607047"},{"id":"arxiv:quant-ph/0310136","type":"article-journal","author":[{"given":"Alexander A.","family":"Razborov"}],"title":"An Upper Bound on the Threshold Quantum Decoherence Rate","issued":{"date-parts":[[2003,10,21]]},"note":"arxivid:quant-ph/0310136\narxiv_version_number:1"},{"id":"arxiv:2411.12805","type":"article-journal","author":[{"given":"Mykhailo","family":"Bilokur"},{"given":"Sarang","family":"Gopalakrishnan"},{"given":"Shayan","family":"Majidy"}],"title":"Thermodynamic limitations on fault-tolerant quantum computing","issued":{"date-parts":[[2024,12,31]]},"note":"arxivid:2411.12805\narxiv_version_number:2"},{"id":"arxiv:quant-ph/9803006","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"5410","abstract":"<jats:p>Quantum key distribution is widely thought to offer unconditional security in communication between two users. Unfortunately, a widely accepted proof of its security in the presence of source, device, and channel noises has been missing. This long-standing problem is solved here by showing that, given fault-tolerant quantum computers, quantum key distribution over an arbitrarily long distance of a realistic noisy channel can be made unconditionally secure. The proof is reduced from a noisy quantum scheme to a noiseless quantum scheme and then from a noiseless quantum scheme to a noiseless classical scheme, which can then be tackled by classical probability theory.</jats:p>","DOI":"10.1126/science.283.5410.2050","page":"2050-2056","source":"Crossref","title":"Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances","volume":"283","author":[{"given":"Hoi-Kwong","family":"Lo","sequence":"first","affiliation":[{"name":"Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, UK."}]},{"given":"H. F.","family":"Chau","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong, China."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[1999,3,26]]},"URL":"http://dx.doi.org/10.1126/science.283.5410.2050","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.283.5410.2050\narxivid:quant-ph/9803006"},{"id":"arxiv:2502.05992","type":"article-journal","author":[{"given":"James","family":"Keppens"},{"given":"Quinten","family":"Eggerickx"},{"given":"Vukan","family":"Levajac"},{"given":"George","family":"Simion"},{"given":"Bart","family":"Sorée"}],"title":"Qudit vs. Qubit: Simulated performance of error correction codes in higher dimensions","issued":{"date-parts":[[2025,9,5]]},"note":"arxivid:2502.05992\narxiv_version_number:2"},{"id":"arxiv:1905.10481","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3307650.3322253","page":"554-566","source":"Crossref","title":"Asymptotic improvements to quantum circuits via qutrits","author":[{"given":"Pranav","family":"Gokhale","sequence":"first","affiliation":[{"name":"University of Chicago"}]},{"given":"Jonathan M.","family":"Baker","sequence":"additional","affiliation":[{"name":"University of Chicago"}]},{"given":"Casey","family":"Duckering","sequence":"additional","affiliation":[{"name":"University of Chicago"}]},{"given":"Natalie C.","family":"Brown","sequence":"additional","affiliation":[{"name":"Georgia Institute of Technology"}]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Duke University"}]},{"given":"Frederic T.","family":"Chong","sequence":"additional","affiliation":[{"name":"University of Chicago"}]}],"event":"ISCA '19: The 46th Annual International Symposium on Computer Architecture","container-title":"Proceedings of the 46th International Symposium on Computer Architecture","original-title":[],"issued":{"date-parts":[[2019,6,22]]},"URL":"http://dx.doi.org/10.1145/3307650.3322253","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3307650.3322253\n10.1145/3307650\narxivid:1905.10481"},{"id":"arxiv:1205.3104","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.2.041021","source":"Crossref","title":"Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes","volume":"2","author":[{"given":"Earl T.","family":"Campbell","sequence":"first","affiliation":[]},{"given":"Hussain","family":"Anwar","sequence":"additional","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2012,12,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.2.041021","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041021","note":"arxivid:1205.3104"},{"id":"arxiv:1406.3055","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.113.230501","source":"Crossref","title":"Enhanced Fault-Tolerant Quantum Computing in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>d</mml:mi></mml:math>-Level Systems","volume":"113","author":[{"given":"Earl T.","family":"Campbell","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.113.230501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230501","note":"arxivid:1406.3055"},{"id":"arxiv:2207.01731","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevd.107.054512","source":"Crossref","title":"Preparations for quantum simulations of quantum chromodynamics in \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>\n dimensions. I. Axial gauge","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0001-7189-0424","authenticated-orcid":true,"given":"Roland C.","family":"Farrell","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8289-1991","authenticated-orcid":true,"given":"Ivan A.","family":"Chernyshev","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5228-8291","authenticated-orcid":true,"given":"Sarah J. M.","family":"Powell","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0794-2389","authenticated-orcid":true,"given":"Nikita A.","family":"Zemlevskiy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3570-2849","authenticated-orcid":true,"given":"Marc","family":"Illa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6502-7106","authenticated-orcid":true,"given":"Martin J.","family":"Savage","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.107.054512","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"054512","note":"update-to:{\"updated\":{\"date-parts\":[[2023,4,4]],\"date-time\":\"2023-04-04T00:00:00Z\",\"timestamp\":1680566400000},\"DOI\":\"10.1103/physrevd.107.054512\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\n{\"updated\":{\"date-parts\":[[2023,4,18]],\"date-time\":\"2023-04-18T00:00:00Z\",\"timestamp\":1681776000000},\"DOI\":\"10.1103/physrevd.107.054512\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2023,4,18]],\"date-time\":\"2023-04-18T00:00:00Z\",\"timestamp\":1681776000000},\"DOI\":\"10.1103/physrevd.107.054512\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2207.01731"},{"id":"arxiv:2209.10781","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevd.107.054513","source":"Crossref","title":"Preparations for quantum simulations of quantum chromodynamics in \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>\n dimensions. II. Single-baryon \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>β</mml:mi></mml:math>\n-decay in real time","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0001-7189-0424","authenticated-orcid":true,"given":"Roland C.","family":"Farrell","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8289-1991","authenticated-orcid":true,"given":"Ivan A.","family":"Chernyshev","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5228-8291","authenticated-orcid":true,"given":"Sarah J. M.","family":"Powell","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0794-2389","authenticated-orcid":true,"given":"Nikita A.","family":"Zemlevskiy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3570-2849","authenticated-orcid":true,"given":"Marc","family":"Illa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6502-7106","authenticated-orcid":true,"given":"Martin J.","family":"Savage","sequence":"additional","affiliation":[]}],"container-title":"Physical Review D","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevD.107.054513","ISSN":["2470-0010","2470-0029"],"container-title-short":"Phys. Rev. D","page":"054513","note":"update-to:{\"updated\":{\"date-parts\":[[2023,4,18]],\"date-time\":\"2023-04-18T00:00:00Z\",\"timestamp\":1681776000000},\"DOI\":\"10.1103/physrevd.107.054513\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2023,4,18]],\"date-time\":\"2023-04-18T00:00:00Z\",\"timestamp\":1681776000000},\"DOI\":\"10.1103/physrevd.107.054513\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2209.10781"},{"id":"arxiv:2311.12003","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/revmodphys.97.021003","source":"Crossref","title":"<i>Colloquium</i>\n: Qudits for decomposing multiqubit gates and realizing quantum algorithms","volume":"97","author":[{"ORCID":"https://orcid.org/0000-0001-5760-441X","authenticated-orcid":true,"given":"Evgeniy O.","family":"Kiktenko","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03v8t4025","id-type":"ROR","asserted-by":"publisher"}],"name":"Russian Quantum Center"},{"id":[{"id":"https://ror.org/019vsm959","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Science and Technology “MISIS"}]},{"ORCID":"https://orcid.org/0000-0002-8321-7103","authenticated-orcid":true,"given":"Anastasiia S.","family":"Nikolaeva","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v8t4025","id-type":"ROR","asserted-by":"publisher"}],"name":"Russian Quantum Center"},{"id":[{"id":"https://ror.org/019vsm959","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Science and Technology “MISIS"}]},{"ORCID":"https://orcid.org/0000-0002-4722-3418","authenticated-orcid":true,"given":"Aleksey K.","family":"Fedorov","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v8t4025","id-type":"ROR","asserted-by":"publisher"}],"name":"Russian Quantum Center"},{"id":[{"id":"https://ror.org/019vsm959","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Science and Technology “MISIS"}]}],"container-title":"Reviews of Modern Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,3]]},"URL":"http://dx.doi.org/10.1103/RevModPhys.97.021003","ISSN":["0034-6861","1539-0756"],"container-title-short":"Rev. Mod. Phys.","page":"021003","note":"arxivid:2311.12003"},{"id":"arxiv:1802.05267","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.8.031084","source":"Crossref","title":"Reinforcement Learning with Neural Networks for Quantum Feedback","volume":"8","author":[{"given":"Thomas","family":"Fösel","sequence":"first","affiliation":[]},{"given":"Petru","family":"Tighineanu","sequence":"additional","affiliation":[]},{"given":"Talitha","family":"Weiss","sequence":"additional","affiliation":[]},{"given":"Florian","family":"Marquardt","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2018,9,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.8.031084","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031084","note":"arxivid:1802.05267"},{"id":"arxiv:1812.08451","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum error correction is widely thought to be the key to fault-tolerant quantum computation. However, determining the most suited encoding for unknown error channels or specific laboratory setups is highly challenging. Here, we present a reinforcement learning framework for optimizing and fault-tolerantly adapting quantum error correction codes. We consider a reinforcement learning agent tasked with modifying a family of surface code quantum memories until a desired logical error rate is reached. Using efficient simulations with about 70 data qubits with arbitrary connectivity, we demonstrate that such a reinforcement learning agent can determine near-optimal solutions, in terms of the number of data qubits, for various error models of interest. Moreover, we show that agents trained on one setting are able to successfully transfer their experience to different settings. This ability for transfer learning showcases the inherent strengths of reinforcement learning and the applicability of our approach for optimization from off-line simulations to on-line laboratory settings.</jats:p>","DOI":"10.22331/q-2019-12-16-215","page":"215","source":"Crossref","title":"Optimizing Quantum Error Correction Codes with Reinforcement Learning","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-7815-7006","authenticated-orcid":false,"given":"Hendrik Poulsen","family":"Nautrup","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria"}]},{"ORCID":"https://orcid.org/0000-0002-3949-981X","authenticated-orcid":false,"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA"}]},{"ORCID":"https://orcid.org/0000-0002-2632-7955","authenticated-orcid":false,"given":"Vedran","family":"Dunjko","sequence":"additional","affiliation":[{"name":"LIACS, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0002-9065-1565","authenticated-orcid":false,"given":"Hans J.","family":"Briegel","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria"},{"name":"Department of Philosophy, University of Konstanz, Konstanz 78457, Germany"}]},{"ORCID":"https://orcid.org/0000-0003-1950-8640","authenticated-orcid":false,"given":"Nicolai","family":"Friis","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria"},{"name":"Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,16]]},"URL":"http://dx.doi.org/10.22331/q-2019-12-16-215","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1812.08451"},{"id":"arxiv:2311.04750","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>In the ongoing race towards experimental implementations of quantum error correction (QEC), finding ways to automatically discover codes and encoding strategies tailored to the qubit hardware platform is emerging as a critical problem. Reinforcement learning (RL) has been identified as a promising approach, but so far it has been severely restricted in terms of scalability. In this work, we significantly expand the power of RL approaches to QEC code discovery. Explicitly, we train an RL agent that automatically discovers both QEC codes and their encoding circuits for a given gate set, qubit connectivity and error model, from scratch. This is enabled by a reward based on the Knill-Laflamme conditions and a vectorized Clifford simulator, showing its effectiveness with up to 25 physical qubits and distance 5 codes, while presenting a roadmap to scale this approach to 100 qubits and distance 10 codes in the near future. We also introduce the concept of a noise-aware meta-agent, which learns to produce encoding strategies simultaneously for a range of noise models, thus leveraging transfer of insights between different situations. Our approach opens the door towards hardware-adapted accelerated discovery of QEC approaches across the full spectrum of quantum hardware platforms of interest.</jats:p>","DOI":"10.1038/s41534-024-00920-y","source":"Crossref","title":"Simultaneous discovery of quantum error correction codes and encoders with a noise-aware reinforcement learning agent","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0003-3338-5130","authenticated-orcid":false,"given":"Jan","family":"Olle","sequence":"first","affiliation":[]},{"given":"Remmy","family":"Zen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5332-213X","authenticated-orcid":false,"given":"Matteo","family":"Puviani","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4566-1753","authenticated-orcid":false,"given":"Florian","family":"Marquardt","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,3]]},"URL":"http://dx.doi.org/10.1038/s41534-024-00920-y","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"126","note":"alternative-id:920\narxivid:2311.04750"},{"id":"arxiv:2412.20380","type":"article-journal","author":[{"given":"Zihao","family":"Wang"},{"given":"Hao","family":"Tang"}],"title":"Artificial Intelligence for Quantum Error Correction: A Comprehensive Review","issued":{"date-parts":[[2024,12,29]]},"note":"arxivid:2412.20380\narxiv_version_number:1"},{"id":"arxiv:2506.11552","type":"article-journal","author":[{"given":"Nico","family":"Meyer"},{"given":"Christopher","family":"Mutschler"},{"given":"Andreas","family":"Maier"},{"given":"Daniel D.","family":"Scherer"}],"title":"Learning Encodings by Maximizing State Distinguishability: Variational Quantum Error Correction","issued":{"date-parts":[[2026,4,10]]},"note":"arxivid:2506.11552\narxiv_version_number:2"},{"id":"arxiv:2503.11783","type":"article-journal","author":[{"given":"Shuwei","family":"Liu"},{"given":"Shiyu","family":"Zhou"},{"given":"Zi-Wen","family":"Liu"},{"given":"Jinmin","family":"Yi"}],"title":"Noise-strength-adapted approximate quantum codes inspired by machine learning","issued":{"date-parts":[[2025,3,25]]},"note":"arxivid:2503.11783\narxiv_version_number:2"},{"id":"arxiv:2501.09622","type":"article-journal","author":[{"given":"Bruno C. A.","family":"Freire"},{"given":"Nicolas","family":"Delfosse"},{"given":"Anthony","family":"Leverrier"}],"title":"Optimizing hypergraph product codes with random walks, simulated annealing and reinforcement learning","issued":{"date-parts":[[2025,6,5]]},"note":"arxivid:2501.09622\narxiv_version_number:3"},{"id":"arxiv:2502.14372","type":"article-journal","author":[{"given":"Austin Yubo","family":"He"},{"given":"Zi-Wen","family":"Liu"}],"title":"Discovering highly efficient low-weight quantum error-correcting codes with reinforcement learning","issued":{"date-parts":[[2025,2,20]]},"note":"arxivid:2502.14372\narxiv_version_number:1"},{"id":"arxiv:1309.1674","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.89.042313","source":"Crossref","title":"Family of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:mo>[</mml:mo><mml:mo>[</mml:mo><mml:mrow><mml:mn>6</mml:mn><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>]</mml:mo><mml:mo>]</mml:mo></mml:mrow></mml:math>codes for practical and scalable adiabatic quantum computation","volume":"89","author":[{"given":"Anand","family":"Ganti","sequence":"first","affiliation":[]},{"given":"Uzoma","family":"Onunkwo","sequence":"additional","affiliation":[]},{"given":"Kevin","family":"Young","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,4,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.89.042313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042313","note":"arxivid:1309.1674"},{"id":"arxiv:1911.01354","type":"article-journal","author":[{"given":"Milad","family":"Marvian"},{"given":"Seth","family":"Lloyd"}],"title":"Robust universal Hamiltonian quantum computing using two-body interactions","issued":{"date-parts":[[2019,11,4]]},"note":"arxivid:1911.01354\narxiv_version_number:1"},{"id":"preset:Campbell16","type":"document","author":[{"family":"Campbell","given":"E."}],"title":"The smallest interesting colour code","note":"Online available at","issued":"2016","URL":"https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/"},{"id":"arxiv:2112.01446","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.3.030319","source":"Crossref","title":"Morphing Quantum Codes","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":true,"given":"Michael","family":"Vasmer","sequence":"first","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,8]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.030319","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030319","note":"arxivid:2112.01446"},{"id":"arxiv:1504.07999","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.117.080501","source":"Crossref","title":"Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations","volume":"117","author":[{"given":"Michael J.","family":"Bremner","sequence":"first","affiliation":[]},{"given":"Ashley","family":"Montanaro","sequence":"additional","affiliation":[]},{"given":"Dan J.","family":"Shepherd","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,8,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.117.080501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080501","note":"arxivid:1504.07999"},{"id":"arxiv:1910.09333","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/jsait.2020.3012914","page":"499-514","source":"Crossref","title":"On Optimality of CSS Codes for Transversal <i>T</i>","volume":"1","author":[{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2084-9717","authenticated-orcid":false,"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6640-1072","authenticated-orcid":false,"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5521-4397","authenticated-orcid":false,"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"IEEE Journal on Selected Areas in Information Theory","original-title":[],"issued":{"date-parts":[[2020,8]]},"URL":"http://dx.doi.org/10.1109/JSAIT.2020.3012914","ISSN":["2641-8770"],"container-title-short":"IEEE J. Sel. Areas Inf. Theory","note":"arxivid:1910.09333"},{"id":"arxiv:1606.01906","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevlett.118.060501","source":"Crossref","title":"Unifying Gate Synthesis and Magic State Distillation","volume":"118","author":[{"given":"Earl T.","family":"Campbell","sequence":"first","affiliation":[]},{"given":"Mark","family":"Howard","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,2,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.118.060501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"060501","note":"arxivid:1606.01906"},{"id":"arxiv:1709.02832","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present several different codes and protocols to distill<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>, controlled-<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>S</mml:mi></mml:math>, and Toffoli (or<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:math>) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>γ</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>1</mml:mn></mml:math>. We also present a Reed-Muller based construction of these codes which obtains a worse<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>γ</mml:mi></mml:math>but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:math>magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of. Several examples, including a Reed-Muller code for<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-to-Toffoli distillation, punctured Reed-Muller codes for<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>512</mml:mn></mml:math>T-gate to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>10</mml:mn></mml:math>Toffoli gate code with distance<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>8</mml:mn></mml:math>as well as triorthogonal codes with parameters<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>887</mml:mn><mml:mo>,</mml:mo><mml:mn>137</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>912</mml:mn><mml:mo>,</mml:mo><mml:mn>112</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>,</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>937</mml:mn><mml:mo>,</mml:mo><mml:mn>87</mml:mn><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math>with very low prefactors in front of the leading order error terms in those codes.</jats:p>","DOI":"10.22331/q-2018-06-07-71","page":"71","source":"Crossref","title":"Codes and Protocols for Distilling<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>, controlled-<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>S</mml:mi></mml:math>, and Toffoli Gates","volume":"2","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA"},{"name":"Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2018,6,7]]},"URL":"http://dx.doi.org/10.22331/q-2018-06-07-71","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1709.02832"},{"id":"arxiv:quant-ph/0301105","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.68.022322","source":"Crossref","title":"Quantum computing in the presence of spontaneous emission by a combined dynamical decoupling and quantum-error-correction strategy","volume":"68","author":[{"given":"K.","family":"Khodjasteh","sequence":"first","affiliation":[]},{"given":"D. A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,8,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.022322","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022322","note":"arxivid:quant-ph/0301105"},{"id":"arxiv:quant-ph/9702029","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.57.127","page":"127-137","source":"Crossref","title":"Theory of fault-tolerant quantum computation","volume":"57","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1998,1,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.57.127","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9702029"},{"id":"arxiv:quant-ph/9809071","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.82.2417","page":"2417-2421","source":"Crossref","title":"Dynamical Decoupling of Open Quantum Systems","volume":"82","author":[{"given":"Lorenza","family":"Viola","sequence":"first","affiliation":[]},{"given":"Emanuel","family":"Knill","sequence":"additional","affiliation":[]},{"given":"Seth","family":"Lloyd","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1999,3,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.82.2417","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9809071"},{"id":"arxiv:1906.08890","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3313276.3316404","page":"515-526","source":"Crossref","title":"Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits","author":[{"given":"Adam Bene","family":"Watts","sequence":"first","affiliation":[{"name":"Massachusetts Institute of Technology, USA"}]},{"given":"Robin","family":"Kothari","sequence":"additional","affiliation":[{"name":"Microsoft Research, USA"}]},{"given":"Luke","family":"Schaeffer","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology, USA"}]},{"given":"Avishay","family":"Tal","sequence":"additional","affiliation":[{"name":"Stanford University, USA"}]}],"event":"STOC '19: 51st Annual ACM SIGACT Symposium on the Theory of Computing","container-title":"Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[2019,6,23]]},"URL":"http://dx.doi.org/10.1145/3313276.3316404","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3313276.3316404\n10.1145/3313276\narxivid:1906.08890"},{"id":"arxiv:2211.06703","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1038/s41567-023-02282-2","page":"219-224","source":"Crossref","title":"Protecting expressive circuits with a quantum error detection code","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0002-1648-6725","authenticated-orcid":false,"given":"Chris N.","family":"Self","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0231-1729","authenticated-orcid":false,"given":"Marcello","family":"Benedetti","sequence":"additional","affiliation":[]},{"given":"David","family":"Amaro","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,5]]},"URL":"http://dx.doi.org/10.1038/s41567-023-02282-2","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2282\narxivid:2211.06703"},{"id":"arxiv:2404.11953","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/jsait.2025.3602446","page":"311-324","source":"Crossref","title":"Tailoring Fault-Tolerance to Quantum Algorithms","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-7754-1345","authenticated-orcid":false,"given":"Zhuangzhuang","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, USA"}]},{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ, USA"}]}],"container-title":"IEEE Journal on Selected Areas in Information Theory","original-title":[],"issued":{"date-parts":[[2025]]},"URL":"http://dx.doi.org/10.1109/JSAIT.2025.3602446","ISSN":["2641-8770"],"container-title-short":"IEEE J. Sel. Areas Inf. Theory","note":"arxivid:2404.11953"},{"id":"arxiv:1705.02329","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.121.050502","source":"Crossref","title":"Quantum Error Correction with Only Two Extra Qubits","volume":"121","author":[{"given":"Rui","family":"Chao","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2018,8,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.121.050502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050502","note":"arxivid:1705.02329"},{"id":"arxiv:2408.14828","type":"article-journal","author":[{"given":"Christopher","family":"Gerhard"},{"given":"Todd A.","family":"Brun"}],"title":"Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code","issued":{"date-parts":[[2026,3,13]]},"note":"arxivid:2408.14828\narxiv_version_number:3"},{"id":"arxiv:2504.21172","type":"article-journal","author":[{"given":"Yuwei","family":"Jin"},{"given":"Zichang","family":"He"},{"given":"Tianyi","family":"Hao"},{"given":"Sivaprasad","family":"Omanakuttan"},{"given":"David","family":"Amaro"},{"given":"Swamit","family":"Tannu"},{"given":"Ruslan","family":"Shaydulin"},{"given":"Marco","family":"Pistoia"}],"title":"Iceberg Beyond the Tip: Co-Compilation of a Quantum Error Detection Code and a Quantum Algorithm","issued":{"date-parts":[[2026,4,29]]},"note":"arxivid:2504.21172\narxiv_version_number:2"},{"id":"arxiv:2602.22211","type":"article-journal","author":[{"given":"Shival","family":"Dasu"},{"given":"Matthew","family":"DeCross"},{"given":"Andrew Y.","family":"Guo"},{"given":"Ali","family":"Lavasani"},{"given":"Jan","family":"Behrends"},{"given":"Asmae","family":"Benhemou"},{"given":"Yi-Hsiang","family":"Chen"},{"given":"Karl","family":"Mayer"},{"given":"Chris N.","family":"Self"},{"given":"Selwyn","family":"Simsek"},{"given":"Basudha","family":"Srivastava"},{"given":"M. S.","family":"Allman"},{"given":"Jake","family":"Arkinstall"},{"given":"Justin G.","family":"Bohnet"},{"given":"Nathaniel Q.","family":"Burdick"},{"given":"J. P.","family":"Campora"},{"given":"Alex","family":"Chernoguzov"},{"given":"Samuel F.","family":"Cooper"},{"given":"Robert D.","family":"Delaney"},{"given":"Joan M.","family":"Dreiling"},{"given":"Brian","family":"Estey"},{"given":"Caroline","family":"Figgatt"},{"given":"Cameron","family":"Foltz"},{"given":"John P.","family":"Gaebler"},{"given":"Alex","family":"Hall"},{"given":"Craig A.","family":"Holliman"},{"given":"Ali A.","family":"Husain"},{"given":"Akhil","family":"Isanaka"},{"given":"Colin J.","family":"Kennedy"},{"given":"Yuga","family":"Kodama"},{"given":"Nikhil","family":"Kotibhaskar"},{"given":"Nathan K.","family":"Lysne"},{"given":"Ivaylo S.","family":"Madjarov"},{"given":"Michael","family":"Mills"},{"given":"Alistair R.","family":"Milne"},{"given":"Brian","family":"Neyenhuis"},{"given":"Annie J.","family":"Park"},{"given":"Anthony","family":"Ransford"},{"given":"Adam P.","family":"Reed"},{"given":"Steven J.","family":"Sanders"},{"given":"Charles H.","family":"Baldwin"},{"given":"David","family":"Hayes"},{"given":"Ben","family":"Criger"},{"given":"Andrew C.","family":"Potter"},{"given":"David","family":"Amaro"}],"title":"Computing with many encoded logical qubits beyond break-even","issued":{"date-parts":[[2026,2,25]]},"note":"arxivid:2602.22211\narxiv_version_number:1"},{"id":"arxiv:1701.01828","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"08","abstract":"<jats:p>Mean king’s problem is a kind of quantum state discrimination problems. In the problem, we try to discriminate eigenstates of noncommutative observables with the help of classical delayed information. The problem has been investigated from the viewpoint of error detection and correction. We construct higher-dimensional quantum error-correcting codes against error corresponding to the noncommutative observables. Any code state of the codes provides a way to discriminate the eigenstates correctly with the classical delayed information.</jats:p>","DOI":"10.1142/s0219749916500489","page":"1650048","source":"Crossref","title":"Solutions to the mean king’s problem: Higher-dimensional quantum error-correcting codes","volume":"14","author":[{"given":"Masakazu","family":"Yoshida","sequence":"first","affiliation":[{"name":"Faculty of Science and Engineering, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, Kyoto 610-0394, Japan"}]},{"given":"Toru","family":"Kuriyama","sequence":"additional","affiliation":[{"name":"Faculty of Science and Engineering, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, Kyoto 610-0394, Japan"}]},{"given":"Jun","family":"Cheng","sequence":"additional","affiliation":[{"name":"Faculty of Science and Engineering, Doshisha University, 1-3 Tataramiyakodani, Kyotanabe-shi, Kyoto 610-0394, Japan"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2016,12]]},"URL":"http://dx.doi.org/10.1142/S0219749916500489","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749916500489\narxivid:1701.01828"},{"id":"arxiv:2408.15936","type":"article-journal","author":[{"given":"Christopher A.","family":"Pattison"},{"given":"Gefen","family":"Baranes"},{"given":"J. Pablo Bonilla","family":"Ataides"},{"given":"Mikhail D.","family":"Lukin"},{"given":"Hengyun","family":"Zhou"}],"title":"Fast quantum interconnects via constant-rate entanglement distillation","issued":{"date-parts":[[2024,8,28]]},"note":"arxivid:2408.15936\narxiv_version_number:1"},{"id":"arxiv:2409.12104","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s42005-025-02136-8","source":"Crossref","title":"Performance of quantum approximate optimization with quantum error detection","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-1723-6568","authenticated-orcid":false,"given":"Zichang","family":"He","sequence":"first","affiliation":[]},{"given":"David","family":"Amaro","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8657-2848","authenticated-orcid":false,"given":"Ruslan","family":"Shaydulin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9002-1128","authenticated-orcid":false,"given":"Marco","family":"Pistoia","sequence":"additional","affiliation":[]}],"container-title":"Communications Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,24]]},"URL":"http://dx.doi.org/10.1038/s42005-025-02136-8","ISSN":["2399-3650"],"container-title-short":"Commun Phys","page":"217","note":"alternative-id:2136\narxivid:2409.12104"},{"id":"arxiv:2501.17447","type":"article-journal","author":[{"given":"Andrew","family":"Cross"},{"given":"Drew","family":"Vandeth"}],"title":"Small Binary Stabilizer Subsystem Codes","issued":{"date-parts":[[2025,1,29]]},"note":"arxivid:2501.17447\narxiv_version_number:1"},{"id":"arxiv:2412.06744","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In the absence of fault tolerant quantum error correction for analog, Hamiltonian quantum computation, error suppression via energy penalties is an effective alternative. We construct families of distance-<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn></mml:math> stabilizer subsystem codes we call “trapezoid codes'', that are tailored for energy-penalty schemes. We identify a family of codes achieving the maximum code rate, and by slightly relaxing this constraint, uncover a broader range of codes with enhanced physical locality, thus increasing their practical applicability. Additionally, we provide an algorithm to map the required qubit connectivity graph into graphs compatible with the locality constraints of quantum hardware. Finally, we provide a systematic framework to evaluate the performance of these codes in terms of code rate, physical locality, graph properties, and penalty gap, enabling an informed selection of error-suppression codes for specific quantum computing applications. We identify the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> family of subsystem codes as optimal in terms of code rate and penalty gap scaling.</jats:p>","DOI":"10.22331/q-2025-08-05-1821","page":"1821","source":"Crossref","title":"Families of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> 2D subsystem stabilizer codes for universal Hamiltonian quantum computation with two-body interactions","volume":"9","author":[{"ORCID":"https://orcid.org/0009-0006-3612-4293","authenticated-orcid":false,"given":"Phattharaporn","family":"Singkanipa","sequence":"first","affiliation":[{"name":"Center for Quantum Information Science & Technology"},{"name":"Department of Physics & Astronomy"}]},{"ORCID":"https://orcid.org/0000-0001-7575-9423","authenticated-orcid":false,"given":"Zihan","family":"Xia","sequence":"additional","affiliation":[{"name":"Center for Quantum Information Science & Technology"},{"name":"Department of Electrical & Computer Engineering"}]},{"ORCID":"https://orcid.org/0000-0002-1671-1515","authenticated-orcid":false,"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[{"name":"Center for Quantum Information Science & Technology"},{"name":"Department of Physics & Astronomy"},{"name":"Department of Electrical & Computer Engineering"},{"name":"Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,5]]},"URL":"http://dx.doi.org/10.22331/q-2025-08-05-1821","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2412.06744"},{"id":"doi:10.1016/j.disc.2007.12.007","type":"journal-article","publisher":"Elsevier BV","issue":"24","DOI":"10.1016/j.disc.2007.12.007","page":"6368-6372","source":"Crossref","title":"Quantum codes from caps","volume":"308","author":[{"given":"Vladimir D.","family":"Tonchev","sequence":"first","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2008,12]]},"URL":"http://dx.doi.org/10.1016/j.disc.2007.12.007","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X07009946"},{"id":"arxiv:0905.1059","type":"article-journal","author":[{"given":"Daniele","family":"Bartoli"},{"given":"Stefano","family":"Marcugini"},{"given":"Fernanda","family":"Pambianco"}],"title":"New quantum caps in PG(4,4)","issued":{"date-parts":[[2010,7,15]]},"note":"arxivid:0905.1059\narxiv_version_number:3"},{"id":"manual:-J.-Bierbrauer-D.-Bartoli-S.-","type":"article-journal","author":[{"family":"Bierbrauer","given":"J."},{"family":"Bartoli","given":"D."},{"family":"Marcugini","given":"S."},{"family":"Pambianco","given":"F."}],"title":"Geometric constructions of quantum codes","volume":"523","container-title":"Error-Correcting Codes, Finite Geometries and Cryptography","issued":"2010","page":"149–154"},{"id":"doi:10.1119/1.16243","type":"journal-article","publisher":"American Association of Physics Teachers (AAPT)","issue":"12","abstract":"<jats:p>It is demonstrated that the premisses of the Einstein–Podolsky–Rosen paper are inconsistent when applied to quantum systems consisting of at least three particles. The demonstration reveals that the EPR program contradicts quantum mechanics even for the cases of perfect correlations. By perfect correlations is meant arrangements by which the result of the measurement on one particle can be predicted with certainty given the outcomes of measurements on the other particles of the system. This incompatibility with quantum mechanics is stronger than the one previously revealed for two-particle systems by Bell’s inequality, where no contradiction arises at the level of perfect correlations. Both spin-correlation and multiparticle interferometry examples are given of suitable three- and four-particle arrangements, both at the gedanken and at the real experiment level.</jats:p>","DOI":"10.1119/1.16243","page":"1131-1143","source":"Crossref","title":"Bell’s theorem without inequalities","volume":"58","author":[{"given":"Daniel M.","family":"Greenberger","sequence":"first","affiliation":[{"name":"Department of Physics, City College of the City University of New York, New York, New York 10031"}]},{"given":"Michael A.","family":"Horne","sequence":"additional","affiliation":[{"name":"Department of Physics, Stonehill College, North Easton, Massachusetts 02357"}]},{"given":"Abner","family":"Shimony","sequence":"additional","affiliation":[{"name":"Departments of Philosophy and Physics, Boston University, Boston, Massachusetts 02215"}]},{"given":"Anton","family":"Zeilinger","sequence":"additional","affiliation":[{"name":"Atominstitut der Österreichischen Universitäten, Schüttelstrasse 115, A-1020 Vienna, Austria"}]}],"container-title":"American Journal of Physics","original-title":[],"language":"en","issued":{"date-parts":[[1990,12,1]]},"URL":"http://dx.doi.org/10.1119/1.16243","ISSN":["0002-9505","1943-2909"]},{"id":"doi:10.1119/1.16503","type":"journal-article","publisher":"American Association of Physics Teachers (AAPT)","issue":"8","abstract":"<jats:p>A gedanken gadget is described, based on an idea of Greenberger, Horne, and Zeilinger, that provides a more powerful demonstration of quantum nonlocality than Bell’s analysis of the Einstein–Podolsky–Rosen experiment.</jats:p>","DOI":"10.1119/1.16503","page":"731-734","source":"Crossref","title":"Quantum mysteries revisited","volume":"58","author":[{"given":"N. David","family":"Mermin","sequence":"first","affiliation":[{"name":"Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501"}]}],"container-title":"American Journal of Physics","original-title":[],"language":"en","issued":{"date-parts":[[1990,8,1]]},"URL":"http://dx.doi.org/10.1119/1.16503","ISSN":["0002-9505","1943-2909"]},{"id":"arxiv:0712.0921","type":"article-journal","author":[{"given":"Daniel M.","family":"Greenberger"},{"given":"Michael A.","family":"Horne"},{"given":"Anton","family":"Zeilinger"}],"title":"Going Beyond Bell's Theorem","issued":{"date-parts":[[2007,12,6]]},"note":"arxivid:0712.0921\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0006088","type":"article-journal","author":[{"given":"E.","family":"Knill"},{"given":"R.","family":"Laflamme"},{"given":"G.","family":"Milburn"}],"title":"Efficient Linear Optics Quantum Computation","issued":{"date-parts":[[2000,6,20]]},"note":"arxivid:quant-ph/0006088\narxiv_version_number:1"},{"id":"doi:10.1103/PhysRevA.32.3266","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.32.3266","page":"3266-3276","source":"Crossref","title":"Reversible logic and quantum computers","volume":"32","author":[{"given":"Asher","family":"Peres","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1985,12,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.32.3266","ISSN":["0556-2791"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:1205.1836","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.86.012333","source":"Crossref","title":"Simplified quantum error detection and correction for superconducting qubits","volume":"86","author":[{"given":"Kyle","family":"Keane","sequence":"first","affiliation":[]},{"given":"Alexander N.","family":"Korotkov","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,7,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.012333","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012333","note":"arxivid:1205.1836"},{"id":"arxiv:2303.17810","type":"journal-article","publisher":"IOP Publishing","issue":"10","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>In this theoretical study, we explore the use of quantum code-based memories to enhance the lifetime of qubits and exceed the break-even point, which is critical for the implementation of fault-tolerant quantum computing. Specifically, we investigate the quantum phase-flip repetition code as a quantum memory and theoretically demonstrate that it can preserve arbitrary quantum information longer than the lifetime of a single idle qubit in a dephasing-time-limited system, e.g. in semiconductor qubits. Our circuit-based analytical calculations show the efficiency of the phase-flip code as a quantum memory in the presence of relaxation, dephasing, and faulty quantum gates. Moreover, we identify the optimal repetition number of quantum error correction cycles required to reach the break-even point by considering the gate error probabilities of current platforms for quantum computing. Our results provide guidelines for developing quantum memories in semiconductor quantum devices.</jats:p>","DOI":"10.1088/1367-2630/acfba5","page":"103004","source":"Crossref","title":"Break-even point of the phase-flip error correcting code","volume":"25","author":[{"given":"Áron","family":"Rozgonyi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4178-5759","authenticated-orcid":true,"given":"Gábor","family":"Széchenyi","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2023,10,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/acfba5","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:2303.17810"},{"id":"arxiv:quant-ph/0408064","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.71.060303","source":"Crossref","title":"High-fidelity<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>Z</mml:mi></mml:math>-measurement error encoding of optical qubits","volume":"71","author":[{"given":"J. L.","family":"O’Brien","sequence":"first","affiliation":[]},{"given":"G. J.","family":"Pryde","sequence":"additional","affiliation":[]},{"given":"A. G.","family":"White","sequence":"additional","affiliation":[]},{"given":"T. C.","family":"Ralph","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,6,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.060303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"060303","note":"arxivid:quant-ph/0408064"},{"id":"arxiv:0709.4539","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.76.062323","source":"Crossref","title":"Distributed quantum computation based on small quantum registers","volume":"76","author":[{"given":"Liang","family":"Jiang","sequence":"first","affiliation":[]},{"given":"Jacob M.","family":"Taylor","sequence":"additional","affiliation":[]},{"given":"Anders S.","family":"Sørensen","sequence":"additional","affiliation":[]},{"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,12,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.062323","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062323","note":"arxivid:0709.4539"},{"id":"arxiv:1807.05572","type":"journal-article","publisher":"Wiley","issue":"5-6","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    Efficient deterministic algorithms are proposed with logarithmic step complexities for the generation of entangled  and  states useful for quantum networks, and an implementation on the IBM quantum computer up to  is demonstrated. Improved quality is then investigated using full quantum tomography for low‐\n                    <jats:italic>N</jats:italic>\n                    GHZ and\n                    <jats:italic>W</jats:italic>\n                    states. This is completed by parity oscillations and histogram distance for large‐\n                    <jats:italic>N</jats:italic>\n                    GHZ and\n                    <jats:italic>W</jats:italic>\n                    states, respectively. Robust states are built with about twice the number of quantum bits which were previously achieved.\n                  </jats:p>","DOI":"10.1002/qute.201900015","source":"Crossref","title":"Efficient Quantum Algorithms for GHZ and\n                    <i>W</i>\n                    States, and Implementation on the IBM Quantum Computer","volume":"2","author":[{"given":"Diogo","family":"Cruz","sequence":"first","affiliation":[{"name":"Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"given":"Romain","family":"Fournier","sequence":"additional","affiliation":[{"name":"Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"given":"Fabien","family":"Gremion","sequence":"additional","affiliation":[{"name":"Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"given":"Alix","family":"Jeannerot","sequence":"additional","affiliation":[{"name":"Communication Theory Laboratory School of Computer and Communication Sciences École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"given":"Kenichi","family":"Komagata","sequence":"additional","affiliation":[{"name":"Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"given":"Tara","family":"Tosic","sequence":"additional","affiliation":[{"name":"Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"given":"Jarla","family":"Thiesbrummel","sequence":"additional","affiliation":[{"name":"Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"given":"Chun Lam","family":"Chan","sequence":"additional","affiliation":[{"name":"Communication Theory Laboratory School of Computer and Communication Sciences École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"given":"Nicolas","family":"Macris","sequence":"additional","affiliation":[{"name":"Communication Theory Laboratory School of Computer and Communication Sciences École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"ORCID":"https://orcid.org/0000-0003-1782-9600","authenticated-orcid":false,"given":"Marc‐André","family":"Dupertuis","sequence":"additional","affiliation":[{"name":"Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"}]},{"ORCID":"https://orcid.org/0000-0002-6816-1391","authenticated-orcid":false,"given":"Clément","family":"Javerzac‐Galy","sequence":"additional","affiliation":[{"name":"Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Lausanne CH 1015 Switzerland"},{"name":"Miraex EPFL Innovation Park Bâtiment C Laussane CH 1015 Switzerland"}]}],"container-title":"Advanced Quantum Technologies","original-title":[],"language":"en","issued":{"date-parts":[[2019,4,23]]},"URL":"http://dx.doi.org/10.1002/qute.201900015","ISSN":["2511-9044","2511-9044"],"container-title-short":"Adv Quantum Tech","page":"1900015","note":"alternative-id:10.1002/qute.201900015\narxivid:1807.05572"},{"id":"arxiv:1910.08980","type":"journal-article","publisher":"American Physical Society (APS)","issue":"26","DOI":"10.1103/physrevlett.125.260505","source":"Crossref","title":"Obstacles to Variational Quantum Optimization from Symmetry Protection","volume":"125","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7973-9665","authenticated-orcid":true,"given":"Alexander","family":"Kliesch","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Koenig","sequence":"additional","affiliation":[]},{"given":"Eugene","family":"Tang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.125.260505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"260505","note":"arxivid:1910.08980"},{"id":"arxiv:1702.08673","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum-enhanced measurements hold the promise to improve high-precision sensing ranging from the definition of time standards to the determination of fundamental constants of nature. However, quantum sensors lose their sensitivity in the presence of noise. To protect them, the use of quantum error-correcting codes has been proposed. Trapped ions are an excellent technological platform for both quantum sensing and quantum error correction. Here we present a quantum error correction scheme that harnesses dissipation to stabilize a trapped-ion qubit. In our approach, always-on couplings to an engineered environment protect the qubit against spin-flips or phase-flips. Our dissipative error correction scheme operates in a continuous manner without the need to perform measurements or feedback operations. We show that the resulting enhanced coherence time translates into a significantly enhanced precision for quantum measurements. Our work constitutes a stepping stone towards the paradigm of self-correcting quantum information processing.</jats:p>","DOI":"10.1038/s41467-017-01895-5","source":"Crossref","title":"Dissipative quantum error correction and application to quantum sensing with trapped ions","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-5217-3064","authenticated-orcid":false,"given":"F.","family":"Reiter","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1337-9163","authenticated-orcid":false,"given":"A. S.","family":"Sørensen","sequence":"additional","affiliation":[]},{"given":"P.","family":"Zoller","sequence":"additional","affiliation":[]},{"given":"C. A.","family":"Muschik","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2017,11,28]]},"URL":"http://dx.doi.org/10.1038/s41467-017-01895-5","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"1822","note":"alternative-id:1895\narxivid:1702.08673"},{"id":"arxiv:2210.14143","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Recent constructions of quantum low-density parity-check (QLDPC) codes provide optimal scaling of the number of logical qubits and the minimum distance in terms of the code length, thereby opening the door to fault-tolerant quantum systems with minimal resource overhead. However, the hardware path from nearest-neighbor-connection-based topological codes to long-range-interaction-demanding QLDPC codes is likely a challenging one. Given the practical difficulty in building a monolithic architecture for quantum systems, such as computers, based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected medium-sized quantum processors. In such a setting, all syndrome measurements and logical operations must be performed through the use of high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. This protocol is applicable beyond the application of DQC since entanglement distribution and purification is a quintessential task of any quantum network. We use the min-sum algorithm (MSA) based iterative decoder with a sequential schedule for distilling <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>3</mml:mn></mml:math>-qubit GHZ states using a rate <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.118</mml:mn></mml:math> family of lifted product QLDPC codes and obtain an input fidelity threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.7974</mml:mn></mml:math> under i.i.d. single-qubit depolarizing noise. This represents the best threshold for a yield of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.118</mml:mn></mml:math> for any GHZ purification protocol. Our results apply to larger size GHZ states as well, where we extend our technical result about a measurement property of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>3</mml:mn></mml:math>-qubit GHZ states to construct a scalable GHZ purification protocol. </jats:p>","DOI":"10.22331/q-2024-01-24-1233","page":"1233","source":"Crossref","title":"Entanglement Purification with Quantum LDPC Codes and Iterative Decoding","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"https://orcid.org/0000-0002-1024-8099","authenticated-orcid":false,"given":"Nithin","family":"Raveendran","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]},{"ORCID":"https://orcid.org/0000-0002-9022-3595","authenticated-orcid":false,"given":"Ankur","family":"Raina","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Sciences, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462066, India"}]},{"ORCID":"https://orcid.org/0000-0003-2365-4106","authenticated-orcid":false,"given":"Bane","family":"Vasić","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,24]]},"URL":"http://dx.doi.org/10.22331/q-2024-01-24-1233","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2210.14143"},{"id":"arxiv:2406.10336","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.134.130604","source":"Crossref","title":"Fast and Accurate Greenberger-Horne-Zeilinger Encoding Using All-to-All Interactions","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0003-3379-310X","authenticated-orcid":true,"given":"Chao","family":"Yin","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,4,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.130604","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130604","note":"arxivid:2406.10336"},{"id":"arxiv:quant-ph/0412168","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.72.012320","source":"Crossref","title":"Microscopic quantum dynamics study on the noise threshold of fault-tolerant quantum error correction","volume":"72","author":[{"given":"Y. C.","family":"Cheng","sequence":"first","affiliation":[]},{"given":"R. J.","family":"Silbey","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,7,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.72.012320","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012320","note":"arxivid:quant-ph/0412168"},{"id":"arxiv:quant-ph/0501049","type":"proceedings-article","publisher":"SPIE","DOI":"10.1117/12.608548","page":"158","source":"Crossref","title":"Continuous quantum error correction","volume":"5842","author":[{"given":"Mohan","family":"Sarovar","sequence":"first","affiliation":[]},{"given":"G. J.","family":"Milburn","sequence":"additional","affiliation":[]}],"event":"SPIE Third International Symposium on Fluctuations and Noise","container-title":"SPIE Proceedings","original-title":[],"editor":[{"given":"Philip R.","family":"Hemmer","sequence":"first","affiliation":[]},{"given":"Julio R.","family":"Gea-Banacloche","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Heszler, Sr.","sequence":"additional","affiliation":[]},{"given":"M. Suhail","family":"Zubairy","sequence":"additional","affiliation":[]}],"issued":{"date-parts":[[2005,5,23]]},"URL":"http://dx.doi.org/10.1117/12.608548","ISSN":["0277-786X"],"note":"arxivid:quant-ph/0501049"},{"id":"arxiv:2110.10378","type":"journal-article","publisher":"IOP Publishing","issue":"6","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Continuous quantum error correction has been found to have certain advantages over discrete quantum error correction, such as a reduction in hardware resources and the elimination of error mechanisms introduced by having entangling gates and ancilla qubits. We propose a machine learning algorithm for continuous quantum error correction that is based on the use of a recurrent neural network to identify bit-flip errors from continuous noisy syndrome measurements. The algorithm is designed to operate on measurement signals deviating from the ideal behavior in which the mean value corresponds to a code syndrome value and the measurement has white noise. We analyze continuous measurements taken from a superconducting architecture using three transmon qubits to identify three significant practical examples of non-ideal behavior, namely auto-correlation at temporal short lags, transient syndrome dynamics after each bit-flip, and drift in the steady-state syndrome values over the course of many experiments. Based on these real-world imperfections, we generate synthetic measurement signals from which to train the recurrent neural network, and then test its proficiency when implementing active error correction, comparing this with a traditional double threshold scheme and a discrete Bayesian classifier. The results show that our machine learning protocol is able to outperform the double threshold protocol across all tests, achieving a final state fidelity comparable to the discrete Bayesian classifier.</jats:p>","DOI":"10.1088/1367-2630/ac66f9","page":"063019","source":"Crossref","title":"Machine learning for continuous quantum error correction on superconducting qubits","volume":"24","author":[{"ORCID":"https://orcid.org/0000-0003-1818-2677","authenticated-orcid":true,"given":"Ian","family":"Convy","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6399-006X","authenticated-orcid":true,"given":"Haoran","family":"Liao","sequence":"additional","affiliation":[]},{"given":"Song","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Sahil","family":"Patel","sequence":"additional","affiliation":[]},{"given":"William P","family":"Livingston","sequence":"additional","affiliation":[]},{"given":"Ho Nam","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"Irfan","family":"Siddiqi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7164-4757","authenticated-orcid":true,"given":"K Birgitta","family":"Whaley","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2022,6,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ac66f9","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:2110.10378"},{"id":"arxiv:2309.03608","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevlett.133.150601","source":"Crossref","title":"Quantum Cellular Automata for Quantum Error Correction and Density Classification","volume":"133","author":[{"ORCID":"https://orcid.org/0000-0003-3281-145X","authenticated-orcid":true,"given":"T. L. M.","family":"Guedes","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0009-0005-5504-9690","authenticated-orcid":true,"given":"D.","family":"Winter","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":true,"given":"M.","family":"Müller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.150601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"150601","note":"arxivid:2309.03608"},{"id":"arxiv:2412.19803","type":"article-journal","author":[{"given":"Shankar","family":"Balasubramanian"},{"given":"Margarita","family":"Davydova"},{"given":"Ethan","family":"Lake"}],"title":"A local automaton for the 2D toric code","issued":{"date-parts":[[2025,6,3]]},"note":"arxivid:2412.19803\narxiv_version_number:2"},{"id":"arxiv:2501.03582","type":"article-journal","author":[{"given":"Hanyan","family":"Cao"},{"given":"Shoukuan","family":"Zhao"},{"given":"Dongyang","family":"Feng"},{"given":"Zisong","family":"Shen"},{"given":"Haisheng","family":"Yan"},{"given":"Tang","family":"Su"},{"given":"Weijie","family":"Sun"},{"given":"Huikai","family":"Xu"},{"given":"Feng","family":"Pan"},{"given":"Haifeng","family":"Yu"},{"given":"Pan","family":"Zhang"}],"title":"Exact Decoding of Repetition Code under Circuit Level Noise","issued":{"date-parts":[[2025,1,7]]},"note":"arxivid:2501.03582\narxiv_version_number:1"},{"id":"arxiv:2505.10162","type":"article-journal","author":[{"given":"Louis","family":"Paletta"},{"given":"Anthony","family":"Leverrier"},{"given":"Mazyar","family":"Mirrahimi"},{"given":"Christophe","family":"Vuillot"}],"title":"High-performance local decoders for defect matching in 1D","issued":{"date-parts":[[2025,9,16]]},"note":"arxivid:2505.10162\narxiv_version_number:3"},{"id":"arxiv:1708.09286","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"4","DOI":"10.1109/tit.2018.2879937","page":"2545-2562","source":"Crossref","title":"Renormalization Group Decoder for a Four-Dimensional Toric Code","volume":"65","author":[{"ORCID":"https://orcid.org/0000-0002-3388-3175","authenticated-orcid":false,"given":"K.","family":"Duivenvoorden","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7211-5515","authenticated-orcid":false,"given":"N. P.","family":"Breuckmann","sequence":"additional","affiliation":[]},{"given":"B. M.","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2019,4]]},"URL":"http://dx.doi.org/10.1109/TIT.2018.2879937","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1708.09286"},{"id":"arxiv:quant-ph/9811068","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.60.1924","page":"1924-1943","source":"Crossref","title":"Experimental realization of a two-bit phase damping quantum code","volume":"60","author":[{"given":"Debbie","family":"Leung","sequence":"first","affiliation":[]},{"given":"Lieven","family":"Vandersypen","sequence":"additional","affiliation":[]},{"given":"Xinlan","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Mark","family":"Sherwood","sequence":"additional","affiliation":[]},{"given":"Constantino","family":"Yannoni","sequence":"additional","affiliation":[]},{"given":"Mark","family":"Kubinec","sequence":"additional","affiliation":[]},{"given":"Isaac","family":"Chuang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1999,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.60.1924","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9811068"},{"id":"arxiv:cs/0012017","type":"article-journal","author":[{"given":"Debbie W.","family":"Leung"}],"title":"Towards Robust Quantum Computation","issued":{"date-parts":[[2000,12,20]]},"note":"arxivid:cs/0012017\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0004030","type":"journal-article","publisher":"Informa UK Limited","issue":"17","DOI":"10.1080/002689700413604","page":"1347-1363","source":"Crossref","title":"A study of quantum error correction by geometric algebra and liquid-state NMR spectroscopy","volume":"98","author":[{"given":"Y.","family":"Sharf","sequence":"first","affiliation":[]},{"given":"D. G.","family":"Cory","sequence":"additional","affiliation":[]},{"given":"S. S.","family":"Somaroo","sequence":"additional","affiliation":[]},{"given":"T. F.","family":"Havel","sequence":"additional","affiliation":[]},{"given":"E.","family":"Knill","sequence":"additional","affiliation":[]},{"given":"R.","family":"Laflamme","sequence":"additional","affiliation":[]},{"given":"W. H.","family":"Zurek","sequence":"additional","affiliation":[]}],"container-title":"Molecular Physics","original-title":[],"language":"en","issued":{"date-parts":[[2000,9]]},"URL":"http://dx.doi.org/10.1080/002689700413604","ISSN":["0026-8976","1362-3028"],"container-title-short":"Molecular Physics","note":"alternative-id:10.1080/002689700413604\narxivid:quant-ph/0004030"},{"id":"arxiv:quant-ph/9802018","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevlett.81.2152","page":"2152-2155","source":"Crossref","title":"Experimental Quantum Error Correction","volume":"81","author":[{"given":"D. G.","family":"Cory","sequence":"first","affiliation":[]},{"given":"M. D.","family":"Price","sequence":"additional","affiliation":[]},{"given":"W.","family":"Maas","sequence":"additional","affiliation":[]},{"given":"E.","family":"Knill","sequence":"additional","affiliation":[]},{"given":"R.","family":"Laflamme","sequence":"additional","affiliation":[]},{"given":"W. H.","family":"Zurek","sequence":"additional","affiliation":[]},{"given":"T. F.","family":"Havel","sequence":"additional","affiliation":[]},{"given":"S. S.","family":"Somaroo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1998,9,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.81.2152","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9802018"},{"id":"arxiv:1108.4842","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevlett.107.160501","source":"Crossref","title":"Demonstration of Sufficient Control for Two Rounds of Quantum Error Correction in a Solid State Ensemble Quantum Information Processor","volume":"107","author":[{"given":"Osama","family":"Moussa","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Baugh","sequence":"additional","affiliation":[]},{"given":"Colm A.","family":"Ryan","sequence":"additional","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2011,10,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.107.160501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"160501","note":"arxivid:1108.4842"},{"id":"arxiv:1109.4821","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.84.034303","source":"Crossref","title":"Experimental quantum error correction with high fidelity","volume":"84","author":[{"given":"Jingfu","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Dorian","family":"Gangloff","sequence":"additional","affiliation":[]},{"given":"Osama","family":"Moussa","sequence":"additional","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,9,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.034303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"034303","note":"arxivid:1109.4821"},{"id":"arxiv:quant-ph/0610038","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.75.012331","source":"Crossref","title":"Using error correction to determine the noise model","volume":"75","author":[{"given":"Martin","family":"Laforest","sequence":"first","affiliation":[]},{"given":"Damien","family":"Simon","sequence":"additional","affiliation":[]},{"given":"Jean-Christian","family":"Boileau","sequence":"additional","affiliation":[]},{"given":"Jonathan","family":"Baugh","sequence":"additional","affiliation":[]},{"given":"Michael J.","family":"Ditty","sequence":"additional","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,1,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.012331","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012331","note":"arxivid:quant-ph/0610038"},{"id":"arxiv:quant-ph/0502042","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.71.052332","source":"Crossref","title":"Demonstration of quantum error correction using linear optics","volume":"71","author":[{"given":"T. B.","family":"Pittman","sequence":"first","affiliation":[]},{"given":"B. C.","family":"Jacobs","sequence":"additional","affiliation":[]},{"given":"J. D.","family":"Franson","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,5,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.052332","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052332","note":"arxivid:quant-ph/0502042"},{"id":"doi:10.1038/nature03074","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7017","DOI":"10.1038/nature03074","page":"602-605","source":"Crossref","title":"Realization of quantum error correction","volume":"432","author":[{"given":"J.","family":"Chiaverini","sequence":"first","affiliation":[]},{"given":"D.","family":"Leibfried","sequence":"additional","affiliation":[]},{"given":"T.","family":"Schaetz","sequence":"additional","affiliation":[]},{"given":"M. D.","family":"Barrett","sequence":"additional","affiliation":[]},{"given":"R. B.","family":"Blakestad","sequence":"additional","affiliation":[]},{"given":"J.","family":"Britton","sequence":"additional","affiliation":[]},{"given":"W. M.","family":"Itano","sequence":"additional","affiliation":[]},{"given":"J. D.","family":"Jost","sequence":"additional","affiliation":[]},{"given":"E.","family":"Knill","sequence":"additional","affiliation":[]},{"given":"C.","family":"Langer","sequence":"additional","affiliation":[]},{"given":"R.","family":"Ozeri","sequence":"additional","affiliation":[]},{"given":"D. J.","family":"Wineland","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2004,12]]},"URL":"http://dx.doi.org/10.1038/nature03074","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature03074"},{"id":"doi:10.1126/science.1203329","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6033","abstract":"<jats:p>An error correction algorithm is applied multiple times to a small quantum system.</jats:p>","DOI":"10.1126/science.1203329","page":"1059-1061","source":"Crossref","title":"Experimental Repetitive Quantum Error Correction","volume":"332","author":[{"given":"Philipp","family":"Schindler","sequence":"first","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria."}]},{"given":"Julio T.","family":"Barreiro","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria."}]},{"given":"Thomas","family":"Monz","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria."}]},{"given":"Volckmar","family":"Nebendahl","sequence":"additional","affiliation":[{"name":"Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria."}]},{"given":"Daniel","family":"Nigg","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria."}]},{"given":"Michael","family":"Chwalla","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria."},{"name":"Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria."}]},{"given":"Markus","family":"Hennrich","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria."}]},{"given":"Rainer","family":"Blatt","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria."},{"name":"Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2011,5,27]]},"URL":"http://dx.doi.org/10.1126/science.1203329","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.1203329"},{"id":"arxiv:2305.03828","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.13.041052","source":"Crossref","title":"A Race-Track Trapped-Ion Quantum Processor","volume":"13","author":[{"ORCID":"https://orcid.org/0009-0000-3162-4615","authenticated-orcid":true,"given":"S. A.","family":"Moses","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3938-7428","authenticated-orcid":true,"given":"C. H.","family":"Baldwin","sequence":"additional","affiliation":[]},{"given":"M. S.","family":"Allman","sequence":"additional","affiliation":[]},{"given":"R.","family":"Ancona","sequence":"additional","affiliation":[]},{"given":"L.","family":"Ascarrunz","sequence":"additional","affiliation":[]},{"given":"C.","family":"Barnes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9791-2345","authenticated-orcid":true,"given":"J.","family":"Bartolotta","sequence":"additional","affiliation":[]},{"given":"B.","family":"Bjork","sequence":"additional","affiliation":[]},{"given":"P.","family":"Blanchard","sequence":"additional","affiliation":[]},{"given":"M.","family":"Bohn","sequence":"additional","affiliation":[]},{"given":"J. G.","family":"Bohnet","sequence":"additional","affiliation":[]},{"given":"N. C.","family":"Brown","sequence":"additional","affiliation":[]},{"given":"N. Q.","family":"Burdick","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5389-5632","authenticated-orcid":true,"given":"W. C.","family":"Burton","sequence":"additional","affiliation":[]},{"given":"S. L.","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"J. P.","family":"Campora","sequence":"additional","affiliation":[]},{"given":"C.","family":"Carron","sequence":"additional","affiliation":[]},{"given":"J.","family":"Chambers","sequence":"additional","affiliation":[]},{"given":"J. W.","family":"Chan","sequence":"additional","affiliation":[]},{"given":"Y. H.","family":"Chen","sequence":"additional","affiliation":[]},{"given":"A.","family":"Chernoguzov","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5957-1356","authenticated-orcid":true,"given":"E.","family":"Chertkov","sequence":"additional","affiliation":[]},{"given":"J.","family":"Colina","sequence":"additional","affiliation":[]},{"given":"J. P.","family":"Curtis","sequence":"additional","affiliation":[]},{"given":"R.","family":"Daniel","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9459-8140","authenticated-orcid":true,"given":"M.","family":"DeCross","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7831-6633","authenticated-orcid":true,"given":"D.","family":"Deen","sequence":"additional","affiliation":[]},{"given":"C.","family":"Delaney","sequence":"additional","affiliation":[]},{"given":"J. M.","family":"Dreiling","sequence":"additional","affiliation":[]},{"given":"C. T.","family":"Ertsgaard","sequence":"additional","affiliation":[]},{"given":"J.","family":"Esposito","sequence":"additional","affiliation":[]},{"given":"B.","family":"Estey","sequence":"additional","affiliation":[]},{"given":"M.","family":"Fabrikant","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8460-4175","authenticated-orcid":true,"given":"C.","family":"Figgatt","sequence":"additional","affiliation":[]},{"given":"C.","family":"Foltz","sequence":"additional","affiliation":[]},{"given":"M.","family":"Foss-Feig","sequence":"additional","affiliation":[]},{"given":"D.","family":"Francois","sequence":"additional","affiliation":[]},{"given":"J. P.","family":"Gaebler","sequence":"additional","affiliation":[]},{"given":"T. M.","family":"Gatterman","sequence":"additional","affiliation":[]},{"given":"C. N.","family":"Gilbreth","sequence":"additional","affiliation":[]},{"given":"J.","family":"Giles","sequence":"additional","affiliation":[]},{"given":"E.","family":"Glynn","sequence":"additional","affiliation":[]},{"given":"A.","family":"Hall","sequence":"additional","affiliation":[]},{"given":"A. M.","family":"Hankin","sequence":"additional","affiliation":[]},{"given":"A.","family":"Hansen","sequence":"additional","affiliation":[]},{"given":"D.","family":"Hayes","sequence":"additional","affiliation":[]},{"given":"B.","family":"Higashi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6671-3482","authenticated-orcid":true,"given":"I. M.","family":"Hoffman","sequence":"additional","affiliation":[]},{"given":"B.","family":"Horning","sequence":"additional","affiliation":[]},{"given":"J. J.","family":"Hout","sequence":"additional","affiliation":[]},{"given":"R.","family":"Jacobs","sequence":"additional","affiliation":[]},{"given":"J.","family":"Johansen","sequence":"additional","affiliation":[]},{"given":"L.","family":"Jones","sequence":"additional","affiliation":[]},{"given":"J.","family":"Karcz","sequence":"additional","affiliation":[]},{"given":"T.","family":"Klein","sequence":"additional","affiliation":[]},{"given":"P.","family":"Lauria","sequence":"additional","affiliation":[]},{"given":"P.","family":"Lee","sequence":"additional","affiliation":[]},{"given":"D.","family":"Liefer","sequence":"additional","affiliation":[]},{"given":"S. T.","family":"Lu","sequence":"additional","affiliation":[]},{"given":"D.","family":"Lucchetti","sequence":"additional","affiliation":[]},{"given":"C.","family":"Lytle","sequence":"additional","affiliation":[]},{"given":"A.","family":"Malm","sequence":"additional","affiliation":[]},{"given":"M.","family":"Matheny","sequence":"additional","affiliation":[]},{"given":"B.","family":"Mathewson","sequence":"additional","affiliation":[]},{"given":"K.","family":"Mayer","sequence":"additional","affiliation":[]},{"given":"D. B.","family":"Miller","sequence":"additional","affiliation":[]},{"given":"M.","family":"Mills","sequence":"additional","affiliation":[]},{"given":"B.","family":"Neyenhuis","sequence":"additional","affiliation":[]},{"given":"L.","family":"Nugent","sequence":"additional","affiliation":[]},{"given":"S.","family":"Olson","sequence":"additional","affiliation":[]},{"given":"J.","family":"Parks","sequence":"additional","affiliation":[]},{"given":"G. N.","family":"Price","sequence":"additional","affiliation":[]},{"given":"Z.","family":"Price","sequence":"additional","affiliation":[]},{"given":"M.","family":"Pugh","sequence":"additional","affiliation":[]},{"given":"A.","family":"Ransford","sequence":"additional","affiliation":[]},{"given":"A. P.","family":"Reed","sequence":"additional","affiliation":[]},{"given":"C.","family":"Roman","sequence":"additional","affiliation":[]},{"given":"M.","family":"Rowe","sequence":"additional","affiliation":[]},{"given":"C.","family":"Ryan-Anderson","sequence":"additional","affiliation":[]},{"given":"S.","family":"Sanders","sequence":"additional","affiliation":[]},{"given":"J.","family":"Sedlacek","sequence":"additional","affiliation":[]},{"given":"P.","family":"Shevchuk","sequence":"additional","affiliation":[]},{"given":"P.","family":"Siegfried","sequence":"additional","affiliation":[]},{"given":"T.","family":"Skripka","sequence":"additional","affiliation":[]},{"given":"B.","family":"Spaun","sequence":"additional","affiliation":[]},{"given":"R. T.","family":"Sprenkle","sequence":"additional","affiliation":[]},{"given":"R. P.","family":"Stutz","sequence":"additional","affiliation":[]},{"given":"M.","family":"Swallows","sequence":"additional","affiliation":[]},{"given":"R. I.","family":"Tobey","sequence":"additional","affiliation":[]},{"given":"A.","family":"Tran","sequence":"additional","affiliation":[]},{"given":"T.","family":"Tran","sequence":"additional","affiliation":[]},{"given":"E.","family":"Vogt","sequence":"additional","affiliation":[]},{"given":"C.","family":"Volin","sequence":"additional","affiliation":[]},{"given":"J.","family":"Walker","sequence":"additional","affiliation":[]},{"given":"A. M.","family":"Zolot","sequence":"additional","affiliation":[]},{"given":"J. M.","family":"Pino","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.13.041052","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041052","note":"arxivid:2305.03828"},{"id":"arxiv:2312.09745","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Encoding information redundantly using quantum error-correcting (QEC) codes allows one to overcome the inherent sensitivity to noise in quantum computers to ultimately achieve large-scale quantum computation. The Steane QEC method involves preparing an auxiliary logical qubit of the same QEC code as used for the data register. The data and auxiliary registers are then coupled with a logical controlled- () gate, enabling a measurement of the auxiliary register to reveal the error syndrome. This study presents the implementation of multiple rounds of fault-tolerant (FT) Steane QEC on a trapped-ion quantum computer. Various QEC codes are employed and the results are compared to a previous experimental approach utilizing flag qubits. Our experimental findings show improved logical fidelities for Steane QEC and accompanying numerical simulations indicate an even larger performance advantage for quantum processors limited by entangling-gate errors. This establishes experimental Steane QEC as a competitive paradigm for FT quantum computing.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.030326","source":"Crossref","title":"Demonstration of Fault-Tolerant Steane Quantum Error Correction","volume":"5","author":[{"given":"Lukas","family":"Postler","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"Universität Innsbruck"}]},{"ORCID":"https://orcid.org/0009-0007-8954-1274","authenticated-orcid":true,"given":"Friederike","family":"Butt","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"Rheinisch-Westfälische Technische Hochschule (RWTH) Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0009-0001-5103-9410","authenticated-orcid":true,"given":"Ivan","family":"Pogorelov","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"Universität Innsbruck"}]},{"ORCID":"https://orcid.org/0000-0001-8401-3981","authenticated-orcid":true,"given":"Christian D.","family":"Marciniak","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"Universität Innsbruck"}]},{"ORCID":"https://orcid.org/0000-0002-7581-2148","authenticated-orcid":true,"given":"Sascha","family":"Heußen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"Rheinisch-Westfälische Technische Hochschule (RWTH) Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0002-3122-2461","authenticated-orcid":true,"given":"Rainer","family":"Blatt","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"Universität Innsbruck"},{"id":[{"id":"https://ror.org/03xxp1119","id-type":"ROR","asserted-by":"publisher"}],"name":"Alpine Quantum Technologies GmbH"},{"id":[{"id":"https://ror.org/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institut für Quantenoptik und Quanteninformation"}]},{"ORCID":"https://orcid.org/0000-0002-9461-9650","authenticated-orcid":true,"given":"Philipp","family":"Schindler","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"Universität Innsbruck"}]},{"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"Rheinisch-Westfälische Technische Hochschule (RWTH) Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":true,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"Rheinisch-Westfälische Technische Hochschule (RWTH) Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0001-7410-4804","authenticated-orcid":true,"given":"Thomas","family":"Monz","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"Universität Innsbruck"},{"id":[{"id":"https://ror.org/03xxp1119","id-type":"ROR","asserted-by":"publisher"}],"name":"Alpine Quantum Technologies GmbH"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,7]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.030326","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030326","note":"arxivid:2312.09745"},{"id":"arxiv:1004.4324","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7315","DOI":"10.1038/nature09416","page":"574-578","source":"Crossref","title":"Preparation and measurement of three-qubit entanglement in a superconducting circuit","volume":"467","author":[{"given":"L.","family":"DiCarlo","sequence":"first","affiliation":[]},{"given":"M. D.","family":"Reed","sequence":"additional","affiliation":[]},{"given":"L.","family":"Sun","sequence":"additional","affiliation":[]},{"given":"B. R.","family":"Johnson","sequence":"additional","affiliation":[]},{"given":"J. M.","family":"Chow","sequence":"additional","affiliation":[]},{"given":"J. M.","family":"Gambetta","sequence":"additional","affiliation":[]},{"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[]},{"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"given":"M. H.","family":"Devoret","sequence":"additional","affiliation":[]},{"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1038/nature09416","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature09416\narxivid:1004.4324"},{"id":"arxiv:1109.4948","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7385","DOI":"10.1038/nature10786","page":"382-385","source":"Crossref","title":"Realization of three-qubit quantum error correction with superconducting circuits","volume":"482","author":[{"given":"M. D.","family":"Reed","sequence":"first","affiliation":[]},{"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]},{"given":"S. E.","family":"Nigg","sequence":"additional","affiliation":[]},{"given":"L.","family":"Sun","sequence":"additional","affiliation":[]},{"given":"L.","family":"Frunzio","sequence":"additional","affiliation":[]},{"given":"S. M.","family":"Girvin","sequence":"additional","affiliation":[]},{"given":"R. J.","family":"Schoelkopf","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2012,2]]},"URL":"http://dx.doi.org/10.1038/nature10786","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature10786\narxivid:1109.4948"},{"id":"arxiv:1411.5542","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum data are susceptible to decoherence induced by the environment and to errors in the hardware processing it. A future fault-tolerant quantum computer will use quantum error correction to actively protect against both. In the smallest error correction codes, the information in one logical qubit is encoded in a two-dimensional subspace of a larger Hilbert space of multiple physical qubits. For each code, a set of non-demolition multi-qubit measurements, termed stabilizers, can discretize and signal physical qubit errors without collapsing the encoded information. Here using a five-qubit superconducting processor, we realize the two parity measurements comprising the stabilizers of the three-qubit repetition code protecting one logical qubit from physical bit-flip errors. While increased physical qubit coherence times and shorter quantum error correction blocks are required to actively safeguard the quantum information, this demonstration is a critical step towards larger codes based on multiple parity measurements.</jats:p>","DOI":"10.1038/ncomms7983","source":"Crossref","title":"Detecting bit-flip errors in a logical qubit using stabilizer measurements","volume":"6","author":[{"given":"D.","family":"Ristè","sequence":"first","affiliation":[]},{"given":"S.","family":"Poletto","sequence":"additional","affiliation":[]},{"given":"M.-Z.","family":"Huang","sequence":"additional","affiliation":[]},{"given":"A.","family":"Bruno","sequence":"additional","affiliation":[]},{"given":"V.","family":"Vesterinen","sequence":"additional","affiliation":[]},{"given":"O.-P.","family":"Saira","sequence":"additional","affiliation":[]},{"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,29]]},"URL":"http://dx.doi.org/10.1038/ncomms7983","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"6983","note":"alternative-id:BFncomms7983\narxivid:1411.5542"},{"id":"arxiv:1508.01388","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Reliable quantum information processing in the face of errors is a major fundamental and technological challenge. Quantum error correction protects quantum states by encoding a logical quantum bit (qubit) in multiple physical qubits. To be compatible with universal fault-tolerant computations, it is essential that states remain encoded at all times and that errors are actively corrected. Here we demonstrate such active error correction on a continuously protected logical qubit using a diamond quantum processor. We encode the logical qubit in three long-lived nuclear spins, repeatedly detect phase errors by non-destructive measurements, and apply corrections by real-time feedback. The actively error-corrected qubit is robust against errors and encoded quantum superposition states are preserved beyond the natural dephasing time of the best physical qubit in the encoding. These results establish a powerful platform to investigate error correction under different types of noise and mark an important step towards fault-tolerant quantum information processing.</jats:p>","DOI":"10.1038/ncomms11526","source":"Crossref","title":"Repeated quantum error correction on a continuously encoded qubit by real-time feedback","volume":"7","author":[{"given":"J.","family":"Cramer","sequence":"first","affiliation":[]},{"given":"N.","family":"Kalb","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Rol","sequence":"additional","affiliation":[]},{"given":"B.","family":"Hensen","sequence":"additional","affiliation":[]},{"given":"M. S.","family":"Blok","sequence":"additional","affiliation":[]},{"given":"M.","family":"Markham","sequence":"additional","affiliation":[]},{"given":"D. J.","family":"Twitchen","sequence":"additional","affiliation":[]},{"given":"R.","family":"Hanson","sequence":"additional","affiliation":[]},{"given":"T. H.","family":"Taminiau","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2016,5,5]]},"URL":"http://dx.doi.org/10.1038/ncomms11526","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"11526","note":"alternative-id:BFncomms11526\narxivid:1508.01388"},{"id":"arxiv:1709.00990","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.97.052313","source":"Crossref","title":"Repetition code of 15 qubits","volume":"97","author":[{"given":"James R.","family":"Wootton","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.052313","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052313","note":"arxivid:1709.00990"},{"id":"arxiv:2004.11037","type":"journal-article","publisher":"IOP Publishing","issue":"4","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Now that ever more sophisticated devices for quantum computing are being developed, we require ever more sophisticated benchmarks. This includes a need to determine how well these devices support the techniques required for quantum error correction. In this paper we introduce the <jats:monospace>topological_codes</jats:monospace> module of Qiskit-Ignis, which is designed to provide the tools necessary to perform such tests. Specifically, we use the <jats:monospace>RepetitionCode</jats:monospace> and <jats:monospace>GraphDecoder</jats:monospace> classes to run tests based on the repetition code and process the results. As an example, data from a 43 qubit code running on IBM’s <jats:italic>Rochester</jats:italic> device is presented.</jats:p>","DOI":"10.1088/2058-9565/aba038","page":"044004","source":"Crossref","title":"Benchmarking near-term devices with quantum error correction","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0003-1943-5306","authenticated-orcid":false,"given":"James R","family":"Wootton","sequence":"first","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2020,7,31]]},"URL":"http://dx.doi.org/10.1088/2058-9565/aba038","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2004.11037"},{"id":"arxiv:1411.7403","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7541","DOI":"10.1038/nature14270","page":"66-69","source":"Crossref","title":"State preservation by repetitive error detection in a superconducting quantum circuit","volume":"519","author":[{"given":"J.","family":"Kelly","sequence":"first","affiliation":[]},{"given":"R.","family":"Barends","sequence":"additional","affiliation":[]},{"given":"A. G.","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"A.","family":"Megrant","sequence":"additional","affiliation":[]},{"given":"E.","family":"Jeffrey","sequence":"additional","affiliation":[]},{"given":"T. C.","family":"White","sequence":"additional","affiliation":[]},{"given":"D.","family":"Sank","sequence":"additional","affiliation":[]},{"given":"J. Y.","family":"Mutus","sequence":"additional","affiliation":[]},{"given":"B.","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Yu","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Z.","family":"Chen","sequence":"additional","affiliation":[]},{"given":"B.","family":"Chiaro","sequence":"additional","affiliation":[]},{"given":"A.","family":"Dunsworth","sequence":"additional","affiliation":[]},{"given":"I.-C.","family":"Hoi","sequence":"additional","affiliation":[]},{"given":"C.","family":"Neill","sequence":"additional","affiliation":[]},{"given":"P. J. J.","family":"O’Malley","sequence":"additional","affiliation":[]},{"given":"C.","family":"Quintana","sequence":"additional","affiliation":[]},{"given":"P.","family":"Roushan","sequence":"additional","affiliation":[]},{"given":"A.","family":"Vainsencher","sequence":"additional","affiliation":[]},{"given":"J.","family":"Wenner","sequence":"additional","affiliation":[]},{"given":"A. N.","family":"Cleland","sequence":"additional","affiliation":[]},{"given":"John M.","family":"Martinis","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2015,3,4]]},"URL":"http://dx.doi.org/10.1038/nature14270","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature14270\narxivid:1411.7403"},{"id":"arxiv:2102.06132","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7867","abstract":"<jats:title>Abstract</jats:title><jats:p>Realizing the potential of quantum computing requires sufficiently low logical error rates<jats:sup>1</jats:sup>. Many applications call for error rates as low as 10<jats:sup>−15</jats:sup> (refs. <jats:sup>2–9</jats:sup>), but state-of-the-art quantum platforms typically have physical error rates near 10<jats:sup>−3</jats:sup> (refs. <jats:sup>10–14</jats:sup>). Quantum error correction<jats:sup>15–17</jats:sup> promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device<jats:sup>18,19</jats:sup> and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.</jats:p>","DOI":"10.1038/s41586-021-03588-y","page":"383-387","source":"Crossref","title":"Exponential suppression of bit or phase errors with cyclic error correction","volume":"595","author":[{"name":"Google Quantum AI","sequence":"first","affiliation":[]},{"given":"Zijun","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Kevin J.","family":"Satzinger","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Atalaya","sequence":"additional","affiliation":[]},{"given":"Alexander N.","family":"Korotkov","sequence":"additional","affiliation":[]},{"given":"Andrew","family":"Dunsworth","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Sank","sequence":"additional","affiliation":[]},{"given":"Chris","family":"Quintana","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9544-141X","authenticated-orcid":false,"given":"Matt","family":"McEwen","sequence":"additional","affiliation":[]},{"given":"Rami","family":"Barends","sequence":"additional","affiliation":[]},{"given":"Paul V.","family":"Klimov","sequence":"additional","affiliation":[]},{"given":"Sabrina","family":"Hong","sequence":"additional","affiliation":[]},{"given":"Cody","family":"Jones","sequence":"additional","affiliation":[]},{"given":"Andre","family":"Petukhov","sequence":"additional","affiliation":[]},{"given":"Dvir","family":"Kafri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3727-7380","authenticated-orcid":false,"given":"Sean","family":"Demura","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8474-6317","authenticated-orcid":false,"given":"Brian","family":"Burkett","sequence":"additional","affiliation":[]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1536-8858","authenticated-orcid":false,"given":"Alexandru","family":"Paler","sequence":"additional","affiliation":[]},{"given":"Harald","family":"Putterman","sequence":"additional","affiliation":[]},{"given":"Igor","family":"Aleiner","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Arute","sequence":"additional","affiliation":[]},{"given":"Kunal","family":"Arya","sequence":"additional","affiliation":[]},{"given":"Ryan","family":"Babbush","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6523-6730","authenticated-orcid":false,"given":"Joseph C.","family":"Bardin","sequence":"additional","affiliation":[]},{"given":"Andreas","family":"Bengtsson","sequence":"additional","affiliation":[]},{"given":"Alexandre","family":"Bourassa","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Broughton","sequence":"additional","affiliation":[]},{"given":"Bob B.","family":"Buckley","sequence":"additional","affiliation":[]},{"given":"David A.","family":"Buell","sequence":"additional","affiliation":[]},{"given":"Nicholas","family":"Bushnell","sequence":"additional","affiliation":[]},{"given":"Benjamin","family":"Chiaro","sequence":"additional","affiliation":[]},{"given":"Roberto","family":"Collins","sequence":"additional","affiliation":[]},{"given":"William","family":"Courtney","sequence":"additional","affiliation":[]},{"given":"Alan R.","family":"Derk","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Eppens","sequence":"additional","affiliation":[]},{"given":"Catherine","family":"Erickson","sequence":"additional","affiliation":[]},{"given":"Edward","family":"Farhi","sequence":"additional","affiliation":[]},{"given":"Brooks","family":"Foxen","sequence":"additional","affiliation":[]},{"given":"Marissa","family":"Giustina","sequence":"additional","affiliation":[]},{"given":"Ami","family":"Greene","sequence":"additional","affiliation":[]},{"given":"Jonathan A.","family":"Gross","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9412-0553","authenticated-orcid":false,"given":"Matthew P.","family":"Harrigan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0521-8378","authenticated-orcid":false,"given":"Sean D.","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"Jeremy","family":"Hilton","sequence":"additional","affiliation":[]},{"given":"Alan","family":"Ho","sequence":"additional","affiliation":[]},{"given":"Trent","family":"Huang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2735-1380","authenticated-orcid":false,"given":"William J.","family":"Huggins","sequence":"additional","affiliation":[]},{"given":"L. B.","family":"Ioffe","sequence":"additional","affiliation":[]},{"given":"Sergei V.","family":"Isakov","sequence":"additional","affiliation":[]},{"given":"Evan","family":"Jeffrey","sequence":"additional","affiliation":[]},{"given":"Zhang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Kostyantyn","family":"Kechedzhi","sequence":"additional","affiliation":[]},{"given":"Seon","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]},{"given":"Fedor","family":"Kostritsa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9804-2185","authenticated-orcid":false,"given":"David","family":"Landhuis","sequence":"additional","affiliation":[]},{"given":"Pavel","family":"Laptev","sequence":"additional","affiliation":[]},{"given":"Erik","family":"Lucero","sequence":"additional","affiliation":[]},{"given":"Orion","family":"Martin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2809-0509","authenticated-orcid":false,"given":"Jarrod R.","family":"McClean","sequence":"additional","affiliation":[]},{"given":"Trevor","family":"McCourt","sequence":"additional","affiliation":[]},{"given":"Xiao","family":"Mi","sequence":"additional","affiliation":[]},{"given":"Kevin C.","family":"Miao","sequence":"additional","affiliation":[]},{"given":"Masoud","family":"Mohseni","sequence":"additional","affiliation":[]},{"given":"Shirin","family":"Montazeri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8497-6363","authenticated-orcid":false,"given":"Wojciech","family":"Mruczkiewicz","sequence":"additional","affiliation":[]},{"given":"Josh","family":"Mutus","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7760-9186","authenticated-orcid":false,"given":"Ofer","family":"Naaman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5548-0051","authenticated-orcid":false,"given":"Matthew","family":"Neeley","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8509-9836","authenticated-orcid":false,"given":"Charles","family":"Neill","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"given":"Murphy Yuezhen","family":"Niu","sequence":"additional","affiliation":[]},{"given":"Thomas E.","family":"O’Brien","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Opremcak","sequence":"additional","affiliation":[]},{"given":"Eric","family":"Ostby","sequence":"additional","affiliation":[]},{"given":"Bálint","family":"Pató","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6549-5441","authenticated-orcid":false,"given":"Nicholas","family":"Redd","sequence":"additional","affiliation":[]},{"given":"Pedram","family":"Roushan","sequence":"additional","affiliation":[]},{"given":"Nicholas C.","family":"Rubin","sequence":"additional","affiliation":[]},{"given":"Vladimir","family":"Shvarts","sequence":"additional","affiliation":[]},{"given":"Doug","family":"Strain","sequence":"additional","affiliation":[]},{"given":"Marco","family":"Szalay","sequence":"additional","affiliation":[]},{"given":"Matthew D.","family":"Trevithick","sequence":"additional","affiliation":[]},{"given":"Benjamin","family":"Villalonga","sequence":"additional","affiliation":[]},{"given":"Theodore","family":"White","sequence":"additional","affiliation":[]},{"given":"Z. Jamie","family":"Yao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0837-1028","authenticated-orcid":false,"given":"Ping","family":"Yeh","sequence":"additional","affiliation":[]},{"given":"Juhwan","family":"Yoo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2585-2424","authenticated-orcid":false,"given":"Adam","family":"Zalcman","sequence":"additional","affiliation":[]},{"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1090-7584","authenticated-orcid":false,"given":"Sergio","family":"Boixo","sequence":"additional","affiliation":[]},{"given":"Vadim","family":"Smelyanskiy","sequence":"additional","affiliation":[]},{"given":"Yu","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6371-6140","authenticated-orcid":false,"given":"Anthony","family":"Megrant","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2596-2121","authenticated-orcid":false,"given":"Julian","family":"Kelly","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2021,7,14]]},"URL":"http://dx.doi.org/10.1038/s41586-021-03588-y","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:3588\narxivid:2102.06132"},{"id":"arxiv:2207.06431","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7949","abstract":"<jats:title>Abstract</jats:title><jats:p>Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction<jats:sup>1,2</jats:sup> offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both logical error probability over 25 cycles and logical error per cycle ((2.914 ± 0.016)% compared to (3.028 ± 0.023)%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a 1.7 × 10<jats:sup>−6</jats:sup> logical error per cycle floor set by a single high-energy event (1.6 × 10<jats:sup>−7</jats:sup> excluding this event). We accurately model our experiment, extracting error budgets that highlight the biggest challenges for future systems. These results mark an experimental demonstration in which quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation.</jats:p>","DOI":"10.1038/s41586-022-05434-1","page":"676-681","source":"Crossref","title":"Suppressing quantum errors by scaling a surface code logical qubit","volume":"614","author":[{"name":"Google Quantum AI","sequence":"first","affiliation":[]},{"given":"Rajeev","family":"Acharya","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9096-254X","authenticated-orcid":false,"given":"Igor","family":"Aleiner","sequence":"additional","affiliation":[]},{"given":"Richard","family":"Allen","sequence":"additional","affiliation":[]},{"given":"Trond I.","family":"Andersen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8765-5621","authenticated-orcid":false,"given":"Markus","family":"Ansmann","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Arute","sequence":"additional","affiliation":[]},{"given":"Kunal","family":"Arya","sequence":"additional","affiliation":[]},{"given":"Abraham","family":"Asfaw","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Atalaya","sequence":"additional","affiliation":[]},{"given":"Ryan","family":"Babbush","sequence":"additional","affiliation":[]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6523-6730","authenticated-orcid":false,"given":"Joseph C.","family":"Bardin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5547-691X","authenticated-orcid":false,"given":"Joao","family":"Basso","sequence":"additional","affiliation":[]},{"given":"Andreas","family":"Bengtsson","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1090-7584","authenticated-orcid":false,"given":"Sergio","family":"Boixo","sequence":"additional","affiliation":[]},{"given":"Gina","family":"Bortoli","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8517-7475","authenticated-orcid":false,"given":"Alexandre","family":"Bourassa","sequence":"additional","affiliation":[]},{"given":"Jenna","family":"Bovaird","sequence":"additional","affiliation":[]},{"given":"Leon","family":"Brill","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Broughton","sequence":"additional","affiliation":[]},{"given":"Bob B.","family":"Buckley","sequence":"additional","affiliation":[]},{"given":"David A.","family":"Buell","sequence":"additional","affiliation":[]},{"given":"Tim","family":"Burger","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8474-6317","authenticated-orcid":false,"given":"Brian","family":"Burkett","sequence":"additional","affiliation":[]},{"given":"Nicholas","family":"Bushnell","sequence":"additional","affiliation":[]},{"given":"Yu","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Zijun","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Ben","family":"Chiaro","sequence":"additional","affiliation":[]},{"given":"Josh","family":"Cogan","sequence":"additional","affiliation":[]},{"given":"Roberto","family":"Collins","sequence":"additional","affiliation":[]},{"given":"Paul","family":"Conner","sequence":"additional","affiliation":[]},{"given":"William","family":"Courtney","sequence":"additional","affiliation":[]},{"given":"Alexander L.","family":"Crook","sequence":"additional","affiliation":[]},{"given":"Ben","family":"Curtin","sequence":"additional","affiliation":[]},{"given":"Dripto M.","family":"Debroy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0869-5110","authenticated-orcid":false,"given":"Alexander","family":"Del Toro Barba","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3727-7380","authenticated-orcid":false,"given":"Sean","family":"Demura","sequence":"additional","affiliation":[]},{"given":"Andrew","family":"Dunsworth","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7134-5733","authenticated-orcid":false,"given":"Daniel","family":"Eppens","sequence":"additional","affiliation":[]},{"given":"Catherine","family":"Erickson","sequence":"additional","affiliation":[]},{"given":"Lara","family":"Faoro","sequence":"additional","affiliation":[]},{"given":"Edward","family":"Farhi","sequence":"additional","affiliation":[]},{"given":"Reza","family":"Fatemi","sequence":"additional","affiliation":[]},{"given":"Leslie","family":"Flores Burgos","sequence":"additional","affiliation":[]},{"given":"Ebrahim","family":"Forati","sequence":"additional","affiliation":[]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"Brooks","family":"Foxen","sequence":"additional","affiliation":[]},{"given":"William","family":"Giang","sequence":"additional","affiliation":[]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[]},{"given":"Dar","family":"Gilboa","sequence":"additional","affiliation":[]},{"given":"Marissa","family":"Giustina","sequence":"additional","affiliation":[]},{"given":"Alejandro","family":"Grajales Dau","sequence":"additional","affiliation":[]},{"given":"Jonathan A.","family":"Gross","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9867-2371","authenticated-orcid":false,"given":"Steve","family":"Habegger","sequence":"additional","affiliation":[]},{"given":"Michael C.","family":"Hamilton","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9412-0553","authenticated-orcid":false,"given":"Matthew P.","family":"Harrigan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0521-8378","authenticated-orcid":false,"given":"Sean D.","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"Oscar","family":"Higgott","sequence":"additional","affiliation":[]},{"given":"Jeremy","family":"Hilton","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7051-5558","authenticated-orcid":false,"given":"Markus","family":"Hoffmann","sequence":"additional","affiliation":[]},{"given":"Sabrina","family":"Hong","sequence":"additional","affiliation":[]},{"given":"Trent","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Ashley","family":"Huff","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2735-1380","authenticated-orcid":false,"given":"William J.","family":"Huggins","sequence":"additional","affiliation":[]},{"given":"Lev B.","family":"Ioffe","sequence":"additional","affiliation":[]},{"given":"Sergei V.","family":"Isakov","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Iveland","sequence":"additional","affiliation":[]},{"given":"Evan","family":"Jeffrey","sequence":"additional","affiliation":[]},{"given":"Zhang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Cody","family":"Jones","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8751-4458","authenticated-orcid":false,"given":"Pavol","family":"Juhas","sequence":"additional","affiliation":[]},{"given":"Dvir","family":"Kafri","sequence":"additional","affiliation":[]},{"given":"Kostyantyn","family":"Kechedzhi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2596-2121","authenticated-orcid":false,"given":"Julian","family":"Kelly","sequence":"additional","affiliation":[]},{"given":"Tanuj","family":"Khattar","sequence":"additional","affiliation":[]},{"given":"Mostafa","family":"Khezri","sequence":"additional","affiliation":[]},{"given":"Mária","family":"Kieferová","sequence":"additional","affiliation":[]},{"given":"Seon","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]},{"given":"Paul V.","family":"Klimov","sequence":"additional","affiliation":[]},{"given":"Andrey R.","family":"Klots","sequence":"additional","affiliation":[]},{"given":"Alexander N.","family":"Korotkov","sequence":"additional","affiliation":[]},{"given":"Fedor","family":"Kostritsa","sequence":"additional","affiliation":[]},{"given":"John Mark","family":"Kreikebaum","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9804-2185","authenticated-orcid":false,"given":"David","family":"Landhuis","sequence":"additional","affiliation":[]},{"given":"Pavel","family":"Laptev","sequence":"additional","affiliation":[]},{"given":"Kim-Ming","family":"Lau","sequence":"additional","affiliation":[]},{"given":"Lily","family":"Laws","sequence":"additional","affiliation":[]},{"given":"Joonho","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Kenny","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Brian J.","family":"Lester","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Lill","sequence":"additional","affiliation":[]},{"given":"Wayne","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Aditya","family":"Locharla","sequence":"additional","affiliation":[]},{"given":"Erik","family":"Lucero","sequence":"additional","affiliation":[]},{"given":"Fionn D.","family":"Malone","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5011-1281","authenticated-orcid":false,"given":"Jeffrey","family":"Marshall","sequence":"additional","affiliation":[]},{"given":"Orion","family":"Martin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2809-0509","authenticated-orcid":false,"given":"Jarrod R.","family":"McClean","sequence":"additional","affiliation":[]},{"given":"Trevor","family":"McCourt","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9544-141X","authenticated-orcid":false,"given":"Matt","family":"McEwen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6371-6140","authenticated-orcid":false,"given":"Anthony","family":"Megrant","sequence":"additional","affiliation":[]},{"given":"Bernardo","family":"Meurer Costa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0507-0211","authenticated-orcid":false,"given":"Xiao","family":"Mi","sequence":"additional","affiliation":[]},{"given":"Kevin C.","family":"Miao","sequence":"additional","affiliation":[]},{"given":"Masoud","family":"Mohseni","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7124-4585","authenticated-orcid":false,"given":"Shirin","family":"Montazeri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9515-6184","authenticated-orcid":false,"given":"Alexis","family":"Morvan","sequence":"additional","affiliation":[]},{"given":"Emily","family":"Mount","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8497-6363","authenticated-orcid":false,"given":"Wojciech","family":"Mruczkiewicz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7760-9186","authenticated-orcid":false,"given":"Ofer","family":"Naaman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5548-0051","authenticated-orcid":false,"given":"Matthew","family":"Neeley","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8509-9836","authenticated-orcid":false,"given":"Charles","family":"Neill","sequence":"additional","affiliation":[]},{"given":"Ani","family":"Nersisyan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9681-6746","authenticated-orcid":false,"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7246-6761","authenticated-orcid":false,"given":"Jiun How","family":"Ng","sequence":"additional","affiliation":[]},{"given":"Anthony","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"Murray","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"Murphy Yuezhen","family":"Niu","sequence":"additional","affiliation":[]},{"given":"Thomas E.","family":"O’Brien","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Opremcak","sequence":"additional","affiliation":[]},{"given":"John","family":"Platt","sequence":"additional","affiliation":[]},{"given":"Andre","family":"Petukhov","sequence":"additional","affiliation":[]},{"given":"Rebecca","family":"Potter","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]},{"given":"Chris","family":"Quintana","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1917-3879","authenticated-orcid":false,"given":"Pedram","family":"Roushan","sequence":"additional","affiliation":[]},{"given":"Nicholas C.","family":"Rubin","sequence":"additional","affiliation":[]},{"given":"Negar","family":"Saei","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8188-364X","authenticated-orcid":false,"given":"Daniel","family":"Sank","sequence":"additional","affiliation":[]},{"given":"Kannan","family":"Sankaragomathi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5865-0813","authenticated-orcid":false,"given":"Kevin J.","family":"Satzinger","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3210-5929","authenticated-orcid":false,"given":"Henry F.","family":"Schurkus","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Schuster","sequence":"additional","affiliation":[]},{"given":"Michael J.","family":"Shearn","sequence":"additional","affiliation":[]},{"given":"Aaron","family":"Shorter","sequence":"additional","affiliation":[]},{"given":"Vladimir","family":"Shvarts","sequence":"additional","affiliation":[]},{"given":"Jindra","family":"Skruzny","sequence":"additional","affiliation":[]},{"given":"Vadim","family":"Smelyanskiy","sequence":"additional","affiliation":[]},{"given":"W. Clarke","family":"Smith","sequence":"additional","affiliation":[]},{"given":"George","family":"Sterling","sequence":"additional","affiliation":[]},{"given":"Doug","family":"Strain","sequence":"additional","affiliation":[]},{"given":"Marco","family":"Szalay","sequence":"additional","affiliation":[]},{"given":"Alfredo","family":"Torres","sequence":"additional","affiliation":[]},{"given":"Guifre","family":"Vidal","sequence":"additional","affiliation":[]},{"given":"Benjamin","family":"Villalonga","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6513-9885","authenticated-orcid":false,"given":"Catherine","family":"Vollgraff Heidweiller","sequence":"additional","affiliation":[]},{"given":"Theodore","family":"White","sequence":"additional","affiliation":[]},{"given":"Cheng","family":"Xing","sequence":"additional","affiliation":[]},{"given":"Z. Jamie","family":"Yao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0837-1028","authenticated-orcid":false,"given":"Ping","family":"Yeh","sequence":"additional","affiliation":[]},{"given":"Juhwan","family":"Yoo","sequence":"additional","affiliation":[]},{"given":"Grayson","family":"Young","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2585-2424","authenticated-orcid":false,"given":"Adam","family":"Zalcman","sequence":"additional","affiliation":[]},{"given":"Yaxing","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3037-2003","authenticated-orcid":false,"given":"Ningfeng","family":"Zhu","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2023,2,22]]},"URL":"http://dx.doi.org/10.1038/s41586-022-05434-1","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:5434\narxivid:2207.06431"},{"id":"arxiv:2211.04728","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","abstract":"<jats:title>Abstract</jats:title><jats:p>The leakage of quantum information out of the two computational states of a qubit into other energy states represents a major challenge for quantum error correction. During the operation of an error-corrected algorithm, leakage builds over time and spreads through multi-qubit interactions. This leads to correlated errors that degrade the exponential suppression of the logical error with scale, thus challenging the feasibility of quantum error correction as a path towards fault-tolerant quantum computation. Here, we demonstrate a distance-3 surface code and distance-21 bit-flip code on a quantum processor for which leakage is removed from all qubits in each cycle. This shortens the lifetime of leakage and curtails its ability to spread and induce correlated errors. We report a tenfold reduction in the steady-state leakage population of the data qubits encoding the logical state and an average leakage population of less than 1 × 10<jats:sup>−3</jats:sup> throughout the entire device. Our leakage removal process efficiently returns the system back to the computational basis. Adding it to a code circuit would prevent leakage from inducing correlated error across cycles. With this demonstration that leakage can be contained, we have resolved a key challenge for practical quantum error correction at scale.</jats:p>","DOI":"10.1038/s41567-023-02226-w","page":"1780-1786","source":"Crossref","title":"Overcoming leakage in quantum error correction","volume":"19","author":[{"ORCID":"https://orcid.org/0000-0002-1904-5917","authenticated-orcid":false,"given":"Kevin C.","family":"Miao","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9544-141X","authenticated-orcid":false,"given":"Matt","family":"McEwen","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Atalaya","sequence":"additional","affiliation":[]},{"given":"Dvir","family":"Kafri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4990-0259","authenticated-orcid":false,"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5264-4025","authenticated-orcid":false,"given":"Andreas","family":"Bengtsson","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Opremcak","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5865-0813","authenticated-orcid":false,"given":"Kevin J.","family":"Satzinger","sequence":"additional","affiliation":[]},{"given":"Zijun","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Paul V.","family":"Klimov","sequence":"additional","affiliation":[]},{"given":"Chris","family":"Quintana","sequence":"additional","affiliation":[]},{"given":"Rajeev","family":"Acharya","sequence":"additional","affiliation":[]},{"given":"Kyle","family":"Anderson","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Ansmann","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Arute","sequence":"additional","affiliation":[]},{"given":"Kunal","family":"Arya","sequence":"additional","affiliation":[]},{"given":"Abraham","family":"Asfaw","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6523-6730","authenticated-orcid":false,"given":"Joseph C.","family":"Bardin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8517-7475","authenticated-orcid":false,"given":"Alexandre","family":"Bourassa","sequence":"additional","affiliation":[]},{"given":"Jenna","family":"Bovaird","sequence":"additional","affiliation":[]},{"given":"Leon","family":"Brill","sequence":"additional","affiliation":[]},{"given":"Bob B.","family":"Buckley","sequence":"additional","affiliation":[]},{"given":"David A.","family":"Buell","sequence":"additional","affiliation":[]},{"given":"Tim","family":"Burger","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8474-6317","authenticated-orcid":false,"given":"Brian","family":"Burkett","sequence":"additional","affiliation":[]},{"given":"Nicholas","family":"Bushnell","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Campero","sequence":"additional","affiliation":[]},{"given":"Ben","family":"Chiaro","sequence":"additional","affiliation":[]},{"given":"Roberto","family":"Collins","sequence":"additional","affiliation":[]},{"given":"Paul","family":"Conner","sequence":"additional","affiliation":[]},{"given":"Alexander L.","family":"Crook","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3856-0615","authenticated-orcid":false,"given":"Ben","family":"Curtin","sequence":"additional","affiliation":[]},{"given":"Dripto M.","family":"Debroy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3727-7380","authenticated-orcid":false,"given":"Sean","family":"Demura","sequence":"additional","affiliation":[]},{"given":"Andrew","family":"Dunsworth","sequence":"additional","affiliation":[]},{"given":"Catherine","family":"Erickson","sequence":"additional","affiliation":[]},{"given":"Reza","family":"Fatemi","sequence":"additional","affiliation":[]},{"given":"Vinicius S.","family":"Ferreira","sequence":"additional","affiliation":[]},{"given":"Leslie Flores","family":"Burgos","sequence":"additional","affiliation":[]},{"given":"Ebrahim","family":"Forati","sequence":"additional","affiliation":[]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3732-168X","authenticated-orcid":false,"given":"Brooks","family":"Foxen","sequence":"additional","affiliation":[]},{"given":"Gonzalo","family":"Garcia","sequence":"additional","affiliation":[]},{"given":"William","family":"Giang","sequence":"additional","affiliation":[]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[]},{"given":"Marissa","family":"Giustina","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3025-7312","authenticated-orcid":false,"given":"Raja","family":"Gosula","sequence":"additional","affiliation":[]},{"given":"Alejandro Grajales","family":"Dau","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":false,"given":"Jonathan A.","family":"Gross","sequence":"additional","affiliation":[]},{"given":"Michael C.","family":"Hamilton","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0521-8378","authenticated-orcid":false,"given":"Sean D.","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"Paula","family":"Heu","sequence":"additional","affiliation":[]},{"given":"Jeremy","family":"Hilton","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7051-5558","authenticated-orcid":false,"given":"Markus R.","family":"Hoffmann","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7421-5635","authenticated-orcid":false,"given":"Sabrina","family":"Hong","sequence":"additional","affiliation":[]},{"given":"Trent","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Ashley","family":"Huff","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Iveland","sequence":"additional","affiliation":[]},{"given":"Evan","family":"Jeffrey","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0435-655X","authenticated-orcid":false,"given":"Zhang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Cody","family":"Jones","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2596-2121","authenticated-orcid":false,"given":"Julian","family":"Kelly","sequence":"additional","affiliation":[]},{"given":"Seon","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Fedor","family":"Kostritsa","sequence":"additional","affiliation":[]},{"given":"John Mark","family":"Kreikebaum","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9804-2185","authenticated-orcid":false,"given":"David","family":"Landhuis","sequence":"additional","affiliation":[]},{"given":"Pavel","family":"Laptev","sequence":"additional","affiliation":[]},{"given":"Lily","family":"Laws","sequence":"additional","affiliation":[]},{"given":"Kenny","family":"Lee","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4297-2508","authenticated-orcid":false,"given":"Brian J.","family":"Lester","sequence":"additional","affiliation":[]},{"given":"Alexander T.","family":"Lill","sequence":"additional","affiliation":[]},{"given":"Wayne","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Aditya","family":"Locharla","sequence":"additional","affiliation":[]},{"given":"Erik","family":"Lucero","sequence":"additional","affiliation":[]},{"given":"Steven","family":"Martin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6371-6140","authenticated-orcid":false,"given":"Anthony","family":"Megrant","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0507-0211","authenticated-orcid":false,"given":"Xiao","family":"Mi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7124-4585","authenticated-orcid":false,"given":"Shirin","family":"Montazeri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9515-6184","authenticated-orcid":false,"given":"Alexis","family":"Morvan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7760-9186","authenticated-orcid":false,"given":"Ofer","family":"Naaman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5548-0051","authenticated-orcid":false,"given":"Matthew","family":"Neeley","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8509-9836","authenticated-orcid":false,"given":"Charles","family":"Neill","sequence":"additional","affiliation":[]},{"given":"Ani","family":"Nersisyan","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7246-6761","authenticated-orcid":false,"given":"Jiun How","family":"Ng","sequence":"additional","affiliation":[]},{"given":"Anthony","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"Murray","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"Rebecca","family":"Potter","sequence":"additional","affiliation":[]},{"given":"Charles","family":"Rocque","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1917-3879","authenticated-orcid":false,"given":"Pedram","family":"Roushan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8930-7081","authenticated-orcid":false,"given":"Kannan","family":"Sankaragomathi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3210-5929","authenticated-orcid":false,"given":"Henry F.","family":"Schurkus","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Schuster","sequence":"additional","affiliation":[]},{"given":"Michael J.","family":"Shearn","sequence":"additional","affiliation":[]},{"given":"Aaron","family":"Shorter","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6035-2812","authenticated-orcid":false,"given":"Noah","family":"Shutty","sequence":"additional","affiliation":[]},{"given":"Vladimir","family":"Shvarts","sequence":"additional","affiliation":[]},{"given":"Jindra","family":"Skruzny","sequence":"additional","affiliation":[]},{"given":"W. Clarke","family":"Smith","sequence":"additional","affiliation":[]},{"given":"George","family":"Sterling","sequence":"additional","affiliation":[]},{"given":"Marco","family":"Szalay","sequence":"additional","affiliation":[]},{"given":"Douglas","family":"Thor","sequence":"additional","affiliation":[]},{"given":"Alfredo","family":"Torres","sequence":"additional","affiliation":[]},{"given":"Theodore","family":"White","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7233-1576","authenticated-orcid":false,"given":"Bryan W. K.","family":"Woo","sequence":"additional","affiliation":[]},{"given":"Z. Jamie","family":"Yao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0837-1028","authenticated-orcid":false,"given":"Ping","family":"Yeh","sequence":"additional","affiliation":[]},{"given":"Juhwan","family":"Yoo","sequence":"additional","affiliation":[]},{"given":"Grayson","family":"Young","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2585-2424","authenticated-orcid":false,"given":"Adam","family":"Zalcman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3037-2003","authenticated-orcid":false,"given":"Ningfeng","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Nicholas","family":"Zobrist","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9681-6746","authenticated-orcid":false,"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[]},{"given":"Vadim","family":"Smelyanskiy","sequence":"additional","affiliation":[]},{"given":"Andre","family":"Petukhov","sequence":"additional","affiliation":[]},{"given":"Alexander N.","family":"Korotkov","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8188-364X","authenticated-orcid":false,"given":"Daniel","family":"Sank","sequence":"additional","affiliation":[]},{"given":"Yu","family":"Chen","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,5]]},"URL":"http://dx.doi.org/10.1038/s41567-023-02226-w","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2226\narxivid:2211.04728"},{"id":"arxiv:2408.13687","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8052","DOI":"10.1038/s41586-024-08449-y","page":"920-926","source":"Crossref","title":"Quantum error correction below the surface code threshold","volume":"638","author":[{"name":"Google Quantum AI and Collaborators","sequence":"first","affiliation":[]},{"given":"Rajeev","family":"Acharya","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2461-0271","authenticated-orcid":false,"given":"Dmitry A.","family":"Abanin","sequence":"additional","affiliation":[]},{"given":"Laleh","family":"Aghababaie-Beni","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9096-254X","authenticated-orcid":false,"given":"Igor","family":"Aleiner","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7806-1626","authenticated-orcid":false,"given":"Trond I.","family":"Andersen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8765-5621","authenticated-orcid":false,"given":"Markus","family":"Ansmann","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Arute","sequence":"additional","affiliation":[]},{"given":"Kunal","family":"Arya","sequence":"additional","affiliation":[]},{"given":"Abraham","family":"Asfaw","sequence":"additional","affiliation":[]},{"given":"Nikita","family":"Astrakhantsev","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Atalaya","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6979-9533","authenticated-orcid":false,"given":"Ryan","family":"Babbush","sequence":"additional","affiliation":[]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[]},{"given":"Brian","family":"Ballard","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6523-6730","authenticated-orcid":false,"given":"Joseph C.","family":"Bardin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3189-9162","authenticated-orcid":false,"given":"Johannes","family":"Bausch","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5264-4025","authenticated-orcid":false,"given":"Andreas","family":"Bengtsson","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Bilmes","sequence":"additional","affiliation":[]},{"given":"Sam","family":"Blackwell","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1090-7584","authenticated-orcid":false,"given":"Sergio","family":"Boixo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-4597-1691","authenticated-orcid":false,"given":"Gina","family":"Bortoli","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8517-7475","authenticated-orcid":false,"given":"Alexandre","family":"Bourassa","sequence":"additional","affiliation":[]},{"given":"Jenna","family":"Bovaird","sequence":"additional","affiliation":[]},{"given":"Leon","family":"Brill","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Broughton","sequence":"additional","affiliation":[]},{"given":"David A.","family":"Browne","sequence":"additional","affiliation":[]},{"given":"Brett","family":"Buchea","sequence":"additional","affiliation":[]},{"given":"Bob B.","family":"Buckley","sequence":"additional","affiliation":[]},{"given":"David A.","family":"Buell","sequence":"additional","affiliation":[]},{"given":"Tim","family":"Burger","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8474-6317","authenticated-orcid":false,"given":"Brian","family":"Burkett","sequence":"additional","affiliation":[]},{"given":"Nicholas","family":"Bushnell","sequence":"additional","affiliation":[]},{"given":"Anthony","family":"Cabrera","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Campero","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9681-6108","authenticated-orcid":false,"given":"Hung-Shen","family":"Chang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7473-6745","authenticated-orcid":false,"given":"Yu","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Zijun","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Ben","family":"Chiaro","sequence":"additional","affiliation":[]},{"given":"Desmond","family":"Chik","sequence":"additional","affiliation":[]},{"given":"Charina","family":"Chou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6711-4585","authenticated-orcid":false,"given":"Jahan","family":"Claes","sequence":"additional","affiliation":[]},{"given":"Agnetta Y.","family":"Cleland","sequence":"additional","affiliation":[]},{"given":"Josh","family":"Cogan","sequence":"additional","affiliation":[]},{"given":"Roberto","family":"Collins","sequence":"additional","affiliation":[]},{"given":"Paul","family":"Conner","sequence":"additional","affiliation":[]},{"given":"William","family":"Courtney","sequence":"additional","affiliation":[]},{"given":"Alexander L.","family":"Crook","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3856-0615","authenticated-orcid":false,"given":"Ben","family":"Curtin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-4853-1051","authenticated-orcid":false,"given":"Sayan","family":"Das","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4917-5234","authenticated-orcid":false,"given":"Alex","family":"Davies","sequence":"additional","affiliation":[]},{"given":"Laura","family":"De Lorenzo","sequence":"additional","affiliation":[]},{"given":"Dripto M.","family":"Debroy","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3727-7380","authenticated-orcid":false,"given":"Sean","family":"Demura","sequence":"additional","affiliation":[]},{"given":"Michel","family":"Devoret","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5174-7310","authenticated-orcid":false,"given":"Agustin","family":"Di Paolo","sequence":"additional","affiliation":[]},{"given":"Paul","family":"Donohoe","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8185-1194","authenticated-orcid":false,"given":"Ilya","family":"Drozdov","sequence":"additional","affiliation":[]},{"given":"Andrew","family":"Dunsworth","sequence":"additional","affiliation":[]},{"given":"Clint","family":"Earle","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-0473-3727","authenticated-orcid":false,"given":"Thomas","family":"Edlich","sequence":"additional","affiliation":[]},{"given":"Alec","family":"Eickbusch","sequence":"additional","affiliation":[]},{"given":"Aviv Moshe","family":"Elbag","sequence":"additional","affiliation":[]},{"given":"Mahmoud","family":"Elzouka","sequence":"additional","affiliation":[]},{"given":"Catherine","family":"Erickson","sequence":"additional","affiliation":[]},{"given":"Lara","family":"Faoro","sequence":"additional","affiliation":[]},{"given":"Edward","family":"Farhi","sequence":"additional","affiliation":[]},{"given":"Vinicius S.","family":"Ferreira","sequence":"additional","affiliation":[]},{"given":"Leslie Flores","family":"Burgos","sequence":"additional","affiliation":[]},{"given":"Ebrahim","family":"Forati","sequence":"additional","affiliation":[]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3732-168X","authenticated-orcid":false,"given":"Brooks","family":"Foxen","sequence":"additional","affiliation":[]},{"given":"Suhas","family":"Ganjam","sequence":"additional","affiliation":[]},{"given":"Gonzalo","family":"Garcia","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Gasca","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2226-1813","authenticated-orcid":false,"given":"Élie","family":"Genois","sequence":"additional","affiliation":[]},{"given":"William","family":"Giang","sequence":"additional","affiliation":[]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[]},{"given":"Dar","family":"Gilboa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3025-7312","authenticated-orcid":false,"given":"Raja","family":"Gosula","sequence":"additional","affiliation":[]},{"given":"Alejandro Grajales","family":"Dau","sequence":"additional","affiliation":[]},{"given":"Dietrich","family":"Graumann","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4034-8744","authenticated-orcid":false,"given":"Alex","family":"Greene","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":false,"given":"Jonathan A.","family":"Gross","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9867-2371","authenticated-orcid":false,"given":"Steve","family":"Habegger","sequence":"additional","affiliation":[]},{"given":"John","family":"Hall","sequence":"additional","affiliation":[]},{"given":"Michael C.","family":"Hamilton","sequence":"additional","affiliation":[]},{"given":"Monica","family":"Hansen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9412-0553","authenticated-orcid":false,"given":"Matthew P.","family":"Harrigan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0521-8378","authenticated-orcid":false,"given":"Sean D.","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"Francisco J. H.","family":"Heras","sequence":"additional","affiliation":[]},{"given":"Stephen","family":"Heslin","sequence":"additional","affiliation":[]},{"given":"Paula","family":"Heu","sequence":"additional","affiliation":[]},{"given":"Oscar","family":"Higgott","sequence":"additional","affiliation":[]},{"given":"Gordon","family":"Hill","sequence":"additional","affiliation":[]},{"given":"Jeremy","family":"Hilton","sequence":"additional","affiliation":[]},{"given":"George","family":"Holland","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7421-5635","authenticated-orcid":false,"given":"Sabrina","family":"Hong","sequence":"additional","affiliation":[]},{"given":"Hsin-Yuan","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Ashley","family":"Huff","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2735-1380","authenticated-orcid":false,"given":"William J.","family":"Huggins","sequence":"additional","affiliation":[]},{"given":"Lev B.","family":"Ioffe","sequence":"additional","affiliation":[]},{"given":"Sergei V.","family":"Isakov","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Iveland","sequence":"additional","affiliation":[]},{"given":"Evan","family":"Jeffrey","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0435-655X","authenticated-orcid":false,"given":"Zhang","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Cody","family":"Jones","sequence":"additional","affiliation":[]},{"given":"Stephen","family":"Jordan","sequence":"additional","affiliation":[]},{"given":"Chaitali","family":"Joshi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8751-4458","authenticated-orcid":false,"given":"Pavol","family":"Juhas","sequence":"additional","affiliation":[]},{"given":"Dvir","family":"Kafri","sequence":"additional","affiliation":[]},{"given":"Hui","family":"Kang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7400-9604","authenticated-orcid":false,"given":"Amir H.","family":"Karamlou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0136-1428","authenticated-orcid":false,"given":"Kostyantyn","family":"Kechedzhi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2596-2121","authenticated-orcid":false,"given":"Julian","family":"Kelly","sequence":"additional","affiliation":[]},{"given":"Trupti","family":"Khaire","sequence":"additional","affiliation":[]},{"given":"Tanuj","family":"Khattar","sequence":"additional","affiliation":[]},{"given":"Mostafa","family":"Khezri","sequence":"additional","affiliation":[]},{"given":"Seon","family":"Kim","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9481-4456","authenticated-orcid":false,"given":"Paul V.","family":"Klimov","sequence":"additional","affiliation":[]},{"given":"Andrey R.","family":"Klots","sequence":"additional","affiliation":[]},{"given":"Bryce","family":"Kobrin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7466-7997","authenticated-orcid":false,"given":"Pushmeet","family":"Kohli","sequence":"additional","affiliation":[]},{"given":"Alexander N.","family":"Korotkov","sequence":"additional","affiliation":[]},{"given":"Fedor","family":"Kostritsa","sequence":"additional","affiliation":[]},{"given":"Robin","family":"Kothari","sequence":"additional","affiliation":[]},{"given":"Borislav","family":"Kozlovskii","sequence":"additional","affiliation":[]},{"given":"John Mark","family":"Kreikebaum","sequence":"additional","affiliation":[]},{"given":"Vladislav D.","family":"Kurilovich","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2002-9495","authenticated-orcid":false,"given":"Nathan","family":"Lacroix","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9804-2185","authenticated-orcid":false,"given":"David","family":"Landhuis","sequence":"additional","affiliation":[]},{"given":"Tiano","family":"Lange-Dei","sequence":"additional","affiliation":[]},{"given":"Brandon W.","family":"Langley","sequence":"additional","affiliation":[]},{"given":"Pavel","family":"Laptev","sequence":"additional","affiliation":[]},{"given":"Kim-Ming","family":"Lau","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3110-3371","authenticated-orcid":false,"given":"Loïck","family":"Le Guevel","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Ledford","sequence":"additional","affiliation":[]},{"given":"Joonho","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Kenny","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Yuri D.","family":"Lensky","sequence":"additional","affiliation":[]},{"given":"Shannon","family":"Leon","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4297-2508","authenticated-orcid":false,"given":"Brian J.","family":"Lester","sequence":"additional","affiliation":[]},{"given":"Wing Yan","family":"Li","sequence":"additional","affiliation":[]},{"given":"Yin","family":"Li","sequence":"additional","affiliation":[]},{"given":"Alexander T.","family":"Lill","sequence":"additional","affiliation":[]},{"given":"Wayne","family":"Liu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8399-0975","authenticated-orcid":false,"given":"William P.","family":"Livingston","sequence":"additional","affiliation":[]},{"given":"Aditya","family":"Locharla","sequence":"additional","affiliation":[]},{"given":"Erik","family":"Lucero","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Lundahl","sequence":"additional","affiliation":[]},{"given":"Aaron","family":"Lunt","sequence":"additional","affiliation":[]},{"given":"Sid","family":"Madhuk","sequence":"additional","affiliation":[]},{"given":"Fionn D.","family":"Malone","sequence":"additional","affiliation":[]},{"given":"Ashley","family":"Maloney","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3908-2694","authenticated-orcid":false,"given":"Salvatore","family":"Mandrà","sequence":"additional","affiliation":[]},{"given":"James","family":"Manyika","sequence":"additional","affiliation":[]},{"given":"Leigh S.","family":"Martin","sequence":"additional","affiliation":[]},{"given":"Orion","family":"Martin","sequence":"additional","affiliation":[]},{"given":"Steven","family":"Martin","sequence":"additional","affiliation":[]},{"given":"Cameron","family":"Maxfield","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2809-0509","authenticated-orcid":false,"given":"Jarrod R.","family":"McClean","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9544-141X","authenticated-orcid":false,"given":"Matt","family":"McEwen","sequence":"additional","affiliation":[]},{"given":"Seneca","family":"Meeks","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6371-6140","authenticated-orcid":false,"given":"Anthony","family":"Megrant","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0507-0211","authenticated-orcid":false,"given":"Xiao","family":"Mi","sequence":"additional","affiliation":[]},{"given":"Kevin C.","family":"Miao","sequence":"additional","affiliation":[]},{"given":"Amanda","family":"Mieszala","sequence":"additional","affiliation":[]},{"given":"Reza","family":"Molavi","sequence":"additional","affiliation":[]},{"given":"Sebastian","family":"Molina","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7124-4585","authenticated-orcid":false,"given":"Shirin","family":"Montazeri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9515-6184","authenticated-orcid":false,"given":"Alexis","family":"Morvan","sequence":"additional","affiliation":[]},{"given":"Ramis","family":"Movassagh","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8497-6363","authenticated-orcid":false,"given":"Wojciech","family":"Mruczkiewicz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7760-9186","authenticated-orcid":false,"given":"Ofer","family":"Naaman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5548-0051","authenticated-orcid":false,"given":"Matthew","family":"Neeley","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8509-9836","authenticated-orcid":false,"given":"Charles","family":"Neill","sequence":"additional","affiliation":[]},{"given":"Ani","family":"Nersisyan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9681-6746","authenticated-orcid":false,"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7246-6761","authenticated-orcid":false,"given":"Jiun How","family":"Ng","sequence":"additional","affiliation":[]},{"given":"Anthony","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"Murray","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"Chia-Hung","family":"Ni","sequence":"additional","affiliation":[]},{"given":"Murphy Yuezhen","family":"Niu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8406-6626","authenticated-orcid":false,"given":"Thomas E.","family":"O’Brien","sequence":"additional","affiliation":[]},{"given":"William D.","family":"Oliver","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Opremcak","sequence":"additional","affiliation":[]},{"given":"Kristoffer","family":"Ottosson","sequence":"additional","affiliation":[]},{"given":"Andre","family":"Petukhov","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Pizzuto","sequence":"additional","affiliation":[]},{"given":"John","family":"Platt","sequence":"additional","affiliation":[]},{"given":"Rebecca","family":"Potter","sequence":"additional","affiliation":[]},{"given":"Orion","family":"Pritchard","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]},{"given":"Chris","family":"Quintana","sequence":"additional","affiliation":[]},{"given":"Ganesh","family":"Ramachandran","sequence":"additional","affiliation":[]},{"given":"Matthew J.","family":"Reagor","sequence":"additional","affiliation":[]},{"given":"John","family":"Redding","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3788-7947","authenticated-orcid":false,"given":"David M.","family":"Rhodes","sequence":"additional","affiliation":[]},{"given":"Gabrielle","family":"Roberts","sequence":"additional","affiliation":[]},{"given":"Eliott","family":"Rosenberg","sequence":"additional","affiliation":[]},{"given":"Emma","family":"Rosenfeld","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1917-3879","authenticated-orcid":false,"given":"Pedram","family":"Roushan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3963-1830","authenticated-orcid":false,"given":"Nicholas C.","family":"Rubin","sequence":"additional","affiliation":[]},{"given":"Negar","family":"Saei","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8188-364X","authenticated-orcid":false,"given":"Daniel","family":"Sank","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8930-7081","authenticated-orcid":false,"given":"Kannan","family":"Sankaragomathi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5865-0813","authenticated-orcid":false,"given":"Kevin J.","family":"Satzinger","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3210-5929","authenticated-orcid":false,"given":"Henry F.","family":"Schurkus","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Schuster","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2401-5691","authenticated-orcid":false,"given":"Andrew W.","family":"Senior","sequence":"additional","affiliation":[]},{"given":"Michael J.","family":"Shearn","sequence":"additional","affiliation":[]},{"given":"Aaron","family":"Shorter","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6035-2812","authenticated-orcid":false,"given":"Noah","family":"Shutty","sequence":"additional","affiliation":[]},{"given":"Vladimir","family":"Shvarts","sequence":"additional","affiliation":[]},{"given":"Shraddha","family":"Singh","sequence":"additional","affiliation":[]},{"given":"Volodymyr","family":"Sivak","sequence":"additional","affiliation":[]},{"given":"Jindra","family":"Skruzny","sequence":"additional","affiliation":[]},{"given":"Spencer","family":"Small","sequence":"additional","affiliation":[]},{"given":"Vadim","family":"Smelyanskiy","sequence":"additional","affiliation":[]},{"given":"W. Clarke","family":"Smith","sequence":"additional","affiliation":[]},{"given":"Rolando D.","family":"Somma","sequence":"additional","affiliation":[]},{"given":"Sofia","family":"Springer","sequence":"additional","affiliation":[]},{"given":"George","family":"Sterling","sequence":"additional","affiliation":[]},{"given":"Doug","family":"Strain","sequence":"additional","affiliation":[]},{"given":"Jordan","family":"Suchard","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1127-2111","authenticated-orcid":false,"given":"Aaron","family":"Szasz","sequence":"additional","affiliation":[]},{"given":"Alex","family":"Sztein","sequence":"additional","affiliation":[]},{"given":"Douglas","family":"Thor","sequence":"additional","affiliation":[]},{"given":"Alfredo","family":"Torres","sequence":"additional","affiliation":[]},{"given":"M. Mert","family":"Torunbalci","sequence":"additional","affiliation":[]},{"given":"Abeer","family":"Vaishnav","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Vargas","sequence":"additional","affiliation":[]},{"given":"Sergey","family":"Vdovichev","sequence":"additional","affiliation":[]},{"given":"Guifre","family":"Vidal","sequence":"additional","affiliation":[]},{"given":"Benjamin","family":"Villalonga","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6513-9885","authenticated-orcid":false,"given":"Catherine Vollgraff","family":"Heidweiller","sequence":"additional","affiliation":[]},{"given":"Steven","family":"Waltman","sequence":"additional","affiliation":[]},{"given":"Shannon X.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Brayden","family":"Ware","sequence":"additional","affiliation":[]},{"given":"Kate","family":"Weber","sequence":"additional","affiliation":[]},{"given":"Travis","family":"Weidel","sequence":"additional","affiliation":[]},{"given":"Theodore","family":"White","sequence":"additional","affiliation":[]},{"given":"Kristi","family":"Wong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7233-1576","authenticated-orcid":false,"given":"Bryan W. K.","family":"Woo","sequence":"additional","affiliation":[]},{"given":"Cheng","family":"Xing","sequence":"additional","affiliation":[]},{"given":"Z. Jamie","family":"Yao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0837-1028","authenticated-orcid":false,"given":"Ping","family":"Yeh","sequence":"additional","affiliation":[]},{"given":"Bicheng","family":"Ying","sequence":"additional","affiliation":[]},{"given":"Juhwan","family":"Yoo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4169-5745","authenticated-orcid":false,"given":"Noureldin","family":"Yosri","sequence":"additional","affiliation":[]},{"given":"Grayson","family":"Young","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2585-2424","authenticated-orcid":false,"given":"Adam","family":"Zalcman","sequence":"additional","affiliation":[]},{"given":"Yaxing","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3037-2003","authenticated-orcid":false,"given":"Ningfeng","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3146-7263","authenticated-orcid":false,"given":"Nicholas","family":"Zobrist","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,9]]},"URL":"http://dx.doi.org/10.1038/s41586-024-08449-y","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:8449\narxivid:2408.13687"},{"id":"arxiv:2309.02863","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41567-024-02696-6","page":"161-167","source":"Crossref","title":"Nishimori transition across the error threshold for constant-depth quantum circuits","volume":"21","author":[{"ORCID":"https://orcid.org/0000-0001-7217-7137","authenticated-orcid":false,"given":"Edward H.","family":"Chen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4513-0795","authenticated-orcid":false,"given":"Guo-Yi","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":false,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5419-5999","authenticated-orcid":false,"given":"Alireza","family":"Seif","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1882-8693","authenticated-orcid":false,"given":"Elisa","family":"Bäumer","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1292-9007","authenticated-orcid":false,"given":"David","family":"Layden","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":false,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-1179-2513","authenticated-orcid":false,"given":"Sarah","family":"Sheldon","sequence":"additional","affiliation":[]},{"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1479-9736","authenticated-orcid":false,"given":"Simon","family":"Trebst","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2566-1388","authenticated-orcid":false,"given":"Abhinav","family":"Kandala","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,16]]},"URL":"http://dx.doi.org/10.1038/s41567-024-02696-6","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2696\narxivid:2309.02863"},{"id":"arxiv:2411.14638","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>The resource overhead required to achieve net computational benefits from quantum error correction (QEC) limits its utility while current systems remain constrained in size, despite exceptional progress in experimental demonstrations. In this paper, we demonstrate that the strategic application of QEC primitives without logical encoding can yield significant advantages on superconducting processors—relative to any alternative error-reduction strategy—while only requiring a modest overhead. We first present a novel protocol for implementing long-range  gates that relies on a unitarily prepared Greenberger-Horne-Zeilinger (GHZ) state as well as a unitary disentangling step; the protocol natively introduces an error-detection process using the disentangled qubits as flags. We demonstrate that it achieves state-of-the-art gate fidelities of over 85% across up to 40 lattice sites, significantly and consistently outperforming the best alternative measurement-based protocol without introducing any additional ancilla qubits. We then apply sparse stabilizer measurements to generate large GHZ states by detecting bit-flip and amplitude-damping errors. Employing this technique in combination with deterministic error suppression, we generate a 75-qubit GHZ state exhibiting genuine multipartite entanglement, the largest reported to date. The generation requires no more than 9 ancilla qubits and the fraction of samples discarded due to errors grows no higher than 78%, far lower than previous discard fractions required for tests using comparable numbers of fully encoded qubits. This work in total represents compelling evidence that adopting QEC primitives on current-generation devices can deliver substantial net benefits.</jats:p>","DOI":"10.1103/prxquantum.6.020331","source":"Crossref","title":"Achieving Computational Gains with Quantum Error-Correction Primitives: Generation of Long-Range Entanglement Enhanced by Error Detection","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-6399-006X","authenticated-orcid":true,"given":"Haoran","family":"Liao","sequence":"first","affiliation":[]},{"given":"Gavin S.","family":"Hartnett","sequence":"additional","affiliation":[]},{"given":"Ashish","family":"Kakkar","sequence":"additional","affiliation":[]},{"given":"Adrian","family":"Tan","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Hush","sequence":"additional","affiliation":[]},{"given":"Pranav S.","family":"Mundada","sequence":"additional","affiliation":[]},{"given":"Michael J.","family":"Biercuk","sequence":"additional","affiliation":[]},{"given":"Yuval","family":"Baum","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,16]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.020331","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020331","note":"arxivid:2411.14638"},{"id":"arxiv:2501.04612","type":"journal-article","publisher":"Springer Science and Business Media LLC","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    Quantum error correction is needed for quantum computers to be capable of executing algorithms using hundreds of logical qubits in a fault-tolerant manner. Recent experiments have progressed towards this by demonstrating sufficiently low error rates for state preservation of a single logical qubit. However, quantum computation algorithms also require that these logical qubits can be entangled and that gate operations can be performed on them. Lattice surgery is a technique that offers a practical approach for implementing such gates, particularly in planar quantum processor layouts. Here we demonstrate lattice surgery between two distance-three repetition-code qubits by splitting a single distance-three surface-code qubit. Using a quantum circuit that is fault-tolerant for bit-flip errors, we achieve an improvement in the value of the decoded\n                    <jats:italic>Z</jats:italic>\n                    <jats:italic>Z</jats:italic>\n                    logical two-qubit observable compared with a similar non-encoded circuit. We therefore demonstrate the functional building blocks needed for lattice-surgery operations on larger-distance codes based on superconducting circuits.\n                  </jats:p>","DOI":"10.1038/s41567-025-03090-6","source":"Crossref","title":"Lattice surgery realized on two distance-three repetition codes with superconducting qubits","author":[{"ORCID":"https://orcid.org/0000-0001-6845-5624","authenticated-orcid":false,"given":"Ilya","family":"Besedin","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0009-0003-6172-1100","authenticated-orcid":false,"given":"Michael","family":"Kerschbaum","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0006-2427-8404","authenticated-orcid":false,"given":"Jonathan","family":"Knoll","sequence":"additional","affiliation":[]},{"given":"Ian","family":"Hesner","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9202-5659","authenticated-orcid":false,"given":"Lukas","family":"Bödeker","sequence":"additional","affiliation":[]},{"given":"Luis","family":"Colmenarez","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0006-1692-1025","authenticated-orcid":false,"given":"Luca","family":"Hofele","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2002-9495","authenticated-orcid":false,"given":"Nathan","family":"Lacroix","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9929-9684","authenticated-orcid":false,"given":"Christoph","family":"Hellings","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0000-2823-8246","authenticated-orcid":false,"given":"François","family":"Swiadek","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Flasby","sequence":"additional","affiliation":[]},{"given":"Mohsen","family":"Bahrami Panah","sequence":"additional","affiliation":[]},{"given":"Dante","family":"Colao Zanuz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":false,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3476-4485","authenticated-orcid":false,"given":"Andreas","family":"Wallraff","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,30]]},"URL":"http://dx.doi.org/10.1038/s41567-025-03090-6","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:3090\narxivid:2501.04612"},{"id":"arxiv:2506.20529","type":"article-journal","author":[{"given":"Alena S.","family":"Kazmina"},{"given":"Artyom M.","family":"Polyanskiy"},{"given":"Elena Yu.","family":"Egorova"},{"given":"Nikolay N.","family":"Abramov"},{"given":"Daria A.","family":"Kalacheva"},{"given":"Viktor B.","family":"Lubsanov"},{"given":"Aleksey N.","family":"Bolgar"},{"given":"Ilya A.","family":"Simakov"}],"title":"Realization of a Quantum Error Detection Code with a Dynamically Reassigned Ancillary Qubit","issued":{"date-parts":[[2025,6,25]]},"note":"arxivid:2506.20529\narxiv_version_number:1"},{"id":"arxiv:2510.09520","type":"article-journal","author":[{"given":"Ali","family":"Javadi-Abhari"},{"given":"Simon","family":"Martiel"},{"given":"Alireza","family":"Seif"},{"given":"Maika","family":"Takita"},{"given":"Ken X.","family":"Wei"}],"title":"Big cats: entanglement in 120 qubits and beyond","issued":{"date-parts":[[2025,10,10]]},"note":"arxivid:2510.09520\narxiv_version_number:1"},{"id":"arxiv:2107.11398","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The storage and processing of quantum information are susceptible to external noise, resulting in computational errors. A powerful method to suppress these effects is quantum error correction. Typically, quantum error correction is executed in discrete rounds, using entangling gates and projective measurement on ancillary qubits to complete each round of error correction. Here we use direct parity measurements to implement a continuous quantum bit-flip correction code in a resource-efficient manner, eliminating entangling gates, ancillary qubits, and their associated errors. An FPGA controller actively corrects errors as they are detected, achieving an average bit-flip detection efficiency of up to 91%. Furthermore, the protocol increases the relaxation time of the protected logical qubit by a factor of 2.7 over the relaxation times of the bare comprising qubits. Our results showcase resource-efficient stabilizer measurements in a multi-qubit architecture and demonstrate how continuous error correction codes can address challenges in realizing a fault-tolerant system.</jats:p>","DOI":"10.1038/s41467-022-29906-0","source":"Crossref","title":"Experimental demonstration of continuous quantum error correction","volume":"13","author":[{"ORCID":"https://orcid.org/0000-0001-8399-0975","authenticated-orcid":false,"given":"William P.","family":"Livingston","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9736-4158","authenticated-orcid":false,"given":"Machiel S.","family":"Blok","sequence":"additional","affiliation":[]},{"given":"Emmanuel","family":"Flurin","sequence":"additional","affiliation":[]},{"given":"Justin","family":"Dressel","sequence":"additional","affiliation":[]},{"given":"Andrew N.","family":"Jordan","sequence":"additional","affiliation":[]},{"given":"Irfan","family":"Siddiqi","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,28]]},"URL":"http://dx.doi.org/10.1038/s41467-022-29906-0","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"2307","note":"alternative-id:29906\narxivid:2107.11398"},{"id":"arxiv:2201.08581","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7924","abstract":"<jats:title>Abstract</jats:title><jats:p>Future large-scale quantum computers will rely on quantum error correction (QEC) to protect the fragile quantum information during computation<jats:sup>1,2</jats:sup>. Among the possible candidate platforms for realizing quantum computing devices, the compatibility with mature nanofabrication technologies of silicon-based spin qubits offers promise to overcome the challenges in scaling up device sizes from the prototypes of today to large-scale computers<jats:sup>3–5</jats:sup>. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems<jats:sup>6–8</jats:sup>. However, the demonstration of QEC, which requires three or more coupled qubits<jats:sup>1</jats:sup>, and involves a three-qubit gate<jats:sup>9–11</jats:sup> or measurement-based feedback, remains an open challenge. Here we demonstrate a three-qubit phase-correcting code in silicon, in which an encoded three-qubit state is protected against any phase-flip error on one of the three qubits. The correction to this encoded state is performed by a three-qubit conditional rotation, which we implement by an efficient single-step resonantly driven iToffoli gate. As expected, the error correction mitigates the errors owing to one-qubit phase-flip, as well as the intrinsic dephasing mainly owing to quasi-static phase noise. These results show successful implementation of QEC and the potential of a silicon-based platform for large-scale quantum computing.</jats:p>","DOI":"10.1038/s41586-022-04986-6","page":"682-686","source":"Crossref","title":"Quantum error correction with silicon spin qubits","volume":"608","author":[{"ORCID":"https://orcid.org/0000-0003-1240-1103","authenticated-orcid":false,"given":"Kenta","family":"Takeda","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9145-0303","authenticated-orcid":false,"given":"Akito","family":"Noiri","sequence":"additional","affiliation":[]},{"given":"Takashi","family":"Nakajima","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2841-8129","authenticated-orcid":false,"given":"Takashi","family":"Kobayashi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7465-0135","authenticated-orcid":false,"given":"Seigo","family":"Tarucha","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,24]]},"URL":"http://dx.doi.org/10.1038/s41586-022-04986-6","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:4986\narxivid:2201.08581"},{"id":"arxiv:2202.11530","type":"article-journal","author":[{"given":"F.","family":"van Riggelen"},{"given":"W. I. L.","family":"Lawrie"},{"given":"M.","family":"Russ"},{"given":"N. W.","family":"Hendrickx"},{"given":"A.","family":"Sammak"},{"given":"M.","family":"Rispler"},{"given":"B. M.","family":"Terhal"},{"given":"G.","family":"Scappucci"},{"given":"M.","family":"Veldhorst"}],"title":"Phase flip code with semiconductor spin qubits","issued":{"date-parts":[[2022,2,23]]},"note":"arxivid:2202.11530\narxiv_version_number:1"},{"id":"arxiv:1309.6424","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7487","DOI":"10.1038/nature12919","page":"204-207","source":"Crossref","title":"Quantum error correction in a solid-state hybrid spin register","volume":"506","author":[{"given":"G.","family":"Waldherr","sequence":"first","affiliation":[]},{"given":"Y.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"S.","family":"Zaiser","sequence":"additional","affiliation":[]},{"given":"M.","family":"Jamali","sequence":"additional","affiliation":[]},{"given":"T.","family":"Schulte-Herbrüggen","sequence":"additional","affiliation":[]},{"given":"H.","family":"Abe","sequence":"additional","affiliation":[]},{"given":"T.","family":"Ohshima","sequence":"additional","affiliation":[]},{"given":"J.","family":"Isoya","sequence":"additional","affiliation":[]},{"given":"J. F.","family":"Du","sequence":"additional","affiliation":[]},{"given":"P.","family":"Neumann","sequence":"additional","affiliation":[]},{"given":"J.","family":"Wrachtrup","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,12]]},"URL":"http://dx.doi.org/10.1038/nature12919","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature12919\narxivid:1309.6424"},{"id":"doi:10.1038/s42005-022-00875-6","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Fault-tolerant quantum memory plays a key role in interfacing quantum computers with quantum networks to construct quantum computer networks. Manipulation of spin quantum memory generally requires a magnetic field, which hinders the integration with superconducting qubits. Completely zero-field operation is desirable for scaling up a quantum computer based on superconducting qubits. Here we demonstrate quantum error correction to protect the nuclear spin of the nitrogen as a quantum memory in a diamond nitrogen-vacancy center with two nuclear spins of the surrounding carbon isotopes under a zero magnetic field. The quantum error correction makes quantum memory resilient against operational or environmental errors without the need for magnetic fields and opens a way toward distributed quantum computation and a quantum internet with memory-based quantum interfaces or quantum repeaters.</jats:p>","DOI":"10.1038/s42005-022-00875-6","source":"Crossref","title":"Quantum error correction of spin quantum memories in diamond under a zero magnetic field","volume":"5","author":[{"given":"Takaya","family":"Nakazato","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4880-1654","authenticated-orcid":false,"given":"Raustin","family":"Reyes","sequence":"additional","affiliation":[]},{"given":"Nobuaki","family":"Imaike","sequence":"additional","affiliation":[]},{"given":"Kazuyasu","family":"Matsuda","sequence":"additional","affiliation":[]},{"given":"Kazuya","family":"Tsurumoto","sequence":"additional","affiliation":[]},{"given":"Yuhei","family":"Sekiguchi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3778-7236","authenticated-orcid":false,"given":"Hideo","family":"Kosaka","sequence":"additional","affiliation":[]}],"container-title":"Communications Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,27]]},"URL":"http://dx.doi.org/10.1038/s42005-022-00875-6","ISSN":["2399-3650"],"container-title-short":"Commun Phys","page":"102","note":"alternative-id:875"},{"id":"arxiv:1309.5452","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1038/nnano.2014.2","page":"171-176","source":"Crossref","title":"Universal control and error correction in multi-qubit spin registers in diamond","volume":"9","author":[{"given":"T. H.","family":"Taminiau","sequence":"first","affiliation":[]},{"given":"J.","family":"Cramer","sequence":"additional","affiliation":[]},{"given":"T.","family":"van der Sar","sequence":"additional","affiliation":[]},{"given":"V. V.","family":"Dobrovitski","sequence":"additional","affiliation":[]},{"given":"R.","family":"Hanson","sequence":"additional","affiliation":[]}],"container-title":"Nature Nanotechnology","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,2]]},"URL":"http://dx.doi.org/10.1038/nnano.2014.2","ISSN":["1748-3387","1748-3395"],"container-title-short":"Nature Nanotech","note":"alternative-id:BFnnano20142\narxivid:1309.5452"},{"id":"arxiv:quant-ph/0512170","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.74.052322","source":"Crossref","title":"Error-correcting codes for adiabatic quantum computation","volume":"74","author":[{"given":"Stephen P.","family":"Jordan","sequence":"first","affiliation":[]},{"given":"Edward","family":"Farhi","sequence":"additional","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,11,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.74.052322","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052322","note":"arxivid:quant-ph/0512170"},{"id":"arxiv:1307.8190","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/ncomms4243","source":"Crossref","title":"Error-corrected quantum annealing with hundreds of qubits","volume":"5","author":[{"given":"Kristen L.","family":"Pudenz","sequence":"first","affiliation":[]},{"given":"Tameem","family":"Albash","sequence":"additional","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,6]]},"URL":"http://dx.doi.org/10.1038/ncomms4243","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"3243","note":"alternative-id:BFncomms4243\narxivid:1307.8190"},{"id":"arxiv:1408.4382","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.91.042302","source":"Crossref","title":"Quantum annealing correction for random Ising problems","volume":"91","author":[{"given":"Kristen L.","family":"Pudenz","sequence":"first","affiliation":[]},{"given":"Tameem","family":"Albash","sequence":"additional","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.042302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042302","note":"arxivid:1408.4382"},{"id":"arxiv:2506.09936","type":"article-journal","author":[{"given":"J. A.","family":"Muniz"},{"given":"D.","family":"Crow"},{"given":"H.","family":"Kim"},{"given":"J. M.","family":"Kindem"},{"given":"W. B.","family":"Cairncross"},{"given":"A.","family":"Ryou"},{"given":"T. C.","family":"Bohdanowicz"},{"given":"C. -A.","family":"Chen"},{"given":"Y.","family":"Ji"},{"given":"A. M. W.","family":"Jones"},{"given":"E.","family":"Megidish"},{"given":"C.","family":"Nishiguchi"},{"given":"M.","family":"Urbanek"},{"given":"L.","family":"Wadleigh"},{"given":"T.","family":"Wilkason"},{"given":"D.","family":"Aasen"},{"given":"K.","family":"Barnes"},{"given":"J. M.","family":"Bello-Rivas"},{"given":"I.","family":"Bloomfield"},{"given":"G.","family":"Booth"},{"given":"A.","family":"Brown"},{"given":"M. O.","family":"Brown"},{"given":"K.","family":"Cassella"},{"given":"G.","family":"Cowan"},{"given":"J.","family":"Epstein"},{"given":"M.","family":"Feldkamp"},{"given":"C.","family":"Griger"},{"given":"Y.","family":"Hassan"},{"given":"A.","family":"Heinz"},{"given":"E.","family":"Halperin"},{"given":"T.","family":"Hofler"},{"given":"F.","family":"Hummel"},{"given":"M.","family":"Jaffe"},{"given":"E.","family":"Kapit"},{"given":"K.","family":"Kotru"},{"given":"J.","family":"Lauigan"},{"given":"J.","family":"Marjanovic"},{"given":"M.","family":"Meredith"},{"given":"M.","family":"McDonald"},{"given":"R.","family":"Morshead"},{"given":"S.","family":"Narayanaswami"},{"given":"K. A.","family":"Pawlak"},{"given":"K. L.","family":"Pudenz"},{"given":"D. Rodríguez","family":"Pérez"},{"given":"P.","family":"Sabharwal"},{"given":"J.","family":"Simon"},{"given":"A.","family":"Smull"},{"given":"M.","family":"Sorensen"},{"given":"D. T.","family":"Stack"},{"given":"M.","family":"Stone"},{"given":"L.","family":"Taneja"},{"given":"R. J. M.","family":"van de Veerdonk"},{"given":"Z.","family":"Vendeiro"},{"given":"R. T.","family":"Weverka"},{"given":"K.","family":"White"},{"given":"T. -Y.","family":"Wu"},{"given":"X.","family":"Xie"},{"given":"E.","family":"Zalys-Geller"},{"given":"X.","family":"Zhang"},{"given":"J.","family":"King"},{"given":"B. J.","family":"Bloom"},{"given":"M. A.","family":"Norcia"}],"title":"Repeated ancilla reuse for logical computation on a neutral atom quantum computer","issued":{"date-parts":[[2025,6,11]]},"note":"arxivid:2506.09936\narxiv_version_number:1"},{"id":"arxiv:2202.11045","type":"article-journal","author":[{"given":"Teague","family":"Tomesh"},{"given":"Pranav","family":"Gokhale"},{"given":"Victory","family":"Omole"},{"given":"Gokul Subramanian","family":"Ravi"},{"given":"Kaitlin N.","family":"Smith"},{"given":"Joshua","family":"Viszlai"},{"given":"Xin-Chuan","family":"Wu"},{"given":"Nikos","family":"Hardavellas"},{"given":"Margaret R.","family":"Martonosi"},{"given":"Frederic T.","family":"Chong"}],"title":"SupermarQ: A Scalable Quantum Benchmark Suite","issued":{"date-parts":[[2022,4,27]]},"note":"arxivid:2202.11045\narxiv_version_number:3"},{"id":"arxiv:1410.3720","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.115.020502","source":"Crossref","title":"From Three-Photon Greenberger-Horne-Zeilinger States to Ballistic Universal Quantum Computation","volume":"115","author":[{"given":"Mercedes","family":"Gimeno-Segovia","sequence":"first","affiliation":[]},{"given":"Pete","family":"Shadbolt","sequence":"additional","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]},{"given":"Terry","family":"Rudolph","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2015,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.115.020502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020502","note":"arxivid:1410.3720"},{"id":"arxiv:1502.07437","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.114.113603","source":"Crossref","title":"Nearly Deterministic Bell Measurement for Multiphoton Qubits and its Application to Quantum Information Processing","volume":"114","author":[{"given":"Seung-Woo","family":"Lee","sequence":"first","affiliation":[]},{"given":"Kimin","family":"Park","sequence":"additional","affiliation":[]},{"given":"Timothy C.","family":"Ralph","sequence":"additional","affiliation":[]},{"given":"Hyunseok","family":"Jeong","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2015,3,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.114.113603","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"113603","note":"arxivid:1502.07437"},{"id":"arxiv:2109.12280","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.3.030309","source":"Crossref","title":"All-Photonic Architecture for Scalable Quantum Computing with Greenberger-Horne-Zeilinger States","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-3112-9101","authenticated-orcid":true,"given":"Srikrishna","family":"Omkar","sequence":"first","affiliation":[]},{"given":"Seok-Hyung","family":"Lee","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1766-6402","authenticated-orcid":true,"given":"Yong Siah","family":"Teo","sequence":"additional","affiliation":[]},{"given":"Seung-Woo","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Hyunseok","family":"Jeong","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,18]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.030309","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030309","note":"arxivid:2109.12280"},{"id":"arxiv:2410.12963","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Photonics provides a viable path to a scalable fault-tolerant quantum computer. The natural framework for this platform is measurement-based quantum computation, where fault-tolerant graph states supersede traditional quantum error-correcting codes. However, the existing formalism for foliation—the construction of fault-tolerant graph states—does not reveal how certain properties, such as single-shot error correction, manifest in the measurement-based setting. We introduce the fault complex, a representation of dynamic quantum error-correction protocols particularly well suited to describe foliation. Our approach enables precise computation of fault tolerance properties of foliated codes and provides insights into circuit-based quantum computation. Analyzing the fault complex yields improved thresholds for three- and four-dimensional toric codes, a generalization of stability experiments, and the existence of single-shot lattice surgery with higher-dimensional topological codes.</jats:p>","DOI":"10.1103/cjb4-l57n","source":"Crossref","title":"Single-shot and measurement-based quantum error correction via fault complexes","volume":"112","author":[{"ORCID":"https://orcid.org/0000-0002-1476-0647","authenticated-orcid":true,"given":"Timo","family":"Hillmann","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"},{"id":[{"id":"https://ror.org/040wg7k59","id-type":"ROR","asserted-by":"publisher"}],"name":"Chalmers University of Technology"}]},{"given":"Guillaume","family":"Dauphinais","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"ORCID":"https://orcid.org/0000-0001-5201-3987","authenticated-orcid":true,"given":"Ilan","family":"Tzitrin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":true,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"},{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute"},{"id":[{"id":"https://ror.org/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Computing, Waterloo"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,9]]},"URL":"http://dx.doi.org/10.1103/cjb4-l57n","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"L040401","note":"arxivid:2410.12963"},{"id":"arxiv:2403.13732","type":"article-journal","author":[{"given":"Ivan","family":"Pogorelov"},{"given":"Friederike","family":"Butt"},{"given":"Lukas","family":"Postler"},{"given":"Christian D.","family":"Marciniak"},{"given":"Philipp","family":"Schindler"},{"given":"Markus","family":"Müller"},{"given":"Thomas","family":"Monz"}],"title":"Experimental fault-tolerant code switching","issued":{"date-parts":[[2024,3,20]]},"note":"arxivid:2403.13732\narxiv_version_number:1"},{"id":"arxiv:1302.1035","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2013.6620283","page":"534-538","source":"Crossref","title":"Leveraging automorphisms of quantum codes for fault-tolerant quantum computation","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Martin","family":"Roetteler","sequence":"additional","affiliation":[]}],"event":"2013 IEEE International Symposium on Information Theory (ISIT)","container-title":"2013 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2013,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2013.6620283","note":"arxivid:1302.1035"},{"id":"arxiv:quant-ph/9704019","type":"article-journal","author":[{"given":"F.","family":"Vatan"},{"given":"V. P.","family":"Roychowdhury"},{"given":"M. P.","family":"Anantram"}],"title":"Spatially Correlated Qubit Errors and Burst-Correcting Quantum Codes","issued":{"date-parts":[[1997,5,16]]},"note":"arxivid:quant-ph/9704019\narxiv_version_number:2"},{"id":"arxiv:2211.00891","type":"article-journal","author":[{"given":"Reza","family":"Dastbasteh"},{"given":"Petr","family":"Lisonek"}],"title":"New quantum codes from self-dual codes over F_4","issued":{"date-parts":[[2022,11,2]]},"note":"arxivid:2211.00891\narxiv_version_number:1"},{"id":"preset:QECDB","type":"document","author":[{"family":"Burton","given":"S."}],"title":"qecdb.org: Quantum Error Correction Database","URL":"https://qecdb.org/"},{"id":"preset:qiskit","type":"document","author":[{"family":"Community","given":"Qiskit"}],"title":"Qiskit QEC framework","URL":"https://github.com/qiskit-community/qiskit-qec"},{"id":"arxiv:0804.2268","type":"journal-article","publisher":"Proceedings of the National Academy of Sciences","issue":"32","abstract":"<jats:p>The fundamental unit for quantum computing is the qubit, an isolated, controllable two-level system. However, for many proposed quantum computer architectures, especially photonic systems, the qubits can be lost or can leak out of the desired two-level systems, posing a significant obstacle for practical quantum computation. Here, we experimentally demonstrate, both in the quantum circuit model and in the one-way quantum computer model, the smallest nontrivial quantum codes to tackle this problem. In the experiment, we encode single-qubit input states into highly entangled multiparticle code words, and we test their ability to protect encoded quantum information from detected 1-qubit loss error. Our results prove in-principle the feasibility of overcoming the qubit loss error by quantum codes.</jats:p>","DOI":"10.1073/pnas.0800740105","page":"11050-11054","source":"Crossref","title":"Experimental quantum coding against qubit loss error","volume":"105","author":[{"given":"Chao-Yang","family":"Lu","sequence":"first","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China; and"}]},{"given":"Wei-Bo","family":"Gao","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China; and"}]},{"given":"Jin","family":"Zhang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China; and"}]},{"given":"Xiao-Qi","family":"Zhou","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China; and"}]},{"given":"Tao","family":"Yang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China; and"}]},{"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China; and"},{"name":"Physikalisches Institut, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany"}]}],"container-title":"Proceedings of the National Academy of Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2008,8,12]]},"URL":"http://dx.doi.org/10.1073/pnas.0800740105","ISSN":["0027-8424","1091-6490"],"container-title-short":"Proc. Natl. Acad. Sci. U.S.A.","note":"alternative-id:10.1073/pnas.0800740105\narxivid:0804.2268"},{"id":"arxiv:1404.5498","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/ncomms4658","source":"Crossref","title":"Experimental demonstration of a graph state quantum error-correction code","volume":"5","author":[{"given":"B. A.","family":"Bell","sequence":"first","affiliation":[]},{"given":"D. A.","family":"Herrera-Martí","sequence":"additional","affiliation":[]},{"given":"M. S.","family":"Tame","sequence":"additional","affiliation":[]},{"given":"D.","family":"Markham","sequence":"additional","affiliation":[]},{"given":"W. J.","family":"Wadsworth","sequence":"additional","affiliation":[]},{"given":"J. G.","family":"Rarity","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2014,4,22]]},"URL":"http://dx.doi.org/10.1038/ncomms4658","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"3658","note":"alternative-id:BFncomms4658\narxivid:1404.5498"},{"id":"arxiv:1912.09410","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","DOI":"10.1038/s41567-020-0920-y","page":"875-880","source":"Crossref","title":"Repeated quantum error detection in a surface code","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0003-2253-6239","authenticated-orcid":false,"given":"Christian Kraglund","family":"Andersen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3220-7044","authenticated-orcid":false,"given":"Ants","family":"Remm","sequence":"additional","affiliation":[]},{"given":"Stefania","family":"Lazar","sequence":"additional","affiliation":[]},{"given":"Sebastian","family":"Krinner","sequence":"additional","affiliation":[]},{"given":"Nathan","family":"Lacroix","sequence":"additional","affiliation":[]},{"given":"Graham J.","family":"Norris","sequence":"additional","affiliation":[]},{"given":"Mihai","family":"Gabureac","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Eichler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3476-4485","authenticated-orcid":false,"given":"Andreas","family":"Wallraff","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,8]]},"URL":"http://dx.doi.org/10.1038/s41567-020-0920-y","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:920\narxivid:1912.09410"},{"id":"arxiv:2102.13071","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41567-021-01423-9","page":"80-86","source":"Crossref","title":"Logical-qubit operations in an error-detecting surface code","volume":"18","author":[{"ORCID":"https://orcid.org/0000-0001-9939-051X","authenticated-orcid":false,"given":"J. F.","family":"Marques","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7124-8933","authenticated-orcid":false,"given":"B. M.","family":"Varbanov","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7100-8013","authenticated-orcid":false,"given":"M. S.","family":"Moreira","sequence":"additional","affiliation":[]},{"given":"H.","family":"Ali","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0332-0280","authenticated-orcid":false,"given":"N.","family":"Muthusubramanian","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9041-2206","authenticated-orcid":false,"given":"C.","family":"Zachariadis","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4800-2518","authenticated-orcid":false,"given":"F.","family":"Battistel","sequence":"additional","affiliation":[]},{"given":"M.","family":"Beekman","sequence":"additional","affiliation":[]},{"given":"N.","family":"Haider","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6379-132X","authenticated-orcid":false,"given":"W.","family":"Vlothuizen","sequence":"additional","affiliation":[]},{"given":"A.","family":"Bruno","sequence":"additional","affiliation":[]},{"given":"B. M.","family":"Terhal","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2924-6085","authenticated-orcid":false,"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,16]]},"URL":"http://dx.doi.org/10.1038/s41567-021-01423-9","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:1423\narxivid:2102.13071"},{"id":"arxiv:2006.03071","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7841","DOI":"10.1038/s41586-020-03079-6","page":"220-224","source":"Crossref","title":"Entangling logical qubits with lattice surgery","volume":"589","author":[{"ORCID":"https://orcid.org/0000-0002-5020-2271","authenticated-orcid":false,"given":"Alexander","family":"Erhard","sequence":"first","affiliation":[]},{"given":"Hendrik","family":"Poulsen Nautrup","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Meth","sequence":"additional","affiliation":[]},{"given":"Lukas","family":"Postler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8001-1487","authenticated-orcid":false,"given":"Roman","family":"Stricker","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Stadler","sequence":"additional","affiliation":[]},{"given":"Vlad","family":"Negnevitsky","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Ringbauer","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9461-9650","authenticated-orcid":false,"given":"Philipp","family":"Schindler","sequence":"additional","affiliation":[]},{"given":"Hans J.","family":"Briegel","sequence":"additional","affiliation":[]},{"given":"Rainer","family":"Blatt","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1950-8640","authenticated-orcid":false,"given":"Nicolai","family":"Friis","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7410-4804","authenticated-orcid":false,"given":"Thomas","family":"Monz","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2021,1,13]]},"URL":"http://dx.doi.org/10.1038/s41586-020-03079-6","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:3079\narxivid:2006.03071"},{"id":"arxiv:2411.11822","type":"article-journal","author":[{"given":"Ben W.","family":"Reichardt"},{"given":"Adam","family":"Paetznick"},{"given":"David","family":"Aasen"},{"given":"Ivan","family":"Basov"},{"given":"Juan M.","family":"Bello-Rivas"},{"given":"Parsa","family":"Bonderson"},{"given":"Rui","family":"Chao"},{"given":"Wim","family":"van Dam"},{"given":"Matthew B.","family":"Hastings"},{"given":"Ryan V.","family":"Mishmash"},{"given":"Andres","family":"Paz"},{"given":"Marcus P.","family":"da Silva"},{"given":"Aarthi","family":"Sundaram"},{"given":"Krysta M.","family":"Svore"},{"given":"Alexander","family":"Vaschillo"},{"given":"Zhenghan","family":"Wang"},{"given":"Matt","family":"Zanner"},{"given":"William B.","family":"Cairncross"},{"given":"Cheng-An","family":"Chen"},{"given":"Daniel","family":"Crow"},{"given":"Hyosub","family":"Kim"},{"given":"Jonathan M.","family":"Kindem"},{"given":"Jonathan","family":"King"},{"given":"Michael","family":"McDonald"},{"given":"Matthew A.","family":"Norcia"},{"given":"Albert","family":"Ryou"},{"given":"Mark","family":"Stone"},{"given":"Laura","family":"Wadleigh"},{"given":"Katrina","family":"Barnes"},{"given":"Peter","family":"Battaglino"},{"given":"Thomas C.","family":"Bohdanowicz"},{"given":"Graham","family":"Booth"},{"given":"Andrew","family":"Brown"},{"given":"Mark O.","family":"Brown"},{"given":"Kayleigh","family":"Cassella"},{"given":"Robin","family":"Coxe"},{"given":"Jeffrey M.","family":"Epstein"},{"given":"Max","family":"Feldkamp"},{"given":"Christopher","family":"Griger"},{"given":"Eli","family":"Halperin"},{"given":"Andre","family":"Heinz"},{"given":"Frederic","family":"Hummel"},{"given":"Matthew","family":"Jaffe"},{"given":"Antonia M. W.","family":"Jones"},{"given":"Eliot","family":"Kapit"},{"given":"Krish","family":"Kotru"},{"given":"Joseph","family":"Lauigan"},{"given":"Ming","family":"Li"},{"given":"Jan","family":"Marjanovic"},{"given":"Eli","family":"Megidish"},{"given":"Matthew","family":"Meredith"},{"given":"Ryan","family":"Morshead"},{"given":"Juan A.","family":"Muniz"},{"given":"Sandeep","family":"Narayanaswami"},{"given":"Ciro","family":"Nishiguchi"},{"given":"Timothy","family":"Paule"},{"given":"Kelly A.","family":"Pawlak"},{"given":"Kristen L.","family":"Pudenz"},{"given":"David Rodríguez","family":"Pérez"},{"given":"Jon","family":"Simon"},{"given":"Aaron","family":"Smull"},{"given":"Daniel","family":"Stack"},{"given":"Miroslav","family":"Urbanek"},{"given":"René J. M.","family":"van de Veerdonk"},{"given":"Zachary","family":"Vendeiro"},{"given":"Robert T.","family":"Weverka"},{"given":"Thomas","family":"Wilkason"},{"given":"Tsung-Yao","family":"Wu"},{"given":"Xin","family":"Xie"},{"given":"Evan","family":"Zalys-Geller"},{"given":"Xiaogang","family":"Zhang"},{"given":"Benjamin J.","family":"Bloom"}],"title":"Fault-tolerant quantum computation with a neutral atom processor","issued":{"date-parts":[[2025,6,9]]},"note":"arxivid:2411.11822\narxiv_version_number:3"},{"id":"arxiv:2405.09035","type":"article-journal","author":[{"given":"Jiaxuan","family":"Zhang"},{"given":"Zhao-Yun","family":"Chen"},{"given":"Yun-Jie","family":"Wang"},{"given":"Bin-Han","family":"Lu"},{"given":"Hai-Feng","family":"Zhang"},{"given":"Jia-Ning","family":"Li"},{"given":"Peng","family":"Duan"},{"given":"Yu-Chun","family":"Wu"},{"given":"Guo-Ping","family":"Guo"}],"title":"Demonstrating a universal logical gate set in error-detecting surface codes on a superconducting quantum processor","issued":{"date-parts":[[2024,11,27]]},"note":"arxivid:2405.09035\narxiv_version_number:3"},{"id":"arxiv:2110.04285","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.128.110504","source":"Crossref","title":"Calibrated Decoders for Experimental Quantum Error Correction","volume":"128","author":[{"ORCID":"https://orcid.org/0000-0001-7217-7137","authenticated-orcid":true,"given":"Edward H.","family":"Chen","sequence":"first","affiliation":[{"name":"IBM Quantum, Almaden Research Center, San Jose, California 95120, USA"}]},{"ORCID":"https://orcid.org/0000-0001-9614-2836","authenticated-orcid":true,"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]},{"given":"Youngseok","family":"Kim","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]},{"given":"Neereja","family":"Sundaresan","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]},{"ORCID":"https://orcid.org/0000-0002-7919-8309","authenticated-orcid":true,"given":"Srikanth","family":"Srinivasan","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]},{"ORCID":"https://orcid.org/0000-0003-2035-5135","authenticated-orcid":true,"given":"Muyuan","family":"Li","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]},{"ORCID":"https://orcid.org/0000-0002-7800-0399","authenticated-orcid":true,"given":"Antonio D.","family":"Córcoles","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]},{"ORCID":"https://orcid.org/0000-0001-9786-8196","authenticated-orcid":true,"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]},{"ORCID":"https://orcid.org/0000-0001-8871-4638","authenticated-orcid":true,"given":"Maika","family":"Takita","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,3,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.128.110504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110504","note":"arxivid:2110.04285"},{"id":"arxiv:2305.13581","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7994","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    To run large-scale algorithms on a quantum computer, error-correcting codes must be able to perform a fundamental set of operations, called logic gates, while isolating the encoded information from noise\n                    <jats:sup>1–8</jats:sup>\n                    . We can complete a universal set of logic gates by producing special resources called magic states\n                    <jats:sup>9–11</jats:sup>\n                    . It is therefore important to produce high-fidelity magic states to conduct algorithms while introducing a minimal amount of noise to the computation. Here we propose and implement a scheme to prepare a magic state on a superconducting qubit array using error correction. We find that our scheme produces better magic states than those that can be prepared using the individual qubits of the device. This demonstrates a fundamental principle of fault-tolerant quantum computing\n                    <jats:sup>12</jats:sup>\n                    , namely, that we can use error correction to improve the quality of logic gates with noisy qubits. Moreover, we show that the yield of magic states can be increased using adaptive circuits, in which the circuit elements are changed depending on the outcome of mid-circuit measurements. This demonstrates an essential capability needed for many error-correction subroutines. We believe that our prototype will be invaluable in the future as it can reduce the number of physical qubits needed to produce high-fidelity magic states in large-scale quantum-computing architectures.\n                  </jats:p>","DOI":"10.1038/s41586-023-06846-3","page":"259-263","source":"Crossref","title":"Encoding a magic state with beyond break-even fidelity","volume":"625","author":[{"given":"Riddhi S.","family":"Gupta","sequence":"first","affiliation":[]},{"given":"Neereja","family":"Sundaresan","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Alexander","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7606-7349","authenticated-orcid":false,"given":"Christopher J.","family":"Wood","sequence":"additional","affiliation":[]},{"given":"Seth T.","family":"Merkel","sequence":"additional","affiliation":[]},{"given":"Michael B.","family":"Healy","sequence":"additional","affiliation":[]},{"given":"Marius","family":"Hillenbrand","sequence":"additional","affiliation":[]},{"given":"Tomas","family":"Jochym-O’Connor","sequence":"additional","affiliation":[]},{"given":"James R.","family":"Wootton","sequence":"additional","affiliation":[]},{"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9786-8196","authenticated-orcid":false,"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8871-4638","authenticated-orcid":false,"given":"Maika","family":"Takita","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":false,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,10]]},"URL":"http://dx.doi.org/10.1038/s41586-023-06846-3","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:6846\narxivid:2305.13581"},{"id":"arxiv:1806.02820","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The color code is both an interesting example of an exactly solved topologically ordered phase of matter and also among the most promising candidate models to realize fault-tolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well as a generating set for all 72 color code twist defects. Our work thus provides an explicit toy model that will help to better understand the abstract theory of domain walls. Secondly, we discover a number of interesting new applications of the cataloged objects for quantum information protocols. These include improved methods for performing quantum computations by code deformation, a new four-qubit error-detecting code, as well as families of new quantum error-correcting codes we call stellated color codes, which encode logical qubits at the same distance as the next best color code, but using approximately half the number of physical qubits. To the best of our knowledge, our new topological codes have the highest encoding rate of local stabilizer codes with bounded-weight stabilizers in two dimensions. Finally, we show how the boundaries and twist defects of the color code are represented by multiple copies of other phases. Indeed, in addition to the well studied comparison between the color code and two copies of the surface code, we also compare the color code to two copies of the three-fermion model. In particular, we find that this analogy offers a very clear lens through which we can view the symmetries of the color code which gives rise to its multitude of domain walls.</jats:p>","DOI":"10.22331/q-2018-10-19-101","page":"101","source":"Crossref","title":"The boundaries and twist defects of the color code and their applications to topological quantum computation","volume":"2","author":[{"given":"Markus S.","family":"Kesselring","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Fernando","family":"Pastawski","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia"},{"name":"Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2018,10,19]]},"URL":"http://dx.doi.org/10.22331/q-2018-10-19-101","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1806.02820"},{"id":"arxiv:quant-ph/9603031","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.54.r1745","page":"R1745-R1748","source":"Crossref","title":"Error prevention scheme with four particles","volume":"54","author":[{"given":"Lev","family":"Vaidman","sequence":"first","affiliation":[]},{"given":"Lior","family":"Goldenberg","sequence":"additional","affiliation":[]},{"given":"Stephen","family":"Wiesner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1996,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.54.R1745","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9603031"},{"id":"arxiv:1612.07330","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7671","DOI":"10.1038/nature23460","page":"172-179","source":"Crossref","title":"Roads towards fault-tolerant universal quantum computation","volume":"549","author":[{"given":"Earl T.","family":"Campbell","sequence":"first","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]},{"given":"Christophe","family":"Vuillot","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2017,9,14]]},"URL":"http://dx.doi.org/10.1038/nature23460","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"updated-by:{\"updated\":{\"date-parts\":[[2018,5,16]],\"date-time\":\"2018-05-16T00:00:00Z\",\"timestamp\":1526428800000},\"DOI\":\"10.1038/s41586-018-0116-0\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nalternative-id:BFnature23460\narxivid:1612.07330"},{"id":"arxiv:2505.20261","type":"article-journal","author":[{"given":"Eric J.","family":"Kuehnke"},{"given":"Kyano","family":"Levi"},{"given":"Joschka","family":"Roffe"},{"given":"Jens","family":"Eisert"},{"given":"Daniel","family":"Miller"}],"title":"Hardware-tailored logical Clifford circuits for stabilizer codes","issued":{"date-parts":[[2025,5,26]]},"note":"arxivid:2505.20261\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0306098","type":"journal-article","publisher":"American Physical Society (APS)","issue":"21","DOI":"10.1103/physrevlett.91.217901","source":"Crossref","title":"All Linear Optical Quantum Memory Based on Quantum Error Correction","volume":"91","author":[{"given":"Robert M.","family":"Gingrich","sequence":"first","affiliation":[]},{"given":"Pieter","family":"Kok","sequence":"additional","affiliation":[]},{"given":"Hwang","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Farrokh","family":"Vatan","sequence":"additional","affiliation":[]},{"given":"Jonathan P.","family":"Dowling","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2003,11,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.91.217901","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"217901","note":"arxivid:quant-ph/0306098"},{"id":"arxiv:1610.03507","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"Quantum fault tolerance in small experiments","issued":{"date-parts":[[2016,10,14]]},"note":"arxivid:1610.03507\narxiv_version_number:2"},{"id":"arxiv:1912.10063","type":"journal-article","publisher":"Rinton Press","issue":"11&12","abstract":"<jats:p>Quantum error correcting codes protect quantum computation from errors caused by decoherence and other noise. Here we study the problem of designing logical operations for quantum error correcting codes. We present an automated procedure that generates logical operations given known encoding and correcting procedures. Our technique is to use variational circuits for learning both the logical gates and the physical operations implementing them. This procedure can be implemented on near-term quantum computers via quantum process tomography. It enables automatic discovery of logical gates from analytically designed error correcting codes and can be extended to error correcting codes found by numerical optimization. We test the procedure by simulating small quantum codes of four to fifteen qubits showing that our procedure finds most logical gates known in the current literature. Additionally, it generates logical gates not found in the current literature for the [[5,1,2]] code, the [[6,3,2]] code, the [[8,3,2]] code, and the [[10,1,2]] code.</jats:p>","DOI":"10.26421/qic22.11-12-3","page":"947-964","source":"Crossref","title":"Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))","volume":"22","author":[{"given":"Hongxiang","family":"Chen","sequence":"first","affiliation":[]},{"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[]},{"given":"Nikolas P.","family":"Breuckmann","sequence":"additional","affiliation":[]},{"given":"Edward","family":"Grant","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2022,8]]},"URL":"http://dx.doi.org/10.26421/QIC22.11-12-3","ISSN":["1533-7146"],"container-title-short":"QIC","note":"arxivid:1912.10063"},{"id":"arxiv:2412.04791","type":"article-journal","author":[{"given":"Divyanshu","family":"Singh"},{"given":"Shiroman","family":"Prakash"}],"title":"Fault-Tolerant Implementation of the Deutsch-Jozsa Algorithm","issued":{"date-parts":[[2025,9,10]]},"note":"arxivid:2412.04791\narxiv_version_number:2"},{"id":"arxiv:2412.07670","type":"article-journal","author":[{"given":"Matt. J.","family":"Bedalov"},{"given":"Matt","family":"Blakely"},{"given":"Peter. D.","family":"Buttler"},{"given":"Caitlin","family":"Carnahan"},{"given":"Frederic T.","family":"Chong"},{"given":"Woo Chang","family":"Chung"},{"given":"Dan C.","family":"Cole"},{"given":"Palash","family":"Goiporia"},{"given":"Pranav","family":"Gokhale"},{"given":"Bettina","family":"Heim"},{"given":"Garrett T.","family":"Hickman"},{"given":"Eric B.","family":"Jones"},{"given":"Ryan A.","family":"Jones"},{"given":"Pradnya","family":"Khalate"},{"given":"Jin-Sung","family":"Kim"},{"given":"Kevin W.","family":"Kuper"},{"given":"Martin T.","family":"Lichtman"},{"given":"Stephanie","family":"Lee"},{"given":"David","family":"Mason"},{"given":"Nathan A.","family":"Neff-Mallon"},{"given":"Thomas W.","family":"Noel"},{"given":"Victory","family":"Omole"},{"given":"Alexander G.","family":"Radnaev"},{"given":"Rich","family":"Rines"},{"given":"Mark","family":"Saffman"},{"given":"Efrat","family":"Shabtai"},{"given":"Mariesa H.","family":"Teo"},{"given":"Bharath","family":"Thotakura"},{"given":"Teague","family":"Tomesh"},{"given":"Angela K.","family":"Tucker"}],"title":"Fault-Tolerant Operation and Materials Science with Neutral Atom Logical Qubits","issued":{"date-parts":[[2024,12,10]]},"note":"arxivid:2412.07670\narxiv_version_number:1"},{"id":"arxiv:1611.06946","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"10","abstract":"<jats:p>We show the fault-tolerant encoding, measurement, and operation of a logical qubit via quantum error detection.</jats:p>","DOI":"10.1126/sciadv.1701074","source":"Crossref","title":"Fault-tolerant quantum error detection","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-5655-9258","authenticated-orcid":false,"given":"Norbert M.","family":"Linke","sequence":"first","affiliation":[{"name":"Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, MD 20742, USA."}]},{"given":"Mauricio","family":"Gutierrez","sequence":"additional","affiliation":[{"name":"Schools of Chemistry and Biochemistry, Computational Science and Engineering, and Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA."}]},{"given":"Kevin A.","family":"Landsman","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, MD 20742, USA."}]},{"given":"Caroline","family":"Figgatt","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, MD 20742, USA."}]},{"ORCID":"https://orcid.org/0000-0003-4659-9138","authenticated-orcid":false,"given":"Shantanu","family":"Debnath","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, MD 20742, USA."}]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":false,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Schools of Chemistry and Biochemistry, Computational Science and Engineering, and Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA."}]},{"given":"Christopher","family":"Monroe","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, MD 20742, USA."},{"name":"IonQ Inc., College Park, MD 20742, USA."}]}],"container-title":"Science Advances","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,6]]},"URL":"http://dx.doi.org/10.1126/sciadv.1701074","ISSN":["2375-2548"],"container-title-short":"Sci. Adv.","page":"e1701074","note":"alternative-id:10.1126/sciadv.1701074\narxivid:1611.06946"},{"id":"arxiv:1705.09259","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.119.180501","source":"Crossref","title":"Experimental Demonstration of Fault-Tolerant State Preparation with Superconducting Qubits","volume":"119","author":[{"given":"Maika","family":"Takita","sequence":"first","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]},{"given":"A. D.","family":"Córcoles","sequence":"additional","affiliation":[]},{"given":"Jerry M.","family":"Chow","sequence":"additional","affiliation":[]},{"given":"Jay M.","family":"Gambetta","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.119.180501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180501","note":"arxivid:1705.09259"},{"id":"arxiv:1705.08957","type":"journal-issue","publisher":"Rinton Press","issue":"11&12","DOI":"10.26421/qic18.11-12","source":"Crossref","title":[],"volume":"18","container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2018,9]]},"URL":"http://dx.doi.org/10.26421/QIC18.11-12","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC","note":"arxivid:1705.08957"},{"id":"arxiv:1806.02359","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.122.080504","source":"Crossref","title":"Fault-Tolerant Logical Gates in the IBM Quantum Experience","volume":"122","author":[{"given":"Robin","family":"Harper","sequence":"first","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,2,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.122.080504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080504","note":"arxivid:1806.02359"},{"id":"arxiv:1709.01866","type":"journal-article","publisher":"IOP Publishing","issue":"3","DOI":"10.1088/2058-9565/aac64e","page":"035010","source":"Crossref","title":"Protecting quantum memories using coherent parity check codes","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-9202-1156","authenticated-orcid":false,"given":"Joschka","family":"Roffe","sequence":"first","affiliation":[]},{"given":"David","family":"Headley","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1293-0761","authenticated-orcid":false,"given":"Nicholas","family":"Chancellor","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4965-0584","authenticated-orcid":false,"given":"Dominic","family":"Horsman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6551-3056","authenticated-orcid":false,"given":"Viv","family":"Kendon","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2018,6,6]]},"URL":"http://dx.doi.org/10.1088/2058-9565/aac64e","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1709.01866"},{"id":"arxiv:2309.05558","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41928-024-01319-5","page":"84-91","source":"Crossref","title":"A real-time, scalable, fast and resource-efficient decoder for a quantum computer","volume":"8","author":[{"given":"Ben","family":"Barber","sequence":"first","affiliation":[]},{"given":"Kenton M.","family":"Barnes","sequence":"additional","affiliation":[]},{"given":"Tomasz","family":"Bialas","sequence":"additional","affiliation":[]},{"given":"Okan","family":"Buğdaycı","sequence":"additional","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0006-9237-0042","authenticated-orcid":false,"given":"Neil I.","family":"Gillespie","sequence":"additional","affiliation":[]},{"given":"Kauser","family":"Johar","sequence":"additional","affiliation":[]},{"given":"Ram","family":"Rajan","sequence":"additional","affiliation":[]},{"given":"Adam W.","family":"Richardson","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2169-3008","authenticated-orcid":false,"given":"Luka","family":"Skoric","sequence":"additional","affiliation":[]},{"given":"Canberk","family":"Topal","sequence":"additional","affiliation":[]},{"given":"Mark L.","family":"Turner","sequence":"additional","affiliation":[]},{"given":"Abbas B.","family":"Ziad","sequence":"additional","affiliation":[]}],"container-title":"Nature Electronics","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,7]]},"URL":"http://dx.doi.org/10.1038/s41928-024-01319-5","ISSN":["2520-1131"],"container-title-short":"Nat Electron","note":"alternative-id:1319\narxivid:2309.05558"},{"id":"arxiv:2410.05202","type":"article-journal","author":[{"given":"Laura","family":"Caune"},{"given":"Luka","family":"Skoric"},{"given":"Nick S.","family":"Blunt"},{"given":"Archibald","family":"Ruban"},{"given":"Jimmy","family":"McDaniel"},{"given":"Joseph A.","family":"Valery"},{"given":"Andrew D.","family":"Patterson"},{"given":"Alexander V.","family":"Gramolin"},{"given":"Joonas","family":"Majaniemi"},{"given":"Kenton M.","family":"Barnes"},{"given":"Tomasz","family":"Bialas"},{"given":"Okan","family":"Buğdaycı"},{"given":"Ophelia","family":"Crawford"},{"given":"György P.","family":"Gehér"},{"given":"Hari","family":"Krovi"},{"given":"Elisha","family":"Matekole"},{"given":"Canberk","family":"Topal"},{"given":"Stefano","family":"Poletto"},{"given":"Michael","family":"Bryant"},{"given":"Kalan","family":"Snyder"},{"given":"Neil I.","family":"Gillespie"},{"given":"Glenn","family":"Jones"},{"given":"Kauser","family":"Johar"},{"given":"Earl T.","family":"Campbell"},{"given":"Alexander D.","family":"Hill"}],"title":"Demonstrating real-time and low-latency quantum error correction with superconducting qubits","issued":{"date-parts":[[2024,10,7]]},"note":"arxivid:2410.05202\narxiv_version_number:1"},{"id":"arxiv:2506.13724","type":"article-journal","author":[{"given":"Bichen","family":"Zhang"},{"given":"Genyue","family":"Liu"},{"given":"Guillaume","family":"Bornet"},{"given":"Sebastian P.","family":"Horvath"},{"given":"Pai","family":"Peng"},{"given":"Shuo","family":"Ma"},{"given":"Shilin","family":"Huang"},{"given":"Shruti","family":"Puri"},{"given":"Jeff D.","family":"Thompson"}],"title":"Leveraging erasure errors in logical qubits with metastable $^{171}$Yb atoms","issued":{"date-parts":[[2025,6,16]]},"note":"arxivid:2506.13724\narxiv_version_number:1"},{"id":"arxiv:2509.13247","type":"article-journal","author":[{"given":"Rich","family":"Rines"},{"given":"Benjamin","family":"Hall"},{"given":"Mariesa H.","family":"Teo"},{"given":"Joshua","family":"Viszlai"},{"given":"Daniel C.","family":"Cole"},{"given":"David","family":"Mason"},{"given":"Cameron","family":"Barker"},{"given":"Matt J.","family":"Bedalov"},{"given":"Matt","family":"Blakely"},{"given":"Tobias","family":"Bothwell"},{"given":"Caitlin","family":"Carnahan"},{"given":"Frederic T.","family":"Chong"},{"given":"Samuel Y.","family":"Eubanks"},{"given":"Brian","family":"Fields"},{"given":"Matthew","family":"Gillette"},{"given":"Palash","family":"Goiporia"},{"given":"Pranav","family":"Gokhale"},{"given":"Garrett T.","family":"Hickman"},{"given":"Marin","family":"Iliev"},{"given":"Eric B.","family":"Jones"},{"given":"Ryan A.","family":"Jones"},{"given":"Kevin W.","family":"Kuper"},{"given":"Stephanie","family":"Lee"},{"given":"Martin T.","family":"Lichtman"},{"given":"Kevin","family":"Loeffler"},{"given":"Nate","family":"Mackintosh"},{"given":"Farhad","family":"Majdeteimouri"},{"given":"Peter T.","family":"Mitchell"},{"given":"Thomas W.","family":"Noel"},{"given":"Ely","family":"Novakoski"},{"given":"Victory","family":"Omole"},{"given":"David","family":"Owusu-Antwi"},{"given":"Alexander G.","family":"Radnaev"},{"given":"Anthony","family":"Reiter"},{"given":"Mark","family":"Saffman"},{"given":"Bharath","family":"Thotakura"},{"given":"Teague","family":"Tomesh"},{"given":"Ilya","family":"Vinogradov"}],"title":"Demonstration of a Logical Architecture Uniting Motion and In-Place Entanglement","issued":{"date-parts":[[2026,4,2]]},"note":"arxivid:2509.13247\narxiv_version_number:2"},{"id":"arxiv:2503.12869","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    Achieving industrial quantum advantage is unlikely without the use of quantum error correction (QEC). Other QEC codes beyond surface code are being experimentally studied, such as color codes and quantum low-density parity check codes, that could benefit from new quantum processing unit (QPU) architectures. We introduce the six-qubit star lattice architecture that offers parallelism and effective local all-to-all connectivity and thus enables hardware-efficient implementation of certain QEC codes. As a first demonstration of this new architecture, we encode two logical qubits in a six-qubit superconducting QPU with a star topology using the\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mtext>⟦</a:mtext>\n                      <a:mn>4</a:mn>\n                      <a:mo>,</a:mo>\n                      <a:mn>2</a:mn>\n                      <a:mo>,</a:mo>\n                      <a:mn>2</a:mn>\n                      <a:mtext>⟧</a:mtext>\n                    </a:math>\n                    code and characterize the logical states with the classical shadow framework. Logical lifetime and logical error rate are measured over repeated quantum error detection cycles for various logical states including a logical Bell state. We measure logical state fidelities above 96% for every cardinal logical state, find logical lifetimes above the best physical element, and logical error-per-cycle values ranging from 0.25(2)% to 0.91(3)%. In future, such star units can be tiled into a lattice to enable QEC codes with high-weight and overlapping stabilizers for improved encoding rates.\n                  </jats:p>","DOI":"10.1103/8m33-wn4g","source":"Crossref","title":"Quantum Error Detection in Qubit-Resonator Star Architecture","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-7557-4493","authenticated-orcid":true,"given":"Florian","family":"Vigneau","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0002-7816-289X","authenticated-orcid":true,"given":"Sourav","family":"Majumder","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0002-0983-9027","authenticated-orcid":true,"given":"Aniket","family":"Rath","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0002-6201-2723","authenticated-orcid":true,"given":"Pedro","family":"Parrado-Rodríguez","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0203dhb90","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0001-5638-6334","authenticated-orcid":true,"given":"Francisco Revson Fernandes","family":"Pereira","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"given":"Hsiang-Sheng","family":"Ku","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"given":"Fedor","family":"Šimkovic","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"given":"Stefan","family":"Pogorzalek","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0001-5762-3856","authenticated-orcid":true,"given":"Tyler","family":"Jones","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0009-0000-2687-699X","authenticated-orcid":true,"given":"Nicola","family":"Wurz","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0002-7277-9856","authenticated-orcid":true,"given":"Michael","family":"Renger","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0002-4215-8688","authenticated-orcid":true,"given":"Jeroen","family":"Verjauw","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"given":"Ping","family":"Yang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"given":"William","family":"Kindel","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0002-1855-4672","authenticated-orcid":true,"given":"Frank","family":"Deppe","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0107a5128","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]},{"ORCID":"https://orcid.org/0000-0003-0721-4880","authenticated-orcid":true,"given":"Johannes","family":"Heinsoo","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0203dhb90","id-type":"ROR","asserted-by":"publisher"}],"name":"IQM Quantum Computers"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,22]]},"URL":"http://dx.doi.org/10.1103/8m33-wn4g","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040369","note":"arxivid:2503.12869"},{"id":"arxiv:2506.22600","type":"article-journal","author":[{"given":"Friederike","family":"Butt"},{"given":"Ivan","family":"Pogorelov"},{"given":"Robert","family":"Freund"},{"given":"Alex","family":"Steiner"},{"given":"Marcel","family":"Meyer"},{"given":"Thomas","family":"Monz"},{"given":"Markus","family":"Müller"}],"title":"Demonstration of measurement-free universal fault-tolerant quantum computation","issued":{"date-parts":[[2025,6,27]]},"note":"arxivid:2506.22600\narxiv_version_number:1"},{"id":"arxiv:2409.04628","type":"article-journal","author":[{"given":"Ben W.","family":"Reichardt"},{"given":"David","family":"Aasen"},{"given":"Rui","family":"Chao"},{"given":"Alex","family":"Chernoguzov"},{"given":"Wim","family":"van Dam"},{"given":"John P.","family":"Gaebler"},{"given":"Dan","family":"Gresh"},{"given":"Dominic","family":"Lucchetti"},{"given":"Michael","family":"Mills"},{"given":"Steven A.","family":"Moses"},{"given":"Brian","family":"Neyenhuis"},{"given":"Adam","family":"Paetznick"},{"given":"Andres","family":"Paz"},{"given":"Peter E.","family":"Siegfried"},{"given":"Marcus P.","family":"da Silva"},{"given":"Krysta M.","family":"Svore"},{"given":"Zhenghan","family":"Wang"},{"given":"Matt","family":"Zanner"}],"title":"Demonstration of quantum computation and error correction with a tesseract code","issued":{"date-parts":[[2024,12,5]]},"note":"arxivid:2409.04628\narxiv_version_number:2"},{"id":"arxiv:2109.12735","type":"article-journal","author":[{"given":"Hanson","family":"Hao"}],"title":"Investigations on Automorphism Groups of Quantum Stabilizer Codes","issued":{"date-parts":[[2021,9,27]]},"note":"arxivid:2109.12735\narxiv_version_number:1"},{"id":"arxiv:2306.12526","type":"article-journal","author":[{"given":"Eric","family":"Kubischta"},{"given":"Ian","family":"Teixeira"},{"given":"J. Maxwell","family":"Silvester"}],"title":"Quantum Weight Enumerators for Real Codes with $X$ and $Z$ Exactly Transversal","issued":{"date-parts":[[2024,2,12]]},"note":"arxivid:2306.12526\narxiv_version_number:2"},{"id":"manual:-I.-Teixeira-private-communic","type":"document","author":[{"family":"Teixeira","given":"I."}],"genre":"private communication,","issued":"2024"},{"id":"arxiv:quant-ph/9605031","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevlett.77.3260","page":"3260-3263","source":"Crossref","title":"Fault-Tolerant Error Correction with Efficient Quantum Codes","volume":"77","author":[{"given":"David P.","family":"DiVincenzo","sequence":"first","affiliation":[]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1996,10,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.77.3260","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9605031"},{"id":"arxiv:quant-ph/9602019","type":"article-journal","author":[{"given":"Raymond","family":"Laflamme"},{"given":"Cesar","family":"Miquel"},{"given":"Juan Pablo","family":"Paz"},{"given":"Wojciech Hubert","family":"Zurek"}],"title":"Perfect Quantum Error Correction Code","issued":{"date-parts":[[1996,2,27]]},"note":"arxivid:quant-ph/9602019\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0410004","type":"journal-article","publisher":"IOP Publishing","issue":"5","DOI":"10.1088/1367-2630/8/5/080","page":"80-80","source":"Crossref","title":"Analytical technique for simplification of the encoder–decoder circuit for a perfect five-qubit error correction","volume":"8","author":[{"given":"Jin-Yuan","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"Che-Ming","family":"Li","sequence":"additional","affiliation":[]},{"given":"Der-San","family":"Chuu","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2006,5,30]]},"URL":"http://dx.doi.org/10.1088/1367-2630/8/5/080","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"alternative-id:S1367-2630(06)15767-5\narxivid:quant-ph/0410004"},{"id":"doi:10.1201/9781420012293","type":"reference-book","publisher":"CRC Press","DOI":"10.1201/9781420012293","source":"Crossref","title":"Quantum Computing","author":[{"given":"Mikio","family":"Nakahara","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2008,3,11]]},"ISBN":["9780429146510"],"URL":"http://dx.doi.org/10.1201/9781420012293","note":"edition-number:0"},{"id":"arxiv:1301.4796","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.87.052305","source":"Crossref","title":"Strategy for implementing stabilizer-based codes on solid-state qubits","volume":"87","author":[{"given":"Tetsufumi","family":"Tanamoto","sequence":"first","affiliation":[]},{"given":"Vladimir M.","family":"Stojanović","sequence":"additional","affiliation":[]},{"given":"Christoph","family":"Bruder","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Becker","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,5,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.052305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052305","note":"arxivid:1301.4796"},{"id":"arxiv:1509.01239","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.93.042333","source":"Crossref","title":"Universal set of dynamically protected gates for bipartite qubit networks: Soft pulse implementation of the [[5,1,3]] quantum error-correcting code","volume":"93","author":[{"given":"Amrit","family":"De","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,4,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.042333","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042333","note":"arxivid:1509.01239"},{"id":"arxiv:2402.17761","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>The realization of large-scale quantum computers requires not only quantum error correction but also fault-tolerant (FT) operations to handle errors that propagate into harmful errors. Recently, flag-based protocols have been introduced that use ancillary qubits to flag harmful errors. However, there is no clear recipe for finding a FT quantum circuit with flag-based protocols, especially when we consider hardware constraints, such as the qubit connectivity and available gate set. This work presents a novel approach to automatically discover compact and hardware-adapted FT quantum circuits to make significant progress towards scalable FT quantum computing. We employ reinforcement learning (RL) as an enabling tool, leveraging a fast, parallelized stabilizer quantum circuit simulator and a nontrivial reward function specifically adapted to the problem. We show that, in the task of FT logical state preparation, RL discovers not only circuits with fewer gates and ancillary qubits than published results but also novel circuits without and with hardware constraints of up to distance-5 codes with 25 physical qubits, and they can be implemented directly in experiments. Furthermore, RL allows for straightforward exploration of different qubit connectivities and the use of transfer learning to accelerate the discovery. More generally, our work sets the framework towards the use of RL or other machine learning techniques for FT quantum circuit discovery with hardware constraints to make real progress towards the realization of large-scale quantum computers, addressing tasks beyond state preparation, including magic state preparation, logical gate synthesis, and syndrome measurement.</jats:p>","DOI":"10.1103/gqpr-dgz7","source":"Crossref","title":"Quantum Circuit Discovery for Fault-Tolerant Logical State Preparation with Reinforcement Learning","volume":"15","author":[{"ORCID":"https://orcid.org/0000-0002-7645-125X","authenticated-orcid":true,"given":"Remmy","family":"Zen","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"}]},{"given":"Jan","family":"Olle","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"}]},{"ORCID":"https://orcid.org/0000-0002-5946-7591","authenticated-orcid":true,"given":"Luis","family":"Colmenarez","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0002-5332-213X","authenticated-orcid":true,"given":"Matteo","family":"Puviani","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"}]},{"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"given":"Florian","family":"Marquardt","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/020as7681","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Science of Light"},{"id":[{"id":"https://ror.org/00f7hpc57","id-type":"ROR","asserted-by":"publisher"}],"name":"Friedrich-Alexander Universität Erlangen-Nürnberg"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,22]]},"URL":"http://dx.doi.org/10.1103/gqpr-dgz7","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041012","note":"arxivid:2402.17761"},{"id":"arxiv:quant-ph/0403025","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.71.022316","source":"Crossref","title":"Universal quantum computation with ideal Clifford gates and noisy ancillas","volume":"71","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Alexei","family":"Kitaev","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,2,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.022316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022316","note":"arxivid:quant-ph/0403025"},{"id":"arxiv:1603.03948","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.6.031039","source":"Crossref","title":"Universal Fault-Tolerant Gates on Concatenated Stabilizer Codes","volume":"6","author":[{"given":"Theodore J.","family":"Yoder","sequence":"first","affiliation":[]},{"given":"Ryuji","family":"Takagi","sequence":"additional","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2016,9,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.6.031039","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031039","note":"arxivid:1603.03948"},{"id":"arxiv:quant-ph/9608028","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.55.4593","page":"4593-4596","source":"Crossref","title":"Conditional generation of error syndromes in fault-tolerant error correction","volume":"55","author":[{"given":"M. B.","family":"Plenio","sequence":"first","affiliation":[]},{"given":"V.","family":"Vedral","sequence":"additional","affiliation":[]},{"given":"P. L.","family":"Knight","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,6,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.55.4593","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9608028"},{"id":"arxiv:quant-ph/0311116","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.69.042314","source":"Crossref","title":"Quantum-error correction on linear-nearest-neighbor qubit arrays","volume":"69","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Charles D.","family":"Hill","sequence":"additional","affiliation":[]},{"given":"Lloyd C. L.","family":"Hollenberg","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,4,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.69.042314","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042314","note":"arxivid:quant-ph/0311116"},{"id":"arxiv:2207.13356","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.107.032403","source":"Crossref","title":"Realizing a class of stabilizer quantum error correction codes using a single ancilla and circular connectivity","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0003-3678-1999","authenticated-orcid":true,"given":"A. V.","family":"Antipov","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5760-441X","authenticated-orcid":true,"given":"E. O.","family":"Kiktenko","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4722-3418","authenticated-orcid":true,"given":"A. K.","family":"Fedorov","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.032403","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032403","note":"arxivid:2207.13356"},{"id":"arxiv:2203.01706","type":"article-journal","author":[{"given":"Chaobin","family":"Liu"}],"title":"Exact performance of the five-qubit code with coherent errors","issued":{"date-parts":[[2022,3,3]]},"note":"arxivid:2203.01706\narxiv_version_number:1"},{"id":"manual:-S.-T.-Flammia-private-commun","type":"document","author":[{"family":"T","given":"S."}],"title":"Flammia, private communication","issued":"2017"},{"id":"arxiv:2409.18175","type":"article-journal","author":[{"given":"Hasan","family":"Sayginel"},{"given":"Stergios","family":"Koutsioumpas"},{"given":"Mark","family":"Webster"},{"given":"Abhishek","family":"Rajput"},{"family":"Dan E Browne","given":""}],"title":"Fault-Tolerant Logical Clifford Gates from Code Automorphisms","issued":{"date-parts":[[2025,5,9]]},"note":"arxivid:2409.18175\narxiv_version_number:3"},{"id":"arxiv:2305.00784","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce57702.2023.00016","page":"63-69","source":"Crossref","title":"Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes","author":[{"given":"Dhruv","family":"Bhatnagar","sequence":"first","affiliation":[{"name":"Delft University of Technology,Quantum &#x0026; Computer Engineering Department,Delft,The Netherlands,2628 CD"}]},{"given":"Matthew","family":"Steinberg","sequence":"additional","affiliation":[{"name":"Delft University of Technology,Quantum &#x0026; Computer Engineering Department,Delft,The Netherlands,2628 CD"}]},{"given":"David","family":"Elkouss","sequence":"additional","affiliation":[{"name":"OIST Graduate University,Networked Quantum Devices Unit,Okinawa,Japan"}]},{"given":"Carmen G.","family":"Almudever","sequence":"additional","affiliation":[{"name":"Technical Universitat Polit&#x00E8;cnica de Val&#x00E8;ncia,Computer Engineering Department,Valencia,Spain"}]},{"given":"Sebastian","family":"Feld","sequence":"additional","affiliation":[{"name":"Delft University of Technology,Quantum &#x0026; Computer Engineering Department,Delft,The Netherlands,2628 CD"}]}],"event":"2023 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2023 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2023,9,17]]},"URL":"http://dx.doi.org/10.1109/QCE57702.2023.00016","note":"arxivid:2305.00784"},{"id":"arxiv:quant-ph/0101034","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/physrevlett.86.5811","page":"5811-5814","source":"Crossref","title":"Benchmarking Quantum Computers: The Five-Qubit Error Correcting Code","volume":"86","author":[{"given":"E.","family":"Knill","sequence":"first","affiliation":[]},{"given":"R.","family":"Laflamme","sequence":"additional","affiliation":[]},{"given":"R.","family":"Martinez","sequence":"additional","affiliation":[]},{"given":"C.","family":"Negrevergne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2001,6,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.86.5811","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/0101034"},{"id":"arxiv:1208.4797","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevlett.109.100503","source":"Crossref","title":"Experimental Implementation of Encoded Logical Qubit Operations in a Perfect Quantum Error Correcting Code","volume":"109","author":[{"given":"Jingfu","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]},{"given":"Dieter","family":"Suter","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,9,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.109.100503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"100503","note":"arxivid:1208.4797"},{"id":"arxiv:1103.2178","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/ncomms1166","source":"Crossref","title":"Experimental magic state distillation for fault-tolerant quantum computing","volume":"2","author":[{"given":"Alexandre M.","family":"Souza","sequence":"first","affiliation":[]},{"given":"Jingfu","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Colm A.","family":"Ryan","sequence":"additional","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2011,1,25]]},"URL":"http://dx.doi.org/10.1038/ncomms1166","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"169","note":"alternative-id:BFncomms1166\narxivid:1103.2178"},{"id":"arxiv:1907.04507","type":"journal-article","publisher":"Oxford University Press (OUP)","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error-correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state and state decoding. To address this challenge, we experimentally realise the [5, 1, 3] code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the [5, 1, 3] code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of $57.1(3)\\%$ while with a high fidelity of $98.6(1)\\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilisers. We further implement logical Pauli operations with a fidelity of $97.2(2)\\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of $74.5(6)\\%$, in total with 92 gates. Our work demonstrates each key aspect of the [5, 1, 3] code and verifies the viability of experimental realisation of quantum error-correcting codes with superconducting qubits.</jats:p>","DOI":"10.1093/nsr/nwab011","source":"Crossref","title":"Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits","volume":"9","author":[{"given":"Ming","family":"Gong","sequence":"first","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Xiao","family":"Yuan","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"},{"name":"Department of Materials, University of Oxford , Oxford OX1 3PH, UK"}]},{"given":"Shiyu","family":"Wang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Yulin","family":"Wu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Youwei","family":"Zhao","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Chen","family":"Zha","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Shaowei","family":"Li","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Zhen","family":"Zhang","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Qi","family":"Zhao","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Yunchao","family":"Liu","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Futian","family":"Liang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Jin","family":"Lin","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Yu","family":"Xu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Hui","family":"Deng","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Hao","family":"Rong","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"He","family":"Lu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"}]},{"given":"Simon C","family":"Benjamin","sequence":"additional","affiliation":[{"name":"Department of Materials, University of Oxford , Oxford OX1 3PH, UK"}]},{"given":"Cheng-Zhi","family":"Peng","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Xiongfeng","family":"Ma","sequence":"additional","affiliation":[{"name":"Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University , Beijing 100084, China"}]},{"given":"Yu-Ao","family":"Chen","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Xiaobo","family":"Zhu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]},{"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China , Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China , Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences , Shanghai 201315, China"}]}],"container-title":"National Science Review","original-title":[],"language":"en","issued":{"date-parts":[[2021,1,21]]},"URL":"http://dx.doi.org/10.1093/nsr/nwab011","ISSN":["2095-5138","2053-714X"],"page":"nwab011","note":"arxivid:1907.04507"},{"id":"arxiv:2208.01863","type":"article-journal","author":[{"given":"C.","family":"Ryan-Anderson"},{"given":"N. C.","family":"Brown"},{"given":"M. S.","family":"Allman"},{"given":"B.","family":"Arkin"},{"given":"G.","family":"Asa-Attuah"},{"given":"C.","family":"Baldwin"},{"given":"J.","family":"Berg"},{"given":"J. G.","family":"Bohnet"},{"given":"S.","family":"Braxton"},{"given":"N.","family":"Burdick"},{"given":"J. P.","family":"Campora"},{"given":"A.","family":"Chernoguzov"},{"given":"J.","family":"Esposito"},{"given":"B.","family":"Evans"},{"given":"D.","family":"Francois"},{"given":"J. P.","family":"Gaebler"},{"given":"T. M.","family":"Gatterman"},{"given":"J.","family":"Gerber"},{"given":"K.","family":"Gilmore"},{"given":"D.","family":"Gresh"},{"given":"A.","family":"Hall"},{"given":"A.","family":"Hankin"},{"given":"J.","family":"Hostetter"},{"given":"D.","family":"Lucchetti"},{"given":"K.","family":"Mayer"},{"given":"J.","family":"Myers"},{"given":"B.","family":"Neyenhuis"},{"given":"J.","family":"Santiago"},{"given":"J.","family":"Sedlacek"},{"given":"T.","family":"Skripka"},{"given":"A.","family":"Slattery"},{"given":"R. P.","family":"Stutz"},{"given":"J.","family":"Tait"},{"given":"R.","family":"Tobey"},{"given":"G.","family":"Vittorini"},{"given":"J.","family":"Walker"},{"given":"D.","family":"Hayes"}],"title":"Implementing Fault-tolerant Entangling Gates on the Five-qubit Code and the Color Code","issued":{"date-parts":[[2022,8,3]]},"note":"arxivid:2208.01863\narxiv_version_number:1"},{"id":"arxiv:2310.12106","type":"article-journal","author":[{"given":"Natalie C.","family":"Brown"},{"given":"John Peter","family":"Campora"},{"given":"Cassandra","family":"Granade"},{"given":"Bettina","family":"Heim"},{"given":"Stefan","family":"Wernli"},{"given":"Ciaran","family":"Ryan-Anderson"},{"given":"Dominic","family":"Lucchetti"},{"given":"Adam","family":"Paetznick"},{"given":"Martin","family":"Roetteler"},{"given":"Krysta","family":"Svore"},{"given":"Alex","family":"Chernoguzov"}],"title":"Advances in compilation for quantum hardware -- A demonstration of magic state distillation and repeat-until-success protocols","issued":{"date-parts":[[2023,10,18]]},"note":"arxivid:2310.12106\narxiv_version_number:1"},{"id":"arxiv:2108.01646","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7916","abstract":"<jats:title>Abstract</jats:title><jats:p>Solid-state spin qubits is a promising platform for quantum computation and quantum networks<jats:sup>1,2</jats:sup>. Recent experiments have demonstrated high-quality control over multi-qubit systems<jats:sup>3–8</jats:sup>, elementary quantum algorithms<jats:sup>8–11</jats:sup> and non-fault-tolerant error correction<jats:sup>12–14</jats:sup>. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations<jats:sup>15–18</jats:sup>. Overcoming imperfections in this way remains an important outstanding challenge for quantum science<jats:sup>15,19–27</jats:sup>. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the five-qubit code with a recently discovered flag protocol that enables fault tolerance using a total of seven qubits<jats:sup>28–30</jats:sup>. We encode the logical qubit using a new protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. Although future improvements in fidelity and the number of qubits will be required to suppress logical error rates below the physical error rates, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards quantum information processing based on solid-state spins.</jats:p>","DOI":"10.1038/s41586-022-04819-6","page":"884-889","source":"Crossref","title":"Fault-tolerant operation of a logical qubit in a diamond quantum processor","volume":"606","author":[{"ORCID":"https://orcid.org/0000-0001-8205-8166","authenticated-orcid":false,"given":"M. H.","family":"Abobeih","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5483-0339","authenticated-orcid":false,"given":"Y.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"J.","family":"Randall","sequence":"additional","affiliation":[]},{"given":"S. J. H.","family":"Loenen","sequence":"additional","affiliation":[]},{"given":"C. E.","family":"Bradley","sequence":"additional","affiliation":[]},{"given":"M.","family":"Markham","sequence":"additional","affiliation":[]},{"given":"D. J.","family":"Twitchen","sequence":"additional","affiliation":[]},{"given":"B. M.","family":"Terhal","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2355-727X","authenticated-orcid":false,"given":"T. H.","family":"Taminiau","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,5]]},"URL":"http://dx.doi.org/10.1038/s41586-022-04819-6","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:4819\narxivid:2108.01646"},{"id":"arxiv:1212.6703","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.88.012311","source":"Crossref","title":"Quantum Kronecker sum-product low-density parity-check codes with finite rate","volume":"88","author":[{"given":"Alexey A.","family":"Kovalev","sequence":"first","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,7,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.88.012311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012311","note":"arxivid:1212.6703"},{"id":"arxiv:2101.09349","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates. In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the well-defined homological formalism introduced by Kitaev for describing CSS surface codes. In particular, we associate a surface code to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>y</mml:mi></mml:math> graph <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>G</mml:mi></mml:math> embedded on <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>y</mml:mi></mml:math> 2D-manifold, in such a way that (1) qubits are associated to the vertices of the graph, (2) stabilizers are associated to faces, (3) twist defects are associated to odd-degree vertices. In this way, we are able to reproduce the variety of surface codes, with and without twists, in the literature and produce new examples. We also calculate and bound various code properties such as the rate and distance in terms of topological graph properties such as genus, systole, and face-width.</jats:p>","DOI":"10.22331/q-2024-07-18-1416","page":"1416","source":"Crossref","title":"A graph-based formalism for surface codes and twists","volume":"8","author":[{"given":"Rahul","family":"Sarkar","sequence":"first","affiliation":[{"name":"Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305"}]},{"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[{"name":"IBM T.J. Watson Research Center, Yorktown Heights, NY"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,18]]},"URL":"http://dx.doi.org/10.22331/q-2024-07-18-1416","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2101.09349"},{"id":"arxiv:2211.03094","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.111.012444","source":"Crossref","title":"Low-overhead quantum error-correction codes with a cyclic topology","volume":"111","author":[{"ORCID":"https://orcid.org/0000-0002-0614-1571","authenticated-orcid":true,"given":"Ilya A.","family":"Simakov","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/019vsm959","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Science and Technology"},{"id":[{"id":"https://ror.org/03v8t4025","id-type":"ROR","asserted-by":"publisher"}],"name":"Russian Quantum Center"},{"id":[{"id":"https://ror.org/00v0z9322","id-type":"ROR","asserted-by":"publisher"}],"name":"Moscow Institute of Physics and Technology"}]},{"ORCID":"https://orcid.org/0000-0001-6845-5624","authenticated-orcid":true,"given":"Ilya S.","family":"Besedin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/019vsm959","id-type":"ROR","asserted-by":"publisher"}],"name":"National University of Science and Technology"},{"id":[{"id":"https://ror.org/03v8t4025","id-type":"ROR","asserted-by":"publisher"}],"name":"Russian Quantum Center"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.111.012444","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012444","note":"arxivid:2211.03094"},{"id":"arxiv:1011.5814","type":"article-journal","author":[{"given":"Sagarmoy","family":"Dutta"},{"given":"Piyush P","family":"Kurur"}],"title":"Quantum Cyclic Code of length dividing $p^{t}+1$","issued":{"date-parts":[[2011,3,25]]},"note":"arxivid:1011.5814\narxiv_version_number:2"},{"id":"preset:GriffithsGS","type":"document","author":[{"family":"Griffiths","given":"R.B."}],"title":"Graph states and graph codes"},{"id":"arxiv:2507.10519","type":"article-journal","author":[{"given":"Shival","family":"Dasu"},{"given":"Simon","family":"Burton"}],"title":"A Classification of Transversal Clifford Gates for Qubit Stabilizer Codes","issued":{"date-parts":[[2025,7,14]]},"note":"arxivid:2507.10519\narxiv_version_number:1"},{"id":"arxiv:1008.0425","type":"article-journal","author":[{"given":"Bilal A.","family":"Shaw"}],"title":"Quantum Steganography and Quantum Error-Correction","issued":{"date-parts":[[2010,8,2]]},"note":"arxivid:1008.0425\narxiv_version_number:1"},{"id":"arxiv:2501.12072","type":"article-journal","author":[{"given":"Harsh","family":"Gupta"},{"given":"Mainak","family":"Bhattacharyya"},{"given":"Ritik","family":"Jain"},{"given":"Ankur","family":"Raina"}],"title":"Fault-tolerant syndrome extraction in [[n,1,3]] non-CSS code family generated using measurements on graph states","issued":{"date-parts":[[2025,11,9]]},"note":"arxivid:2501.12072\narxiv_version_number:2"},{"id":"arxiv:2506.14688","type":"article-journal","author":[{"given":"Shival","family":"Dasu"},{"given":"Simon","family":"Burton"},{"given":"Karl","family":"Mayer"},{"given":"David","family":"Amaro"},{"given":"Justin A.","family":"Gerber"},{"given":"Kevin","family":"Gilmore"},{"given":"Dan","family":"Gresh"},{"given":"Davide","family":"DelVento"},{"given":"Andrew C.","family":"Potter"},{"given":"David","family":"Hayes"}],"title":"Breaking even with magic: demonstration of a high-fidelity logical non-Clifford gate","issued":{"date-parts":[[2025,6,17]]},"note":"arxivid:2506.14688\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0402171","type":"article-journal","author":[{"given":"E.","family":"Knill"}],"title":"Fault-Tolerant Postselected Quantum Computation: Schemes","issued":{"date-parts":[[2004,2,23]]},"note":"arxivid:quant-ph/0402171\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0404104","type":"article-journal","author":[{"given":"E.","family":"Knill"}],"title":"Fault-Tolerant Postselected Quantum Computation: Threshold Analysis","issued":{"date-parts":[[2004,4,19]]},"note":"arxivid:quant-ph/0404104\narxiv_version_number:1"},{"id":"arxiv:1907.00310","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/tqe.2020.3023419","page":"1-17","source":"Crossref","title":"Logical Clifford Synthesis for Stabilizer Codes","volume":"1","author":[{"ORCID":"https://orcid.org/0000-0002-2369-3159","authenticated-orcid":false,"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2084-9717","authenticated-orcid":false,"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2006-1030","authenticated-orcid":false,"given":"Swanand","family":"Kadhe","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5521-4397","authenticated-orcid":false,"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Quantum Engineering","original-title":[],"issued":{"date-parts":[[2020]]},"URL":"http://dx.doi.org/10.1109/TQE.2020.3023419","ISSN":["2689-1808"],"container-title-short":"IEEE Trans. Quantum Eng.","note":"arxivid:1907.00310"},{"id":"arxiv:2306.16608","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Quantum phase estimation (QPE) serves as a building block of many different quantum algorithms and finds important applications in computational chemistry problems. Despite the rapid development of quantum hardware, experimental demonstration of QPE for chemistry problems remains challenging due to its large circuit depth and the lack of quantum resources to protect the hardware from noise with fully fault-tolerant protocols. In the present work, we take a step towards fault-tolerant quantum computing by demonstrating a QPE algorithm on a Quantinuum trapped-ion computer. We employ a Bayesian approach to QPE and introduce a routine for optimal parameter selection, which we combine with a <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\"><a:mrow><a:mo>[[</a:mo><a:mi>n</a:mi><a:mo>+</a:mo><a:mn>2</a:mn><a:mo>,</a:mo><a:mi>n</a:mi><a:mo>,</a:mo><a:mn>2</a:mn><a:mo>]]</a:mo></a:mrow></a:math> quantum error detection code carefully tailored to the hardware capabilities. As a simple quantum chemistry example, we take a hydrogen molecule represented by a two-qubit Hamiltonian and estimate its ground state energy using our QPE protocol. In the experiment, we use the quantum circuits containing as many as 920 physical two-qubit gates to estimate the ground state energy within <b:math xmlns:b=\"http://www.w3.org/1998/Math/MathML\"><b:mrow><b:mn>6</b:mn><b:mo>×</b:mo><b:msup><b:mn>10</b:mn><b:mrow><b:mo>−</b:mo><b:mn>3</b:mn></b:mrow></b:msup></b:mrow></b:math> hartree of the exact value.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevresearch.6.013221","source":"Crossref","title":"Demonstrating Bayesian quantum phase estimation with quantum error detection","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-9994-1200","authenticated-orcid":true,"given":"Kentaro","family":"Yamamoto","sequence":"first","affiliation":[{"name":"Quantinuum K.K., Otemachi Financial City Grand Cube 3F, 1-9-2 Otemachi, Chiyoda-ku, Tokyo, Japan"}]},{"given":"Samuel","family":"Duffield","sequence":"additional","affiliation":[{"name":"Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom"}]},{"given":"Yuta","family":"Kikuchi","sequence":"additional","affiliation":[{"name":"Quantinuum K.K., Otemachi Financial City Grand Cube 3F, 1-9-2 Otemachi, Chiyoda-ku, Tokyo, Japan"},{"name":"Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan"}]},{"given":"David","family":"Muñoz Ramo","sequence":"additional","affiliation":[{"name":"Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2024,2,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.6.013221","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013221","note":"arxivid:2306.16608"},{"id":"arxiv:1702.01155","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.96.032341","source":"Crossref","title":"Fault tolerance with bare ancillary qubits for a [[7,1,3]] code","volume":"96","author":[{"given":"Muyuan","family":"Li","sequence":"first","affiliation":[]},{"given":"Mauricio","family":"Gutiérrez","sequence":"additional","affiliation":[]},{"given":"Stanley E.","family":"David","sequence":"additional","affiliation":[]},{"given":"Alonzo","family":"Hernandez","sequence":"additional","affiliation":[]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,9,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.96.032341","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032341","note":"arxivid:1702.01155"},{"id":"doi:10.1098/rsta.2011.0494","type":"journal-article","publisher":"The Royal Society","issue":"1975","abstract":"<jats:p>In this introductory article on the subject of quantum error correction and fault-tolerant quantum computation, we review three important ingredients that enter known constructions for fault-tolerant quantum computation, namely quantum codes, error discretization and transversal quantum gates. Taken together, they provide a ground on which the theory of quantum error correction can be developed and fault-tolerant quantum information protocols can be built.</jats:p>","DOI":"10.1098/rsta.2011.0494","page":"4541-4565","source":"Crossref","title":"Key ideas in quantum error correction","volume":"370","author":[{"given":"Robert","family":"Raussendorf","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University of British Columbia, Vancouver, Canada"}]}],"container-title":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2012,9,28]]},"URL":"http://dx.doi.org/10.1098/rsta.2011.0494","ISSN":["1364-503X","1471-2962"],"container-title-short":"Phil. Trans. R. Soc. A.","note":"alternative-id:10.1098/rsta.2011.0494"},{"id":"arxiv:1306.4532","type":"journal-article","publisher":"Open Publishing Association","DOI":"10.4204/eptcs.171.4","page":"33-49","source":"Crossref","title":"Verifying the Steane code with Quantomatic","volume":"171","author":[{"given":"Ross","family":"Duncan","sequence":"first","affiliation":[{"name":"University of Strathclyde, Glasgow, UK"}]},{"given":"Maxime","family":"Lucas","sequence":"additional","affiliation":[{"name":"Université Libre de Bruxelles, Brussels, Belgium"}]}],"container-title":"Electronic Proceedings in Theoretical Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,27]]},"URL":"http://dx.doi.org/10.4204/EPTCS.171.4","ISSN":["2075-2180"],"container-title-short":"Electron. Proc. Theor. Comput. Sci.","note":"arxivid:1306.4532"},{"id":"doi:10.1103/PhysRevLett.77.793","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.77.793","page":"793-797","source":"Crossref","title":"Error Correcting Codes in Quantum Theory","volume":"77","author":[{"given":"A. M.","family":"Steane","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1996,7,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.77.793","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett."},{"id":"arxiv:quant-ph/9601029","type":"journal-article","publisher":"The Royal Society","issue":"1954","DOI":"10.1098/rspa.1996.0136","page":"2551-2577","source":"Crossref","title":"Multiple-particle interference and quantum error correction","volume":"452","container-title":"Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[1996,11,8]]},"URL":"http://dx.doi.org/10.1098/rspa.1996.0136","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. Lond. A","note":"alternative-id:10.1098/rspa.1996.0136\narxivid:quant-ph/9601029"},{"id":"arxiv:2504.01083","type":"article-journal","author":[{"given":"Andrea","family":"Rodriguez-Blanco"},{"given":"Ho Nam","family":"Nguyen"},{"given":"K. Birgitta","family":"Whaley"}],"title":"Fault-tolerant correction-ready encoding of the [[7,1,3]] Steane code on a 2D grid","issued":{"date-parts":[[2025,4,1]]},"note":"arxivid:2504.01083\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0411206","type":"article-journal","author":[{"given":"Austin G.","family":"Fowler"}],"title":"Constructing arbitrary Steane code single logical qubit fault-tolerant gates","issued":{"date-parts":[[2010,12,20]]},"note":"arxivid:quant-ph/0411206\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0506126","type":"article-journal","author":[{"given":"Austin G.","family":"Fowler"}],"title":"Towards Large-Scale Quantum Computation","issued":{"date-parts":[[2005,6,16]]},"note":"arxivid:quant-ph/0506126\narxiv_version_number:1"},{"id":"arxiv:1303.4291","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.87.032320","source":"Crossref","title":"Non-fault-tolerant<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>T</mml:mi></mml:math>gates for the [7,1,3] quantum error-correction code","volume":"87","author":[{"given":"Yaakov S.","family":"Weinstein","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,3,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.032320","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032320","note":"arxivid:1303.4291"},{"id":"doi:10.1038/srep19578","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>The seven-qubit quantum error-correcting code originally proposed by Steane is one of the best known quantum codes. The Steane code has a desirable property that most basic operations can be performed easily in a fault-tolerant manner. A major obstacle to fault-tolerant quantum computation with the Steane code is fault-tolerant preparation of encoded states, which requires large computational resources. Here we propose efficient state preparation methods for zero and magic states encoded with the Steane code, where the zero state is one of the computational basis states and the magic state allows us to achieve universality in fault-tolerant quantum computation. The methods minimize resource overheads for the fault-tolerant state preparation and therefore reduce necessary resources for quantum computation with the Steane code. Thus, the present results will open a new possibility for efficient fault-tolerant quantum computation.</jats:p>","DOI":"10.1038/srep19578","source":"Crossref","title":"Minimizing resource overheads for fault-tolerant preparation of encoded states of the Steane code","volume":"6","author":[{"given":"Hayato","family":"Goto","sequence":"first","affiliation":[]}],"container-title":"Scientific Reports","original-title":[],"language":"en","issued":{"date-parts":[[2016,1,27]]},"URL":"http://dx.doi.org/10.1038/srep19578","ISSN":["2045-2322"],"container-title-short":"Sci Rep","page":"19578","note":"updated-by:{\"updated\":{\"date-parts\":[[2022,2,9]],\"date-time\":\"2022-02-09T00:00:00Z\",\"timestamp\":1644364800000},\"DOI\":\"10.1038/s41598-022-06633-6\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nalternative-id:BFsrep19578"},{"id":"arxiv:2401.10982","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Fault-tolerant architectures aim to reduce the noise of a quantum computation. Despite such architectures being well studied a detailed understanding of how noise is transformed in a fault-tolerant primitive such as magic state injection is currently lacking. We use numerical simulations of logical process tomography on a fault-tolerant gadget that implements a logical <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mi>Z</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>&amp;#x03C0;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math> gate using magic state injection, to understand how noise characteristics at the physical level are transformed into noise characteristics at the logical level. We show how, in this gadget, a significant phase (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math>) bias can arise in the logical noise, even with unbiased noise at the physical level. While the magic state injection gadget intrinsically induces biased noise, with extant phase bias being further amplified at the logical level, we identify noisy error correction circuits as a key limiting factor in the circuits studied on the magnitude of this logical noise bias. Our approach provides a framework for assessing the detailed noise characteristics, as well as the overall performance, of fault-tolerant logical primitives.</jats:p>","DOI":"10.22331/q-2025-06-24-1779","page":"1779","source":"Crossref","title":"Logical Noise Bias in Magic State Injection","volume":"9","author":[{"given":"Nicholas","family":"Fazio","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia."}]},{"given":"Robin","family":"Harper","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia."}]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia."}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,24]]},"URL":"http://dx.doi.org/10.22331/q-2025-06-24-1779","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2401.10982"},{"id":"arxiv:1101.1950","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.84.012323","source":"Crossref","title":"Fidelity of an encoded [7,1,3] logical zero","volume":"84","author":[{"given":"Yaakov S.","family":"Weinstein","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,7,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.84.012323","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012323","note":"arxivid:1101.1950"},{"id":"arxiv:1109.1714","type":"article-journal","author":[{"given":"Sidney D.","family":"Buchbinder"},{"given":"Channing L.","family":"Huang"},{"given":"Yaakov S.","family":"Weinstein"}],"title":"Encoding an Arbitrary State in a [7,1,3] Quantum Error Correction Code","issued":{"date-parts":[[2011,9,8]]},"note":"arxivid:1109.1714\narxiv_version_number:1"},{"id":"arxiv:1111.3930","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.86.052336","source":"Crossref","title":"Use of Shor states for the [7,1,3] quantum error-correcting code","volume":"86","author":[{"given":"Yaakov S.","family":"Weinstein","sequence":"first","affiliation":[]},{"given":"Sidney D.","family":"Buchbinder","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,11,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.052336","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052336","note":"arxivid:1111.3930"},{"id":"arxiv:2307.13296","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Logical qubits can be protected from decoherence by performing quantum error-correction (QEC) cycles repeatedly. Algorithms for fault-tolerant QEC must be compiled to the specific hardware platform under consideration in order to practically realize a quantum memory that operates for in principle arbitrary long times. All circuit components must be assumed as noisy unless specific assumptions about the form of the noise are made. Modern QEC schemes are challenging to implement experimentally in physical architectures where in-sequence measurements and feed forward of classical information cannot be reliably executed fast enough or even at all. Here we provide a novel scheme to perform QEC cycles without the need of measuring qubits that is fully fault-tolerant with respect to all components used in the circuit. Our scheme can be used for any low-distance CSS code since its only requirement towards the underlying code is a transversal  gate. Similarly to Steane-type EC, we coherently copy errors to a logical auxiliary qubit but then apply a coherent feedback operation from the auxiliary system to the logical data qubit. The logical auxiliary qubit is prepared fault tolerantly without measurements, too. We benchmark logical failure rates of the scheme in comparison to a flag-qubit-based EC cycle. We map out a parameter region where our scheme is feasible and estimate physical error rates necessary to achieve the break-even point of beneficial QEC with our scheme. We outline how our scheme could be implemented in ion traps and with neutral atoms in a tweezer array. For recently demonstrated capabilities of atom shuttling and native multiatom Rydberg gates, we achieve moderate circuit depths and beneficial performance of our scheme while not breaking fault tolerance. These results thereby enable practical fault-tolerant QEC in hardware architectures that do not support midcircuit measurements.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.010333","source":"Crossref","title":"Measurement-Free Fault-Tolerant Quantum Error Correction in Near-Term Devices","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-7581-2148","authenticated-orcid":true,"given":"Sascha","family":"Heußen","sequence":"first","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen 52056, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich 52425, Germany"}]},{"ORCID":"https://orcid.org/0009-0001-0180-4225","authenticated-orcid":true,"given":"David F.","family":"Locher","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen 52056, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich 52425, Germany"}]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":true,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen 52056, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich 52425, Germany"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,2,27]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.010333","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010333","note":"arxivid:2307.13296"},{"id":"arxiv:quant-ph/9605011","type":"article-journal","author":[{"given":"Peter W.","family":"Shor"}],"title":"Fault-tolerant quantum computation","issued":{"date-parts":[[1997,3,5]]},"note":"arxivid:quant-ph/9605011\narxiv_version_number:2"},{"id":"arxiv:0706.1382","type":"article-journal","author":[{"given":"Bei","family":"Zeng"},{"given":"Andrew","family":"Cross"},{"given":"Isaac L.","family":"Chuang"}],"title":"Transversality versus Universality for Additive Quantum Codes","issued":{"date-parts":[[2007,9,11]]},"note":"arxivid:0706.1382\narxiv_version_number:3"},{"id":"arxiv:1509.03239","type":"article-journal","author":[{"given":"Sergey","family":"Bravyi"},{"given":"Andrew","family":"Cross"}],"title":"Doubled Color Codes","issued":{"date-parts":[[2015,9,10]]},"note":"arxivid:1509.03239\narxiv_version_number:1"},{"id":"arxiv:1304.3709","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.111.090505","source":"Crossref","title":"Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction","volume":"111","author":[{"given":"Adam","family":"Paetznick","sequence":"first","affiliation":[]},{"given":"Ben W.","family":"Reichardt","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2013,8,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.111.090505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090505","note":"arxivid:1304.3709"},{"id":"arxiv:1403.2734","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.113.080501","source":"Crossref","title":"Fault-Tolerant Conversion between the Steane and Reed-Muller Quantum Codes","volume":"113","author":[{"given":"Jonas T.","family":"Anderson","sequence":"first","affiliation":[]},{"given":"Guillaume","family":"Duclos-Cianci","sequence":"additional","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,8,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.113.080501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080501","note":"arxivid:1403.2734"},{"id":"arxiv:1703.03860","type":"journal-article","publisher":"IOP Publishing","issue":"11","DOI":"10.1088/1751-8121/aaad13","page":"115305","source":"Crossref","title":"Fault-tolerant conversion between adjacent Reed–Muller quantum codes based on gauge fixing","volume":"51","author":[{"ORCID":"https://orcid.org/0000-0001-5420-3880","authenticated-orcid":false,"given":"Dong-Xiao","family":"Quan","sequence":"first","affiliation":[]},{"given":"Li-Li","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Chang-Xing","family":"Pei","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8326-8912","authenticated-orcid":false,"given":"Barry C","family":"Sanders","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2018,2,19]]},"URL":"http://dx.doi.org/10.1088/1751-8121/aaad13","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1703.03860"},{"id":"arxiv:2210.14074","type":"article-journal","author":[{"given":"Darren","family":"Banfield"},{"given":"Heather","family":"Leitch"},{"given":"Alastair","family":"Kay"}],"title":"Implementing Clifford Gates on Stabilizer Codes via Measurement","issued":{"date-parts":[[2025,2,7]]},"note":"arxivid:2210.14074\narxiv_version_number:3"},{"id":"doi:10.1088/2058-9565/abc6f4","type":"journal-article","publisher":"IOP Publishing","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Steane’s seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as ‘flagged’ syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Steane’s error-correction method, and also conforms to planar geometry constraints. We further show that Steane’s code and some others can be error-corrected with no extra qubits, provided there are at least two code blocks. The rough idea is that two seven-qubit codewords can be temporarily joined into a twelve-qubit code, freeing two qubits for flagged syndrome measurement.</jats:p>","DOI":"10.1088/2058-9565/abc6f4","page":"015007","source":"Crossref","title":"Fault-tolerant quantum error correction for Steane’s seven-qubit color code with few or no extra qubits","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-4934-8732","authenticated-orcid":false,"given":"Ben W","family":"Reichardt","sequence":"first","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2020,11,24]]},"URL":"http://dx.doi.org/10.1088/2058-9565/abc6f4","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol."},{"id":"arxiv:2505.09133","type":"journal-article","publisher":"American Physical Society (APS)","DOI":"10.1103/m7j3-5sk6","source":"Crossref","title":"Quantum error-corrected computation of molecular energies","author":[{"family":"Anonymous","sequence":"first","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,3]]},"URL":"http://dx.doi.org/10.1103/m7j3-5sk6","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","note":"arxivid:2505.09133"},{"id":"arxiv:1403.5426","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6194","abstract":"<jats:title>Fault-tolerant quantum computing</jats:title>\n          <jats:p>\n            Quantum states can be delicate. Attempts to process and manipulate quantum states can destroy the encoded information. Nigg\n            <jats:italic>et al.</jats:italic>\n            encoded the quantum state of a single qubit (in this case, a trapped ion) over the global properties of a series of trapped ions. These so-called stabilizers protected the information against noise sources that can degrade the single qubit. The protocol provides a route to fault-tolerant quantum computing.\n          </jats:p>\n          <jats:p>\n            <jats:italic>Science</jats:italic>\n            , this issue p.\n            <jats:related-article xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"doi\" issue=\"6194\" page=\"302\" related-article-type=\"in-this-issue\" vol=\"345\" xlink:href=\"10.1126/science.1253742\">302</jats:related-article>\n          </jats:p>","DOI":"10.1126/science.1253742","page":"302-305","source":"Crossref","title":"Quantum computations on a topologically encoded qubit","volume":"345","author":[{"given":"D.","family":"Nigg","sequence":"first","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M.","family":"Müller","sequence":"additional","affiliation":[{"name":"Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain."}]},{"given":"E. A.","family":"Martinez","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"P.","family":"Schindler","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M.","family":"Hennrich","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"T.","family":"Monz","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."}]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[{"name":"Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain."}]},{"given":"R.","family":"Blatt","sequence":"additional","affiliation":[{"name":"Institut für Experimentalphysik, Universität Innsbruck, A-6020 Innsbruck, Austria."},{"name":"Institut für Quantenoptik und Quanteninformation der Österreichischen Akademie der Wissenschaften, A-6020 Innsbruck, Austria."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2014,7,18]]},"URL":"http://dx.doi.org/10.1126/science.1253742","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.1253742\narxivid:1403.5426"},{"id":"arxiv:2107.07505","type":"article-journal","author":[{"given":"C.","family":"Ryan-Anderson"},{"given":"J. G.","family":"Bohnet"},{"given":"K.","family":"Lee"},{"given":"D.","family":"Gresh"},{"given":"A.","family":"Hankin"},{"given":"J. P.","family":"Gaebler"},{"given":"D.","family":"Francois"},{"given":"A.","family":"Chernoguzov"},{"given":"D.","family":"Lucchetti"},{"given":"N. C.","family":"Brown"},{"given":"T. M.","family":"Gatterman"},{"given":"S. K.","family":"Halit"},{"given":"K.","family":"Gilmore"},{"given":"J.","family":"Gerber"},{"given":"B.","family":"Neyenhuis"},{"given":"D.","family":"Hayes"},{"given":"R. P.","family":"Stutz"}],"title":"Realization of real-time fault-tolerant quantum error correction","issued":{"date-parts":[[2021,7,15]]},"note":"arxivid:2107.07505\narxiv_version_number:1"},{"id":"doi:10.1103/Physics.14.184","type":"journal-article","publisher":"American Physical Society (APS)","DOI":"10.1103/physics.14.184","source":"Crossref","title":"Real-Time Error Correction for Quantum Computing","volume":"14","author":[{"given":"Philip","family":"Ball","sequence":"first","affiliation":[]}],"container-title":"Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,23]]},"URL":"http://dx.doi.org/10.1103/Physics.14.184","ISSN":["1943-2879"],"container-title-short":"Physics","page":"184"},{"id":"arxiv:2111.12654","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7911","DOI":"10.1038/s41586-022-04721-1","page":"675-680","source":"Crossref","title":"Demonstration of fault-tolerant universal quantum gate operations","volume":"605","author":[{"given":"Lukas","family":"Postler","sequence":"first","affiliation":[]},{"given":"Sascha","family":"Heuβen","sequence":"additional","affiliation":[]},{"given":"Ivan","family":"Pogorelov","sequence":"additional","affiliation":[]},{"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Feldker","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Meth","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8401-3981","authenticated-orcid":false,"given":"Christian D.","family":"Marciniak","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8001-1487","authenticated-orcid":false,"given":"Roman","family":"Stricker","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Ringbauer","sequence":"additional","affiliation":[]},{"given":"Rainer","family":"Blatt","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9461-9650","authenticated-orcid":false,"given":"Philipp","family":"Schindler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":false,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7410-4804","authenticated-orcid":false,"given":"Thomas","family":"Monz","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,25]]},"URL":"http://dx.doi.org/10.1038/s41586-022-04721-1","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:4721\narxivid:2111.12654"},{"id":"arxiv:2404.08616","type":"article-journal","author":[{"given":"Karl","family":"Mayer"},{"given":"Ciarán","family":"Ryan-Anderson"},{"given":"Natalie","family":"Brown"},{"given":"Elijah","family":"Durso-Sabina"},{"given":"Charles H.","family":"Baldwin"},{"given":"David","family":"Hayes"},{"given":"Joan M.","family":"Dreiling"},{"given":"Cameron","family":"Foltz"},{"given":"John P.","family":"Gaebler"},{"given":"Thomas M.","family":"Gatterman"},{"given":"Justin A.","family":"Gerber"},{"given":"Kevin","family":"Gilmore"},{"given":"Dan","family":"Gresh"},{"given":"Nathan","family":"Hewitt"},{"given":"Chandler V.","family":"Horst"},{"given":"Jacob","family":"Johansen"},{"given":"Tanner","family":"Mengle"},{"given":"Michael","family":"Mills"},{"given":"Steven A.","family":"Moses"},{"given":"Peter E.","family":"Siegfried"},{"given":"Brian","family":"Neyenhuis"},{"given":"Juan","family":"Pino"},{"given":"Russell","family":"Stutz"}],"title":"Benchmarking logical three-qubit quantum Fourier transform encoded in the Steane code on a trapped-ion quantum computer","issued":{"date-parts":[[2024,4,12]]},"note":"arxivid:2404.08616\narxiv_version_number:1"},{"id":"arxiv:2404.16728","type":"article-journal","author":[{"given":"C.","family":"Ryan-Anderson"},{"given":"N. C.","family":"Brown"},{"given":"C. H.","family":"Baldwin"},{"given":"J. M.","family":"Dreiling"},{"given":"C.","family":"Foltz"},{"given":"J. P.","family":"Gaebler"},{"given":"T. M.","family":"Gatterman"},{"given":"N.","family":"Hewitt"},{"given":"C.","family":"Holliman"},{"given":"C. V.","family":"Horst"},{"given":"J.","family":"Johansen"},{"given":"D.","family":"Lucchetti"},{"given":"T.","family":"Mengle"},{"given":"M.","family":"Matheny"},{"given":"Y.","family":"Matsuoka"},{"given":"K.","family":"Mayer"},{"given":"M.","family":"Mills"},{"given":"S. A.","family":"Moses"},{"given":"B.","family":"Neyenhuis"},{"given":"J.","family":"Pino"},{"given":"P.","family":"Siegfried"},{"given":"R. P.","family":"Stutz"},{"given":"J.","family":"Walker"},{"given":"D.","family":"Hayes"}],"title":"High-fidelity and Fault-tolerant Teleportation of a Logical Qubit using Transversal Gates and Lattice Surgery on a Trapped-ion Quantum Computer","issued":{"date-parts":[[2024,4,25]]},"note":"arxivid:2404.16728\narxiv_version_number:1"},{"id":"arxiv:2506.14169","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Preparation of high-fidelity logical magic states has remained as a necessary but daunting step towards building a large-scale fault-tolerant quantum computer. One approach is to fault-tolerantly prepare a magic state in one code and then switch to another, a method known as code switching. We experimentally demonstrate this protocol on an ion-trap quantum processor, yielding a logical magic state encoded in an error-correcting code with state-of-the-art logical fidelity. Our experiment is based on the first demonstration of code switching between color codes, from the fifteen-qubit quantum Reed-Muller code to the seven-qubit Steane code. We prepare an encoded magic state in the Steane code with 82.58% probability, with an infidelity of at most <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mn>5.1</a:mn><a:mo stretchy=\"false\">(</a:mo><a:mn>2.7</a:mn><a:mo stretchy=\"false\">)</a:mo><a:mo>×</a:mo><a:msup><a:mn>10</a:mn><a:mrow><a:mo>−</a:mo><a:mn>4</a:mn></a:mrow></a:msup></a:math>. The reported infidelity is lower than the leading infidelity of the physical operations utilized in the protocol by a factor of at least 2.7, indicating the quantum processor is below the pseudothreshold. Furthermore, we create two copies of the magic state in the same quantum processor and perform a logical Bell basis measurement for a sample-efficient certification of the encoded magic state. The high-fidelity magic state can be combined with the already-demonstrated fault-tolerant Clifford gates, state preparation, and measurement of the 2D color code, completing a universal set of fault-tolerant computational primitives with logical error rates equal or better than the physical two-qubit error rate.</jats:p>","DOI":"10.1103/dck4-x9c2","source":"Crossref","title":"Experimental Demonstration of High-Fidelity Logical Magic States from Code Switching","volume":"15","author":[{"ORCID":"https://orcid.org/0000-0002-8699-1452","authenticated-orcid":true,"given":"Lucas","family":"Daguerre","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05rrcem69","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California"}]},{"ORCID":"https://orcid.org/0000-0001-8134-948X","authenticated-orcid":true,"given":"Robin","family":"Blume-Kohout","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01apwpt12","id-type":"ROR","asserted-by":"publisher"}],"name":"Sandia National Laboratories"}]},{"given":"Natalie C.","family":"Brown","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"given":"David","family":"Hayes","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0000-0001-7689-3157","authenticated-orcid":true,"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05rrcem69","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,14]]},"URL":"http://dx.doi.org/10.1103/dck4-x9c2","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041008","note":"arxivid:2506.14169"},{"id":"arxiv:2603.04584","type":"article-journal","author":[{"given":"Michael A.","family":"Perlin"},{"given":"Zichang","family":"He"},{"given":"Anthony Alexiades","family":"Armenakas"},{"given":"Pablo","family":"Andres-Martinez"},{"given":"Tianyi","family":"Hao"},{"given":"Dylan","family":"Herman"},{"given":"Yuwei","family":"Jin"},{"given":"Karl","family":"Mayer"},{"given":"Chris","family":"Self"},{"given":"David","family":"Amaro"},{"given":"Ciaran","family":"Ryan-Anderson"},{"given":"Ruslan","family":"Shaydulin"}],"title":"Fault-tolerant execution of error-corrected quantum algorithms","issued":{"date-parts":[[2026,3,4]]},"note":"arxivid:2603.04584\narxiv_version_number:1"},{"id":"arxiv:2312.03982","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7997","abstract":"<jats:title>Abstract</jats:title><jats:p>Suppressing errors is the central challenge for useful quantum computing<jats:sup>1</jats:sup>, requiring quantum error correction (QEC)<jats:sup>2–6</jats:sup>for large-scale processing. However, the overhead in the realization of error-corrected ‘logical’ qubits, in which information is encoded across many physical qubits for redundancy<jats:sup>2–4</jats:sup>, poses substantial challenges to large-scale logical quantum computing. Here we report the realization of a programmable quantum processor based on encoded logical qubits operating with up to 280 physical qubits. Using logical-level control and a zoned architecture in reconfigurable neutral-atom arrays<jats:sup>7</jats:sup>, our system combines high two-qubit gate fidelities<jats:sup>8</jats:sup>, arbitrary connectivity<jats:sup>7,9</jats:sup>, as well as fully programmable single-qubit rotations and mid-circuit readout<jats:sup>10–15</jats:sup>. Operating this logical processor with various types of encoding, we demonstrate improvement of a two-qubit logic gate by scaling surface-code<jats:sup>6</jats:sup>distance from<jats:italic>d</jats:italic> = 3 to<jats:italic>d</jats:italic> = 7, preparation of colour-code qubits with break-even fidelities<jats:sup>5</jats:sup>, fault-tolerant creation of logical Greenberger–Horne–Zeilinger (GHZ) states and feedforward entanglement teleportation, as well as operation of 40 colour-code qubits. Finally, using 3D [[8,3,2]] code blocks<jats:sup>16,17</jats:sup>, we realize computationally complex sampling circuits<jats:sup>18</jats:sup>with up to 48 logical qubits entangled with hypercube connectivity<jats:sup>19</jats:sup>with 228 logical two-qubit gates and 48 logical CCZ gates<jats:sup>20</jats:sup>. We find that this logical encoding substantially improves algorithmic performance with error detection, outperforming physical-qubit fidelities at both cross-entropy benchmarking and quantum simulations of fast scrambling<jats:sup>21,22</jats:sup>. These results herald the advent of early error-corrected quantum computation and chart a path towards large-scale logical processors.</jats:p>","DOI":"10.1038/s41586-023-06927-3","page":"58-65","source":"Crossref","title":"Logical quantum processor based on reconfigurable atom arrays","volume":"626","author":[{"ORCID":"https://orcid.org/0000-0002-9934-9530","authenticated-orcid":false,"given":"Dolev","family":"Bluvstein","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8986-1103","authenticated-orcid":false,"given":"Simon J.","family":"Evered","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5294-4941","authenticated-orcid":false,"given":"Alexandra A.","family":"Geim","sequence":"additional","affiliation":[]},{"given":"Sophie H.","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2148-8856","authenticated-orcid":false,"given":"Hengyun","family":"Zhou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3470-1369","authenticated-orcid":false,"given":"Tom","family":"Manovitz","sequence":"additional","affiliation":[]},{"given":"Sepehr","family":"Ebadi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5298-3112","authenticated-orcid":false,"given":"Madelyn","family":"Cain","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0605-8791","authenticated-orcid":false,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4766-7967","authenticated-orcid":false,"given":"Dominik","family":"Hangleiter","sequence":"additional","affiliation":[]},{"given":"J. Pablo","family":"Bonilla Ataides","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5775-9542","authenticated-orcid":false,"given":"Nishad","family":"Maskara","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7706-5927","authenticated-orcid":false,"given":"Iris","family":"Cong","sequence":"additional","affiliation":[]},{"given":"Xun","family":"Gao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0002-8337-0762","authenticated-orcid":false,"given":"Pedro","family":"Sales Rodriguez","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Karolyshyn","sequence":"additional","affiliation":[]},{"given":"Giulia","family":"Semeghini","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":false,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2935-2363","authenticated-orcid":false,"given":"Markus","family":"Greiner","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9786-0538","authenticated-orcid":false,"given":"Vladan","family":"Vuletić","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8658-1007","authenticated-orcid":false,"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,6]]},"URL":"http://dx.doi.org/10.1038/s41586-023-06927-3","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:6927\narxivid:2312.03982"},{"id":"arxiv:2506.20661","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8095","DOI":"10.1038/s41586-025-09848-5","page":"39-46","source":"Crossref","title":"A fault-tolerant neutral-atom architecture for universal quantum computation","volume":"649","author":[{"ORCID":"https://orcid.org/0000-0002-9934-9530","authenticated-orcid":false,"given":"Dolev","family":"Bluvstein","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5294-4941","authenticated-orcid":false,"given":"Alexandra A.","family":"Geim","sequence":"additional","affiliation":[]},{"given":"Sophie H.","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8986-1103","authenticated-orcid":false,"given":"Simon J.","family":"Evered","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5518-7907","authenticated-orcid":false,"given":"J. Pablo","family":"Bonilla Ataides","sequence":"additional","affiliation":[]},{"given":"Gefen","family":"Baranes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2748-7333","authenticated-orcid":false,"given":"Andi","family":"Gu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3470-1369","authenticated-orcid":false,"given":"Tom","family":"Manovitz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2384-0208","authenticated-orcid":false,"given":"Muqing","family":"Xu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0605-8791","authenticated-orcid":false,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[]},{"given":"Shayan","family":"Majidy","sequence":"additional","affiliation":[]},{"given":"Christian","family":"Kokail","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5775-9542","authenticated-orcid":false,"given":"Nishad","family":"Maskara","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0005-2464-5676","authenticated-orcid":false,"given":"Elias C.","family":"Trapp","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0005-7037-5240","authenticated-orcid":false,"given":"Luke M.","family":"Stewart","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2773-2554","authenticated-orcid":false,"given":"Simon","family":"Hollerith","sequence":"additional","affiliation":[]},{"given":"Hengyun","family":"Zhou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":false,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1655-9151","authenticated-orcid":false,"given":"Susanne F.","family":"Yelin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2935-2363","authenticated-orcid":false,"given":"Markus","family":"Greiner","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9786-0538","authenticated-orcid":false,"given":"Vladan","family":"Vuletić","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5298-3112","authenticated-orcid":false,"given":"Madelyn","family":"Cain","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8658-1007","authenticated-orcid":false,"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,10]]},"URL":"http://dx.doi.org/10.1038/s41586-025-09848-5","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"updated-by:{\"DOI\":\"10.1038/s41586-026-10108-3\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2026,1,19]],\"date-time\":\"2026-01-19T00:00:00Z\",\"timestamp\":1768780800000}}\nalternative-id:9848\narxivid:2506.20661"},{"id":"arxiv:0811.2639","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.80.042308","source":"Crossref","title":"Entanglement purification with double selection","volume":"80","author":[{"given":"Keisuke","family":"Fujii","sequence":"first","affiliation":[]},{"given":"Katsuji","family":"Yamamoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,10,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.042308","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042308","note":"arxivid:0811.2639"},{"id":"arxiv:2306.17686","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Topological color codes are widely acknowledged as promising candidates for fault-tolerant quantum computing. Neither a two-dimensional nor a three-dimensional topology, however, can provide a universal gate set {, , }, with the  gate missing in the two-dimensional and the  gate in the three-dimensional case. These complementary shortcomings of the isolated topologies may be overcome in a combined approach, by switching between a two- and a three-dimensional code while maintaining the logical state. In this work, we construct resource-optimized deterministic and nondeterministic code-switching protocols for two- and three-dimensional distance-three color codes using fault-tolerant quantum circuits based on flag qubits. Deterministic protocols allow for the fault-tolerant implementation of logical gates on an encoded quantum state, while nondeterministic protocols may be used for the fault-tolerant preparation of magic states. Taking the error rates of state-of-the-art trapped-ion quantum processors as a reference, we find a logical failure probability of 3% for deterministic logical gates, which cannot be realized transversally in the respective code. By replacing the three-dimensional distance-three color code in the protocol for magic state preparation with the morphed code introduced in Vasmer and Kubica [PRX Quantum 3, 030319 (2022)], we reduce the logical failure rates by 2 orders of magnitude, thus rendering it a viable method for magic state preparation on near-term quantum processors. Our results demonstrate that code switching enables the fault-tolerant and deterministic implementation of a universal gate set under realistic conditions, and thereby provide a practical avenue to advance universal, fault-tolerant quantum computing and enable quantum algorithms on first, error-corrected logical qubits.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020345","source":"Crossref","title":"Fault-Tolerant Code-Switching Protocols for Near-Term Quantum Processors","volume":"5","author":[{"ORCID":"https://orcid.org/0009-0007-8954-1274","authenticated-orcid":true,"given":"Friederike","family":"Butt","sequence":"first","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0002-7581-2148","authenticated-orcid":true,"given":"Sascha","family":"Heußen","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich"}]},{"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich"}]},{"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,28]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020345","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020345","note":"arxivid:2306.17686"},{"id":"arxiv:1309.3310","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.112.010505","source":"Crossref","title":"Using Concatenated Quantum Codes for Universal Fault-Tolerant Quantum Gates","volume":"112","author":[{"given":"Tomas","family":"Jochym-O’Connor","sequence":"first","affiliation":[]},{"given":"Raymond","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,1,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.112.010505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"010505","note":"arxivid:1309.3310"},{"id":"arxiv:1612.04795","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.</jats:p>","DOI":"10.22331/q-2017-04-25-2","page":"2","source":"Crossref","title":"The surface code with a twist","volume":"1","author":[{"ORCID":"https://orcid.org/0000-0001-9614-2836","authenticated-orcid":false,"given":"Theodore J.","family":"Yoder","sequence":"first","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology"}]},{"given":"Isaac H.","family":"Kim","sequence":"additional","affiliation":[{"name":"IBM, Thomas J. Watson Research Center"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2017,4,25]]},"URL":"http://dx.doi.org/10.22331/q-2017-04-25-2","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1612.04795"},{"id":"arxiv:2010.06628","type":"article-journal","author":[{"given":"Andrew J.","family":"Landahl"}],"title":"The surface code on the rhombic dodecahedron","issued":{"date-parts":[[2020,10,13]]},"note":"arxivid:2010.06628\narxiv_version_number:1"},{"id":"arxiv:1906.11394","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2022.3170846","page":"5955-5974","source":"Crossref","title":"Quantum Pin Codes","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0002-3445-0179","authenticated-orcid":false,"given":"Christophe","family":"Vuillot","sequence":"first","affiliation":[{"name":"QuTech, TU Delft, Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0002-7211-5515","authenticated-orcid":false,"given":"Nikolas P.","family":"Breuckmann","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University College London, London, U.K."}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2022.3170846","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1906.11394"},{"id":"arxiv:2007.07929","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Magic state distillation uses special codes to suppress errors in input states, which are often tailored to a Clifford-twirled error model. We present detailed measurement sequences for magic state distillation protocols which can suppress arbitrary errors on any part of a protocol, assuming the independence of errors across qubits. Provided with input magic states, our protocol operates on a two-dimensional square grid by measurements of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi><mml:mi>Z</mml:mi></mml:math> on horizontal pairs of qubits, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>X</mml:mi></mml:math> on vertical pairs, and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:math> on single qubits.</jats:p>","DOI":"10.22331/q-2021-01-20-383","page":"383","source":"Crossref","title":"Measurement sequences for magic state distillation","volume":"5","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[{"name":"Microsoft Quantum, Redmond, Washington, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Santa Barbara, California, USA"},{"name":"Microsoft Quantum, Redmond, Washington, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,1,20]]},"URL":"http://dx.doi.org/10.22331/q-2021-01-20-383","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2007.07929"},{"id":"arxiv:2603.15610","type":"article-journal","author":[{"given":"Shixin","family":"Wu"},{"given":"Dawei","family":"Zhong"},{"given":"Todd A.","family":"Brun"},{"given":"Daniel A.","family":"Lidar"}],"title":"Universal Weakly Fault-Tolerant Quantum Computation via Code Switching in the [[8,3,2]] Code","issued":{"date-parts":[[2026,4,6]]},"note":"arxivid:2603.15610\narxiv_version_number:2"},{"id":"arxiv:2507.20387","type":"article-journal","author":[{"given":"Jacob S.","family":"Nelson"},{"given":"Andrew J.","family":"Landahl"},{"given":"Andrew D.","family":"Baczewski"}],"title":"A small and interesting architecture for early fault-tolerant quantum computers","issued":{"date-parts":[[2025,7,27]]},"note":"arxivid:2507.20387\narxiv_version_number:1"},{"id":"arxiv:2309.09893","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"29","abstract":"<jats:p>Fault-tolerant operations based on stabilizer codes are the state of the art in suppressing error rates in quantum computations. Most such codes do not permit a straightforward implementation of non-Clifford logical operations, which are necessary to define a universal gate set. As a result, implementations of these operations must use either error-correcting codes with more complicated error correction procedures or gate teleportation and magic states, which are prepared at the logical level, increasing overhead to a degree that precludes near-term implementation. Here, we implement a small quantum algorithm, one-qubit addition, fault-tolerantly on a trapped-ion quantum computer, using the<jats:inline-formula><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><mml:mrow><mml:mo mathvariant=\"double-struck\">[[</mml:mo><mml:mn>8</mml:mn><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:mn>2</mml:mn><mml:mo mathvariant=\"double-struck\">]]</mml:mo></mml:mrow></mml:math></jats:inline-formula>color code. By removing unnecessary error correction circuits and using low-overhead techniques for fault-tolerant preparation and measurement, we reduce the number of error-prone two-qubit gates and measurements to 36. We observe arithmetic errors with a rate of ∼1.1 × 10<jats:sup>−3</jats:sup>for the fault-tolerant circuit and ∼9.5 × 10<jats:sup>−3</jats:sup>for the unencoded circuit.</jats:p>","DOI":"10.1126/sciadv.ado9024","source":"Crossref","title":"Fault-tolerant one-bit addition with the smallest interesting color code","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0002-5483-0339","authenticated-orcid":true,"given":"Yang","family":"Wang","sequence":"first","affiliation":[{"name":"QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft, Netherlands."},{"name":"3. Physikalisches Institut, ZAQuant University of Stuttgart, Allmandring 13, 70569 Stuttgart, Germany."}]},{"given":"Selwyn","family":"Simsek","sequence":"additional","affiliation":[{"name":"Quantinuum Terrington House, 13–15 Hills Road, Cambridge CB2 1NL, UK."}]},{"ORCID":"https://orcid.org/0009-0004-6338-4221","authenticated-orcid":true,"given":"Thomas M.","family":"Gatterman","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"ORCID":"https://orcid.org/0000-0002-3140-4935","authenticated-orcid":true,"given":"Justin A.","family":"Gerber","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8230-7000","authenticated-orcid":true,"given":"Kevin","family":"Gilmore","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"given":"Dan","family":"Gresh","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"ORCID":"https://orcid.org/0000-0002-1021-0795","authenticated-orcid":true,"given":"Nathan","family":"Hewitt","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"ORCID":"https://orcid.org/0009-0007-5208-7338","authenticated-orcid":true,"given":"Chandler V.","family":"Horst","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9865-3338","authenticated-orcid":true,"given":"Mitchell","family":"Matheny","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"given":"Tanner","family":"Mengle","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"given":"Brian","family":"Neyenhuis","sequence":"additional","affiliation":[{"name":"Quantinuum 303 South Technology Ct., Broomfield, CO 80021, USA."}]},{"ORCID":"https://orcid.org/0000-0001-9959-6462","authenticated-orcid":true,"given":"Ben","family":"Criger","sequence":"additional","affiliation":[{"name":"Quantinuum Terrington House, 13–15 Hills Road, Cambridge CB2 1NL, UK."},{"name":"Institute for Globally Distributed Open Research and Education (IGDORE)."}]}],"container-title":"Science Advances","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,19]]},"URL":"http://dx.doi.org/10.1126/sciadv.ado9024","ISSN":["2375-2548"],"container-title-short":"Sci. Adv.","page":"eado9024","note":"alternative-id:10.1126/sciadv.ado9024\narxivid:2309.09893"},{"id":"arxiv:2309.08663","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.109.062438","source":"Crossref","title":"Implementing fault-tolerant non-Clifford gates using the [[8,3,2]] color code","volume":"109","author":[{"ORCID":"https://orcid.org/0009-0000-0478-2220","authenticated-orcid":true,"given":"Daniel","family":"Honciuc Menendez","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03dbr7087","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Toronto"}]},{"ORCID":"https://orcid.org/0009-0000-8217-4108","authenticated-orcid":true,"given":"Annie","family":"Ray","sequence":"additional","affiliation":[{"name":"University of Waterloo"},{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"}]},{"ORCID":"https://orcid.org/0000-0002-6711-5924","authenticated-orcid":true,"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"name":"University of Waterloo"},{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.109.062438","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062438","note":"arxivid:2309.08663"},{"id":"arxiv:quant-ph/9607030","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.56.76","page":"76-82","source":"Crossref","title":"Efficient computations of encodings for quantum error correction","volume":"56","author":[{"given":"Richard","family":"Cleve","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,7,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.56.76","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9607030"},{"id":"arxiv:2504.11444","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce65121.2025.00027","page":"158-168","source":"Crossref","title":"Fault Tolerant Quantum Simulation via Symplectic Transvections","author":[{"given":"Zhuangzhuang","family":"Chen","sequence":"first","affiliation":[{"name":"University of Arizona,Dept. of Elec. Comp. Engg.,Tucson,USA"}]},{"given":"Jack Owen","family":"Weinberg","sequence":"additional","affiliation":[{"name":"University of Arizona,Dept. of Elec. Comp. Engg.,Tucson,USA"}]},{"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"University of Arizona,Dept. of Elec. Comp. Engg.,Tucson,USA"}]}],"event":"2025 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2025 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2025,8,30]]},"URL":"http://dx.doi.org/10.1109/QCE65121.2025.00027","note":"arxivid:2504.11444"},{"id":"arxiv:0704.2122","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.101.090501","source":"Crossref","title":"Nonadditive Quantum Error-Correcting Code","volume":"101","author":[{"given":"Sixia","family":"Yu","sequence":"first","affiliation":[]},{"given":"Qing","family":"Chen","sequence":"additional","affiliation":[]},{"given":"C. H.","family":"Lai","sequence":"additional","affiliation":[]},{"given":"C. H.","family":"Oh","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2008,8,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.101.090501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090501","note":"arxivid:0704.2122"},{"id":"doi:10.1103/PhysRevA.52.R2493","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.52.r2493","page":"R2493-R2496","source":"Crossref","title":"Scheme for reducing decoherence in quantum computer memory","volume":"52","author":[{"given":"Peter W.","family":"Shor","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1995,10,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.52.R2493","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A"},{"id":"arxiv:2303.17211","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevresearch.5.043137","source":"Crossref","title":"Measurement-free fault-tolerant logical-zero-state encoding of the distance-three nine-qubit surface code in a one-dimensional qubit array","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0003-1145-6620","authenticated-orcid":true,"given":"Hayato","family":"Goto","sequence":"first","affiliation":[]},{"given":"Yinghao","family":"Ho","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3462-8288","authenticated-orcid":true,"given":"Taro","family":"Kanao","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.5.043137","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"043137","note":"arxivid:2303.17211"},{"id":"arxiv:2104.01205","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevapplied.16.024057","source":"Crossref","title":"Demonstration of Shor Encoding on a Trapped-Ion Quantum Computer","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0002-0462-9258","authenticated-orcid":true,"given":"Nhung H.","family":"Nguyen","sequence":"first","affiliation":[{"name":"Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Muyuan","family":"Li","sequence":"additional","affiliation":[{"name":"Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, North Carolina 27708, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3365-4433","authenticated-orcid":true,"given":"Alaina M.","family":"Green","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3673-9985","authenticated-orcid":true,"given":"C.","family":"Huerta Alderete","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Yingyue","family":"Zhu","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0003-0019-256X","authenticated-orcid":true,"given":"Daiwei","family":"Zhu","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":true,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Departments of Electrical and Computer Engineering, Chemistry, and Physics, Duke University, Durham, North Carolina 27708, USA"}]},{"given":"Norbert M.","family":"Linke","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA"}]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.16.024057","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"024057","note":"arxivid:2104.01205"},{"id":"arxiv:2105.05068","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevlett.127.240501","source":"Crossref","title":"Optimizing Stabilizer Parities for Improved Logical Qubit Memories","volume":"127","author":[{"given":"Dripto M.","family":"Debroy","sequence":"first","affiliation":[{"name":"Department of Physics, Duke University, Durham, North Carolina 27708, USA"}]},{"ORCID":"https://orcid.org/0000-0003-2405-9106","authenticated-orcid":true,"given":"Laird","family":"Egan","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, Center for Quantum Information and Computer Science, and Departments of Physics and Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0002-2977-2747","authenticated-orcid":true,"given":"Crystal","family":"Noel","sequence":"additional","affiliation":[{"name":"Department of Physics, Duke University, Durham, North Carolina 27708, USA"},{"name":"Joint Quantum Institute, Center for Quantum Information and Computer Science, and Departments of Physics and Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA"}]},{"given":"Andrew","family":"Risinger","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, Center for Quantum Information and Computer Science, and Departments of Physics and Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Daiwei","family":"Zhu","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, Center for Quantum Information and Computer Science, and Departments of Physics and Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Debopriyo","family":"Biswas","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, Center for Quantum Information and Computer Science, and Departments of Physics and Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA"}]},{"ORCID":"https://orcid.org/0000-0003-1942-9977","authenticated-orcid":true,"given":"Marko","family":"Cetina","sequence":"additional","affiliation":[{"name":"Department of Physics, Duke University, Durham, North Carolina 27708, USA"},{"name":"Joint Quantum Institute, Center for Quantum Information and Computer Science, and Departments of Physics and Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA"}]},{"given":"Chris","family":"Monroe","sequence":"additional","affiliation":[{"name":"Department of Physics, Duke University, Durham, North Carolina 27708, USA"},{"name":"Joint Quantum Institute, Center for Quantum Information and Computer Science, and Departments of Physics and Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA"},{"name":"IonQ, College Park, Maryland 20740, USA"}]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Department of Physics, Duke University, Durham, North Carolina 27708, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.127.240501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240501","note":"arxivid:2105.05068"},{"id":"arxiv:2009.06242","type":"journal-article","publisher":"Proceedings of the National Academy of Sciences","issue":"36","abstract":"<jats:title>Significance</jats:title>\n          <jats:p>Quantum teleportation and quantum error correction play crucial roles in fault-tolerant quantum computing. Here, we implemented error-correctable quantum teleportation to manipulate a logical qubit and observed the protection of quantum information. Our work presents a useful technology for scalable quantum computing and can serve as a quantum simulator for holographic quantum gravity.</jats:p>","DOI":"10.1073/pnas.2026250118","source":"Crossref","title":"Quantum teleportation of physical qubits into logical code spaces","volume":"118","author":[{"given":"Yi-Han","family":"Luo","sequence":"first","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Ming-Cheng","family":"Chen","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Manuel","family":"Erhard","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, A-1090 Vienna, Austria;"},{"name":"Vienna Center for Quantum Science and Technology, Faculty of Physics, University of Vienna, A-1090 Vienna, Austria;"}]},{"given":"Han-Sen","family":"Zhong","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Dian","family":"Wu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Hao-Yang","family":"Tang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Qi","family":"Zhao","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Xi-Lin","family":"Wang","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Keisuke","family":"Fujii","sequence":"additional","affiliation":[{"name":"Division of Advanced Electronics and Optical Science, Graduate School of Engineering Science, Osaka University, Toyonaka 560-8531, Japan;"}]},{"given":"Li","family":"Li","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Nai-Le","family":"Liu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[{"name":"NTT Basic Research Laboratories, NTT Research Center for Theoretical Quantum Physics, NTT Corporation, Kanagawa 243-0198, Japan;"},{"name":"National Institute of Informatics, Tokyo 101-8430, Japan"}]},{"given":"William J.","family":"Munro","sequence":"additional","affiliation":[{"name":"NTT Basic Research Laboratories, NTT Research Center for Theoretical Quantum Physics, NTT Corporation, Kanagawa 243-0198, Japan;"},{"name":"National Institute of Informatics, Tokyo 101-8430, Japan"}]},{"given":"Chao-Yang","family":"Lu","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]},{"given":"Anton","family":"Zeilinger","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, A-1090 Vienna, Austria;"},{"name":"Vienna Center for Quantum Science and Technology, Faculty of Physics, University of Vienna, A-1090 Vienna, Austria;"}]},{"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[{"name":"Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;"},{"name":"Chinese Academy of Sciences Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, Hefei, Anhui 230026, China;"}]}],"container-title":"Proceedings of the National Academy of Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2021,9,3]]},"URL":"http://dx.doi.org/10.1073/pnas.2026250118","ISSN":["0027-8424","1091-6490"],"container-title-short":"Proc. Natl. Acad. Sci. U.S.A.","page":"e2026250118","note":"alternative-id:10.1073/pnas.2026250118\narxivid:2009.06242"},{"id":"arxiv:2203.07979","type":"journal-article","publisher":"Optica Publishing Group","issue":"2","abstract":"<jats:p>The all-photonic quantum repeater (APQR) is a promising repeater scheme to realize long-distance quantum communication. For a practical APQR, an indispensable requirement is the robustness of the repeater graph state (RGS) against photon loss. We propose a new loss-tolerant scheme by applying the generalized Shor code to RGS, which can be experimentally demonstrated with current technology. Experimentally, we first prepare and verify the nine-qubit Shor code. Then, by applying the generalized Shor code to APQR and preparing a simplified encoded RGS with the structure of <jats:inline-formula>\n      <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n\t<mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t    <mml:mn>1</mml:mn>\n\t  </mml:mrow>\n\t</mml:mrow>\n\t<mml:mo>×</mml:mo>\n\t<mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n\t    <mml:mn>2</mml:mn>\n\t  </mml:mrow>\n\t</mml:mrow>\n      </mml:math>\n    </jats:inline-formula> based on the Shor code state, the effectiveness of our loss-tolerant scheme and the loss tolerance of the encoded RGS are respectively verified. Our results make an essential step toward a practical APQR and enrich the research of quantum error correction code.</jats:p>","DOI":"10.1364/optica.439170","page":"152","source":"Crossref","title":"Loss-tolerant all-photonic quantum repeater with generalized Shor code","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0001-5687-5576","authenticated-orcid":true,"given":"Rui","family":"Zhang","sequence":"first","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Li-Zheng","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Zheng-Da","family":"Li","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yue-Yang","family":"Fei","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Xu-Fei","family":"Yin","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Li","family":"Li","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Nai-Le","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yingqiu","family":"Mao","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Yu-Ao","family":"Chen","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]},{"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[{"name":"University of Science and Technology of China"},{"name":"Shanghai Research Center for Quantum Sciences"}]}],"container-title":"Optica","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,2]]},"URL":"http://dx.doi.org/10.1364/OPTICA.439170","ISSN":["2334-2536"],"container-title-short":"Optica","note":"arxivid:2203.07979"},{"id":"arxiv:quant-ph/9810055","type":"article-journal","author":[{"given":"Michael H.","family":"Freedman"},{"given":"David A.","family":"Meyer"}],"title":"Projective plane and planar quantum codes","issued":{"date-parts":[[1998,10,18]]},"note":"arxivid:quant-ph/9810055\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0508131","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.95.230504","source":"Crossref","title":"Stabilizer Formalism for Operator Quantum Error Correction","volume":"95","author":[{"given":"David","family":"Poulin","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,12,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.95.230504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230504","note":"arxivid:quant-ph/0508131"},{"id":"doi:10.1137/S0895480104441085","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","DOI":"10.1137/s0895480104441085","page":"363-370","source":"Crossref","title":"Classification of Self-Orthogonal Codes over \\boldmath$\\F_3$ and \\boldmath$\\F_4$","volume":"19","author":[{"given":"Iliya","family":"Bouyukliev","sequence":"first","affiliation":[]},{"given":"Patric R. J.","family":"Östergard","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2005,1]]},"URL":"http://dx.doi.org/10.1137/S0895480104441085","ISSN":["0895-4801","1095-7146"],"container-title-short":"SIAM J. Discrete Math.","note":"alternative-id:10.1137/S0895480104441085"},{"id":"arxiv:1404.3747","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.90.062320","source":"Crossref","title":"Low-distance surface codes under realistic quantum noise","volume":"90","author":[{"given":"Yu","family":"Tomita","sequence":"first","affiliation":[]},{"given":"Krysta M.","family":"Svore","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.062320","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062320","note":"arxivid:1404.3747"},{"id":"arxiv:1708.08683","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.97.012318","source":"Crossref","title":"Measurement-free implementations of small-scale surface codes for quantum-dot qubits","volume":"97","author":[{"given":"H. Ekmel","family":"Ercan","sequence":"first","affiliation":[]},{"given":"Joydip","family":"Ghosh","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Crow","sequence":"additional","affiliation":[]},{"given":"Vickram N.","family":"Premakumar","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Joynt","sequence":"additional","affiliation":[]},{"given":"Mark","family":"Friesen","sequence":"additional","affiliation":[]},{"given":"S. N.","family":"Coppersmith","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,1,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.012318","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012318","note":"arxivid:1708.08683"},{"id":"arxiv:2112.03708","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7911","DOI":"10.1038/s41586-022-04566-8","page":"669-674","source":"Crossref","title":"Realizing repeated quantum error correction in a distance-three surface code","volume":"605","author":[{"ORCID":"https://orcid.org/0000-0002-4388-071X","authenticated-orcid":false,"given":"Sebastian","family":"Krinner","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2002-9495","authenticated-orcid":false,"given":"Nathan","family":"Lacroix","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3220-7044","authenticated-orcid":false,"given":"Ants","family":"Remm","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5174-7310","authenticated-orcid":false,"given":"Agustin","family":"Di Paolo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2226-1813","authenticated-orcid":false,"given":"Elie","family":"Genois","sequence":"additional","affiliation":[]},{"given":"Catherine","family":"Leroux","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9929-9684","authenticated-orcid":false,"given":"Christoph","family":"Hellings","sequence":"additional","affiliation":[]},{"given":"Stefania","family":"Lazar","sequence":"additional","affiliation":[]},{"given":"Francois","family":"Swiadek","sequence":"additional","affiliation":[]},{"given":"Johannes","family":"Herrmann","sequence":"additional","affiliation":[]},{"given":"Graham J.","family":"Norris","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2253-6239","authenticated-orcid":false,"given":"Christian Kraglund","family":"Andersen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":false,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0591-2012","authenticated-orcid":false,"given":"Alexandre","family":"Blais","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5109-5009","authenticated-orcid":false,"given":"Christopher","family":"Eichler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3476-4485","authenticated-orcid":false,"given":"Andreas","family":"Wallraff","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,25]]},"URL":"http://dx.doi.org/10.1038/s41586-022-04566-8","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:4566\narxivid:2112.03708"},{"id":"arxiv:2112.13505","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.129.030501","source":"Crossref","title":"Realization of an Error-Correcting Surface Code with Superconducting Qubits","volume":"129","author":[{"given":"Youwei","family":"Zhao","sequence":"first","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Yangsen","family":"Ye","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"He-Liang","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Yiming","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Dachao","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Huijie","family":"Guan","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Qingling","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Zuolin","family":"Wei","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Tan","family":"He","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Sirui","family":"Cao","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Fusheng","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Tung-Hsun","family":"Chung","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Hui","family":"Deng","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Daojin","family":"Fan","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Ming","family":"Gong","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Cheng","family":"Guo","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Shaojun","family":"Guo","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Lianchen","family":"Han","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Na","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Shaowei","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Yuan","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Futian","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Jin","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Haoran","family":"Qian","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Hao","family":"Rong","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Hong","family":"Su","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Lihua","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Shiyu","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Yulin","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Yu","family":"Xu","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Chong","family":"Ying","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Jiale","family":"Yu","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Chen","family":"Zha","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Kaili","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Yong-Heng","family":"Huo","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Chao-Yang","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Cheng-Zhi","family":"Peng","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"ORCID":"https://orcid.org/0000-0001-5974-9452","authenticated-orcid":true,"given":"Xiaobo","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]},{"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[{"name":"Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Shanghai Branch, CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China"},{"name":"Shanghai Research Center for Quantum Sciences, Shanghai 201315, China"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.129.030501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030501","note":"arxivid:2112.13505"},{"id":"doi:10.1103/Physics.15.103","type":"journal-article","publisher":"American Physical Society (APS)","DOI":"10.1103/physics.15.103","source":"Crossref","title":"Error-Correcting Surface Codes Get Experimental Vetting","volume":"15","author":[{"given":"Luigi","family":"Frunzio","sequence":"first","affiliation":[]},{"given":"Shraddha","family":"Singh","sequence":"additional","affiliation":[]}],"container-title":"Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,11]]},"URL":"http://dx.doi.org/10.1103/Physics.15.103","ISSN":["1943-2879"],"container-title-short":"Physics","page":"103"},{"id":"arxiv:2305.15972","type":"article-journal","author":[{"given":"Yangsen","family":"Ye"},{"given":"Tan","family":"He"},{"given":"He-Liang","family":"Huang"},{"given":"Zuolin","family":"Wei"},{"given":"Yiming","family":"Zhang"},{"given":"Youwei","family":"Zhao"},{"given":"Dachao","family":"Wu"},{"given":"Qingling","family":"Zhu"},{"given":"Huijie","family":"Guan"},{"given":"Sirui","family":"Cao"},{"given":"Fusheng","family":"Chen"},{"given":"Tung-Hsun","family":"Chung"},{"given":"Hui","family":"Deng"},{"given":"Daojin","family":"Fan"},{"given":"Ming","family":"Gong"},{"given":"Cheng","family":"Guo"},{"given":"Shaojun","family":"Guo"},{"given":"Lianchen","family":"Han"},{"given":"Na","family":"Li"},{"given":"Shaowei","family":"Li"},{"given":"Yuan","family":"Li"},{"given":"Futian","family":"Liang"},{"given":"Jin","family":"Lin"},{"given":"Haoran","family":"Qian"},{"given":"Hao","family":"Rong"},{"given":"Hong","family":"Su"},{"given":"Shiyu","family":"Wang"},{"given":"Yulin","family":"Wu"},{"given":"Yu","family":"Xu"},{"given":"Chong","family":"Ying"},{"given":"Jiale","family":"Yu"},{"given":"Chen","family":"Zha"},{"given":"Kaili","family":"Zhang"},{"given":"Yong-Heng","family":"Huo"},{"given":"Chao-Yang","family":"Lu"},{"given":"Cheng-Zhi","family":"Peng"},{"given":"Xiaobo","family":"Zhu"},{"given":"Jian-Wei","family":"Pan"}],"title":"Logical Magic State Preparation with Fidelity Beyond the Distillation Threshold on a Superconducting Quantum Processor","issued":{"date-parts":[[2023,5,31]]},"note":"arxivid:2305.15972\narxiv_version_number:2"},{"id":"arxiv:2502.17722","type":"article-journal","author":[{"given":"Ants","family":"Remm"},{"given":"Nathan","family":"Lacroix"},{"given":"Lukas","family":"Bödeker"},{"given":"Elie","family":"Genois"},{"given":"Christoph","family":"Hellings"},{"given":"François","family":"Swiadek"},{"given":"Graham J.","family":"Norris"},{"given":"Christopher","family":"Eichler"},{"given":"Alexandre","family":"Blais"},{"given":"Markus","family":"Müller"},{"given":"Sebastian","family":"Krinner"},{"given":"Andreas","family":"Wallraff"}],"title":"Experimentally Informed Decoding of Stabilizer Codes Based on Syndrome Correlations","issued":{"date-parts":[[2025,2,24]]},"note":"arxivid:2502.17722\narxiv_version_number:1"},{"id":"arxiv:1710.01378","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1088/1367-2630/aab341","page":"043038","source":"Crossref","title":"Simulating the performance of a distance-3 surface code in a linear ion trap","volume":"20","author":[{"given":"Colin J","family":"Trout","sequence":"first","affiliation":[]},{"given":"Muyuan","family":"Li","sequence":"additional","affiliation":[]},{"given":"Mauricio","family":"Gutiérrez","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2184-7553","authenticated-orcid":false,"given":"Yukai","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Sheng-Tao","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Luming","family":"Duan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":false,"given":"Kenneth R","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2018,4,19]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aab341","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1710.01378"},{"id":"arxiv:1910.08495","type":"article-journal","author":[{"given":"Dripto M.","family":"Debroy"},{"given":"Muyuan","family":"Li"},{"given":"Shilin","family":"Huang"},{"given":"Kenneth R.","family":"Brown"}],"title":"Logical Performance of 9 Qubit Compass Codes in Ion Traps with Crosstalk Errors","issued":{"date-parts":[[2020,3,15]]},"note":"arxivid:1910.08495\narxiv_version_number:2"},{"id":"arxiv:1612.08208","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevapplied.8.034021","source":"Crossref","title":"Scalable Quantum Circuit and Control for a Superconducting Surface Code","volume":"8","author":[{"given":"R.","family":"Versluis","sequence":"first","affiliation":[]},{"given":"S.","family":"Poletto","sequence":"additional","affiliation":[]},{"given":"N.","family":"Khammassi","sequence":"additional","affiliation":[]},{"given":"B.","family":"Tarasinski","sequence":"additional","affiliation":[]},{"given":"N.","family":"Haider","sequence":"additional","affiliation":[]},{"given":"D. J.","family":"Michalak","sequence":"additional","affiliation":[]},{"given":"A.","family":"Bruno","sequence":"additional","affiliation":[]},{"given":"K.","family":"Bertels","sequence":"additional","affiliation":[]},{"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2017,9,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.8.034021","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"034021","note":"arxivid:1612.08208"},{"id":"arxiv:1703.04136","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>We present a density-matrix simulation of the quantum memory and computing performance of the distance-3 logical qubit Surface-17, following a recently proposed quantum circuit and using experimental error parameters for transmon qubits in a planar circuit QED architecture. We use this simulation to optimize components of the QEC scheme (e.g., trading off stabilizer measurement infidelity for reduced cycle time) and to investigate the benefits of feedback harnessing the fundamental asymmetry of relaxation-dominated error in the constituent transmons. A lower-order approximate calculation extends these predictions to the distance-5 Surface-49. These results clearly indicate error rates below the fault-tolerance threshold of the surface code, and the potential for Surface-17 to perform beyond the break-even point of quantum memory. However, Surface-49 is required to surpass the break-even point of computation at state-of-the-art qubit relaxation times and readout speeds.</jats:p>","DOI":"10.1038/s41534-017-0039-x","source":"Crossref","title":"Density-matrix simulation of small surface codes under current and projected experimental noise","volume":"3","author":[{"given":"T. E.","family":"O’Brien","sequence":"first","affiliation":[]},{"given":"B.","family":"Tarasinski","sequence":"additional","affiliation":[]},{"given":"L.","family":"DiCarlo","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2017,9,25]]},"URL":"http://dx.doi.org/10.1038/s41534-017-0039-x","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"39","note":"alternative-id:39\narxivid:1703.04136"},{"id":"arxiv:2002.07119","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Leakage outside of the qubit computational subspace, present in many leading experimental platforms, constitutes a threatening error for quantum error correction (QEC) for qubits. We develop a leakage-detection scheme via Hidden Markov models (HMMs) for transmon-based implementations of the surface code. By performing realistic density-matrix simulations of the distance-3 surface code (Surface-17), we observe that leakage is sharply projected and leads to an increase in the surface-code defect probability of neighboring stabilizers. Together with the analog readout of the ancilla qubits, this increase enables the accurate detection of the time and location of leakage. We restore the logical error rate below the memory break-even point by post-selecting out leakage, discarding less than half of the data for the given noise parameters. Leakage detection via HMMs opens the prospect for near-term QEC demonstrations, targeted leakage reduction and leakage-aware decoding and is applicable to other experimental platforms.</jats:p>","DOI":"10.1038/s41534-020-00330-w","source":"Crossref","title":"Leakage detection for a transmon-based surface code","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-7124-8933","authenticated-orcid":false,"given":"Boris Mihailov","family":"Varbanov","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4800-2518","authenticated-orcid":false,"given":"Francesco","family":"Battistel","sequence":"additional","affiliation":[]},{"given":"Brian Michael","family":"Tarasinski","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9892-3759","authenticated-orcid":false,"given":"Viacheslav Petrovych","family":"Ostroukh","sequence":"additional","affiliation":[]},{"given":"Thomas Eugene","family":"O’Brien","sequence":"additional","affiliation":[]},{"given":"Leonardo","family":"DiCarlo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0218-6614","authenticated-orcid":false,"given":"Barbara Maria","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,14]]},"URL":"http://dx.doi.org/10.1038/s41534-020-00330-w","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"102","note":"alternative-id:330\narxivid:2002.07119"},{"id":"arxiv:0901.1968","type":"article-journal","author":[{"given":"Sixia","family":"Yu"},{"given":"Juergen","family":"Bierbrauer"},{"given":"Ying","family":"Dong"},{"given":"Qing","family":"Chen"},{"given":"C. H.","family":"Oh"}],"title":"All the stabilizer codes of distance 3","issued":{"date-parts":[[2011,7,29]]},"note":"arxivid:0901.1968\narxiv_version_number:3"},{"id":"arxiv:2009.01244","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge <jats:italic>c</jats:italic> = <jats:italic>n</jats:italic> ≤ 12, and find many interesting examples. Among them is a non-chiral <jats:italic>E</jats:italic><jats:sub>8</jats:sub> theory, which is based on the root lattice of <jats:italic>E</jats:italic><jats:sub>8</jats:sub> understood as an even self-dual Lorentzian lattice. By analyzing all graphs with <jats:italic>n</jats:italic> ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.</jats:p>","DOI":"10.1007/jhep03(2021)160","source":"Crossref","title":"Quantum stabilizer codes, lattices, and CFTs","volume":"2021","author":[{"ORCID":"https://orcid.org/0000-0001-5762-6774","authenticated-orcid":false,"given":"Anatoly","family":"Dymarsky","sequence":"first","affiliation":[]},{"given":"Alfred","family":"Shapere","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,3]]},"URL":"http://dx.doi.org/10.1007/JHEP03(2021)160","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"160","note":"alternative-id:15117\narxivid:2009.01244"},{"id":"arxiv:2205.12998","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.106.022432","source":"Crossref","title":"Quantum error correction in a time-dependent transverse-field Ising model","volume":"106","author":[{"given":"Yifan","family":"Hong","sequence":"first","affiliation":[{"name":"Department of Physics, University of Colorado, Boulder, Colorado 80309, USA"},{"name":"Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA"}]},{"given":"Jeremy T.","family":"Young","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Colorado, Boulder, Colorado 80309, USA"},{"name":"Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA"},{"name":"JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309, USA"}]},{"given":"Adam M.","family":"Kaufman","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Colorado, Boulder, Colorado 80309, USA"},{"name":"JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Colorado 80309, USA"}]},{"given":"Andrew","family":"Lucas","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Colorado, Boulder, Colorado 80309, USA"},{"name":"Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.106.022432","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022432","note":"arxivid:2205.12998"},{"id":"arxiv:quant-ph/0703182","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2007.4557325","page":"816-820","source":"Crossref","title":"Constructions of Quantum Convolutional Codes","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Martin","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"2007 IEEE International Symposium on Information Theory","container-title":"2007 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2007,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2007.4557325","note":"arxivid:quant-ph/0703182"},{"id":"arxiv:1105.0649","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2012.2220520","page":"1198-1210","source":"Crossref","title":"Minimal-Memory, Noncatastrophic, Polynomial-Depth Quantum Convolutional Encoders","volume":"59","author":[{"given":"Monireh","family":"Houshmand","sequence":"first","affiliation":[]},{"given":"Saied","family":"Hosseini-Khayat","sequence":"additional","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2013,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2012.2220520","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1105.0649"},{"id":"arxiv:1011.5535","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2011.6034166","page":"450-454","source":"Crossref","title":"Examples of minimal-memory, non-catastrophic quantum convolutional encoders","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]},{"given":"Monireh","family":"Houshmand","sequence":"additional","affiliation":[]},{"given":"Saied","family":"Hosseini-Khayat","sequence":"additional","affiliation":[]}],"event":"2011 IEEE International Symposium on Information Theory - ISIT","container-title":"2011 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2011,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2011.6034166","note":"arxivid:1011.5535"},{"id":"arxiv:2206.13040","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.107.062416","source":"Crossref","title":"Transforming collections of Pauli operators into equivalent collections of Pauli operators over minimal registers","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0003-3893-9278","authenticated-orcid":true,"given":"Lane G.","family":"Gunderman","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.062416","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062416","note":"arxivid:2206.13040"},{"id":"arxiv:quant-ph/9712029","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.58.905","page":"905-909","source":"Crossref","title":"Quantum convolutional error-correcting codes","volume":"58","author":[{"given":"H. F.","family":"Chau","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1998,8,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.58.905","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9712029"},{"id":"arxiv:quant-ph/9806032","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.60.1966","page":"1966-1974","source":"Crossref","title":"Good quantum-convolutional error-correction codes and their decoding algorithm exist","volume":"60","author":[{"given":"H. F.","family":"Chau","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1999,9,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.60.1966","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9806032"},{"id":"arxiv:quant-ph/0304189","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.91.177902","source":"Crossref","title":"Description of a Quantum Convolutional Code","volume":"91","author":[{"given":"Harold","family":"Ollivier","sequence":"first","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2003,10,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.91.177902","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"177902","note":"arxivid:quant-ph/0304189"},{"id":"arxiv:quant-ph/0602129","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2006.261956","source":"Crossref","title":"Non-catastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Martin","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"2006 IEEE International Symposium on Information Theory","container-title":"2006 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2006,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2006.261956","note":"arxivid:quant-ph/0602129"},{"id":"arxiv:1004.5179","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tc.2010.226","page":"299-312","source":"Crossref","title":"Minimal-Memory Requirements for Pearl-Necklace Encoders of Quantum Convolutional Codes","volume":"61","author":[{"given":"Monireh","family":"Houshmand","sequence":"first","affiliation":[]},{"given":"Saied","family":"Hosseini-Khayat","sequence":"additional","affiliation":[]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Computers","original-title":[],"issued":{"date-parts":[[2012,3]]},"URL":"http://dx.doi.org/10.1109/TC.2010.226","ISSN":["0018-9340"],"container-title-short":"IEEE Trans. Comput.","note":"arxivid:1004.5179"},{"id":"arxiv:0903.3894","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.79.062325","source":"Crossref","title":"Quantum-shift-register circuits","volume":"79","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,6,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.062325","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062325","note":"arxivid:0903.3894"},{"id":"arxiv:quant-ph/0703181","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2005.1523493","page":"1018-1022","source":"Crossref","title":"Quantum block and convolutional codes from self-orthogonal product codes","author":[{"given":"M.","family":"Grassl","sequence":"first","affiliation":[]},{"given":"M.","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"Proceedings. International Symposium on Information Theory, 2005. ISIT 2005.","container-title":"Proceedings. International Symposium on Information Theory, 2005. ISIT 2005.","original-title":[],"issued":{"date-parts":[[2005]]},"URL":"http://dx.doi.org/10.1109/ISIT.2005.1523493","note":"arxivid:quant-ph/0703181"},{"id":"doi:10.1017/CBO9781139034807.011","type":"book-chapter","publisher":"Cambridge University Press","DOI":"10.1017/cbo9781139034807.011","page":"231-260","source":"Crossref","title":"Quantum convolutional codes","author":[{"given":"Mark","family":"Wilde","sequence":"first","affiliation":[]}],"container-title":"Quantum Error Correction","original-title":[],"issued":{"date-parts":[[2013,9,12]]},"URL":"http://dx.doi.org/10.1017/CBO9781139034807.011","note":"edition-number:1"},{"id":"arxiv:1902.07395","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/spawc.2019.8815487","source":"Crossref","title":"Quantum convolutional data-syndrome codes","author":[{"given":"Weilei","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Alexei","family":"Ashikhmin","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Woolls","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"event":"2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)","container-title":"2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)","original-title":[],"issued":{"date-parts":[[2019,7]]},"URL":"http://dx.doi.org/10.1109/SPAWC.2019.8815487","note":"arxivid:1902.07395"},{"id":"arxiv:quant-ph/0701037","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"},{"given":"Andreas","family":"Klappenecker"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Quantum Convolutional Codes Derived From Reed-Solomon and Reed-Muller Codes","issued":{"date-parts":[[2007,1,22]]},"note":"arxivid:quant-ph/0701037\narxiv_version_number:2"},{"id":"arxiv:1503.02372","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/access.2015.2405533","page":"146-176","source":"Crossref","title":"The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure","volume":"3","author":[{"given":"Zunaira","family":"Babar","sequence":"first","affiliation":[]},{"given":"Panagiotis","family":"Botsinis","sequence":"additional","affiliation":[]},{"given":"Dimitrios","family":"Alanis","sequence":"additional","affiliation":[]},{"family":"Soon Xin Ng","sequence":"additional","affiliation":[]},{"given":"Lajos","family":"Hanzo","sequence":"additional","affiliation":[]}],"container-title":"IEEE Access","original-title":[],"issued":{"date-parts":[[2015]]},"URL":"http://dx.doi.org/10.1109/ACCESS.2015.2405533","ISSN":["2169-3536"],"container-title-short":"IEEE Access","note":"arxivid:1503.02372"},{"id":"arxiv:0712.2888","type":"article-journal","author":[{"given":"David","family":"Poulin"},{"given":"Jean-Pierre","family":"Tillich"},{"given":"Harold","family":"Ollivier"}],"title":"Quantum serial turbo-codes","issued":{"date-parts":[[2009,6,10]]},"note":"arxivid:0712.2888\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0512041","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.74.032304","source":"Crossref","title":"Trellises for stabilizer codes: Definition and uses","volume":"74","author":[{"given":"Harold","family":"Ollivier","sequence":"first","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,9,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.74.032304","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032304","note":"arxivid:quant-ph/0512041"},{"id":"arxiv:1502.00910","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tvt.2014.2328638","page":"866-875","source":"Crossref","title":"EXIT-Chart-Aided Near-Capacity Quantum Turbo Code Design","volume":"64","author":[{"given":"Zunaira","family":"Babar","sequence":"first","affiliation":[]},{"given":"Soon Xin","family":"Ng","sequence":"additional","affiliation":[]},{"given":"Lajos","family":"Hanzo","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Vehicular Technology","original-title":[],"issued":{"date-parts":[[2015,3]]},"URL":"http://dx.doi.org/10.1109/TVT.2014.2328638","ISSN":["0018-9545","1939-9359"],"container-title-short":"IEEE Trans. Veh. Technol.","note":"arxivid:1502.00910"},{"id":"doi:10.1007/978-3-540-70583-3_25","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-540-70583-3_25","page":"298-310","source":"Crossref","title":"Interacting Quantum Observables","author":[{"given":"Bob","family":"Coecke","sequence":"first","affiliation":[]},{"given":"Ross","family":"Duncan","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[null]]},"ISBN":["9783540705826","9783540705833"],"URL":"http://dx.doi.org/10.1007/978-3-540-70583-3_25","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:0906.4725","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1088/1367-2630/13/4/043016","page":"043016","source":"Crossref","title":"Interacting quantum observables: categorical algebra and diagrammatics","volume":"13","author":[{"given":"Bob","family":"Coecke","sequence":"first","affiliation":[]},{"given":"Ross","family":"Duncan","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2011,4,14]]},"URL":"http://dx.doi.org/10.1088/1367-2630/13/4/043016","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:0906.4725"},{"id":"arxiv:2009.07752","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.102.052409","source":"Crossref","title":"Extended flag gadgets for low-overhead circuit verification","volume":"102","author":[{"given":"Dripto M.","family":"Debroy","sequence":"first","affiliation":[]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,11,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.052409","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052409","note":"arxivid:2009.07752"},{"id":"arxiv:2407.06583","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Edwin","family":"Tham"}],"title":"Low-cost noise reduction for Clifford circuits","issued":{"date-parts":[[2024,7,9]]},"note":"arxivid:2407.06583\narxiv_version_number:1"},{"id":"arxiv:2504.13356","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We propose several optimizations of the CliNR partial error correction scheme which implements Clifford circuits by consuming a resource state. Errors are corrected by measuring a sequence of Pauli operators that we refer to as the verification sequence. We first propose a global optimization algorithm searching for a verification sequence resulting in a low logical error rate using tabu search. Then, we introduce a proxy for the logical error rate which is easier to evaluate and we design a two-step optimization algorithm. First, a verification sequence minimizing the proxy is computed, then this sequence is refined by reintroducing the logical error rate. Finally, we identify a large group of automorphisms of the search space which preserve the proxy and we use this symmetry to reduce the size of the search space. This results in a 168 <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x00D7;</mml:mo></mml:math> (respectively 20,160 <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x00D7;</mml:mo></mml:math>) reduction of the size of the search space for the optimization of verification sequences with three (respectively four) Pauli operators. Our numerical simulations for 20-qubit Clifford circuits with size 400 under the ion chain model show that our optimization algorithms improve the performance of CliNR by 25% and that the two-step optimization achieves the same results as the global optimization with 64% fewer evaluations of the logical error rate. Finally, we perform experiments on a 36-qubit trapped ion quantum computer, without mid-circuit measurements, showing that the CZNR variant of CliNR is at breakeven.</jats:p>","DOI":"10.22331/q-2025-08-27-1829","page":"1829","source":"Crossref","title":"Optimized Clifford Noise Reduction: Theory, Simulations and Experiments","volume":"9","author":[{"given":"Edwin","family":"Tham","sequence":"first","affiliation":[{"name":"IonQ Inc."}]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[{"name":"IonQ Inc."}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,27]]},"URL":"http://dx.doi.org/10.22331/q-2025-08-27-1829","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2504.13356"},{"id":"doi:10.1109/ISIT.2009.5205650","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2009.5205650","page":"804-808","source":"Crossref","title":"New decoding algorithms for a class of subsystem codes and generalized shor codes","author":[{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[]},{"given":"Martin","family":"Rotteler","sequence":"additional","affiliation":[]}],"event":"2009 IEEE International Symposium on Information Theory - ISIT","container-title":"2009 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2009,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2009.5205650"},{"id":"arxiv:1310.5291","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/physrevlett.112.250501","source":"Crossref","title":"Ultrafast and Fault-Tolerant Quantum Communication across Long Distances","volume":"112","author":[{"given":"Sreraman","family":"Muralidharan","sequence":"first","affiliation":[]},{"given":"Jungsang","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Norbert","family":"Lütkenhaus","sequence":"additional","affiliation":[]},{"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2014,6,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.112.250501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"250501","note":"arxivid:1310.5291"},{"id":"arxiv:2409.13801","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>We formulate code concatenation as the action of a unitary quantum circuit on an expanding tree geometry and find that for certain classes of gates, applied identically at each node, a binary tree circuit encodes a single logical qubit with code distance that grows exponentially in the depth of the tree. When there is noise in the bulk or at the end of this encoding circuit, the system undergoes a phase transition between a coding phase, where an optimal decoder can successfully recover logical information, and a noncoding phase. Leveraging the tree structure, we combine the formalism of “tensor enumerators” from quantum coding theory with standard recursive techniques for classical spin models on the Bethe lattice to explore these phases. In the presence of bulk errors, the coding phase is a type of spin glass, characterized by a distribution of failure probabilities. When the errors are heralded, the recursion relation is exactly solvable, giving us an analytic handle on the phase diagram.</jats:p>","DOI":"10.1103/physrevresearch.7.023086","source":"Crossref","title":"Dynamically generated concatenated codes and their phase diagrams","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0003-4229-8282","authenticated-orcid":true,"given":"Grace M.","family":"Sommers","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00hx57361","id-type":"ROR","asserted-by":"publisher"}],"name":"Princeton University"}]},{"ORCID":"https://orcid.org/0000-0003-1008-5178","authenticated-orcid":true,"given":"David A.","family":"Huse","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00hx57361","id-type":"ROR","asserted-by":"publisher"}],"name":"Princeton University"}]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2025,4,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.7.023086","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023086","note":"arxivid:2409.13801"},{"id":"arxiv:0707.0903","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.77.012310","source":"Crossref","title":"Loss-tolerant operations in parity-code linear optics quantum computing","volume":"77","author":[{"given":"A. J. F.","family":"Hayes","sequence":"first","affiliation":[]},{"given":"A.","family":"Gilchrist","sequence":"additional","affiliation":[]},{"given":"T. C.","family":"Ralph","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,1,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.012310","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012310","note":"arxivid:0707.0903"},{"id":"arxiv:0908.3932","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.82.022323","source":"Crossref","title":"Fault tolerance in parity-state linear optical quantum computing","volume":"82","author":[{"given":"A. J. F.","family":"Hayes","sequence":"first","affiliation":[]},{"given":"H. L.","family":"Haselgrove","sequence":"additional","affiliation":[]},{"given":"Alexei","family":"Gilchrist","sequence":"additional","affiliation":[]},{"given":"T. C.","family":"Ralph","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,8,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.82.022323","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022323","note":"arxivid:0908.3932"},{"id":"arxiv:cond-mat/0403712","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevb.71.024505","source":"Crossref","title":"Protected qubits and Chern-Simons theories in Josephson junction arrays","volume":"71","author":[{"given":"B.","family":"Douçot","sequence":"first","affiliation":[]},{"given":"M. V.","family":"Feigel’man","sequence":"additional","affiliation":[]},{"given":"L. B.","family":"Ioffe","sequence":"additional","affiliation":[]},{"given":"A. S.","family":"Ioselevich","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2005,1,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.71.024505","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"024505","note":"arxivid:cond-mat/0403712"},{"id":"doi:10.1088/0034-4885/75/7/072001","type":"journal-article","publisher":"IOP Publishing","issue":"7","DOI":"10.1088/0034-4885/75/7/072001","page":"072001","source":"Crossref","title":"Physical implementation of protected qubits","volume":"75","author":[{"given":"B","family":"Douçot","sequence":"first","affiliation":[]},{"given":"L B","family":"Ioffe","sequence":"additional","affiliation":[]}],"container-title":"Reports on Progress in Physics","original-title":[],"issued":{"date-parts":[[2012,6,28]]},"URL":"http://dx.doi.org/10.1088/0034-4885/75/7/072001","ISSN":["0034-4885","1361-6633"],"container-title-short":"Rep. Prog. Phys."},{"id":"arxiv:0802.2295","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/nphys1151","page":"48-53","source":"Crossref","title":"Superconducting nanocircuits for topologically protected qubits","volume":"5","author":[{"given":"Sergey","family":"Gladchenko","sequence":"first","affiliation":[]},{"given":"David","family":"Olaya","sequence":"additional","affiliation":[]},{"given":"Eva","family":"Dupont-Ferrier","sequence":"additional","affiliation":[]},{"given":"Benoit","family":"Douçot","sequence":"additional","affiliation":[]},{"given":"Lev B.","family":"Ioffe","sequence":"additional","affiliation":[]},{"given":"Michael E.","family":"Gershenson","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2008,11,30]]},"URL":"http://dx.doi.org/10.1038/nphys1151","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nature Phys","note":"alternative-id:BFnphys1151\narxivid:0802.2295"},{"id":"arxiv:2207.06805","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Measurement-based quantum computing (MBQC) in linear optical systems is promising for near-future quantum computing architecture. However, the nondeterministic nature of entangling operations and photon losses hinder the large-scale generation of graph states and introduce logical errors. In this work, we propose a linear optical topological MBQC protocol employing multiphoton qubits based on the parity encoding, which turns out to be highly photon-loss tolerant and resource-efficient even under the effects of nonideal entangling operations that unavoidably corrupt nearby qubits. For the realistic error analysis, we introduce a Bayesian methodology, in conjunction with the stabilizer formalism, to track errors caused by such detrimental effects. We additionally suggest a graph-theoretical optimization scheme for the process of constructing an arbitrary graph state, which greatly reduces its resource overhead. Notably, we show that our protocol is advantageous over several other existing approaches in terms of the fault-tolerance and resource overhead.</jats:p>","DOI":"10.1038/s41534-023-00705-9","source":"Crossref","title":"Parity-encoding-based quantum computing with Bayesian error tracking","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0002-1207-2752","authenticated-orcid":false,"given":"Seok-Hyung","family":"Lee","sequence":"first","affiliation":[]},{"given":"Srikrishna","family":"Omkar","sequence":"additional","affiliation":[]},{"given":"Yong Siah","family":"Teo","sequence":"additional","affiliation":[]},{"given":"Hyunseok","family":"Jeong","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2023,4,24]]},"URL":"http://dx.doi.org/10.1038/s41534-023-00705-9","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"39","note":"alternative-id:705\narxivid:2207.06805"},{"id":"arxiv:0711.1556","type":"article-journal","author":[{"given":"Andrew W.","family":"Cross"},{"given":"David P.","family":"DiVincenzo"},{"given":"Barbara M.","family":"Terhal"}],"title":"A comparative code study for quantum fault-tolerance","issued":{"date-parts":[[2009,2,6]]},"note":"arxivid:0711.1556\narxiv_version_number:2"},{"id":"doi:10.1109/ISIT.2014.6874892","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2014.6874892","source":"Crossref","title":"Robust quantum error syndrome extraction by classical coding","author":[{"given":"Alexei","family":"Ashikhmin","sequence":"first","affiliation":[]},{"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"event":"2014 IEEE International Symposium on Information Theory (ISIT)","container-title":"2014 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2014.6874892"},{"id":"doi:10.1109/ISIT.2015.7282628","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2015.7282628","page":"1114-1118","source":"Crossref","title":"Global stabilizer quantum error correction with combinatorial arrays","author":[{"given":"Yuichiro","family":"Fujiwara","sequence":"first","affiliation":[]}],"event":"2015 IEEE International Symposium on Information Theory (ISIT)","container-title":"2015 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2015,6]]},"URL":"http://dx.doi.org/10.1109/ISIT.2015.7282628"},{"id":"doi:10.1109/ISIT.2016.7541704","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2016.7541704","source":"Crossref","title":"Correction of data and syndrome errors by stabilizer codes","author":[{"given":"Alexei","family":"Ashikhmin","sequence":"first","affiliation":[]},{"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"event":"2016 IEEE International Symposium on Information Theory (ISIT)","container-title":"2016 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2016,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2016.7541704"},{"id":"arxiv:2302.01527","type":"article-journal","author":[{"given":"Andrew","family":"Nemec"}],"title":"Quantum Data-Syndrome Codes: Subsystem and Impure Code Constructions","issued":{"date-parts":[[2023,2,3]]},"note":"arxivid:2302.01527\narxiv_version_number:1"},{"id":"arxiv:2510.08402","type":"article-journal","author":[{"given":"Benjamin","family":"Anker"},{"given":"Milad","family":"Marvian"}],"title":"Universal Fault Tolerance with Non-Transversal Clifford Gates","issued":{"date-parts":[[2025,10,9]]},"note":"arxivid:2510.08402\narxiv_version_number:1"},{"id":"arxiv:2209.09267","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.130.200601","source":"Crossref","title":"Learning Logical Pauli Noise in Quantum Error Correction","volume":"130","author":[{"ORCID":"https://orcid.org/0000-0002-3889-528X","authenticated-orcid":true,"given":"Thomas","family":"Wagner","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0659-6699","authenticated-orcid":true,"given":"Hermann","family":"Kampermann","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4661-2267","authenticated-orcid":true,"given":"Dagmar","family":"Bruß","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8009-0549","authenticated-orcid":true,"given":"Martin","family":"Kliesch","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2023,5,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.130.200601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200601","note":"arxivid:2209.09267"},{"id":"arxiv:1712.07067","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/aac54f","page":"063010","source":"Crossref","title":"Fermion-to-qubit mappings with varying resource requirements for quantum simulation","volume":"20","author":[{"given":"Mark","family":"Steudtner","sequence":"first","affiliation":[]},{"given":"Stephanie","family":"Wehner","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2018,6,7]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aac54f","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1712.07067"},{"id":"arxiv:1810.02681","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.99.022308","source":"Crossref","title":"Quantum codes for quantum simulation of fermions on a square lattice of qubits","volume":"99","author":[{"given":"Mark","family":"Steudtner","sequence":"first","affiliation":[]},{"given":"Stephanie","family":"Wehner","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,2,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.022308","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022308","note":"arxivid:1810.02681"},{"id":"doi:10.1002/spe.4380240306","type":"journal-article","publisher":"Wiley","issue":"3","abstract":"<jats:title>Abstract</jats:title><jats:p>A new method (the ‘binary indexed tree’) is presented for maintaining the cumulative frequencies which are needed to support dynamic arithmetic data compression. It is based on a decomposition of the cumulative frequencies into portions which parallel the binary representation of the index of the table element (or symbol). The operations to traverse the data structure are based on the binary coding of the index. In comparison with previous methods, the binary indexed tree is faster, using more compact data and simpler code. The access time for all operations is either constant or proportional to the logarithm of the table size. In conjunction with the compact data structure, this makes the new method particularly suitable for large symbol alphabets.</jats:p>","DOI":"10.1002/spe.4380240306","page":"327-336","source":"Crossref","title":"A new data structure for cumulative frequency tables","volume":"24","author":[{"given":"Peter M.","family":"Fenwick","sequence":"first","affiliation":[]}],"container-title":"Software: Practice and Experience","original-title":[],"language":"en","issued":{"date-parts":[[1994,3]]},"URL":"http://dx.doi.org/10.1002/spe.4380240306","ISSN":["0038-0644","1097-024X"],"container-title-short":"Softw Pract Exp","note":"alternative-id:10.1002/spe.4380240306"},{"id":"arxiv:1701.07072","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.95.032332","source":"Crossref","title":"Operator locality in the quantum simulation of fermionic models","volume":"95","author":[{"given":"Vojtěch","family":"Havlíček","sequence":"first","affiliation":[]},{"given":"Matthias","family":"Troyer","sequence":"additional","affiliation":[]},{"given":"James D.","family":"Whitfield","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,3,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.95.032332","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032332","note":"arxivid:1701.07072"},{"id":"arxiv:1904.09912","type":"journal-article","publisher":"Quanta","abstract":"<jats:p>Representations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of superfluous branches. The usual Jordan–Wigner construction also may be formally obtained in this approach by bringing the process up to trivial qubit chain (trunk). The methods can also be used for effective simulation of some quantum circuits corresponding to the binary tree structure. The modeling of more general qubit trees, as well as the relationship with the mapping used in the Bravyi–Kitaev transformation, are also briefly discussed.\r\nQuanta 2022; 11: 97–114.</jats:p>","DOI":"10.12743/quanta.v11i1.199","page":"97-114","source":"Crossref","title":"Clifford Algebras, Spin Groups and Qubit Trees","volume":"11","author":[{"given":"Alexander Yurievich","family":"Vlasov","sequence":"first","affiliation":[]}],"container-title":"Quanta","original-title":[],"issued":{"date-parts":[[2022,12,1]]},"URL":"http://dx.doi.org/10.12743/quanta.v11i1.199","ISSN":["1314-7374"],"container-title-short":"Quanta","note":"arxivid:1904.09912"},{"id":"doi:10.1002/qua.24969","type":"journal-article","publisher":"Wiley","issue":"19","abstract":"<jats:p>Quantum chemistry is an important area of application for quantum computation. In particular, quantum algorithms applied to the electronic structure problem promise exact, efficient methods for determination of the electronic energy of atoms and molecules. The Bravyi–Kitaev transformation is a method of mapping the occupation state of a fermionic system onto qubits. This transformation maps the Hamiltonian of<jats:italic>n</jats:italic>interacting fermions to an<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/qua24969-math-0001.png\" xlink:title=\"urn:x-wiley:00207608:media:qua24969:qua24969-math-0001\"/>‐local Hamiltonian of<jats:italic>n</jats:italic>qubits. This is an improvement in locality over the Jordan–Wigner transformation, which results in an<jats:italic>O</jats:italic>(<jats:italic>n</jats:italic>)‐local qubit Hamiltonian. We present the Bravyi–Kitaev transformation in detail, introducing the sets of qubits which must be acted on to change occupancy and parity of states in the occupation number basis. We give recursive definitions of these sets and of the transformation and inverse transformation matrices, which relate the occupation number basis and the Bravyi–Kitaev basis. We then compare the use of the Jordan–Wigner and Bravyi–Kitaev Hamiltonians for the quantum simulation of methane using the STO‐6G basis. © 2015 Wiley Periodicals, Inc.</jats:p>","DOI":"10.1002/qua.24969","page":"1431-1441","source":"Crossref","title":"The<scp>B</scp>ravyi–<scp>K</scp>itaev transformation: Properties and applications","volume":"115","author":[{"given":"Andrew","family":"Tranter","sequence":"first","affiliation":[{"name":"Department of Physics Imperial College London South Kensington Campus London SW7 2AZ United Kingdom"},{"name":"Centre for Computational Science University College London 20 Gordon Street London WC1H 0AJ United Kingdom"}]},{"given":"Sarah","family":"Sofia","sequence":"additional","affiliation":[{"name":"Photovoltaics Research Laboratory Massachusetts Institute of Technology Cambridge Massachusetts 02139"},{"name":"Department of Physics Haverford College 370 Lancaster Ave. Haverford Pennsylvania 19041"}]},{"given":"Jake","family":"Seeley","sequence":"additional","affiliation":[{"name":"Department of Physics Haverford College 370 Lancaster Ave. Haverford Pennsylvania 19041"},{"name":"Earth and Planetary Science University of California Berkeley, 307 McCone Hall Berkeley California 94720‐4767"}]},{"given":"Michael","family":"Kaicher","sequence":"additional","affiliation":[{"name":"Theoretical Physics Saarland University 66123 Saarbrücken Germany"}]},{"given":"Jarrod","family":"McClean","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Chemical Biology Harvard University Cambridge Massachusetts 02138"}]},{"given":"Ryan","family":"Babbush","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Chemical Biology Harvard University Cambridge Massachusetts 02138"}]},{"given":"Peter V.","family":"Coveney","sequence":"additional","affiliation":[{"name":"Centre for Computational Science University College London 20 Gordon Street London WC1H 0AJ United Kingdom"}]},{"given":"Florian","family":"Mintert","sequence":"additional","affiliation":[{"name":"Department of Physics Imperial College London South Kensington Campus London SW7 2AZ United Kingdom"}]},{"given":"Frank","family":"Wilhelm","sequence":"additional","affiliation":[{"name":"Theoretical Physics Saarland University 66123 Saarbrücken Germany"}]},{"given":"Peter J.","family":"Love","sequence":"additional","affiliation":[{"name":"Department of Physics Haverford College 370 Lancaster Ave. Haverford Pennsylvania 19041"}]}],"container-title":"International Journal of Quantum Chemistry","original-title":[],"language":"en","issued":{"date-parts":[[2015,7]]},"URL":"http://dx.doi.org/10.1002/qua.24969","ISSN":["0020-7608","1097-461X"],"container-title-short":"Int J of Quantum Chemistry","note":"alternative-id:10.1002/qua.24969"},{"id":"arxiv:1910.10746","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">⌈</mml:mo><mml:msub><mml:mi>log</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⌉</mml:mo></mml:math>qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>log</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>qubits on average. We apply it to the problem of learning<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that one can determine individual elements of all<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>-fermion RDMs in parallel, to precision<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>ϵ</mml:mi></mml:math>, by repeating a single quantum circuit for<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>≲</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mi>k</mml:mi></mml:msup><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>times. This result is based on a method we develop here that allows one to determine individual elements of all<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math>-qubit RDMs in parallel, to precision<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>ϵ</mml:mi></mml:math>, by repeating a single quantum circuit for<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>≲</mml:mo><mml:msup><mml:mn>3</mml:mn><mml:mi>k</mml:mi></mml:msup><mml:msup><mml:mi>ϵ</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>times, independent of the system size. This improves over existing schemes for determining qubit RDMs.</jats:p>","DOI":"10.22331/q-2020-06-04-276","page":"276","source":"Crossref","title":"Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0003-0435-655X","authenticated-orcid":false,"given":"Zhang","family":"Jiang","sequence":"first","affiliation":[{"name":"Google Research, Venice, CA 90291"}]},{"ORCID":"https://orcid.org/0000-0002-6618-9622","authenticated-orcid":false,"given":"Amir","family":"Kalev","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742-2420, USA"}]},{"ORCID":"https://orcid.org/0000-0002-8497-6363","authenticated-orcid":false,"given":"Wojciech","family":"Mruczkiewicz","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291"}]},{"ORCID":"https://orcid.org/0000-0002-9681-6746","authenticated-orcid":false,"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,4]]},"URL":"http://dx.doi.org/10.22331/q-2020-06-04-276","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1910.10746"},{"id":"arxiv:2210.08411","type":"journal-article","publisher":"Stichting SciPost","issue":"1","abstract":"<jats:p>Utilizing the framework of <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of d=2,3,4,5,6,7, allowing correction of any <jats:inline-formula><jats:alternatives><jats:tex-math>\\lfloor \\frac{d-1}{2} \\rfloor</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo stretchy=\"false\" form=\"prefix\">⌊</mml:mo><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mo stretchy=\"false\" form=\"postfix\">⌋</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>-qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of d=3,4,5. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically.</jats:p>","DOI":"10.21468/scipostphys.16.1.033","source":"Crossref","title":"Error-correcting codes for fermionic quantum simulation","volume":"16","author":[{"given":"Yu-An","family":"Chen","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02v51f717","id-type":"ROR","asserted-by":"publisher"}],"name":"Peking University"}]},{"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"}]},{"given":"Yijia","family":"Xu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2024,1,26]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.16.1.033","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"033","note":"arxivid:2210.08411"},{"id":"arxiv:1909.10552","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.101.155105","source":"Crossref","title":"Tensor network approach to two-dimensional bosonization","volume":"101","author":[{"ORCID":"https://orcid.org/0000-0002-4283-536X","authenticated-orcid":true,"given":"Sujeet K.","family":"Shukla","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1740-6889","authenticated-orcid":true,"given":"Tyler D.","family":"Ellison","sequence":"additional","affiliation":[]},{"given":"Lukasz","family":"Fidkowski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2020,4,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.101.155105","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155105","note":"arxivid:1909.10552"},{"id":"arxiv:2404.07727","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","abstract":"<jats:p>We present a locality preserving unitary mapping from fermions to qubits on a two-dimensional torus while accounting for the mapping of topological sectors. Extending the work of Shukla  [], an explicit intertwiner is constructed in the form of a projected entangled pair operator. By encoding the information about the charge sectors (and if applicable the twisted boundary conditions) in ancillary qubit(s), the intertwiner becomes a unitary operator which exchanges boundary conditions and charge sectors.</jats:p>","DOI":"10.1103/physrevb.111.115126","source":"Crossref","title":"Local Jordan-Wigner transformations and topological sectors on the torus","volume":"111","author":[{"ORCID":"https://orcid.org/0009-0003-0866-3511","authenticated-orcid":true,"given":"Oliver","family":"O'Brien","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Cambridge"}]},{"ORCID":"https://orcid.org/0000-0002-1364-4863","authenticated-orcid":true,"given":"Laurens","family":"Lootens","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Cambridge"},{"id":[{"id":"https://ror.org/00cv9y106","id-type":"ROR","asserted-by":"publisher"}],"name":"Ghent University"}]},{"ORCID":"https://orcid.org/0000-0003-0270-5592","authenticated-orcid":true,"given":"Frank","family":"Verstraete","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Cambridge"},{"id":[{"id":"https://ror.org/00cv9y106","id-type":"ROR","asserted-by":"publisher"}],"name":"Ghent University"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.111.115126","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"115126","note":"arxivid:2404.07727"},{"id":"arxiv:1810.05274","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.1.033033","source":"Crossref","title":"Superfast encodings for fermionic quantum simulation","volume":"1","author":[{"given":"Kanav","family":"Setia","sequence":"first","affiliation":[]},{"given":"Sergey","family":"Bravyi","sequence":"additional","affiliation":[]},{"given":"Antonio","family":"Mezzacapo","sequence":"additional","affiliation":[]},{"given":"James D.","family":"Whitfield","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2019,10,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.1.033033","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033033","note":"arxivid:1810.05274"},{"id":"arxiv:1812.08190","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevapplied.12.064041","source":"Crossref","title":"Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations","volume":"12","author":[{"ORCID":"https://orcid.org/0000-0003-0435-655X","authenticated-orcid":true,"given":"Zhang","family":"Jiang","sequence":"first","affiliation":[]},{"given":"Jarrod","family":"McClean","sequence":"additional","affiliation":[]},{"given":"Ryan","family":"Babbush","sequence":"additional","affiliation":[]},{"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.12.064041","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"064041","note":"arxivid:1812.08190"},{"id":"arxiv:cond-mat/0409485","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevlett.95.176407","source":"Crossref","title":"Fermions without Fermion Fields","volume":"95","author":[{"given":"R. C.","family":"Ball","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,10,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.95.176407","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"176407","note":"arxivid:cond-mat/0409485"},{"id":"arxiv:cond-mat/0508353","type":"journal-article","publisher":"IOP Publishing","issue":"09","abstract":"<jats:p>We show how to map local fermionic problems onto local spin problems on a lattice in any\ndimension. The main idea is to introduce auxiliary degrees of freedom, represented by\nMajorana fermions, which allow us to extend the Jordan–Wigner transformation to\ndimensions higher than one. We also discuss the implications of our results in the numerical\ninvestigation of fermionic systems.</jats:p>","DOI":"10.1088/1742-5468/2005/09/p09012","page":"P09012-P09012","source":"Crossref","title":"Mapping local Hamiltonians of fermions to local Hamiltonians of spins","volume":"2005","author":[{"given":"F","family":"Verstraete","sequence":"first","affiliation":[]},{"given":"J I","family":"Cirac","sequence":"additional","affiliation":[]}],"container-title":"Journal of Statistical Mechanics: Theory and Experiment","original-title":[],"issued":{"date-parts":[[2005,9,1]]},"URL":"http://dx.doi.org/10.1088/1742-5468/2005/09/P09012","ISSN":["1742-5468"],"container-title-short":"J. Stat. Mech.","note":"arxivid:cond-mat/0508353"},{"id":"arxiv:2003.07125","type":"article-journal","author":[{"given":"Johannes","family":"Bausch"},{"given":"Toby","family":"Cubitt"},{"given":"Charles","family":"Derby"},{"given":"Joel","family":"Klassen"}],"title":"Mitigating Errors in Local Fermionic Encodings","issued":{"date-parts":[[2020,6,15]]},"note":"arxivid:2003.07125\narxiv_version_number:2"},{"id":"preset:Derby23","type":"thesis","author":[{"family":"Derby","given":"C."}],"title":"Compact fermion to qubit mappings for quantum simulation","genre":"PhD thesis,","publisher":"UCL (University College London","issued":"2023"},{"id":"arxiv:2003.06939","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevb.104.035118","source":"Crossref","title":"Compact fermion to qubit mappings","volume":"104","author":[{"ORCID":"https://orcid.org/0000-0001-9256-8151","authenticated-orcid":true,"given":"Charles","family":"Derby","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1438-3172","authenticated-orcid":true,"given":"Joel","family":"Klassen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3189-9162","authenticated-orcid":true,"given":"Johannes","family":"Bausch","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5087-9346","authenticated-orcid":true,"given":"Toby","family":"Cubitt","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2021,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.104.035118","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"035118","note":"arxivid:2003.06939"},{"id":"arxiv:2205.15256","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Determining the ground and excited state properties of materials is considered one of the most promising applications of quantum computers. On near-term hardware, the limiting constraint on such simulations is the requisite circuit depths and qubit numbers, which currently lie well beyond near-term capabilities. Here we develop a quantum algorithm which reduces the estimated cost of material simulations. For example, we obtain a circuit depth improvement by up to 6 orders of magnitude for a Trotter layer of time-dynamics simulation in the transition-metal oxide SrVO<jats:sub>3</jats:sub>compared with the best previous quantum algorithms. We achieve this by introducing a collection of connected techniques, including highly localised and physically compact representations of materials Hamiltonians in the Wannier basis, a hybrid fermion-to-qubit mapping, and an efficient circuit compiler. Combined together, these methods leverage locality of materials Hamiltonians and result in a design that generates quantum circuits with depth independent of the system’s size. Although the requisite resources for the quantum simulation of materials are still beyond current hardware, our results show that realistic simulation of specific properties may be feasible without necessarily requiring fully scalable, fault-tolerant quantum computers, providing quantum algorithm design incorporates deeper understanding of the target materials and applications.</jats:p>","DOI":"10.1038/s41467-023-43479-6","source":"Crossref","title":"Towards near-term quantum simulation of materials","volume":"15","author":[{"ORCID":"https://orcid.org/0000-0002-4102-4741","authenticated-orcid":false,"given":"Laura","family":"Clinton","sequence":"first","affiliation":[]},{"given":"Toby","family":"Cubitt","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6883-6785","authenticated-orcid":false,"given":"Brian","family":"Flynn","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9182-1415","authenticated-orcid":false,"given":"Filippo Maria","family":"Gambetta","sequence":"additional","affiliation":[]},{"given":"Joel","family":"Klassen","sequence":"additional","affiliation":[]},{"given":"Ashley","family":"Montanaro","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7899-6619","authenticated-orcid":false,"given":"Stephen","family":"Piddock","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1161-1328","authenticated-orcid":false,"given":"Raul A.","family":"Santos","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5675-6059","authenticated-orcid":false,"given":"Evan","family":"Sheridan","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,24]]},"URL":"http://dx.doi.org/10.1038/s41467-023-43479-6","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"211","note":"alternative-id:43479\narxivid:2205.15256"},{"id":"arxiv:2101.10735","type":"article-journal","author":[{"given":"Charles","family":"Derby"},{"given":"Joel","family":"Klassen"}],"title":"A Compact Fermion to Qubit Mapping Part 2: Alternative Lattice Geometries","issued":{"date-parts":[[2021,1,26]]},"note":"arxivid:2101.10735\narxiv_version_number:1"},{"id":"arxiv:2009.11860","type":"article-journal","author":[{"given":"Riley W.","family":"Chien"},{"given":"James D.","family":"Whitfield"}],"title":"Custom fermionic codes for quantum simulation","issued":{"date-parts":[[2020,9,24]]},"note":"arxivid:2009.11860\narxiv_version_number:1"},{"id":"arxiv:1807.07081","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.100.245127","source":"Crossref","title":"Bosonization in three spatial dimensions and a 2-form gauge theory","volume":"100","author":[{"ORCID":"https://orcid.org/0000-0002-8810-9355","authenticated-orcid":true,"given":"Yu-An","family":"Chen","sequence":"first","affiliation":[]},{"given":"Anton","family":"Kapustin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.100.245127","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"245127","note":"arxivid:1807.07081"},{"id":"arxiv:2309.15903","type":"article-journal","author":[{"given":"Lukasz","family":"Fidkowski"},{"given":"Matthew B.","family":"Hastings"}],"title":"Pumping Chirality in Three Dimensions","issued":{"date-parts":[[2024,11,18]]},"note":"arxivid:2309.15903\narxiv_version_number:2"},{"id":"arxiv:1911.00017","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.2.033527","source":"Crossref","title":"Exact bosonization in arbitrary dimensions","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-8810-9355","authenticated-orcid":true,"given":"Yu-An","family":"Chen","sequence":"first","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033527","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033527","note":"arxivid:1911.00017"},{"id":"arxiv:2002.12026","type":"article-journal","author":[{"given":"Wilbur","family":"Shirley"}],"title":"Fractonic order and emergent fermionic gauge theory","issued":{"date-parts":[[2020,8,31]]},"note":"arxivid:2002.12026\narxiv_version_number:2"},{"id":"arxiv:2002.11345","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevresearch.2.023353","source":"Crossref","title":"Jordan-Wigner dualities for translation-invariant Hamiltonians in any dimension: Emergent fermions in fracton topological order","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"first","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,6,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.023353","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023353","note":"arxivid:2002.11345"},{"id":"arxiv:2508.20167","type":"article-journal","author":[{"given":"Lei","family":"Su"},{"given":"Ivar","family":"Martin"}],"title":"Bosonization and Kramers-Wannier dualities in general dimensions","issued":{"date-parts":[[2025,11,13]]},"note":"arxivid:2508.20167\narxiv_version_number:2"},{"id":"arxiv:2304.00028","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.132.016604","source":"Crossref","title":"Fracton Self-Statistics","volume":"132","author":[{"ORCID":"https://orcid.org/0000-0001-5513-8857","authenticated-orcid":true,"given":"Hao","family":"Song","sequence":"first","affiliation":[{"name":"CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada"}]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":true,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Department of Physics, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA"}]},{"given":"Wilbur","family":"Shirley","sequence":"additional","affiliation":[{"name":"School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA"},{"name":"Department of Physics, California Institute of Technology, Pasadena, California 91125, USA"},{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"Michael","family":"Hermele","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.132.016604","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"016604","note":"arxivid:2304.00028"},{"id":"arxiv:2212.09731","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.4.030314","source":"Crossref","title":"Bonsai Algorithm: Grow Your Own Fermion-to-Qubit Mappings","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-7108-1811","authenticated-orcid":true,"given":"Aaron","family":"Miller","sequence":"first","affiliation":[]},{"given":"Zoltán","family":"Zimborás","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9818-2372","authenticated-orcid":true,"given":"Stefan","family":"Knecht","sequence":"additional","affiliation":[]},{"given":"Sabrina","family":"Maniscalco","sequence":"additional","affiliation":[]},{"given":"Guillermo","family":"García-Pérez","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,1]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.030314","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030314","note":"arxivid:2212.09731"},{"id":"arxiv:2311.07409","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>In  electronic structure simulations, fermion-to-qubit mappings represent the initial encoding step from the problem of fermions into a problem of qubits. This work introduces a physically inspired method for constructing mappings that significantly simplify entanglement requirements when one is simulating states of interest. The presence of electronic excitations drives the construction of our mappings, reducing correlations for target states in the qubit space. To benchmark our method, we simulate ground-states of small molecules and observe an enhanced performance when compared with classical and quantum variational approaches from prior research using conventional mappings. In particular, on the quantum side, our mappings require a reduced number of entangling layers to achieve accuracy for <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mi>LiH</a:mi></a:math>, <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><d:msub><d:mrow><d:mi mathvariant=\"normal\">H</d:mi></d:mrow><d:mn>2</d:mn></d:msub></d:math>, <h:math xmlns:h=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><h:mo stretchy=\"false\">(</h:mo><h:msub><h:mrow><h:mi mathvariant=\"normal\">H</h:mi></h:mrow><h:mn>2</h:mn></h:msub><h:msub><h:mo stretchy=\"false\">)</h:mo><h:mn>2</h:mn></h:msub></h:math>, <n:math xmlns:n=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><n:msubsup><n:mrow><n:mi mathvariant=\"normal\">H</n:mi></n:mrow><n:mn>4</n:mn><n:mo>≠</n:mo></n:msubsup></n:math> stretching, and benzene’s <r:math xmlns:r=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><r:mi>π</r:mi></r:math> system using the RY hardware-efficient ansatz. In addition, our mappings also provide an enhanced ground-state simulation performance in the density matrix renormalization group algorithm for the <u:math xmlns:u=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><u:msub><u:mrow><u:mi mathvariant=\"normal\">N</u:mi></u:mrow><u:mn>2</u:mn></u:msub></u:math> molecule.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.030333","source":"Crossref","title":"Reducing Entanglement with Physically Inspired Fermion-To-Qubit Mappings","volume":"5","author":[{"ORCID":"https://orcid.org/0009-0000-7593-2417","authenticated-orcid":true,"given":"Teodor","family":"Parella-Dilmé","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03kpps236","id-type":"ROR","asserted-by":"publisher"}],"name":"The Barcelona Institute of Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0001-7817-9571","authenticated-orcid":true,"given":"Korbinian","family":"Kottmann","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03kpps236","id-type":"ROR","asserted-by":"publisher"}],"name":"The Barcelona Institute of Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0001-7070-1433","authenticated-orcid":true,"given":"Leonardo","family":"Zambrano","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03kpps236","id-type":"ROR","asserted-by":"publisher"}],"name":"The Barcelona Institute of Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0002-5644-8985","authenticated-orcid":true,"given":"Luke","family":"Mortimer","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03kpps236","id-type":"ROR","asserted-by":"publisher"}],"name":"The Barcelona Institute of Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0002-4156-2048","authenticated-orcid":true,"given":"Jakob S.","family":"Kottmann","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03g5ew477","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Computer Science"},{"id":[{"id":"https://ror.org/03p14d497","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Augsburg"},{"id":[{"id":"https://ror.org/03p14d497","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Augsburg"}]},{"given":"Antonio","family":"Acín","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03kpps236","id-type":"ROR","asserted-by":"publisher"}],"name":"The Barcelona Institute of Science and Technology"},{"id":[{"id":"https://ror.org/0371hy230","id-type":"ROR","asserted-by":"publisher"}],"name":"ICREA—Institucio Catalana de Recerca i Estudis Avancats"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,14]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.030333","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030333","note":"arxivid:2311.07409"},{"id":"arxiv:2302.01862","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present a general strategy for mapping fermionic systems to quantum hardware with square qubit connectivity which yields low-depth quantum circuits, counted in the number of native two-qubit fSIM gates. We achieve this by leveraging novel operator decomposition and circuit compression techniques paired with specifically chosen low-depth fermion-to-qubit mappings and allow for a high degree of gate cancellations and parallelism. Our mappings retain the flexibility to simultaneously optimize for qubit counts or qubit operator weights and can be used to investigate arbitrary fermionic lattice geometries. We showcase our approach by investigating the tight-binding model, the Fermi-Hubbard model as well as the multi-orbital Hubbard-Kanamori model. We report unprecedentedly low circuit depths per single Trotter layer with up to a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>70</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> improvement upon previous state-of-the-art. Our compression technique also results in significant reduction of two-qubit gates. We find the lowest gate-counts when applying the XYZ-formalism to the DK mapping. Additionally, we show that our decomposition and compression formalism produces favourable circuits even when no native parameterized two-qubit gates are available.</jats:p>","DOI":"10.22331/q-2024-04-30-1327","page":"1327","source":"Crossref","title":"Low-depth simulations of fermionic systems on square-grid quantum hardware","volume":"8","author":[{"given":"Manuel G.","family":"Algaba","sequence":"first","affiliation":[{"name":"IQM, Nymphenburgerstr. 86, 80636 Munich, Germany"}]},{"given":"P. V.","family":"Sriluckshmy","sequence":"additional","affiliation":[{"name":"IQM, Nymphenburgerstr. 86, 80636 Munich, Germany"}]},{"given":"Martin","family":"Leib","sequence":"additional","affiliation":[{"name":"IQM, Nymphenburgerstr. 86, 80636 Munich, Germany"}]},{"given":"Fedor","family":"Šimkovic IV","sequence":"additional","affiliation":[{"name":"IQM, Nymphenburgerstr. 86, 80636 Munich, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,30]]},"URL":"http://dx.doi.org/10.22331/q-2024-04-30-1327","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2302.01862"},{"id":"doi:10.48550/arXiv.2403.17794","type":"article-journal","categories":["Quantum Physics (quant-ph)","Emerging Technologies (cs.ET)","FOS: Physical sciences","FOS: Physical sciences","FOS: Computer and information sciences","FOS: Computer and information sciences"],"author":[{"family":"Liu","given":"Yuhao"},{"family":"Che","given":"Shize"},{"family":"Zhou","given":"Junyu"},{"family":"Shi","given":"Yunong"},{"family":"Li","given":"Gushu"}],"issued":{"date-parts":[[2024]]},"abstract":"This paper introduces Fermihedral, a compiler framework focusing on discovering the optimal Fermion-to-qubit encoding for targeted Fermionic Hamiltonians. Fermion-to-qubit encoding is a crucial step in harnessing quantum computing for efficient simulation of Fermionic quantum systems. Utilizing Pauli algebra, Fermihedral redefines complex constraints and objectives of Fermion-to-qubit encoding into a Boolean Satisfiability problem which can then be solved with high-performance solvers. To accommodate larger-scale scenarios, this paper proposed two new strategies that yield approximate optimal solutions mitigating the overhead from the exponentially large number of clauses. Evaluation across diverse Fermionic systems highlights the superiority of Fermihedral, showcasing substantial reductions in implementation costs, gate counts, and circuit depth in the compiled circuits. Real-system experiments on IonQ's device affirm its effectiveness, notably enhancing simulation accuracy.","container-title":"arXiv","DOI":"10.48550/ARXIV.2403.17794","publisher":"arXiv","title":"Fermihedral: On the Optimal Compilation for Fermion-to-Qubit Encoding","URL":"https://arxiv.org/abs/2403.17794","version":"2","note":"copyright:arXiv.org perpetual, non-exclusive license"},{"id":"arxiv:2509.00147","type":"article-journal","author":[{"given":"Ruby","family":"Wei"},{"given":"Aqua","family":"Chung"},{"given":"Luke","family":"Coffman"},{"given":"Su-Kuan","family":"Chu"},{"given":"Xun","family":"Gao"}],"title":"High-Distance Error-Correcting Codes for Fermion-to-Qubit Mappings in 2D and 3D","issued":{"date-parts":[[2025,8,29]]},"note":"arxivid:2509.00147\narxiv_version_number:1"},{"id":"arxiv:2110.10280","type":"article-journal","author":[{"given":"Andrew J.","family":"Landahl"},{"given":"Benjamin C. A.","family":"Morrison"}],"title":"Logical fermions for fault-tolerant quantum simulation","issued":{"date-parts":[[2023,7,7]]},"note":"arxivid:2110.10280\narxiv_version_number:3"},{"id":"doi:10.1007/978-3-662-02781-3_9","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-662-02781-3_9","page":"109-129","source":"Crossref","title":"Über das Paulische Äquivalenzverbot","author":[{"given":"P.","family":"Jordan","sequence":"first","affiliation":[]},{"given":"E. P.","family":"Wigner","sequence":"additional","affiliation":[]}],"container-title":"The Collected Works of Eugene Paul Wigner","original-title":[],"language":"en","issued":{"date-parts":[[1993]]},"ISBN":["9783642081545","9783662027813"],"URL":"http://dx.doi.org/10.1007/978-3-662-02781-3_9","note":"publisher-location:Berlin, Heidelberg"},{"id":"arxiv:quant-ph/0108146","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.65.042323","source":"Crossref","title":"Simulating physical phenomena by quantum networks","volume":"65","author":[{"given":"R.","family":"Somma","sequence":"first","affiliation":[]},{"given":"G.","family":"Ortiz","sequence":"additional","affiliation":[]},{"given":"J. E.","family":"Gubernatis","sequence":"additional","affiliation":[]},{"given":"E.","family":"Knill","sequence":"additional","affiliation":[]},{"given":"R.","family":"Laflamme","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2002,4,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.65.042323","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042323","note":"arxivid:quant-ph/0108146"},{"id":"arxiv:2211.04501","type":"article-journal","author":[{"given":"Brent","family":"Harrison"},{"given":"Dylan","family":"Nelson"},{"given":"Daniel","family":"Adamiak"},{"given":"James","family":"Whitfield"}],"title":"Reducing the qubit requirement of Jordan-Wigner encodings of $N$-mode, $K$-fermion systems from $N$ to $\\lceil \\log_2 {N \\choose K} \\rceil$","issued":{"date-parts":[[2023,8,15]]},"note":"arxivid:2211.04501\narxiv_version_number:4"},{"id":"arxiv:2110.12792","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Simulating fermionic systems on a quantum computer requires a high-performing mapping of fermionic states to qubits. A characteristic of an efficient mapping is its ability to translate local fermionic interactions into local qubit interactions, leading to easy-to-simulate qubit Hamiltonians.<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>A</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi></mml:math> fermion-qubit mappings must use a numbering scheme for the fermionic modes in order for translation to qubit operations. We make a distinction between the unordered labelling of fermions and the ordered labelling of the qubits. This separation shines light on a new way to design fermion-qubit mappings by making use of the enumeration scheme for the fermionic modes. The purpose of this paper is to demonstrate that this concept permits notions of fermion-qubit mappings that are <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>o</mml:mi><mml:mi>p</mml:mi><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>l</mml:mi></mml:math> with regard to any cost function one might choose. Our main example is the minimisation of the average number of Pauli matrices in the Jordan-Wigner transformations of Hamiltonians for fermions interacting in square lattice arrangements. In choosing the best ordering of fermionic modes for the Jordan-Wigner transformation, and unlike other popular modifications, our prescription does not cost additional resources such as ancilla qubits.We demonstrate how Mitchison and Durbin's enumeration pattern minimises the average Pauli weight of Jordan-Wigner transformations of systems interacting in square lattices. This leads to qubit Hamiltonians consisting of terms with average Pauli weights 13.9% shorter than previously known. By adding only two ancilla qubits we introduce a new class of fermion-qubit mappings, and reduce the average Pauli weight of Hamiltonian terms by 37.9% compared to previous methods. For <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math>-mode fermionic systems in cellular arrangements, we find enumeration patterns which result in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>n</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math> improvement in average Pauli weight over naïve schemes.</jats:p>","DOI":"10.22331/q-2023-10-18-1145","page":"1145","source":"Crossref","title":"Discovering optimal fermion-qubit mappings through algorithmic enumeration","volume":"7","author":[{"ORCID":"https://orcid.org/0009-0007-3562-6010","authenticated-orcid":false,"given":"Mitchell","family":"Chiew","sequence":"first","affiliation":[{"name":"DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB30WA, UK"}]},{"ORCID":"https://orcid.org/0000-0001-8390-3034","authenticated-orcid":false,"given":"Sergii","family":"Strelchuk","sequence":"additional","affiliation":[{"name":"DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB30WA, UK"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,18]]},"URL":"http://dx.doi.org/10.22331/q-2023-10-18-1145","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2110.12792"},{"id":"arxiv:2510.05099","type":"article-journal","author":[{"given":"Nathan","family":"Constantinides"},{"given":"Jeffery","family":"Yu"},{"given":"Dhruv","family":"Devulapalli"},{"given":"Ali","family":"Fahimniya"},{"given":"Luke","family":"Schaeffer"},{"given":"Andrew M.","family":"Childs"},{"given":"Michael J.","family":"Gullans"},{"given":"Alexander","family":"Schuckert"},{"given":"Alexey V.","family":"Gorshkov"}],"title":"Low-depth fermion routing without ancillas","issued":{"date-parts":[[2025,10,28]]},"note":"arxivid:2510.05099\narxiv_version_number:2"},{"id":"arxiv:2210.03727","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevb.107.125121","source":"Crossref","title":"Gapped lineon and fracton models on graphs","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0002-5926-1059","authenticated-orcid":true,"given":"Pranay","family":"Gorantla","sequence":"first","affiliation":[{"name":"Physics Department, Princeton University, Princeton, New Jersey 08544, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3826-3837","authenticated-orcid":true,"given":"Ho Tat","family":"Lam","sequence":"additional","affiliation":[{"name":"Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA"}]},{"ORCID":"https://orcid.org/0000-0003-3897-046X","authenticated-orcid":true,"given":"Nathan","family":"Seiberg","sequence":"additional","affiliation":[{"name":"School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA"}]},{"ORCID":"https://orcid.org/0000-0003-1294-2786","authenticated-orcid":true,"given":"Shu-Heng","family":"Shao","sequence":"additional","affiliation":[{"name":"C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794, USA"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.107.125121","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"125121","note":"arxivid:2210.03727"},{"id":"arxiv:2209.07987","type":"journal-article","publisher":"Stichting SciPost","issue":"5","abstract":"<jats:p>\n                    We study unusual gapped topological phases where they admit\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>\\mathbb{Z}_N</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:msub>\n                            <mml:mi>ℤ</mml:mi>\n                            <mml:mi>N</mml:mi>\n                          </mml:msub>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    fractional excitations in the same manner as topologically ordered phases, yet their ground state degeneracy depends on the local geometry of the system. Placing such phases on 2D lattice, composed of an arbitrary connected graph and 1D line, we find that the fusion rules of quasiparticle excitations are described by the Laplacian of the graph and that the number of superselection sectors is related to the kernel of the Laplacian. Based on this analysis, we further show that the ground state degeneracy is given by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>\\bigl[N\\times \\prod_{i}\\text{gcd}(N, p_i)\\bigr]^2</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:mrow>\n                            <mml:mo minsize=\"1.2\" maxsize=\"1.2\" stretchy=\"false\" form=\"prefix\">[</mml:mo>\n                            <mml:mi>N</mml:mi>\n                            <mml:mo>×</mml:mo>\n                            <mml:msub>\n                              <mml:mo>∏</mml:mo>\n                              <mml:mi>i</mml:mi>\n                            </mml:msub>\n                            <mml:mtext mathvariant=\"normal\">gcd</mml:mtext>\n                            <mml:mrow>\n                              <mml:mo stretchy=\"true\" form=\"prefix\">(</mml:mo>\n                              <mml:mi>N</mml:mi>\n                              <mml:mo>,</mml:mo>\n                              <mml:msub>\n                                <mml:mi>p</mml:mi>\n                                <mml:mi>i</mml:mi>\n                              </mml:msub>\n                              <mml:mo stretchy=\"true\" form=\"postfix\">)</mml:mo>\n                            </mml:mrow>\n                            <mml:msup>\n                              <mml:mo minsize=\"1.2\" maxsize=\"1.2\" stretchy=\"false\" form=\"postfix\">]</mml:mo>\n                              <mml:mn>2</mml:mn>\n                            </mml:msup>\n                          </mml:mrow>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>p_i</jats:tex-math>\n                        <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                          <mml:msub>\n                            <mml:mi>p</mml:mi>\n                            <mml:mi>i</mml:mi>\n                          </mml:msub>\n                        </mml:math>\n                      </jats:alternatives>\n                    </jats:inline-formula>\n                    ’s are invariant factors of the Laplacian that are greater than one and gcd stands for the greatest common divisor. We also discuss braiding statistics between quasiparticle excitations.\n                  </jats:p>","DOI":"10.21468/scipostphys.14.5.106","source":"Crossref","title":"Anisotropic higher rank $\\mathbb{Z}_N$ topological phases on graphs","volume":"14","author":[{"given":"Hiromi","family":"Ebisu","sequence":"first","affiliation":[{"name":"Rutgers, The State University of New Jersey"}]},{"given":"Bo","family":"Han","sequence":"additional","affiliation":[{"name":"Weizmann Institute of Science"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2023,5,11]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.14.5.106","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"106","note":"arxivid:2209.07987"},{"id":"arxiv:cond-mat/0404182","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.94.040402","source":"Crossref","title":"Quantum Glassiness in Strongly Correlated Clean Systems: An Example of Topological Overprotection","volume":"94","author":[{"given":"Claudio","family":"Chamon","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2005,1,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.94.040402","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"040402","note":"arxivid:cond-mat/0404182"},{"id":"arxiv:1006.4871","type":"journal-article","publisher":"Elsevier BV","issue":"4","DOI":"10.1016/j.aop.2010.11.002","page":"839-866","source":"Crossref","title":"Topological order in an exactly solvable 3D spin model","volume":"326","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Bernhard","family":"Leemhuis","sequence":"additional","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2011,4]]},"URL":"http://dx.doi.org/10.1016/j.aop.2010.11.002","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491610001910\narxivid:1006.4871"},{"id":"arxiv:2303.05267","type":"article-journal","author":[{"given":"Jian","family":"Zhao"},{"given":"Yu-Chun","family":"Wu"},{"given":"Guo-Ping","family":"Guo"}],"title":"Quantum memory error correction computation based on Chamon model","issued":{"date-parts":[[2023,3,9]]},"note":"arxivid:2303.05267\narxiv_version_number:1"},{"id":"arxiv:2206.12791","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevb.107.035136","source":"Crossref","title":"Emergent fermionic gauge theory and foliated fracton order in the Chamon model","volume":"107","author":[{"given":"Wilbur","family":"Shirley","sequence":"first","affiliation":[{"name":"School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA"},{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"Xu","family":"Liu","sequence":"additional","affiliation":[{"name":"Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, California 90095, USA"}]},{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2023,1,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.107.035136","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"035136","note":"arxivid:2206.12791"},{"id":"arxiv:1805.06899","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.99.155118","source":"Crossref","title":"Twisted fracton models in three dimensions","volume":"99","author":[{"given":"Hao","family":"Song","sequence":"first","affiliation":[]},{"given":"Abhinav","family":"Prem","sequence":"additional","affiliation":[]},{"given":"Sheng-Jie","family":"Huang","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,4,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.99.155118","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155118","note":"arxivid:1805.06899"},{"id":"arxiv:1901.08061","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevresearch.2.013303","source":"Crossref","title":"Parallelized quantum error correction with fracton topological codes","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.013303","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013303","note":"arxivid:1901.08061"},{"id":"arxiv:2112.05122","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.129.230502","source":"Crossref","title":"Optimal Thresholds for Fracton Codes and Random Spin Models with Subsystem Symmetry","volume":"129","author":[{"ORCID":"https://orcid.org/0000-0001-5513-8857","authenticated-orcid":true,"given":"Hao","family":"Song","sequence":"first","affiliation":[]},{"given":"Janik","family":"Schönmeier-Kromer","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5768-1631","authenticated-orcid":true,"given":"Ke","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Oscar","family":"Viyuela","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7274-2842","authenticated-orcid":true,"given":"Lode","family":"Pollet","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2746-5062","authenticated-orcid":true,"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,11,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.129.230502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230502","note":"arxivid:2112.05122"},{"id":"arxiv:1806.08633","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevb.99.115123","source":"Crossref","title":"Foliated fracton order in the checkerboard model","volume":"99","author":[{"given":"Wilbur","family":"Shirley","sequence":"first","affiliation":[]},{"given":"Kevin","family":"Slagle","sequence":"additional","affiliation":[]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2019,3,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.99.115123","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"115123","note":"arxivid:1806.08633"},{"id":"arxiv:2412.14320","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/wg39-vjwc","source":"Crossref","title":"Planon-modular fracton orders","volume":"112","author":[{"ORCID":"https://orcid.org/0009-0005-6218-505X","authenticated-orcid":true,"given":"Evan","family":"Wickenden","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"},{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"}]},{"ORCID":"https://orcid.org/0000-0003-1123-1157","authenticated-orcid":true,"given":"Marvin","family":"Qi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"},{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"},{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"}]},{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02smfhw86","id-type":"ROR","asserted-by":"publisher"}],"name":"Virginia Tech"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"Caltech"}]},{"ORCID":"https://orcid.org/0000-0003-2594-1176","authenticated-orcid":true,"given":"Michael","family":"Hermele","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"},{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,15]]},"URL":"http://dx.doi.org/10.1103/wg39-vjwc","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"115129","note":"arxivid:2412.14320"},{"id":"arxiv:1108.2051","type":"journal-article","publisher":"Informa UK Limited","issue":"1-3","DOI":"10.1080/14786435.2011.609152","page":"304-323","source":"Crossref","title":"Topological quantum glassiness","volume":"92","author":[{"given":"Claudio","family":"Castelnovo","sequence":"first","affiliation":[]},{"given":"Claudio","family":"Chamon","sequence":"additional","affiliation":[]}],"container-title":"Philosophical Magazine","original-title":[],"language":"en","issued":{"date-parts":[[2012,1]]},"URL":"http://dx.doi.org/10.1080/14786435.2011.609152","ISSN":["1478-6435","1478-6443"],"container-title-short":"Philosophical Magazine","note":"alternative-id:10.1080/14786435.2011.609152\narxivid:1108.2051"},{"id":"arxiv:1310.4507","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.89.075119","source":"Crossref","title":"Bifurcation in entanglement renormalization group flow of a gapped spin model","volume":"89","author":[{"given":"Jeongwan","family":"Haah","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.89.075119","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"075119","note":"arxivid:1310.4507"},{"id":"arxiv:1909.12304","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.2.033021","source":"Crossref","title":"Bifurcating entanglement-renormalization group flows of fracton stabilizer models","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"first","affiliation":[]},{"given":"Pratyush","family":"Sarkar","sequence":"additional","affiliation":[]},{"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033021","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033021","note":"arxivid:1909.12304"},{"id":"arxiv:2308.00138","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevb.109.205125","source":"Crossref","title":"Boundaries and defects in the cubic code","volume":"109","author":[{"ORCID":"https://orcid.org/0009-0008-0368-0095","authenticated-orcid":true,"given":"Cory T.","family":"Aitchison","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems"}]},{"ORCID":"https://orcid.org/0000-0001-8978-4531","authenticated-orcid":true,"given":"Daniel","family":"Bulmash","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for Theory of Quantum Matter"}]},{"ORCID":"https://orcid.org/0000-0001-6348-4135","authenticated-orcid":true,"given":"Arpit","family":"Dua","sequence":"additional","affiliation":[{"name":"Department of Physics"},{"name":"Institute for Quantum Information and Matter"}]},{"ORCID":"https://orcid.org/0000-0002-8069-7754","authenticated-orcid":true,"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems"}]},{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":true,"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.109.205125","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"205125","note":"arxivid:2308.00138"},{"id":"arxiv:1812.02101","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2019.168014","page":"168014","source":"Crossref","title":"Haah codes on general three-manifolds","volume":"412","author":[{"ORCID":"https://orcid.org/0000-0002-9876-3991","authenticated-orcid":false,"given":"Kevin T.","family":"Tian","sequence":"first","affiliation":[]},{"given":"Eric","family":"Samperton","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,1]]},"URL":"http://dx.doi.org/10.1016/j.aop.2019.168014","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491619302696\narxivid:1812.02101"},{"id":"arxiv:1902.04543","type":"article-journal","author":[{"given":"Kevin T.","family":"Tian"},{"given":"Zhenghan","family":"Wang"}],"title":"Generalized Haah Codes and Fracton Models","issued":{"date-parts":[[2019,10,12]]},"note":"arxivid:1902.04543\narxiv_version_number:4"},{"id":"arxiv:2012.04068","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2021.3119384","page":"213-229","source":"Crossref","title":"Quantum LDPC Codes With Almost Linear Minimum Distance","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0002-6772-7867","authenticated-orcid":false,"given":"Pavel","family":"Panteleev","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2695-3179","authenticated-orcid":false,"given":"Gleb","family":"Kalachev","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3119384","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2012.04068"},{"id":"arxiv:1703.02973","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevb.96.165105","source":"Crossref","title":"Fractons from partons","volume":"96","author":[{"given":"Timothy H.","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"Gábor B.","family":"Halász","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.96.165105","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"165105","note":"arxivid:1703.02973"},{"id":"arxiv:1707.02308","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/physrevlett.119.257202","source":"Crossref","title":"Fracton Topological Phases from Strongly Coupled Spin Chains","volume":"119","author":[{"given":"Gábor B.","family":"Halász","sequence":"first","affiliation":[]},{"given":"Timothy H.","family":"Hsieh","sequence":"additional","affiliation":[]},{"given":"Leon","family":"Balents","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,12,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.119.257202","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"257202","note":"arxivid:1707.02308"},{"id":"arxiv:1803.10426","type":"journal-article","publisher":"Stichting SciPost","issue":"1","abstract":"<jats:p>Fracton models exhibit a variety of exotic properties and lie beyond\nthe conventional framework of gapped topological order. In , we\ngeneralized the notion of gapped phase to one of foliated\nfracton phase by allowing the addition of layers of gapped\ntwo-dimensional resources in the adiabatic evolution between gapped\nthree-dimensional models. Moreover, we showed that the X-cube model is a\nfixed point of one such phase. In this paper, according to this\ndefinition, we look for universal properties of such phases which remain\ninvariant throughout the entire phase. We propose multi-partite\nentanglement quantities, generalizing the proposal of topological\nentanglement entropy designed for conventional topological phases. We\npresent arguments for the universality of these quantities and show that\nthey attain non-zero constant value in non-trivial foliated fracton\nphases.</jats:p>","DOI":"10.21468/scipostphys.6.1.015","source":"Crossref","title":"Universal entanglement signatures of foliated fracton phases","volume":"6","author":[{"given":"Wilbur","family":"Shirley","sequence":"first","affiliation":[{"name":"California Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-8036-3447","authenticated-orcid":false,"given":"Kevin","family":"Slagle","sequence":"additional","affiliation":[{"name":"University of Toronto"}]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[{"name":"California Institute of Technology"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2019,1,31]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.6.1.015","container-title-short":"SciPost Phys.","page":"015","note":"arxivid:1803.10426"},{"id":"arxiv:quant-ph/0502086","type":"article-journal","author":[{"given":"T.","family":"Camara"},{"given":"H.","family":"Ollivier"},{"given":"J. -P.","family":"Tillich"}],"title":"Constructions and performance of classes of quantum LDPC codes","issued":{"date-parts":[[2005,4,21]]},"note":"arxivid:quant-ph/0502086\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0005008","type":"article-journal","author":[{"given":"Alexei","family":"Ashikhmin"},{"given":"Emanuel","family":"Knill"}],"title":"Nonbinary Quantum Stabilizer Codes","issued":{"date-parts":[[2000,5,1]]},"note":"arxivid:quant-ph/0005008\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0503236","type":"article-journal","author":[{"given":"Lars Eirik","family":"Danielsen"}],"title":"On Self-Dual Quantum Codes, Graphs, and Boolean Functions","issued":{"date-parts":[[2005,3,31]]},"note":"arxivid:quant-ph/0503236\narxiv_version_number:1"},{"id":"arxiv:1706.08823","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n          </jats:title>\n          <jats:p>We describe how to introduce dynamics for the holographic states and codes introduced by Pastawski, Yoshida, Harlow and Preskill. This task requires the definition of a continuous limit of the kinematical Hilbert space which we argue may be achieved via the <jats:italic>semicontinuous limit</jats:italic> of Jones. Dynamics is then introduced by building a unitary representation of a group known as Thompson’s group <jats:italic>T</jats:italic>, which is closely related to the conformal group conf (ℝ<jats:sup>1<jats:italic>,</jats:italic>1</jats:sup>). The bulk Hilbert space is realised as a special subspace of the semicontinuous limit Hilbert space spanned by a class of distinguished states which can be assigned a discrete bulk geometry. The analogue of the group of large bulk diffeomorphisms is given by a unitary representation of the <jats:italic>Ptolemy group Pt</jats:italic> , on the bulk Hilbert space thus realising a toy model of the AdS/CFT correspondence which we call the <jats:italic>Pt /T</jats:italic> correspondence.</jats:p>","DOI":"10.1007/jhep04(2020)154","source":"Crossref","title":"Dynamics for holographic codes","volume":"2020","author":[{"given":"Tobias J.","family":"Osborne","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1853-3924","authenticated-orcid":false,"given":"Deniz E.","family":"Stiegemann","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,4]]},"URL":"http://dx.doi.org/10.1007/JHEP04(2020)154","ISSN":["1029-8479","1029-8479"],"container-title-short":"J high energy phys","page":"154","note":"alternative-id:12879\narxivid:1706.08823"},{"id":"arxiv:2112.12468","type":"article-journal","author":[{"given":"Marika","family":"Taylor"},{"given":"Charles","family":"Woodward"}],"title":"Holography, cellulations and error correcting codes","issued":{"date-parts":[[2023,1,6]]},"note":"arxivid:2112.12468\narxiv_version_number:2"},{"id":"arxiv:1611.05841","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/jhep04(2017)093","source":"Crossref","title":"Living on the edge: a toy model for holographic reconstruction of algebras with centers","volume":"2017","author":[{"given":"William","family":"Donnelly","sequence":"first","affiliation":[]},{"given":"Donald","family":"Marolf","sequence":"additional","affiliation":[]},{"given":"Ben","family":"Michel","sequence":"additional","affiliation":[]},{"given":"Jason","family":"Wien","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,4]]},"URL":"http://dx.doi.org/10.1007/JHEP04(2017)093","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"93","note":"alternative-id:5879\narxivid:1611.05841"},{"id":"arxiv:1810.05338","type":"article-journal","author":[{"given":"Daniel","family":"Harlow"},{"given":"Hirosi","family":"Ooguri"}],"title":"Symmetries in quantum field theory and quantum gravity","issued":{"date-parts":[[2019,6,6]]},"note":"arxivid:1810.05338\narxiv_version_number:2"},{"id":"arxiv:2209.12903","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>Holography has taught us that spacetime is emergent and its properties depend on the entanglement structure of the dual theory. In this paper, we describe how changes in the entanglement due to a local projective measurement (LPM) on a subregion <jats:italic>A</jats:italic> of the boundary theory modify the bulk dual spacetime. We find that LPMs destroy portions of the bulk geometry, yielding post-measurement bulk spacetimes dual to the complementary unmeasured region <jats:italic>A</jats:italic><jats:sup><jats:italic>c</jats:italic></jats:sup> that are cut off by end-of-the-world branes. Using a bulk calculation in <jats:italic>AdS</jats:italic><jats:sub>3</jats:sub> and tensor network models of holography (in particular, the HaPPY code and random tensor networks), we show that the portions of the bulk geometry that are preserved after the measurement depend on the size of <jats:italic>A</jats:italic> and the state we project onto. The post-measurement bulk dual to <jats:italic>A</jats:italic><jats:sup><jats:italic>c</jats:italic></jats:sup> includes regions that were originally part of the entanglement wedge of <jats:italic>A</jats:italic> prior to measurement. This suggests that LPMs performed on a boundary subregion <jats:italic>A</jats:italic> teleport part of the bulk information originally encoded in <jats:italic>A</jats:italic> into the complementary region <jats:italic>A</jats:italic><jats:sup><jats:italic>c</jats:italic></jats:sup>. In semiclassical holography an arbitrary amount of bulk information can be teleported in this way, while in tensor network models the teleported information is upper-bounded by the amount of entanglement shared between <jats:italic>A</jats:italic> and <jats:italic>A</jats:italic><jats:sup><jats:italic>c</jats:italic></jats:sup> due to finite-<jats:italic>N</jats:italic> effects. When <jats:italic>A</jats:italic> is the union of two disjoint subregions, the measurement triggers an entangled/disentangled phase transition between the remaining two unmeasured subregions, corresponding to a connected/disconnected phase transition in the bulk description. Our results shed new light on the effects of measurement on the entanglement structure of holographic theories and give insight on how bulk information can be manipulated from the boundary theory. They could also represent a first step towards a holographic description of measurement-induced phase transitions.</jats:p>","DOI":"10.1007/jhep12(2022)124","source":"Crossref","title":"Holographic measurement and bulk teleportation","volume":"2022","author":[{"ORCID":"https://orcid.org/0000-0002-0950-653X","authenticated-orcid":false,"given":"Stefano","family":"Antonini","sequence":"first","affiliation":[]},{"given":"Gregory","family":"Bentsen","sequence":"additional","affiliation":[]},{"given":"ChunJun","family":"Cao","sequence":"additional","affiliation":[]},{"given":"Jonathan","family":"Harper","sequence":"additional","affiliation":[]},{"given":"Shao-Kai","family":"Jian","sequence":"additional","affiliation":[]},{"given":"Brian","family":"Swingle","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2022,12,21]]},"URL":"http://dx.doi.org/10.1007/JHEP12(2022)124","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"124","note":"alternative-id:19861\narxivid:2209.12903"},{"id":"arxiv:2008.10206","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.102.062417","source":"Crossref","title":"Decoding holographic codes with an integer optimization decoder","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0002-8838-6299","authenticated-orcid":true,"given":"Robert J.","family":"Harris","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8383-3715","authenticated-orcid":true,"given":"Elliot","family":"Coupe","sequence":"additional","affiliation":[]},{"given":"Nathan A.","family":"McMahon","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6019-966X","authenticated-orcid":true,"given":"Gavin K.","family":"Brennen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2300-1394","authenticated-orcid":true,"given":"Thomas M.","family":"Stace","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.062417","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062417","note":"arxivid:2008.10206"},{"id":"arxiv:2408.06232","type":"article-journal","author":[{"given":"Junyu","family":"Fan"},{"given":"Matthew","family":"Steinberg"},{"given":"Alexander","family":"Jahn"},{"given":"Chunjun","family":"Cao"},{"given":"Sebastian","family":"Feld"}],"title":"Biased-Noise Thresholds of Zero-Rate Holographic Codes with Tensor-Network Decoding","issued":{"date-parts":[[2025,12,31]]},"note":"arxivid:2408.06232\narxiv_version_number:3"},{"id":"arxiv:1704.04229","type":"journal-article","publisher":"American Physical Society (APS)","issue":"14","DOI":"10.1103/physrevlett.119.141602","source":"Crossref","title":"Hyperinvariant Tensor Networks and Holography","volume":"119","author":[{"given":"Glen","family":"Evenbly","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.119.141602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"141602","note":"arxivid:1704.04229"},{"id":"arxiv:2407.11926","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We introduce a new class of qubit codes that we call Evenbly codes, building on a previous proposal of hyperinvariant tensor networks. Its tensor network description consists of local, non-perfect tensors describing CSS codes interspersed with Hadamard gates, placed on a hyperbolic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math> geometry with even <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>q</mml:mi><mml:mo>&amp;#x2265;</mml:mo><mml:mn>4</mml:mn></mml:math>, yielding an infinitely large class of subsystem codes. We construct an example for a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo></mml:math> manifold and describe strategies of logical gauge fixing that lead to different rates <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:math> and distances <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, which we calculate analytically, finding distances which range from <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>&amp;#x223C;</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>. Investigating threshold performance under erasure, depolarizing, and pure Pauli noise channels, we find that the code exhibits a depolarizing noise threshold of about 19.1% in the code-capacity model and 50% for pure Pauli and erasure channels under suitable gauges. We also test a constant-rate version with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0.125</mml:mn></mml:math>, finding excellent error resilience (about 40%) under the erasure channel. Recovery rates for these and other settings are studied both under an optimal decoder as well as a more efficient but non-optimal greedy decoder. We also consider generalizations beyond the CSS tensor construction, compute error rates and thresholds for other hyperbolic geometries, and discuss the relationship to holographic bulk/boundary dualities. Our work indicates that Evenbly codes may show promise for practical quantum computing applications.</jats:p>","DOI":"10.22331/q-2025-08-08-1826","page":"1826","source":"Crossref","title":"Far from Perfect: Quantum Error Correction with (Hyperinvariant) Evenbly Codes","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0003-3348-7380","authenticated-orcid":false,"given":"Matthew","family":"Steinberg","sequence":"first","affiliation":[{"name":"QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands"},{"name":"Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0009-0000-3765-2748","authenticated-orcid":false,"given":"Junyu","family":"Fan","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands"},{"name":"Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0002-8838-6299","authenticated-orcid":false,"given":"Robert J.","family":"Harris","sequence":"additional","affiliation":[{"name":"ARC Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD, 4072, Australia"}]},{"ORCID":"https://orcid.org/0000-0003-2023-2768","authenticated-orcid":false,"given":"David","family":"Elkouss","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands"},{"name":"Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan"}]},{"ORCID":"https://orcid.org/0000-0003-2782-1469","authenticated-orcid":false,"given":"Sebastian","family":"Feld","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands"},{"name":"Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0002-7142-0059","authenticated-orcid":false,"given":"Alexander","family":"Jahn","sequence":"additional","affiliation":[{"name":"Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,8]]},"URL":"http://dx.doi.org/10.22331/q-2025-08-08-1826","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2407.11926"},{"id":"arxiv:2109.13481","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math> gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math>-stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math>-rotation through <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03C0;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:msup><mml:mn>2</mml:mn><mml:mi>l</mml:mi></mml:msup></mml:math>, that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>l</mml:mi></mml:math>. The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of non-degenerate stabilizer codes.</jats:p>","DOI":"10.22331/q-2022-09-08-802","page":"802","source":"Crossref","title":"Designing the Quantum Channels Induced by Diagonal Gates","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-2699-3966","authenticated-orcid":false,"given":"Jingzhen","family":"Hu","sequence":"first","affiliation":[{"name":"Department of Mathematics, Duke University, Durham, NC 27708, USA"}]},{"ORCID":"https://orcid.org/0000-0002-5073-9431","authenticated-orcid":false,"given":"Qingzhong","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Duke University, Durham, NC 27708, USA"}]},{"ORCID":"https://orcid.org/0000-0003-2084-9717","authenticated-orcid":false,"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Duke University, Durham, NC 27708, USA"},{"name":"Department of Electrical and Computer Engineering, Department of Computer Science, Duke University, NC 27708, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,8]]},"URL":"http://dx.doi.org/10.22331/q-2022-09-08-802","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2109.13481"},{"id":"arxiv:1302.3240","type":"article-journal","author":[{"given":"Andrew J.","family":"Landahl"},{"given":"Chris","family":"Cesare"}],"title":"Complex instruction set computing architecture for performing accurate quantum $Z$ rotations with less magic","issued":{"date-parts":[[2013,10,15]]},"note":"arxivid:1302.3240\narxiv_version_number:2"},{"id":"arxiv:2206.04871","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.110.022444","source":"Crossref","title":"Secure multiparty quantum computation protocol for quantum circuits: The exploitation of triply even quantum error-correcting codes","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0002-8583-1123","authenticated-orcid":true,"given":"Petr A.","family":"Mishchenko","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00berct97","id-type":"ROR","asserted-by":"publisher"}],"name":"NTT"}]},{"ORCID":"https://orcid.org/0000-0002-6832-9940","authenticated-orcid":true,"given":"Keita","family":"Xagawa","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00berct97","id-type":"ROR","asserted-by":"publisher"}],"name":"NTT"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.022444","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022444","note":"arxivid:2206.04871"},{"id":"arxiv:1210.3388","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.87.042305","source":"Crossref","title":"Multilevel distillation of magic states for quantum computing","volume":"87","author":[{"given":"Cody","family":"Jones","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,4,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.87.042305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042305","note":"arxivid:1210.3388"},{"id":"arxiv:2001.04887","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit44484.2020.9174408","page":"1891-1896","source":"Crossref","title":"Classical Coding Problem from Transversal T Gates","author":[{"given":"Narayanan","family":"Rengaswamy","sequence":"first","affiliation":[]},{"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"event":"2020 IEEE International Symposium on Information Theory (ISIT)","container-title":"2020 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2020,6]]},"URL":"http://dx.doi.org/10.1109/ISIT44484.2020.9174408","note":"arxivid:2001.04887"},{"id":"arxiv:2412.08586","type":"article-journal","author":[{"given":"Elena","family":"Berardini"},{"given":"Reza","family":"Dastbasteh"},{"given":"Josu Etxezarreta","family":"Martinez"},{"given":"Shreyas","family":"Jain"},{"given":"Olatz Sanz","family":"Larrarte"}],"title":"Asymptotically good CSS-T codes and a new construction of triorthogonal codes","issued":{"date-parts":[[2025,6,20]]},"note":"arxivid:2412.08586\narxiv_version_number:2"},{"id":"arxiv:2305.06423","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/jsait.2025.3583217","page":"199-204","source":"Crossref","title":"CSS-T Codes From Reed-Muller Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-9386-9837","authenticated-orcid":false,"given":"Jessalyn","family":"Bolkema","sequence":"first","affiliation":[{"name":"Mathematics Department, California State University, Carson, CA, USA"}]},{"given":"Emma","family":"Andrade","sequence":"additional","affiliation":[{"name":"Department of Mathematics for cal state Fresno, California State University at Fresno, Fresno, CA, USA"}]},{"ORCID":"https://orcid.org/0009-0008-6062-4903","authenticated-orcid":false,"given":"Thomas","family":"Dexter","sequence":"additional","affiliation":[{"name":"Texas A&#x0026;M University at College Station, College Station, TX, USA"}]},{"given":"Harrison","family":"Eggers","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA"}]},{"ORCID":"https://orcid.org/0009-0007-0678-460X","authenticated-orcid":false,"given":"Victoria L.","family":"Fisher","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA"}]},{"given":"Luke","family":"Szramowsky","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA"}]},{"ORCID":"https://orcid.org/0000-0002-5764-569X","authenticated-orcid":false,"given":"Felice","family":"Manganiello","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA"}]}],"container-title":"IEEE Journal on Selected Areas in Information Theory","original-title":[],"issued":{"date-parts":[[2025]]},"URL":"http://dx.doi.org/10.1109/JSAIT.2025.3583217","ISSN":["2641-8770"],"container-title-short":"IEEE J. Sel. Areas Inf. Theory","note":"arxivid:2305.06423"},{"id":"arxiv:2310.16504","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","DOI":"10.1007/s10623-024-01415-9","page":"2801-2823","source":"Crossref","title":"Structure of CSS and CSS-T quantum codes","volume":"92","author":[{"given":"Elena","family":"Berardini","sequence":"first","affiliation":[]},{"given":"Alessio","family":"Caminata","sequence":"additional","affiliation":[]},{"given":"Alberto","family":"Ravagnani","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,24]]},"URL":"http://dx.doi.org/10.1007/s10623-024-01415-9","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:1415\narxivid:2310.16504"},{"id":"arxiv:2210.14066","type":"article-journal","author":[{"given":"Stergios","family":"Koutsioumpas"},{"given":"Darren","family":"Banfield"},{"given":"Alastair","family":"Kay"}],"title":"The Smallest Code with Transversal T","issued":{"date-parts":[[2022,10,25]]},"note":"arxivid:2210.14066\narxiv_version_number:1"},{"id":"arxiv:2408.13130","type":"article-journal","author":[{"given":"Thomas R.","family":"Scruby"},{"given":"Arthur","family":"Pesah"},{"given":"Mark","family":"Webster"}],"title":"Quantum Rainbow Codes: Achieving Linear Rate, Growing Distance and Transversal Non-Clifford Gates with Generalised Colour Codes","issued":{"date-parts":[[2025,12,15]]},"note":"arxivid:2408.13130\narxiv_version_number:3"},{"id":"arxiv:2510.05708","type":"article-journal","author":[{"given":"Dawei","family":"Jiao"},{"given":"Mahdi","family":"Bayanifar"},{"given":"Alexei","family":"Ashikhmin"},{"given":"Olav","family":"Tirkkonen"}],"title":"Low Overhead Universal Quantum Computation with Triorthogonal Codes","issued":{"date-parts":[[2025,11,5]]},"note":"arxivid:2510.05708\narxiv_version_number:2"},{"id":"arxiv:2107.09684","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.106.012437","source":"Crossref","title":"Classification of small triorthogonal codes","volume":"106","author":[{"given":"Sepehr","family":"Nezami","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1087-6853","authenticated-orcid":true,"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.106.012437","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012437","note":"arxivid:2107.09684"},{"id":"arxiv:2307.14425","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.109.042416","source":"Crossref","title":"Code conversion with the quantum Golay code for a universal transversal gate set","volume":"109","author":[{"ORCID":"https://orcid.org/0000-0002-3136-8751","authenticated-orcid":true,"given":"Matthew","family":"Sullivan","sequence":"first","affiliation":[{"name":"Department of Physics, Brookhaven National Laboratory, Upton, New York 11973, USA"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.109.042416","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042416","note":"arxivid:2307.14425"},{"id":"arxiv:2408.07764","type":"article-journal","author":[{"given":"Adam","family":"Wills"},{"given":"Min-Hsiu","family":"Hsieh"},{"given":"Hayata","family":"Yamasaki"}],"title":"Constant-Overhead Magic State Distillation","issued":{"date-parts":[[2024,8,21]]},"note":"arxivid:2408.07764\narxiv_version_number:2"},{"id":"arxiv:2003.02717","type":"journal-article","publisher":"The Royal Society","issue":"2241","abstract":"<jats:p>\n            The ternary Golay code—one of the first and most beautiful classical error-correcting codes discovered—naturally gives rise to an 11-qutrit quantum error correcting code. We apply this code to magic state distillation, a leading approach to fault-tolerant quantum computing. We find that the 11-qutrit Golay code can distil the ‘most magic’ qutrit state—an eigenstate of the qutrit Fourier transform known as the\n            <jats:italic>strange state</jats:italic>\n            —with cubic error suppression and a remarkably high threshold. It also distils the ‘second-most magic’ qutrit state, the Norell state, with quadratic error suppression and an equally high threshold to depolarizing noise.\n          </jats:p>","DOI":"10.1098/rspa.2020.0187","source":"Crossref","title":"Magic state distillation with the ternary Golay code","volume":"476","author":[{"ORCID":"https://orcid.org/0000-0003-2033-3091","authenticated-orcid":true,"given":"Shiroman","family":"Prakash","sequence":"first","affiliation":[{"name":"Department of Physics and Computer Science, Dayalbagh Educational Institute, Dayalbagh, Agra 282005, India"}]}],"container-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2020,9]]},"URL":"http://dx.doi.org/10.1098/rspa.2020.0187","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. A.","page":"20200187","note":"alternative-id:10.1098/rspa.2020.0187\narxivid:2003.02717"},{"id":"arxiv:1106.2190","type":"article-journal","author":[{"given":"Adam","family":"Paetznick"},{"given":"Ben W.","family":"Reichardt"}],"title":"Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code","issued":{"date-parts":[[2013,4,12]]},"note":"arxivid:1106.2190\narxiv_version_number:2"},{"id":"arxiv:2503.14660","type":"article-journal","author":[{"given":"Mark","family":"Webster"},{"given":"Stergios","family":"Koutsioumpas"},{"family":"Dan E Browne","given":""}],"title":"Heuristic and Optimal Synthesis of CNOT and Clifford Circuits","issued":{"date-parts":[[2025,3,18]]},"note":"arxivid:2503.14660\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0411036","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s11128-005-7654-8","page":"251-264","source":"Crossref","title":"Quantum Universality from Magic States Distillation Applied to CSS Codes","volume":"4","author":[{"given":"Ben W.","family":"Reichardt","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2005,8]]},"URL":"http://dx.doi.org/10.1007/s11128-005-7654-8","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:7654\narxivid:quant-ph/0411036"},{"id":"arxiv:quant-ph/0207119","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.68.042322","source":"Crossref","title":"Overhead and noise threshold of fault-tolerant quantum error correction","volume":"68","author":[{"given":"Andrew M.","family":"Steane","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,10,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.042322","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042322","note":"arxivid:quant-ph/0207119"},{"id":"manual:-B.-Reichardt-and-Y.-Ouyang.-","type":"manuscript","author":[{"family":"Reichardt","given":"B."},{"family":"Ouyang","given":"Y."}],"note":"Unpublished (2006"},{"id":"arxiv:quant-ph/0612004","type":"article-journal","author":[{"given":"Ben W.","family":"Reichardt"}],"title":"Error-detection-based quantum fault tolerance against discrete Pauli noise","issued":{"date-parts":[[2006,11,30]]},"note":"arxivid:quant-ph/0612004\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0607065","type":"article-journal","author":[{"given":"Rodney Doyle","family":"Van Meter"}],"title":"Architecture of a Quantum Multicomputer Optimized for Shor's Factoring Algorithm","issued":{"date-parts":[[2006,7,11]]},"note":"arxivid:quant-ph/0607065\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0701043","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"12","DOI":"10.1109/tc.2007.70775","page":"1643-1653","source":"Crossref","title":"Communication Links for Distributed Quantum Computation","volume":"56","author":[{"given":"Rod","family":"Van Meter","sequence":"first","affiliation":[]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]},{"given":"W.","family":"Munro","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Computers","original-title":[],"issued":{"date-parts":[[2007,12]]},"URL":"http://dx.doi.org/10.1109/TC.2007.70775","ISSN":["0018-9340"],"container-title-short":"IEEE Trans. Comput.","note":"arxivid:quant-ph/0701043"},{"id":"arxiv:1607.02579","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.117.070501","source":"Crossref","title":"Foliated Quantum Error-Correcting Codes","volume":"117","author":[{"given":"A.","family":"Bolt","sequence":"first","affiliation":[]},{"given":"G.","family":"Duclos-Cianci","sequence":"additional","affiliation":[]},{"given":"D.","family":"Poulin","sequence":"additional","affiliation":[]},{"given":"T. M.","family":"Stace","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2016,8,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.117.070501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070501","note":"arxivid:1607.02579"},{"id":"arxiv:1705.05602","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.95.062316","source":"Crossref","title":"General scheme for preparation of different topological states on cluster states","volume":"95","author":[{"given":"Mohammad Hossein","family":"Zarei","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,6,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.95.062316","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062316","note":"arxivid:1705.05602"},{"id":"arxiv:1811.11780","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.2.033305","source":"Crossref","title":"Universal fault-tolerant measurement-based quantum computation","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[]},{"given":"Sam","family":"Roberts","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033305","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033305","note":"arxivid:1811.11780"},{"id":"arxiv:quant-ph/0510135","type":"journal-article","publisher":"Elsevier BV","issue":"9","DOI":"10.1016/j.aop.2006.01.012","page":"2242-2270","source":"Crossref","title":"A fault-tolerant one-way quantum computer","volume":"321","author":[{"given":"R.","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"J.","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"K.","family":"Goyal","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2006,9]]},"URL":"http://dx.doi.org/10.1016/j.aop.2006.01.012","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491606000236\narxivid:quant-ph/0510135"},{"id":"arxiv:quant-ph/0610082","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevlett.98.190504","source":"Crossref","title":"Fault-Tolerant Quantum Computation with High Threshold in Two Dimensions","volume":"98","author":[{"given":"Robert","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"Jim","family":"Harrington","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,5,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.98.190504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"190504","note":"arxivid:quant-ph/0610082"},{"id":"arxiv:quant-ph/0307130","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.69.062311","source":"Crossref","title":"Multiparty entanglement in graph states","volume":"69","author":[{"given":"M.","family":"Hein","sequence":"first","affiliation":[]},{"given":"J.","family":"Eisert","sequence":"additional","affiliation":[]},{"given":"H. J.","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,6,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.69.062311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062311","note":"arxivid:quant-ph/0307130"},{"id":"arxiv:2409.20183","type":"document","categories":["Quantum computing","Graph theory","Entanglement","Local complementation","Theory of computation → Quantum information theory"],"language":"en","author":[{"family":"Claudet","given":"Nathan"},{"family":"Perdrix","given":"Simon"}],"contributor":[{"family":"Beyersdorff","given":"Olaf"},{"family":"Pilipczuk","given":"Michał"},{"family":"Pimentel","given":"Elaine"},{"family":"Thắng","given":"Nguyễn Kim"}],"editor":[{"family":"Beyersdorff","given":"Olaf"},{"family":"Pilipczuk","given":"Michał"},{"family":"Pimentel","given":"Elaine"},{"family":"Thắng","given":"Nguyễn Kim"}],"issued":{"date-parts":[[2025]]},"abstract":"Stabilizer states form a ubiquitous family of quantum states that can be graphically represented through the graph state formalism. A fundamental property of graph states is that applying a local complementation - a well-known and extensively studied graph transformation - results in a graph that represents the same entanglement as the original. In other words, the corresponding graph states are LU-equivalent. This property served as the cornerstone for capturing non-trivial quantum properties in a simple graphical manner, in the study of quantum entanglement but also for developing protocols and models based on graph states and stabilizer states, such as measurement-based quantum computing, secret sharing, error correction, entanglement distribution... However, local complementation fails short to fully characterise entanglement: there exist pairs of graph states that are LU-equivalent but cannot be transformed one into the other using local complementations. Only few is known about the equivalence of graph states beyond local complementation. We introduce a generalisation of local complementation which graphically characterises the LU-equivalence of graph states. We use this characterisation to show the existence of a strict infinite hierarchy of equivalences of graph states. Our approach is based on minimal local sets, which are subsets of vertices that are known to cover any graph, and to be invariant under local complementation and even LU-equivalence. We use these structures to provide a type to each vertex of a graph, leading to a natural standard form in which the LU-equivalence can be exhibited and captured by means of generalised local complementation.","container-title":"LIPIcs, Volume 327, STACS 2025","DOI":"10.4230/LIPICS.STACS.2025.27","volume":"327","number":"27","page":"27:1-27:18","page-first":"27:1","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"Local Equivalence of Stabilizer States: A Graphical Characterisation","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.27","note":"copyright:Creative Commons Attribution 4.0 International license\narxivid:2409.20183"},{"id":"arxiv:0709.1266","type":"article-journal","author":[{"given":"Zhengfeng","family":"Ji"},{"given":"Jianxin","family":"Chen"},{"given":"Zhaohui","family":"Wei"},{"given":"Mingsheng","family":"Ying"}],"title":"The LU-LC conjecture is false","issued":{"date-parts":[[2008,6,30]]},"note":"arxivid:0709.1266\narxiv_version_number:2"},{"id":"arxiv:0812.4625","type":"journal-article","publisher":"Elsevier BV","issue":"26","DOI":"10.1016/j.physleta.2009.04.055","page":"2219-2225","source":"Crossref","title":"Entanglement in eight-qubit graph states","volume":"373","author":[{"given":"Adán","family":"Cabello","sequence":"first","affiliation":[]},{"given":"Antonio J.","family":"López-Tarrida","sequence":"additional","affiliation":[]},{"given":"Pilar","family":"Moreno","sequence":"additional","affiliation":[]},{"given":"José R.","family":"Portillo","sequence":"additional","affiliation":[]}],"container-title":"Physics Letters A","original-title":[],"language":"en","issued":{"date-parts":[[2009,6]]},"URL":"http://dx.doi.org/10.1016/j.physleta.2009.04.055","ISSN":["0375-9601"],"container-title-short":"Physics Letters A","note":"alternative-id:S0375960109005398\narxivid:0812.4625"},{"id":"arxiv:1910.03969","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation – the graph operation that links all local-Clifford equivalent graph states – allows us to classify all stabiliser states by their entanglement. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We provide programs to compute these orbits, along with our data for each of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>587</mml:mn></mml:math> orbits up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>9</mml:mn></mml:math> qubits and a means to visualise them. We find direct links between the connectivity of certain orbits with the entanglement properties of their component graph states. Furthermore, we observe the correlations between graph-theoretical orbit properties, such as diameter and colourability, with Schmidt measure and preparation complexity and suggest potential applications. It is well known that graph theory and quantum entanglement have strong interplay – our exploration deepens this relationship, providing new tools with which to probe the nature of entanglement.</jats:p>","DOI":"10.22331/q-2020-08-07-305","page":"305","source":"Crossref","title":"Mapping graph state orbits under local complementation","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0002-8923-7180","authenticated-orcid":false,"given":"Jeremy C.","family":"Adcock","sequence":"first","affiliation":[{"name":"Quantum Engineering Technology (QET) Labs, H. H. Wills Physics Laboratory & Department of Electrical & Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK"}]},{"ORCID":"https://orcid.org/0000-0002-4445-734X","authenticated-orcid":false,"given":"Sam","family":"Morley-Short","sequence":"additional","affiliation":[{"name":"Quantum Engineering Technology (QET) Labs, H. H. Wills Physics Laboratory & Department of Electrical & Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK"}]},{"ORCID":"https://orcid.org/0000-0003-2479-7424","authenticated-orcid":false,"given":"Axel","family":"Dahlberg","sequence":"additional","affiliation":[{"name":"QuTech - TU Delft, Lorentzweg 1, 2628CJ Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0002-3429-4890","authenticated-orcid":false,"given":"Joshua W.","family":"Silverstone","sequence":"additional","affiliation":[{"name":"Quantum Engineering Technology (QET) Labs, H. H. Wills Physics Laboratory & Department of Electrical & Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,8,7]]},"URL":"http://dx.doi.org/10.22331/q-2020-08-07-305","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1910.03969"},{"id":"arxiv:quant-ph/0004051","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.86.910","page":"910-913","source":"Crossref","title":"Persistent Entanglement in Arrays of Interacting Particles","volume":"86","author":[{"given":"Hans J.","family":"Briegel","sequence":"first","affiliation":[]},{"given":"Robert","family":"Raussendorf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2001,1,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.86.910","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/0004051"},{"id":"arxiv:1905.06928","type":"journal-article","publisher":"IOP Publishing","issue":"34","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Correlations in multiparticle systems are constrained by restrictions from quantum mechanics. A prominent example for these restrictions are monogamy relations, limiting the amount of entanglement between pairs of particles in a three-particle system. A powerful tool to study correlation constraints is the notion of sector lengths. These quantify, for different <jats:italic>k</jats:italic>, the amount of <jats:italic>k</jats:italic>-partite correlations in a quantum state in a basis-independent manner. We derive tight bounds on the sector lengths in multi-qubit states and highlight applications of these bounds to entanglement detection, monogamy relations and the <jats:italic>n</jats:italic>-representability problem. For the case of two- and three qubits we characterize the possible sector lengths completely and prove a symmetrized version of strong subadditivity for the linear entropy.</jats:p>","DOI":"10.1088/1751-8121/ab7f0a","page":"345302","source":"Crossref","title":"Characterizing quantum states via sector lengths","volume":"53","author":[{"ORCID":"https://orcid.org/0000-0003-3002-9878","authenticated-orcid":false,"given":"N","family":"Wyderka","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6033-0867","authenticated-orcid":false,"given":"O","family":"Gühne","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2020,7,31]]},"URL":"http://dx.doi.org/10.1088/1751-8121/ab7f0a","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1905.06928"},{"id":"arxiv:1908.04220","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The existence of correlations between the parts of a quantum system on the one hand, and entanglement between them on the other, are different properties. Yet, one intuitively would identify strong<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>-party correlations with<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>-party entanglement in an<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>-partite quantum state. If the local systems are qubits, this intuition is confirmed: The state with the strongest<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>-party correlations is the Greenberger-Horne-Zeilinger (GHZ) state, which does have genuine multipartite entanglement. However, for high-dimensional local systems the state with strongest<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>-party correlations may be a tensor product of Bell states, that is, partially separable. We show this by introducing several novel tools for handling the Bloch representation.</jats:p>","DOI":"10.22331/q-2020-02-10-229","page":"229","source":"Crossref","title":"Maximum<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math>-body correlations do not in general imply genuine multipartite entanglement","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-9428-5593","authenticated-orcid":false,"given":"Christopher","family":"Eltschka","sequence":"first","affiliation":[{"name":"Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany"}]},{"ORCID":"https://orcid.org/0000-0002-9410-5043","authenticated-orcid":false,"given":"Jens","family":"Siewert","sequence":"additional","affiliation":[{"name":"Departamento de Química Física, Universidad del País Vasco UPV/EHU, E-48080 Bilbao, Spain"},{"name":"IKERBASQUE Basque Foundation for Science, E-48013 Bilbao, Spain"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,2,10]]},"URL":"http://dx.doi.org/10.22331/q-2020-02-10-229","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1908.04220"},{"id":"arxiv:2105.12752","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce52317.2021.00057","page":"378-384","source":"Crossref","title":"GraphStateVis: Interactive Visual Analysis of Qubit Graph States and their Stabilizer Groups","author":[{"given":"Matthias","family":"Miller","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Miller","sequence":"additional","affiliation":[]}],"event":"2021 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2021 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2021,10]]},"URL":"http://dx.doi.org/10.1109/QCE52317.2021.00057","note":"arxivid:2105.12752"},{"id":"arxiv:2207.07665","type":"journal-article","publisher":"IOP Publishing","issue":"33","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>The Shor–Laflamme distribution (SLD) of a quantum state is a collection of local unitary invariants that quantify <jats:italic>k</jats:italic>-body correlations. We show that the SLD of graph states can be derived by solving a graph-theoretical problem. In this way, the mean and variance of the SLD are obtained as simple functions of efficiently computable graph properties. Furthermore, this formulation enables us to derive closed expressions of SLDs for some graph state families. For cluster states, we observe that the SLD is very similar to a binomial distribution, and we argue that this property is typical for graph states in general. Finally, we derive an SLD-based entanglement criterion from the purity criterion and apply it to derive meaningful noise thresholds for entanglement. Our new entanglement criterion is easy to use and also applies to the case of higher-dimensional qudits. In the bigger picture, our results foster the understanding both of quantum error-correcting codes, where a closely related notion of SLDs plays an important role, and of the geometry of quantum states, where SLDs are known as sector length distributions.</jats:p>","DOI":"10.1088/1751-8121/ace8d4","page":"335303","source":"Crossref","title":"Shor–Laflamme distributions of graph states and noise robustness of entanglement","volume":"56","author":[{"ORCID":"https://orcid.org/0000-0003-2100-5612","authenticated-orcid":true,"given":"Daniel","family":"Miller","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5176-3073","authenticated-orcid":false,"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5690-1981","authenticated-orcid":true,"given":"Ivano","family":"Tavernelli","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0659-6699","authenticated-orcid":false,"given":"Hermann","family":"Kampermann","sequence":"additional","affiliation":[]},{"given":"Dagmar","family":"Bruß","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3002-9878","authenticated-orcid":true,"given":"Nikolai","family":"Wyderka","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2023,7,31]]},"URL":"http://dx.doi.org/10.1088/1751-8121/ace8d4","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:2207.07665"},{"id":"arxiv:quant-ph/0605014","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.74.042343","source":"Crossref","title":"Potential and limits to cluster-state quantum computing using probabilistic gates","volume":"74","author":[{"given":"D.","family":"Gross","sequence":"first","affiliation":[]},{"given":"K.","family":"Kieling","sequence":"additional","affiliation":[]},{"given":"J.","family":"Eisert","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,10,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.74.042343","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042343","note":"arxivid:quant-ph/0605014"},{"id":"arxiv:2212.04834","type":"article-journal","author":[{"given":"Tom J.","family":"Bell"},{"given":"Love A.","family":"Pettersson"},{"given":"Stefano","family":"Paesani"}],"title":"Optimising graph codes for measurement-based loss tolerance","issued":{"date-parts":[[2022,12,9]]},"note":"arxivid:2212.04834\narxiv_version_number:1"},{"id":"arxiv:1011.5464","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.83.042314","source":"Crossref","title":"Optimal preparation of graph states","volume":"83","author":[{"given":"Adán","family":"Cabello","sequence":"first","affiliation":[]},{"given":"Lars Eirik","family":"Danielsen","sequence":"additional","affiliation":[]},{"given":"Antonio J.","family":"López-Tarrida","sequence":"additional","affiliation":[]},{"given":"José R.","family":"Portillo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,4,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.042314","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042314","note":"arxivid:1011.5464"},{"id":"arxiv:quant-ph/0108067","type":"article-journal","author":[{"given":"Robert","family":"Raussendorf"},{"given":"Hans","family":"Briegel"}],"title":"Computational model underlying the one-way quantum computer","issued":{"date-parts":[[2002,3,12]]},"note":"arxivid:quant-ph/0108067\narxiv_version_number:2"},{"id":"arxiv:0808.1782","type":"journal-article","publisher":"IOP Publishing","issue":"8","DOI":"10.1088/1367-2630/11/8/083032","page":"083032","source":"Crossref","title":"Architectural design for a topological cluster state quantum computer","volume":"11","author":[{"given":"Simon J","family":"Devitt","sequence":"first","affiliation":[]},{"given":"Austin G","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"Ashley M","family":"Stephens","sequence":"additional","affiliation":[]},{"given":"Andrew D","family":"Greentree","sequence":"additional","affiliation":[]},{"given":"Lloyd C L","family":"Hollenberg","sequence":"additional","affiliation":[]},{"given":"William J","family":"Munro","sequence":"additional","affiliation":[]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2009,8,24]]},"URL":"http://dx.doi.org/10.1088/1367-2630/11/8/083032","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:0808.1782"},{"id":"arxiv:quant-ph/0503130","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.73.032308","source":"Crossref","title":"Simple proof of fault tolerance in the graph-state model","volume":"73","author":[{"given":"Panos","family":"Aliferis","sequence":"first","affiliation":[]},{"given":"Debbie W.","family":"Leung","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,3,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.032308","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032308","note":"arxivid:quant-ph/0503130"},{"id":"arxiv:quant-ph/0401101","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/j.nuclphysb.2004.07.003","page":"462-480","source":"Crossref","title":"Phase structure of the random-plaquette <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" altimg=\"si1.gif\" overflow=\"scroll\"><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> gauge model: accuracy threshold for a toric quantum memory","volume":"697","author":[{"given":"Takuya","family":"Ohno","sequence":"first","affiliation":[]},{"given":"Gaku","family":"Arakawa","sequence":"additional","affiliation":[]},{"given":"Ikuo","family":"Ichinose","sequence":"additional","affiliation":[]},{"given":"Tetsuo","family":"Matsui","sequence":"additional","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[2004,10]]},"URL":"http://dx.doi.org/10.1016/j.nuclphysb.2004.07.003","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"alternative-id:S055032130400481X\narxivid:quant-ph/0401101"},{"id":"arxiv:cond-mat/0501372","type":"journal-article","publisher":"IOP Publishing","issue":"17","DOI":"10.1088/0305-4470/38/17/004","page":"3751-3774","source":"Crossref","title":"Exact location of the multicritical point for finite-dimensional spin glasses: a conjecture","volume":"38","author":[{"given":"Koujin","family":"Takeda","sequence":"first","affiliation":[]},{"given":"Tomohiro","family":"Sasamoto","sequence":"additional","affiliation":[]},{"given":"Hidetoshi","family":"Nishimori","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[2005,4,13]]},"URL":"http://dx.doi.org/10.1088/0305-4470/38/17/004","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen.","note":"alternative-id:S0305-4470(05)93241-1\narxivid:cond-mat/0501372"},{"id":"arxiv:quant-ph/0503126","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7030","DOI":"10.1038/nature03347","page":"169-176","source":"Crossref","title":"Experimental one-way quantum computing","volume":"434","author":[{"given":"P.","family":"Walther","sequence":"first","affiliation":[]},{"given":"K. J.","family":"Resch","sequence":"additional","affiliation":[]},{"given":"T.","family":"Rudolph","sequence":"additional","affiliation":[]},{"given":"E.","family":"Schenck","sequence":"additional","affiliation":[]},{"given":"H.","family":"Weinfurter","sequence":"additional","affiliation":[]},{"given":"V.","family":"Vedral","sequence":"additional","affiliation":[]},{"given":"M.","family":"Aspelmeyer","sequence":"additional","affiliation":[]},{"given":"A.","family":"Zeilinger","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2005,3]]},"URL":"http://dx.doi.org/10.1038/nature03347","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature03347\narxivid:quant-ph/0503126"},{"id":"arxiv:0906.2233","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/physrevlett.103.160401","source":"Crossref","title":"Experimental Entanglement and Nonlocality of a Two-Photon Six-Qubit Cluster State","volume":"103","author":[{"given":"Raino","family":"Ceccarelli","sequence":"first","affiliation":[]},{"given":"Giuseppe","family":"Vallone","sequence":"additional","affiliation":[]},{"given":"Francesco","family":"De Martini","sequence":"additional","affiliation":[]},{"given":"Paolo","family":"Mataloni","sequence":"additional","affiliation":[]},{"given":"Adán","family":"Cabello","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,10,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.103.160401","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"160401","note":"arxivid:0906.2233"},{"id":"arxiv:1202.5459","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7386","DOI":"10.1038/nature10770","page":"489-494","source":"Crossref","title":"Experimental demonstration of topological error correction","volume":"482","author":[{"given":"Xing-Can","family":"Yao","sequence":"first","affiliation":[]},{"given":"Tian-Xiong","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Hao-Ze","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Wei-Bo","family":"Gao","sequence":"additional","affiliation":[]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Raussendorf","sequence":"additional","affiliation":[]},{"given":"Zeng-Bing","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Nai-Le","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Chao-Yang","family":"Lu","sequence":"additional","affiliation":[]},{"given":"You-Jin","family":"Deng","sequence":"additional","affiliation":[]},{"given":"Yu-Ao","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2012,2]]},"URL":"http://dx.doi.org/10.1038/nature10770","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature10770\narxivid:1202.5459"},{"id":"arxiv:0705.4165","type":"journal-article","publisher":"IOP Publishing","issue":"8","DOI":"10.1088/0034-4885/70/8/r03","page":"1381-1424","source":"Crossref","title":"Entanglement purification and quantum error correction","volume":"70","author":[{"given":"W","family":"Dür","sequence":"first","affiliation":[]},{"given":"H J","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Reports on Progress in Physics","original-title":[],"issued":{"date-parts":[[2007,7,17]]},"URL":"http://dx.doi.org/10.1088/0034-4885/70/8/R03","ISSN":["0034-4885","1361-6633"],"container-title-short":"Rep. Prog. Phys.","note":"alternative-id:S0034-4885(07)89056-7\narxivid:0705.4165"},{"id":"doi:10.5281/zenodo.2582616","type":"book","author":[{"literal":"Sammorley-Short"}],"issued":{"date-parts":[[2019,3,4]]},"abstract":"Extended GSC to prime-power qudit graph states","DOI":"10.5281/ZENODO.2582616","publisher":"Zenodo","title":"sammorley-short/gsc: v2.0","URL":"https://zenodo.org/record/2582616","version":"v2.0","note":"copyright:Open Access"},{"id":"manual:-S.-Morley-Short.-GSC.-url-ht","type":"book","author":[{"family":"Morley-Short","given":"S."}],"title":"GSC","publisher":"\\url{https://github.com/sammorley-short/gsc/tree/master}","issued":"2021"},{"id":"arxiv:1805.05305","type":"journal-article","publisher":"The Royal Society","issue":"2123","abstract":"<jats:p>Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool—for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph<jats:italic>G</jats:italic>′ is a<jats:italic>vertex-minor</jats:italic>of another graph<jats:italic>G</jats:italic>. The vertex-minor problem is, in general,<jats:inline-formula><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"rsta20170325im1.gif\"/></jats:inline-formula>-Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the<jats:italic>rank-width</jats:italic>which equals the<jats:italic>Schmidt-rank width</jats:italic>of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time<jats:italic>O</jats:italic>(|<jats:italic>G</jats:italic>|<jats:sup>3</jats:sup>), where |<jats:italic>G</jats:italic>| is the size of the graph<jats:italic>G</jats:italic>, whenever the rank-width of<jats:italic>G</jats:italic>and the size of<jats:italic>G</jats:italic>′ are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information.</jats:p><jats:p>This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.</jats:p>","DOI":"10.1098/rsta.2017.0325","page":"20170325","source":"Crossref","title":"Transforming graph states using single-qubit operations","volume":"376","author":[{"ORCID":"https://orcid.org/0000-0003-2479-7424","authenticated-orcid":false,"given":"Axel","family":"Dahlberg","sequence":"first","affiliation":[]},{"given":"Stephanie","family":"Wehner","sequence":"additional","affiliation":[]}],"container-title":"Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,28]]},"URL":"http://dx.doi.org/10.1098/rsta.2017.0325","ISSN":["1364-503X","1471-2962"],"container-title-short":"Phil. Trans. R. Soc. A.","note":"alternative-id:10.1098/rsta.2017.0325\narxivid:1805.05305"},{"id":"arxiv:1805.05306","type":"article-journal","author":[{"given":"Axel","family":"Dahlberg"},{"given":"Jonas","family":"Helsen"},{"given":"Stephanie","family":"Wehner"}],"title":"How to transform graph states using single-qubit operations: computational complexity and algorithms","issued":{"date-parts":[[2018,5,15]]},"note":"arxivid:1805.05306\narxiv_version_number:2"},{"id":"arxiv:1805.04559","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum communication between distant parties is based on suitable instances of shared entanglement. For efficiency reasons, in an anticipated quantum network beyond point-to-point communication, it is preferable that many parties can communicate simultaneously over the underlying infrastructure; however, bottlenecks in the network may cause delays. Sharing of multi-partite entangled states between parties offers a solution, allowing for parallel quantum communication. Specifically for the two-pair problem, the butterfly network provides the first instance of such an advantage in a bottleneck scenario. In this paper, we propose a more general method for establishing EPR pairs in arbitrary networks. The main difference from standard repeater network approaches is that we use a graph state instead of maximally entangled pairs to achieve long-distance simultaneous communication. We demonstrate how graph-theoretic tools, and specifically local complementation, help decrease the number of required measurements compared to usual methods applied in repeater schemes. We examine other examples of network architectures, where deploying local complementation techniques provides an advantage. We finally consider the problem of extracting graph states for quantum communication via local Clifford operations and Pauli measurements, and discuss that while the general problem is known to be NP-complete, interestingly, for specific classes of structured resources, polynomial time algorithms can be identified.</jats:p>","DOI":"10.1038/s41534-019-0191-6","source":"Crossref","title":"Quantum network routing and local complementation","volume":"5","author":[{"given":"F.","family":"Hahn","sequence":"first","affiliation":[]},{"given":"A.","family":"Pappa","sequence":"additional","affiliation":[]},{"given":"J.","family":"Eisert","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2019,9,6]]},"URL":"http://dx.doi.org/10.1038/s41534-019-0191-6","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"76","note":"alternative-id:191\narxivid:1805.04559"},{"id":"arxiv:2211.01376","type":"article-journal","author":[{"given":"Ruben","family":"Verresen"},{"given":"Umberto","family":"Borla"},{"given":"Ashvin","family":"Vishwanath"},{"given":"Sergej","family":"Moroz"},{"given":"Ryan","family":"Thorngren"}],"title":"Higgs Condensates are Symmetry-Protected Topological Phases: I. Discrete Symmetries","issued":{"date-parts":[[2024,3,15]]},"note":"arxivid:2211.01376\narxiv_version_number:2"},{"id":"arxiv:2101.09310","type":"article-journal","author":[{"given":"Sara","family":"Bartolucci"},{"given":"Patrick","family":"Birchall"},{"given":"Hector","family":"Bombin"},{"given":"Hugo","family":"Cable"},{"given":"Chris","family":"Dawson"},{"given":"Mercedes","family":"Gimeno-Segovia"},{"given":"Eric","family":"Johnston"},{"given":"Konrad","family":"Kieling"},{"given":"Naomi","family":"Nickerson"},{"given":"Mihir","family":"Pant"},{"given":"Fernando","family":"Pastawski"},{"given":"Terry","family":"Rudolph"},{"given":"Chris","family":"Sparrow"}],"title":"Fusion-based quantum computation","issued":{"date-parts":[[2021,1,22]]},"note":"arxivid:2101.09310\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0407255","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.71.062313","source":"Crossref","title":"Long-range quantum entanglement in noisy cluster states","volume":"71","author":[{"given":"Robert","family":"Raussendorf","sequence":"first","affiliation":[]},{"given":"Sergey","family":"Bravyi","sequence":"additional","affiliation":[]},{"given":"Jim","family":"Harrington","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,6,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.062313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062313","note":"arxivid:quant-ph/0407255"},{"id":"arxiv:1307.1740","type":"article-journal","author":[{"given":"Austin G.","family":"Fowler"}],"title":"Minimum weight perfect matching of fault-tolerant topological quantum error correction in average $O(1)$ parallel time","issued":{"date-parts":[[2014,10,10]]},"note":"arxivid:1307.1740\narxiv_version_number:3"},{"id":"doi:10.4153/CJM-1965-045-4","type":"journal-article","publisher":"Canadian Mathematical Society","abstract":"<jats:p>A <jats:italic>graph G</jats:italic> for purposes here is a finite set of elements called <jats:italic>vertices</jats:italic> and a finite set of elements called <jats:italic>edges</jats:italic> such that each edge <jats:italic>meets</jats:italic> exactly two vertices, called the <jats:italic>end-points</jats:italic> of the edge. An edge is said to <jats:italic>join</jats:italic> its end-points.</jats:p><jats:p>A <jats:italic>matching</jats:italic> in <jats:italic>G</jats:italic> is a subset of its edges such that no two meet the same vertex. We describe an efficient algorithm for finding in a given graph a matching of maximum cardinality. This problem was posed and partly solved by C. Berge; see Sections 3.7 and 3.8.</jats:p>","DOI":"10.4153/cjm-1965-045-4","page":"449-467","source":"Crossref","title":"Paths, Trees, and Flowers","volume":"17","author":[{"given":"Jack","family":"Edmonds","sequence":"first","affiliation":[]}],"container-title":"Canadian Journal of Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[1965]]},"URL":"http://dx.doi.org/10.4153/CJM-1965-045-4","ISSN":["0008-414X","1496-4279"],"container-title-short":"Can. j. math.","note":"alternative-id:S0008414X00039419"},{"id":"doi:10.6028/jres.069B.013","type":"journal-article","publisher":"National Institute of Standards and Technology (NIST)","issue":"1 and 2","DOI":"10.6028/jres.069b.013","page":"125","source":"Crossref","title":"Maximum matching and a polyhedron with 0,1-vertices","volume":"69B","author":[{"given":"Jack","family":"Edmonds","sequence":"first","affiliation":[]}],"container-title":"Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1965,1]]},"URL":"http://dx.doi.org/10.6028/jres.069B.013","ISSN":["0022-4340"],"container-title-short":"J. RES. NATL. BUR. STAN. SECT. B. MATH. MATH. PHYS."},{"id":"arxiv:quant-ph/0509060","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.96.020501","source":"Crossref","title":"Noise Thresholds for Optical Quantum Computers","volume":"96","author":[{"given":"Christopher M.","family":"Dawson","sequence":"first","affiliation":[]},{"given":"Henry L.","family":"Haselgrove","sequence":"additional","affiliation":[]},{"given":"Michael A.","family":"Nielsen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,1,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.96.020501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020501","note":"arxivid:quant-ph/0509060"},{"id":"arxiv:1005.2915","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.82.032332","source":"Crossref","title":"Photonic implementation for the topological cluster-state quantum computer","volume":"82","author":[{"given":"David A.","family":"Herrera-Martí","sequence":"first","affiliation":[]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"David","family":"Jennings","sequence":"additional","affiliation":[]},{"given":"Terry","family":"Rudolph","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,9,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.82.032332","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032332","note":"arxivid:1005.2915"},{"id":"arxiv:1005.2456","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.105.200502","source":"Crossref","title":"Fault Tolerant Quantum Computation with Very High Threshold for Loss Errors","volume":"105","author":[{"given":"Sean D.","family":"Barrett","sequence":"first","affiliation":[]},{"given":"Thomas M.","family":"Stace","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,11,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.105.200502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200502","note":"arxivid:1005.2456"},{"id":"arxiv:1504.01444","type":"article-journal","author":[{"given":"Keisuke","family":"Fujii"}],"title":"Quantum Computation with Topological Codes: from qubit to topological fault-tolerance","issued":{"date-parts":[[2015,4,7]]},"note":"arxivid:1504.01444\narxiv_version_number:1"},{"id":"arxiv:1110.5460","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/ncomms2043","source":"Crossref","title":"Blind topological measurement-based quantum computation","volume":"3","author":[{"given":"Tomoyuki","family":"Morimae","sequence":"first","affiliation":[]},{"given":"Keisuke","family":"Fujii","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2012,9,4]]},"URL":"http://dx.doi.org/10.1038/ncomms2043","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"1036","note":"alternative-id:BFncomms2043\narxivid:1110.5460"},{"id":"arxiv:2505.01978","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1038/s41567-026-03179-6","page":"430-438","source":"Crossref","title":"One- and two-dimensional cluster states for topological phase simulation and measurement-based quantum computation","volume":"22","author":[{"ORCID":"https://orcid.org/0009-0004-3342-6814","authenticated-orcid":false,"given":"Tao","family":"Jiang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0009-0002-7968-7070","authenticated-orcid":false,"given":"Jianbin","family":"Cai","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-1225-585X","authenticated-orcid":false,"given":"Junxiang","family":"Huang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-6762-9695","authenticated-orcid":false,"given":"Naibin","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Yukun","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Jiahao","family":"Bei","sequence":"additional","affiliation":[]},{"given":"Guoqing","family":"Cai","sequence":"additional","affiliation":[]},{"given":"Sirui","family":"Cao","sequence":"additional","affiliation":[]},{"given":"Fusheng","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Jiang","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Kefu","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0003-7731-338X","authenticated-orcid":false,"given":"Xiawei","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Xiqing","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Zhe","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9481-4686","authenticated-orcid":false,"given":"Zhiyuan","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Zihua","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Wenhao","family":"Chu","sequence":"additional","affiliation":[]},{"given":"Hui","family":"Deng","sequence":"additional","affiliation":[]},{"given":"Zhibin","family":"Deng","sequence":"additional","affiliation":[]},{"given":"Pei","family":"Ding","sequence":"additional","affiliation":[]},{"given":"Xun","family":"Ding","sequence":"additional","affiliation":[]},{"given":"Zhuzhengqi","family":"Ding","sequence":"additional","affiliation":[]},{"given":"Shuai","family":"Dong","sequence":"additional","affiliation":[]},{"given":"Bo","family":"Fan","sequence":"additional","affiliation":[]},{"given":"Daojin","family":"Fan","sequence":"additional","affiliation":[]},{"given":"Yuanhao","family":"Fu","sequence":"additional","affiliation":[]},{"given":"Dongxin","family":"Gao","sequence":"additional","affiliation":[]},{"given":"Lei","family":"Ge","sequence":"additional","affiliation":[]},{"given":"Jiacheng","family":"Gui","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1185-4805","authenticated-orcid":false,"given":"Cheng","family":"Guo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0001-3068-2335","authenticated-orcid":false,"given":"Shaojun","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Xiaoyang","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Lianchen","family":"Han","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2086-7085","authenticated-orcid":false,"given":"Tan","family":"He","sequence":"additional","affiliation":[]},{"given":"Linyin","family":"Hong","sequence":"additional","affiliation":[]},{"given":"Yisen","family":"Hu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5121-3028","authenticated-orcid":false,"given":"He-Liang","family":"Huang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0310-3078","authenticated-orcid":false,"given":"Yong-Heng","family":"Huo","sequence":"additional","affiliation":[]},{"given":"Zuokai","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Honghong","family":"Jin","sequence":"additional","affiliation":[]},{"given":"Yunxiang","family":"Leng","sequence":"additional","affiliation":[]},{"given":"Dayu","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7920-0835","authenticated-orcid":false,"given":"Dongdong","family":"Li","sequence":"additional","affiliation":[]},{"given":"Fangyu","family":"Li","sequence":"additional","affiliation":[]},{"given":"Jiaqi","family":"Li","sequence":"additional","affiliation":[]},{"given":"Jinjin","family":"Li","sequence":"additional","affiliation":[]},{"given":"Junyan","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0009-5556-8510","authenticated-orcid":false,"given":"Junyun","family":"Li","sequence":"additional","affiliation":[]},{"given":"Na","family":"Li","sequence":"additional","affiliation":[]},{"given":"Shaowei","family":"Li","sequence":"additional","affiliation":[]},{"given":"Wei","family":"Li","sequence":"additional","affiliation":[]},{"given":"Yuhuai","family":"Li","sequence":"additional","affiliation":[]},{"given":"Yuan","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9843-0559","authenticated-orcid":false,"given":"Futian","family":"Liang","sequence":"additional","affiliation":[]},{"given":"Xuelian","family":"Liang","sequence":"additional","affiliation":[]},{"given":"Nanxing","family":"Liao","sequence":"additional","affiliation":[]},{"given":"Jin","family":"Lin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0003-1745-5917","authenticated-orcid":false,"given":"Weiping","family":"Lin","sequence":"additional","affiliation":[]},{"given":"Dailin","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Hongxiu","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Maliang","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Xinyu","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Xuemeng","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Yancheng","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Haoxin","family":"Lou","sequence":"additional","affiliation":[]},{"given":"Yuwei","family":"Ma","sequence":"additional","affiliation":[]},{"given":"Lingxin","family":"Meng","sequence":"additional","affiliation":[]},{"given":"Hao","family":"Mou","sequence":"additional","affiliation":[]},{"given":"Kailiang","family":"Nan","sequence":"additional","affiliation":[]},{"given":"Binghan","family":"Nie","sequence":"additional","affiliation":[]},{"given":"Meijuan","family":"Nie","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-1357-4457","authenticated-orcid":false,"given":"Jie","family":"Ning","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0887-5154","authenticated-orcid":false,"given":"Le","family":"Niu","sequence":"additional","affiliation":[]},{"given":"Wenyi","family":"Peng","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0001-0928-6776","authenticated-orcid":false,"given":"Haoran","family":"Qian","sequence":"additional","affiliation":[]},{"given":"Hao","family":"Rong","sequence":"additional","affiliation":[]},{"given":"Tao","family":"Rong","sequence":"additional","affiliation":[]},{"given":"Huiyan","family":"Shen","sequence":"additional","affiliation":[]},{"given":"Qiong","family":"Shen","sequence":"additional","affiliation":[]},{"given":"Hong","family":"Su","sequence":"additional","affiliation":[]},{"given":"Feifan","family":"Su","sequence":"additional","affiliation":[]},{"given":"Chenyin","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Liangchao","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Tianzuo","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Yingxiu","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Yimeng","family":"Tan","sequence":"additional","affiliation":[]},{"given":"Jun","family":"Tan","sequence":"additional","affiliation":[]},{"given":"Longyue","family":"Tang","sequence":"additional","affiliation":[]},{"given":"Wenbing","family":"Tu","sequence":"additional","affiliation":[]},{"given":"Jiafei","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Biao","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Chang","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Chen","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Chu","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Jian","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Liangyuan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Rui","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Shengtao","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Xiaomin","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Xinzhe","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Xunxun","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Yeru","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0006-0032-8059","authenticated-orcid":false,"given":"Zuolin","family":"Wei","sequence":"additional","affiliation":[]},{"given":"Jiazhou","family":"Wei","sequence":"additional","affiliation":[]},{"given":"Dachao","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Gang","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Jin","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Yulin","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Shiyong","family":"Xie","sequence":"additional","affiliation":[]},{"given":"Lianjie","family":"Xin","sequence":"additional","affiliation":[]},{"given":"Yu","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Chun","family":"Xue","sequence":"additional","affiliation":[]},{"given":"Kai","family":"Yan","sequence":"additional","affiliation":[]},{"given":"Weifeng","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Xinpeng","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Yang","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Yangsen","family":"Ye","sequence":"additional","affiliation":[]},{"given":"Zhenping","family":"Ye","sequence":"additional","affiliation":[]},{"given":"Chong","family":"Ying","sequence":"additional","affiliation":[]},{"given":"Jiale","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Qinjing","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Wenhu","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Xiangdong","family":"Zeng","sequence":"additional","affiliation":[]},{"given":"Chen","family":"Zha","sequence":"additional","affiliation":[]},{"given":"Shaoyu","family":"Zhan","sequence":"additional","affiliation":[]},{"given":"Feifei","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Haibin","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Kaili","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Wen","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Yiming","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Yongzhuo","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Lixiang","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Guming","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Peng","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Xintao","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Youwei","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Zhong","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Luyuan","family":"Zheng","sequence":"additional","affiliation":[]},{"given":"Fei","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Na","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Shifeng","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Shuang","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Zhengxiao","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Chengjun","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Qingling","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Guihong","family":"Zou","sequence":"additional","affiliation":[]},{"given":"Haonan","family":"Zou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3482-3091","authenticated-orcid":false,"given":"Qiang","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8227-9177","authenticated-orcid":false,"given":"Chao-Yang","family":"Lu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4753-5243","authenticated-orcid":false,"given":"Cheng-Zhi","family":"Peng","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0205-6545","authenticated-orcid":false,"given":"Xiao","family":"Yuan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5534-2177","authenticated-orcid":false,"given":"Ming","family":"Gong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5974-9452","authenticated-orcid":false,"given":"Xiaobo","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6100-5142","authenticated-orcid":false,"given":"Jian-Wei","family":"Pan","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,3]]},"URL":"http://dx.doi.org/10.1038/s41567-026-03179-6","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:3179\narxivid:2505.01978"},{"id":"arxiv:1008.2048","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.82.060301","source":"Crossref","title":"Topological one-way quantum computation on verified logical cluster states","volume":"82","author":[{"given":"Keisuke","family":"Fujii","sequence":"first","affiliation":[]},{"given":"Katsuji","family":"Yamamoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,12,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.82.060301","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"060301","note":"arxivid:1008.2048"},{"id":"arxiv:1008.3752","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/physrevlett.105.250503","source":"Crossref","title":"Fault-Tolerant Topological One-Way Quantum Computation with Probabilistic Two-Qubit Gates","volume":"105","author":[{"given":"Keisuke","family":"Fujii","sequence":"first","affiliation":[]},{"given":"Yuuki","family":"Tokunaga","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,12,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.105.250503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"250503","note":"arxivid:1008.3752"},{"id":"arxiv:quant-ph/0507036","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.97.120501","source":"Crossref","title":"Loss Tolerance in One-Way Quantum Computation via Counterfactual Error Correction","volume":"97","author":[{"given":"Michael","family":"Varnava","sequence":"first","affiliation":[]},{"given":"Daniel E.","family":"Browne","sequence":"additional","affiliation":[]},{"given":"Terry","family":"Rudolph","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,9,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.97.120501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120501","note":"arxivid:quant-ph/0507036"},{"id":"arxiv:1309.7207","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum communication holds promise for unconditionally secure transmission of secret messages and faithful transfer of unknown quantum states. Photons appear to be the medium of choice for quantum communication. Owing to photon losses, robust quantum communication over long lossy channels requires quantum repeaters. It is widely believed that a necessary and highly demanding requirement for quantum repeaters is the existence of matter quantum memories. Here we show that such a requirement is, in fact, unnecessary by introducing the concept of all-photonic quantum repeaters based on flying qubits. In particular, we present a protocol based on photonic cluster-state machine guns and a loss-tolerant measurement equipped with local high-speed active feedforwards. We show that, with such all-photonic quantum repeaters, the communication efficiency scales polynomially with the channel distance. Our result paves a new route towards quantum repeaters with efficient single-photon sources rather than matter quantum memories.</jats:p>","DOI":"10.1038/ncomms7787","source":"Crossref","title":"All-photonic quantum repeaters","volume":"6","author":[{"given":"Koji","family":"Azuma","sequence":"first","affiliation":[]},{"given":"Kiyoshi","family":"Tamaki","sequence":"additional","affiliation":[]},{"given":"Hoi-Kwong","family":"Lo","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,15]]},"URL":"http://dx.doi.org/10.1038/ncomms7787","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"6787","note":"alternative-id:BFncomms7787\narxivid:1309.7207"},{"id":"arxiv:1603.01353","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.95.012304","source":"Crossref","title":"Rate-distance tradeoff and resource costs for all-optical quantum repeaters","volume":"95","author":[{"given":"Mihir","family":"Pant","sequence":"first","affiliation":[]},{"given":"Hari","family":"Krovi","sequence":"additional","affiliation":[]},{"given":"Dirk","family":"Englund","sequence":"additional","affiliation":[]},{"given":"Saikat","family":"Guha","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,1,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.95.012304","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012304","note":"arxivid:1603.01353"},{"id":"arxiv:2503.14003","type":"article-journal","author":[{"given":"Róbert","family":"Trényi"},{"given":"Simeon","family":"Ball"},{"given":"David G.","family":"Glynn"},{"given":"Marcos","family":"Curty"}],"title":"Geometrical constructions of purity testing protocols and their applications to quantum communication","issued":{"date-parts":[[2025,3,18]]},"note":"arxivid:2503.14003\narxiv_version_number:1"},{"id":"arxiv:0801.1544","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/focs.2006.68","page":"249-260","source":"Crossref","title":"Secure Multiparty Quantum Computation with (Only) a Strict Honest Majority","author":[{"given":"Michael","family":"Ben-Or","sequence":"first","affiliation":[]},{"given":"Claude","family":"Crepeau","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[]},{"given":"Avinatan","family":"Hassidim","sequence":"additional","affiliation":[]},{"given":"Adam","family":"Smith","sequence":"additional","affiliation":[]}],"event":"2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)","container-title":"2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)","original-title":[],"issued":{"date-parts":[[2006]]},"URL":"http://dx.doi.org/10.1109/FOCS.2006.68","note":"arxivid:0801.1544"},{"id":"arxiv:2009.03921","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3406325.3451005","page":"1276-1288","source":"Crossref","title":"Fiber bundle codes: breaking the\n            <i>n</i>\n            <sup>1/2</sup>\n            polylog(\n            <i>n</i>\n            ) barrier for Quantum LDPC codes","author":[{"given":"Matthew B.","family":"Hastings","sequence":"first","affiliation":[{"name":"Station Q, USA / Microsoft Quantum, USA"}]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[{"name":"Station Q, USA / Microsoft Quantum, USA"}]},{"given":"Ryan","family":"O'Donnell","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University, USA"}]}],"event":"STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing","container-title":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[2021,6,15]]},"URL":"http://dx.doi.org/10.1145/3406325.3451005","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3406325.3451005\n10.1145/3406325\narxivid:2009.03921"},{"id":"arxiv:2012.09271","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2021.3097347","page":"6653-6674","source":"Crossref","title":"Balanced Product Quantum Codes","volume":"67","author":[{"ORCID":"https://orcid.org/0000-0002-7211-5515","authenticated-orcid":false,"given":"Nikolas P.","family":"Breuckmann","sequence":"first","affiliation":[]},{"given":"Jens N.","family":"Eberhardt","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2021,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3097347","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2012.09271"},{"id":"arxiv:2308.07915","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8005","abstract":"<jats:title>Abstract</jats:title><jats:p>The accumulation of physical errors<jats:sup>1–3</jats:sup> prevents the execution of large-scale algorithms in current quantum computers. Quantum error correction<jats:sup>4</jats:sup> promises a solution by encoding <jats:italic>k</jats:italic> logical qubits onto a larger number <jats:italic>n</jats:italic> of physical qubits, such that the physical errors are suppressed enough to allow running a desired computation with tolerable fidelity. Quantum error correction becomes practically realizable once the physical error rate is below a threshold value that depends on the choice of quantum code, syndrome measurement circuit and decoding algorithm<jats:sup>5</jats:sup>. We present an end-to-end quantum error correction protocol that implements fault-tolerant memory on the basis of a family of low-density parity-check codes<jats:sup>6</jats:sup>. Our approach achieves an error threshold of 0.7% for the standard circuit-based noise model, on par with the surface code<jats:sup>7–10</jats:sup> that for 20 years was the leading code in terms of error threshold. The syndrome measurement cycle for a length-<jats:italic>n</jats:italic> code in our family requires <jats:italic>n</jats:italic> ancillary qubits and a depth-8 circuit with CNOT gates, qubit initializations and measurements. The required qubit connectivity is a degree-6 graph composed of two edge-disjoint planar subgraphs. In particular, we show that 12 logical qubits can be preserved for nearly 1 million syndrome cycles using 288 physical qubits in total, assuming the physical error rate of 0.1%, whereas the surface code would require nearly 3,000 physical qubits to achieve said performance. Our findings bring demonstrations of a low-overhead fault-tolerant quantum memory within the reach of near-term quantum processors.</jats:p>","DOI":"10.1038/s41586-024-07107-7","page":"778-782","source":"Crossref","title":"High-threshold and low-overhead fault-tolerant quantum memory","volume":"627","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9786-8196","authenticated-orcid":false,"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]},{"given":"Jay M.","family":"Gambetta","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7381-4556","authenticated-orcid":false,"given":"Dmitri","family":"Maslov","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Rall","sequence":"additional","affiliation":[]},{"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2024,3,27]]},"URL":"http://dx.doi.org/10.1038/s41586-024-07107-7","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:7107\narxivid:2308.07915"},{"id":"arxiv:2503.22071","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>\n                    We propose a model for quantum computing with long chains of trapped ions and we design quantum error correction schemes for this model. The main components of a quantum error correction scheme are the quantum code and a quantum circuit called the syndrome extraction circuit, which is executed to perform error correction with this code. In this work, we design syndrome extraction circuits tailored to our ion chain model, a syndrome extraction tuning protocol to optimize these circuits, and we construct new quantum codes that outperform the state-of-the-art for chains of about\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mn>50</mml:mn>\n                    </mml:math>\n                    qubits. To establish a baseline under the ion chain model, we simulate the performance of surface codes and bivariate bicycle (BB) codes equipped with our optimized syndrome extraction circuits. Then, we propose a new variant of BB codes defined by weight-five measurements, that we refer to as BB5 codes and we identify BB5 codes that achieve a better minimum distance than any BB codes with the same number of logical qubits and data qubits, such as a\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mo stretchy=\"false\">[</mml:mo>\n                      <mml:mo stretchy=\"false\">[</mml:mo>\n                      <mml:mn>48</mml:mn>\n                      <mml:mo>,</mml:mo>\n                      <mml:mn>4</mml:mn>\n                      <mml:mo>,</mml:mo>\n                      <mml:mn>7</mml:mn>\n                      <mml:mo stretchy=\"false\">]</mml:mo>\n                      <mml:mo stretchy=\"false\">]</mml:mo>\n                    </mml:math>\n                    BB5 code. For a physical error rate of\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:msup>\n                        <mml:mn>10</mml:mn>\n                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                          <mml:mo>&amp;#x2212;</mml:mo>\n                          <mml:mn>3</mml:mn>\n                        </mml:mrow>\n                      </mml:msup>\n                    </mml:math>\n                    , the\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mo stretchy=\"false\">[</mml:mo>\n                      <mml:mo stretchy=\"false\">[</mml:mo>\n                      <mml:mn>48</mml:mn>\n                      <mml:mo>,</mml:mo>\n                      <mml:mn>4</mml:mn>\n                      <mml:mo>,</mml:mo>\n                      <mml:mn>7</mml:mn>\n                      <mml:mo stretchy=\"false\">]</mml:mo>\n                      <mml:mo stretchy=\"false\">]</mml:mo>\n                    </mml:math>\n                    BB5 code achieves a logical error rate per logical qubit of\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mn>5</mml:mn>\n                      <mml:mo>&amp;#x22C5;</mml:mo>\n                      <mml:msup>\n                        <mml:mn>10</mml:mn>\n                        <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                          <mml:mo>&amp;#x2212;</mml:mo>\n                          <mml:mn>5</mml:mn>\n                        </mml:mrow>\n                      </mml:msup>\n                    </mml:math>\n                    , which is four times smaller than the best BB code in our baseline family. It also achieves the same logical error rate per logical qubit as the distance-7 surface code but using four times fewer physical qubits per logical qubit.\n                  </jats:p>","DOI":"10.22331/q-2025-11-27-1920","page":"1920","source":"Crossref","title":"Quantum error correction for long chains of trapped ions","volume":"9","author":[{"given":"Min","family":"Ye","sequence":"first","affiliation":[{"name":"IonQ Inc."}]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[{"name":"IonQ Inc."}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,27]]},"URL":"http://dx.doi.org/10.22331/q-2025-11-27-1920","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2503.22071"},{"id":"arxiv:2411.03302","type":"article-journal","author":[{"given":"Ryan","family":"Tiew"},{"given":"Nikolas P.","family":"Breuckmann"}],"title":"Low-Overhead Entangling Gates from Generalised Dehn Twists","issued":{"date-parts":[[2024,11,5]]},"note":"arxivid:2411.03302\narxiv_version_number:1"},{"id":"arxiv:2407.16336","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.134.090602","source":"Crossref","title":"Lowering Connectivity Requirements for Bivariate Bicycle Codes Using Morphing Circuits","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0002-0776-886X","authenticated-orcid":true,"given":"Mackenzie H.","family":"Shaw","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04wf30j82","id-type":"ROR","asserted-by":"publisher"}],"name":"QuTech"},{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"}]},{"ORCID":"https://orcid.org/0000-0003-0218-6614","authenticated-orcid":true,"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04wf30j82","id-type":"ROR","asserted-by":"publisher"}],"name":"QuTech"},{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.090602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090602","note":"arxivid:2407.16336"},{"id":"arxiv:2407.18393","type":"article-journal","author":[{"given":"Andrew W.","family":"Cross"},{"given":"Zhiyang","family":"He"},{"given":"Patrick J.","family":"Rall"},{"given":"Theodore J.","family":"Yoder"}],"title":"Improved QLDPC Surgery: Logical Measurements and Bridging Codes","issued":{"date-parts":[[2025,10,27]]},"note":"arxivid:2407.18393\narxiv_version_number:4"},{"id":"arxiv:2403.18901","type":"article-journal","author":[{"given":"Anqi","family":"Gong"},{"given":"Sebastian","family":"Cammerer"},{"given":"Joseph M.","family":"Renes"}],"title":"Toward Low-latency Iterative Decoding of QLDPC Codes Under Circuit-Level Noise","issued":{"date-parts":[[2024,3,27]]},"note":"arxivid:2403.18901\narxiv_version_number:1"},{"id":"arxiv:2406.14527","type":"article-journal","author":[{"given":"Stasiu","family":"Wolanski"},{"given":"Ben","family":"Barber"}],"title":"Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes","issued":{"date-parts":[[2025,1,3]]},"note":"arxivid:2406.14527\narxiv_version_number:2"},{"id":"arxiv:2504.13043","type":"article-journal","author":[{"given":"John","family":"Blue"},{"given":"Harshil","family":"Avlani"},{"given":"Zhiyang","family":"He"},{"given":"Liu","family":"Ziyin"},{"given":"Isaac L.","family":"Chuang"}],"title":"Machine Learning Decoding of Circuit-Level Noise for Bivariate Bicycle Codes","issued":{"date-parts":[[2025,4,17]]},"note":"arxivid:2504.13043\narxiv_version_number:1"},{"id":"arxiv:2506.03094","type":"article-journal","author":[{"given":"Theodore J.","family":"Yoder"},{"given":"Eddie","family":"Schoute"},{"given":"Patrick","family":"Rall"},{"given":"Emily","family":"Pritchett"},{"given":"Jay M.","family":"Gambetta"},{"given":"Andrew W.","family":"Cross"},{"given":"Malcolm","family":"Carroll"},{"given":"Michael E.","family":"Beverland"}],"title":"Tour de gross: A modular quantum computer based on bivariate bicycle codes","issued":{"date-parts":[[2025,6,3]]},"note":"arxivid:2506.03094\narxiv_version_number:1"},{"id":"arxiv:2510.21600","type":"article-journal","author":[{"given":"Thilo","family":"Maurer"},{"given":"Markus","family":"Bühler"},{"given":"Michael","family":"Kröner"},{"given":"Frank","family":"Haverkamp"},{"given":"Tristan","family":"Müller"},{"given":"Drew","family":"Vandeth"},{"given":"Blake R.","family":"Johnson"}],"title":"Real-time decoding of the gross code memory with FPGAs","issued":{"date-parts":[[2025,10,24]]},"note":"arxivid:2510.21600\narxiv_version_number:1"},{"id":"arxiv:2411.03202","type":"article-journal","author":[{"given":"Samuel","family":"Stein"},{"given":"Shifan","family":"Xu"},{"given":"Andrew W.","family":"Cross"},{"given":"Theodore J.","family":"Yoder"},{"given":"Ali","family":"Javadi-Abhari"},{"given":"Chenxu","family":"Liu"},{"given":"Kun","family":"Liu"},{"given":"Zeyuan","family":"Zhou"},{"given":"Charles","family":"Guinn"},{"given":"Yufei","family":"Ding"},{"given":"Yongshan","family":"Ding"},{"given":"Ang","family":"Li"}],"title":"Architectures for Heterogeneous Quantum Error Correction Codes","issued":{"date-parts":[[2024,11,10]]},"note":"arxivid:2411.03202\narxiv_version_number:3"},{"id":"arxiv:2404.18809","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.111.022433","source":"Crossref","title":"Architecture for fast implementation of quantum low-density parity-check codes with optimized Rydberg gates","volume":"111","author":[{"given":"C.","family":"Poole","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01y2jtd41","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Wisconsin-Madison"}]},{"given":"T. M.","family":"Graham","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01y2jtd41","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Wisconsin-Madison"}]},{"ORCID":"https://orcid.org/0000-0002-9316-1596","authenticated-orcid":true,"given":"M. A.","family":"Perlin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049ykdj13","id-type":"ROR","asserted-by":"publisher"}],"name":"Infleqtion, Inc."}]},{"given":"M.","family":"Otten","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01y2jtd41","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Wisconsin-Madison"}]},{"ORCID":"https://orcid.org/0000-0001-6398-2097","authenticated-orcid":true,"given":"M.","family":"Saffman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01y2jtd41","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Wisconsin-Madison"},{"id":[{"id":"https://ror.org/049ykdj13","id-type":"ROR","asserted-by":"publisher"}],"name":"Infleqtion, Inc."}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.111.022433","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022433","note":"arxivid:2404.18809"},{"id":"arxiv:2510.05211","type":"article-journal","author":[{"given":"Zijian","family":"Liang"},{"given":"Yu-An","family":"Chen"}],"title":"Self-dual bivariate bicycle codes with transversal Clifford gates","issued":{"date-parts":[[2026,1,11]]},"note":"arxivid:2510.05211\narxiv_version_number:2"},{"id":"arxiv:2510.06159","type":"article-journal","author":[{"given":"Qian","family":"Xu"},{"given":"Hengyun","family":"Zhou"},{"given":"Dolev","family":"Bluvstein"},{"given":"Madelyn","family":"Cain"},{"given":"Marcin","family":"Kalinowski"},{"given":"John","family":"Preskill"},{"given":"Mikhail D.","family":"Lukin"},{"given":"Nishad","family":"Maskara"}],"title":"Batched high-rate logical operations for quantum LDPC codes","issued":{"date-parts":[[2025,10,7]]},"note":"arxivid:2510.06159\narxiv_version_number:1"},{"id":"arxiv:2406.19151","type":"article-journal","author":[{"given":"Lukas","family":"Voss"},{"given":"Sim Jian","family":"Xian"},{"given":"Tobias","family":"Haug"},{"given":"Kishor","family":"Bharti"}],"title":"Multivariate Bicycle Codes","issued":{"date-parts":[[2025,2,20]]},"note":"arxivid:2406.19151\narxiv_version_number:4"},{"id":"arxiv:2408.10001","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>\n                    Quantum computing is deemed to require error correction at scale to mitigate physical noise by reducing it to lower noise levels while operating on encoded logical qubits. Popular quantum error correction schemes include CSS code, of which surface codes provide regular mappings onto 2D planes suitable for contemporary quantum devices together with known transversal logical gates. Recently, qLDPC codes have been proposed as a means to provide denser encoding with the class of bivariate bicycle (BB) codes promising feasible design for devices.This work contributes a novel subclass of BB codes suitable for quantum error correction. This subclass employs\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mi>c</mml:mi>\n                      <mml:mi>o</mml:mi>\n                      <mml:mi>p</mml:mi>\n                      <mml:mi>r</mml:mi>\n                      <mml:mi>i</mml:mi>\n                      <mml:mi>m</mml:mi>\n                      <mml:mi>e</mml:mi>\n                      <mml:mi>s</mml:mi>\n                    </mml:math>\n                    and the product\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mi>x</mml:mi>\n                      <mml:mi>y</mml:mi>\n                    </mml:math>\n                    of the two generating variables\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mi>x</mml:mi>\n                    </mml:math>\n                    and\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mi>y</mml:mi>\n                    </mml:math>\n                    to construct polynomials, rather than using\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mi>x</mml:mi>\n                    </mml:math>\n                    and\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mi>y</mml:mi>\n                    </mml:math>\n                    separately as in vanilla BB codes. In contrast to vanilla BB codes, where parameters remain unknown prior to code discovery, the rate of the proposed code can be determined beforehand by specifying a factor polynomial as an input to the numerical search algorithm. Using this coprime-BB construction, we found a number of surprisingly short to medium-length codes that were previously unknown. We also propose a layout on cold atom arrays tailored for coprime-BB codes. The proposed layout reduces both move time for short to medium-length codes and the number of moves of atoms to perform syndrome extractions. We consider an error model with global laser noise on cold atoms, and simulations show that our proposed layout achieves significant improvements over prior work across the simulated codes.\n                  </jats:p>","DOI":"10.22331/q-2026-02-23-2009","page":"2009","source":"Crossref","title":"Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms","volume":"10","author":[{"given":"Ming","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Computer Science, North Carolina State University, Raleigh, North Carolina, USA"}]},{"given":"Frank","family":"Mueller","sequence":"additional","affiliation":[{"name":"Department of Computer Science, North Carolina State University, Raleigh, North Carolina, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,2,23]]},"URL":"http://dx.doi.org/10.22331/q-2026-02-23-2009","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2408.10001"},{"id":"arxiv:2602.20158","type":"article-journal","author":[{"given":"Zijian","family":"Liang"},{"given":"Yu-An","family":"Chen"}],"title":"Generalized $\\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes","issued":{"date-parts":[[2026,2,23]]},"note":"arxivid:2602.20158\narxiv_version_number:1"},{"id":"arxiv:0803.0272","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.80.052312","source":"Crossref","title":"High-threshold universal quantum computation on the surface code","volume":"80","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Ashley M.","family":"Stephens","sequence":"additional","affiliation":[]},{"given":"Peter","family":"Groszkowski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,11,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.052312","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052312","note":"arxivid:0803.0272"},{"id":"arxiv:2404.17676","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Geometric locality is an important theoretical and practical factor for quantum low-density parity-check (qLDPC) codes that affects code performance and ease of physical realization. For device architectures restricted to two-dimensional (2D) local gates, naively implementing the high-rate codes suitable for low-overhead fault-tolerant quantum computing incurs prohibitive overhead. In this work, we present an error-correction protocol built on a bilayer architecture that aims to reduce operational overheads when restricted to 2D local gates by measuring some generators less frequently than others. We investigate the family of bivariate-bicycle qLDPC codes and show that they are well suited for a parallel syndrome-measurement scheme using fast routing with local operations and classical communication (LOCC). Through circuit-level simulations, we find that in some parameter regimes, bivariate-bicycle codes implemented with this protocol have logical error rates comparable to the surface code while using fewer physical qubits.</jats:p>","DOI":"10.1103/prxquantum.6.010306","source":"Crossref","title":"Toward a 2D Local Implementation of Quantum Low-Density Parity-Check Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-7586-2786","authenticated-orcid":true,"given":"Noah","family":"Berthusen","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology (NIST)"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0002-2612-308X","authenticated-orcid":true,"given":"Dhruv","family":"Devulapalli","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology (NIST)"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0002-5613-1443","authenticated-orcid":true,"given":"Eddie","family":"Schoute","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01e41cf67","id-type":"ROR","asserted-by":"publisher"}],"name":"Los Alamos National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-9903-837X","authenticated-orcid":true,"given":"Andrew M.","family":"Childs","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology (NIST)"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"College Park"}]},{"ORCID":"https://orcid.org/0000-0003-3974-2987","authenticated-orcid":true,"given":"Michael J.","family":"Gullans","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology (NIST)"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0003-0509-3421","authenticated-orcid":true,"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology (NIST)"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/04xz38214","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Quantum Institute"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology (NIST)"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0009-0002-2948-3395","authenticated-orcid":true,"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology (NIST)"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"College Park"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,9]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.010306","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010306","note":"arxivid:2404.17676"},{"id":"arxiv:2312.08813","type":"article-journal","author":[{"given":"Craig","family":"Gidney"},{"given":"Cody","family":"Jones"}],"title":"New circuits and an open source decoder for the color code","issued":{"date-parts":[[2023,12,14]]},"note":"arxivid:2312.08813\narxiv_version_number:1"},{"id":"arxiv:2503.01738","type":"article-journal","author":[{"given":"Stergios","family":"Koutsioumpas"},{"given":"Hasan","family":"Sayginel"},{"given":"Mark","family":"Webster"},{"family":"Dan E Browne","given":""}],"title":"Automorphism Ensemble Decoding of Quantum LDPC Codes","issued":{"date-parts":[[2025,3,3]]},"note":"arxivid:2503.01738\narxiv_version_number:1"},{"id":"arxiv:2602.22770","type":"article-journal","author":[{"given":"Kaavya","family":"Sahay"},{"given":"Dominic J.","family":"Williamson"},{"given":"Benjamin J.","family":"Brown"}],"title":"A matching decoder for bivariate bicycle codes","issued":{"date-parts":[[2026,2,26]]},"note":"arxivid:2602.22770\narxiv_version_number:1"},{"id":"arxiv:2407.03973","type":"article-journal","author":[{"given":"Jens Niklas","family":"Eberhardt"},{"given":"Vincent","family":"Steffan"}],"title":"Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes","issued":{"date-parts":[[2024,7,4]]},"note":"arxivid:2407.03973\narxiv_version_number:1"},{"id":"arxiv:2505.09684","type":"journal-article","publisher":"Springer Science and Business Media LLC","DOI":"10.1038/s41567-025-03157-4","source":"Crossref","title":"Demonstration of low-overhead quantum error correction codes","author":[{"ORCID":"https://orcid.org/0000-0003-0307-5457","authenticated-orcid":false,"given":"Ke","family":"Wang","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6087-5138","authenticated-orcid":false,"given":"Zhide","family":"Lu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4710-1752","authenticated-orcid":false,"given":"Chuanyu","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Gongyu","family":"Liu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0006-6444-2416","authenticated-orcid":false,"given":"Jiachen","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Yanzhe","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1137-4660","authenticated-orcid":false,"given":"Yaozu","family":"Wu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8546-4255","authenticated-orcid":false,"given":"Shibo","family":"Xu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1318-8367","authenticated-orcid":false,"given":"Xuhao","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0789-8207","authenticated-orcid":false,"given":"Feitong","family":"Jin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4138-8983","authenticated-orcid":false,"given":"Yu","family":"Gao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0001-8858-2193","authenticated-orcid":false,"given":"Ziqi","family":"Tan","sequence":"additional","affiliation":[]},{"given":"Zhengyi","family":"Cui","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-3884-9997","authenticated-orcid":false,"given":"Ning","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0008-2838-9552","authenticated-orcid":false,"given":"Yiren","family":"Zou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0002-0599-5361","authenticated-orcid":false,"given":"Aosai","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6589-3706","authenticated-orcid":false,"given":"Tingting","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0003-5289-419X","authenticated-orcid":false,"given":"Fanhao","family":"Shen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0001-3631-6774","authenticated-orcid":false,"given":"Jiarun","family":"Zhong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0007-9164-714X","authenticated-orcid":false,"given":"Zehang","family":"Bao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8219-2311","authenticated-orcid":false,"given":"Zitian","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Yihang","family":"Han","sequence":"additional","affiliation":[]},{"given":"Yiyang","family":"He","sequence":"additional","affiliation":[]},{"given":"Jiayuan","family":"Shen","sequence":"additional","affiliation":[]},{"given":"Han","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Jia-Nan","family":"Yang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1089-9251","authenticated-orcid":false,"given":"Zixuan","family":"Song","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4076-1608","authenticated-orcid":false,"given":"Jinfeng","family":"Deng","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1852-4094","authenticated-orcid":false,"given":"Hang","family":"Dong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7370-4525","authenticated-orcid":false,"given":"Zheng-Zhi","family":"Sun","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7137-5390","authenticated-orcid":false,"given":"Weikang","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0002-5606-8824","authenticated-orcid":false,"given":"Qi","family":"Ye","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3441-4140","authenticated-orcid":false,"given":"Si","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Yixuan","family":"Ma","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2761-5083","authenticated-orcid":false,"given":"Pei-Xin","family":"Shen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8908-4228","authenticated-orcid":false,"given":"Pengfei","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3914-4979","authenticated-orcid":false,"given":"Hekang","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1093-3405","authenticated-orcid":false,"given":"Qiujiang","family":"Guo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6369-8010","authenticated-orcid":false,"given":"Zhen","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1853-9392","authenticated-orcid":false,"given":"Chao","family":"Song","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5380-4297","authenticated-orcid":false,"given":"H.","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1042-4646","authenticated-orcid":false,"given":"Dong-Ling","family":"Deng","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,22]]},"URL":"http://dx.doi.org/10.1038/s41567-025-03157-4","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:3157\narxivid:2505.09684"},{"id":"arxiv:1206.2656","type":"article-journal","author":[{"given":"Alain","family":"Couvreur"},{"given":"Nicolas","family":"Delfosse"},{"given":"Gilles","family":"Zémor"}],"title":"A Construction of Quantum LDPC Codes from Cayley Graphs","issued":{"date-parts":[[2013,12,17]]},"note":"arxivid:1206.2656\narxiv_version_number:3"},{"id":"arxiv:quant-ph/0304161","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2004.834737","page":"2315-2330","source":"Crossref","title":"Sparse-Graph Codes for Quantum Error Correction","volume":"50","author":[{"given":"D.J.C.","family":"MacKay","sequence":"first","affiliation":[]},{"given":"G.","family":"Mitchison","sequence":"additional","affiliation":[]},{"given":"P.L.","family":"McFadden","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2004,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2004.834737","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0304161"},{"id":"arxiv:2206.06557","type":"article-journal","author":[{"given":"Shouzhen","family":"Gu"},{"given":"Christopher A.","family":"Pattison"},{"given":"Eugene","family":"Tang"}],"title":"An efficient decoder for a linear distance quantum LDPC code","issued":{"date-parts":[[2022,6,14]]},"note":"arxivid:2206.06557\narxiv_version_number:1"},{"id":"arxiv:2401.02911","type":"journal-article","publisher":"IOP Publishing","issue":"4","abstract":"<jats:title>Abstract</jats:title><jats:p>We use the recently introduced lifted product to construct a family of quantum low density parity check codes (QLDPC codes). The codes we obtain can be viewed as stacks of surface codes that are interconnected, leading to the name lift-connected surface (LCS) codes. LCS codes offer a wide range of parameters—a particularly striking feature is that they show interesting properties that are favorable compared to the standard surface code. For example, already at moderate numbers of physical qubits in the order of tens, LCS codes of equal size have lower logical error rate or similarly, require fewer qubits for a fixed target logical error rate. We present and analyze the construction and provide numerical simulation results for the logical error rate under code capacity and phenomenological noise. These results show that LCS codes attain thresholds that are comparable to corresponding (non-connected) copies of surface codes, while the logical error rate can be orders of magnitude lower, even for representatives with the same parameters. This provides a code family showing the potential of modern product constructions at already small qubit numbers. Their amenability to 3D-local connectivity renders them particularly relevant for near-term implementations.</jats:p>","DOI":"10.1088/2058-9565/ad5eb6","page":"045012","source":"Crossref","title":"Lift-connected surface codes","volume":"9","author":[{"ORCID":"https://orcid.org/0009-0002-8811-1401","authenticated-orcid":true,"given":"Josias","family":"Old","sequence":"first","affiliation":[]},{"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":false,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2024,7,17]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ad5eb6","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2401.02911"},{"id":"arxiv:2008.09495","type":"article-journal","author":[{"given":"Tali","family":"Kaufman"},{"given":"Ran J.","family":"Tessler"}],"title":"New Cosystolic Expanders from Tensors Imply Explicit Quantum LDPC Codes with $Ω(\\sqrt{n}\\log^kn)$ Distance","issued":{"date-parts":[[2020,11,16]]},"note":"arxivid:2008.09495\narxiv_version_number:2"},{"id":"arxiv:2507.16797","type":"article-journal","author":[{"given":"Esther Xiaozhen","family":"Fu"},{"given":"Han","family":"Zheng"},{"given":"Zimu","family":"Li"},{"given":"Zi-Wen","family":"Liu"}],"title":"No-go theorems for logical gates on product quantum codes","issued":{"date-parts":[[2025,10,16]]},"note":"arxivid:2507.16797\narxiv_version_number:2"},{"id":"arxiv:2510.06760","type":"article-journal","author":[{"given":"Louis","family":"Golowich"},{"given":"Kathleen","family":"Chang"},{"given":"Guanyu","family":"Zhu"}],"title":"Constant-Overhead Addressable Gates via Single-Shot Code Switching","issued":{"date-parts":[[2025,10,8]]},"note":"arxivid:2510.06760\narxiv_version_number:1"},{"id":"arxiv:1311.0885","type":"article-journal","author":[{"given":"Sergey","family":"Bravyi"},{"given":"Matthew B.","family":"Hastings"}],"title":"Homological Product Codes","issued":{"date-parts":[[2013,11,4]]},"note":"arxivid:1311.0885\narxiv_version_number:1"},{"id":"arxiv:1807.09783","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.</jats:p>","DOI":"10.22331/q-2019-02-04-120","page":"120","source":"Crossref","title":"Fault-tolerant gates via homological product codes","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-8265-4774","authenticated-orcid":false,"given":"Tomas","family":"Jochym-O'Connor","sequence":"first","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics, Institute for Quantum Information & Matter, California Institute of Technology, Pasadena, CA 91125, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,2,4]]},"URL":"http://dx.doi.org/10.22331/q-2019-02-04-120","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1807.09783"},{"id":"arxiv:2009.14226","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.</jats:p>","DOI":"10.22331/q-2021-03-10-406","page":"406","source":"Crossref","title":"Union-Find Decoders For Homological Product Codes","volume":"5","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Matthew B.","family":"Hastings","sequence":"additional","affiliation":[{"name":"Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA"},{"name":"Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,3,10]]},"URL":"http://dx.doi.org/10.22331/q-2021-03-10-406","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2009.14226"},{"id":"arxiv:2204.10812","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The promise of high-rate low-density parity check (LDPC) codes to substantially reduce the overhead of fault-tolerant quantum computation depends on constructing efficient, fault-tolerant implementations of logical gates on such codes. Transversal gates are the simplest type of fault-tolerant gate, but the potential of transversal gates on LDPC codes has hitherto been largely neglected. We investigate the transversal gates that can be implemented in hypergraph product codes, a class of LDPC codes. Our analysis is aided by the construction of a symplectic canonical basis for the logical operators of hypergraph product codes, a result that may be of independent interest. We show that in these codes transversal gates can implement Hadamard (up to logical SWAP gates) and control-Z on all logical qubits. Moreover, we show that sequences of transversal operations, interleaved with error correction, allow implementation of entangling gates between arbitrary pairs of logical qubits in the same code block. We thereby demonstrate that transversal gates can be used as the basis for universal quantum computing on LDPC codes, when supplemented with state injection.</jats:p>","DOI":"10.22331/q-2023-10-24-1153","page":"1153","source":"Crossref","title":"Partitioning qubits in hypergraph product codes to implement logical gates","volume":"7","author":[{"given":"Armanda O.","family":"Quintavalle","sequence":"first","affiliation":[{"name":"Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom"},{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Paul","family":"Webster","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"Michael","family":"Vasmer","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,10,24]]},"URL":"http://dx.doi.org/10.22331/q-2023-10-24-1153","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2204.10812"},{"id":"arxiv:2601.18879","type":"article-journal","author":[{"given":"Feroz Ahmed","family":"Mian"},{"given":"Owen","family":"Gwilliam"},{"given":"Stefan","family":"Krastanov"}],"title":"Multivariate Multicycle Codes for Complete Single-Shot Decoding","issued":{"date-parts":[[2026,1,26]]},"note":"arxivid:2601.18879\narxiv_version_number:1"},{"id":"arxiv:2410.05171","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum error correction is a fundamental primitive of fault-tolerant quantum computing. But in order for error correction to proceed, one must first prepare the codespace of the underlying error-correcting code. A popular method for encoding quantum low-density parity-check codes is transversal initialization, where one begins in a product state and measures a set of stabilizer generators. In the presence of measurement errors however, this procedure is generically not fault-tolerant, and so one typically needs to repeat the measurements many times, resulting in a deep initialization circuit. We present a protocol that prepares the codespace of constant-rate hypergraph product codes in constant depth with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt><mml:mo stretchy=\"false\">)</mml:mo></mml:math> spatial overhead, and we show that the protocol is robust even in the presence of measurement errors. Our construction is inspired by dimension-jumping in topological codes and leverages two properties that arise from the homological product of codes. We provide some improvements to lower the spatial overhead and discuss applications to fault-tolerant architectures.</jats:p>","DOI":"10.22331/q-2025-10-07-1879","page":"1879","source":"Crossref","title":"Single-shot preparation of hypergraph product codes via dimension jump","volume":"9","author":[{"given":"Yifan","family":"Hong","sequence":"first","affiliation":[{"name":"Joint Quantum Institute & Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD 20742, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,7]]},"URL":"http://dx.doi.org/10.22331/q-2025-10-07-1879","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2410.05171"},{"id":"arxiv:1909.07424","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevx.11.011023","source":"Crossref","title":"Fault-Tolerant Gates on Hypergraph Product Codes","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0002-2027-6448","authenticated-orcid":true,"given":"Anirudh","family":"Krishna","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2021,2,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.11.011023","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011023","note":"arxivid:1909.07424"},{"id":"arxiv:2411.17050","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Starting with an explicit framework for designing logical Clifford circuits for CSS codes, we construct logical gates for Hypergraph Product Codes. We first derive symplectic matrices for CNOT, CZ, Phase, and Hadamard operators, which together generate the Clifford group. This enables us to design explicit transformations that result in targeted logical gates for arbitrary codes in this family. As a concrete example, we give logical circuits for the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>18</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">]</mml:mo></mml:math> toric code.</jats:p>","DOI":"10.22331/q-2025-08-29-1842","page":"1842","source":"Crossref","title":"Targeted Clifford logical gates for hypergraph product codes","volume":"9","author":[{"given":"Adway","family":"Patra","sequence":"first","affiliation":[{"name":"Department of ECE and Institute for Systems Research, University of Maryland, College Park, MD 20742"}]},{"ORCID":"https://orcid.org/0000-0002-8972-4413","authenticated-orcid":false,"given":"Alexander","family":"Barg","sequence":"additional","affiliation":[{"name":"Department of ECE and Institute for Systems Research, University of Maryland, College Park, MD 20742"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,29]]},"URL":"http://dx.doi.org/10.22331/q-2025-08-29-1842","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2411.17050"},{"id":"arxiv:2105.02370","type":"article-journal","author":[{"given":"Armanda O.","family":"Quintavalle"},{"given":"Earl T.","family":"Campbell"}],"title":"ReShape: a decoder for hypergraph product codes","issued":{"date-parts":[[2022,7,13]]},"note":"arxivid:2105.02370\narxiv_version_number:2"},{"id":"arxiv:2109.14599","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Michael E.","family":"Beverland"},{"given":"Maxime A.","family":"Tremblay"}],"title":"Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes","issued":{"date-parts":[[2021,9,29]]},"note":"arxivid:2109.14599\narxiv_version_number:1"},{"id":"arxiv:2206.03122","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/prxquantum.4.020332","source":"Crossref","title":"Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-9880-5218","authenticated-orcid":true,"given":"Oscar","family":"Higgott","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7211-5515","authenticated-orcid":true,"given":"Nikolas P.","family":"Breuckmann","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,5,30]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.4.020332","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020332","note":"arxivid:2206.03122"},{"id":"arxiv:2208.01002","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We propose a decoder for the correction of erasures with hypergraph product codes, which form one of the most popular families of quantum LDPC codes. Our numerical simulations show that this decoder provides a close approximation of the maximum likelihood decoder that can be implemented in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>N</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> bit operations where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>N</mml:mi></mml:math> is the length of the quantum code. A probabilistic version of this decoder can be implemented in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>N</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1.5</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> bit operations.</jats:p>","DOI":"10.22331/q-2024-08-27-1450","page":"1450","source":"Crossref","title":"Fast erasure decoder for hypergraph product codes","volume":"8","author":[{"given":"Nicholas","family":"Connolly","sequence":"first","affiliation":[{"name":"Inria, Paris, France"},{"name":"Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan"}]},{"given":"Vivien","family":"Londe","sequence":"additional","affiliation":[{"name":"Microsoft, Paris, France"}]},{"given":"Anthony","family":"Leverrier","sequence":"additional","affiliation":[{"name":"Inria, Paris, France"}]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[{"name":"Microsoft Quantum, Redmond, Washington 98052, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,27]]},"URL":"http://dx.doi.org/10.22331/q-2024-08-27-1450","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2208.01002"},{"id":"arxiv:2411.08177","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/allerton63246.2024.10735275","page":"1-8","source":"Crossref","title":"Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided Decimation","author":[{"given":"Mert","family":"Gökduman","sequence":"first","affiliation":[{"name":"Duke University,Duke Quantum Center,Durham,NC,USA"}]},{"given":"Hanwen","family":"Yao","sequence":"additional","affiliation":[{"name":"Duke University,Duke Quantum Center,Durham,NC,USA"}]},{"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[{"name":"Duke University,Duke Quantum Center,Durham,NC,USA"}]}],"event":"2024 60th Annual Allerton Conference on Communication, Control, and Computing (Allerton)","container-title":"2024 60th Annual Allerton Conference on Communication, Control, and Computing","original-title":[],"issued":{"date-parts":[[2024,9,24]]},"URL":"http://dx.doi.org/10.1109/Allerton63246.2024.10735275","note":"arxivid:2411.08177"},{"id":"arxiv:2412.08817","type":"article-journal","author":[{"given":"Hanwen","family":"Yao"},{"given":"Mert","family":"Gökduman"},{"given":"Henry D.","family":"Pfister"}],"title":"Cluster Decomposition for Improved Erasure Decoding of Quantum LDPC Codes","issued":{"date-parts":[[2024,12,11]]},"note":"arxivid:2412.08817\narxiv_version_number:1"},{"id":"arxiv:2308.15520","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Unlike the surface code, quantum low-density parity-check (QLDPC) codes can have a finite encoding rate, potentially lowering the error correction overhead. However, finite-rate QLDPC codes have nonlocal stabilizers, making it difficult to design stabilizer measurement circuits that are low-depth and do not decrease the effective distance. Here, we demonstrate that a popular family of finite-rate QLDPC codes, hypergraph product codes, has the convenient property of distance-robustness: any stabilizer measurement circuit preserves the effective distance. In particular, we prove the depth-optimal circuit in [Tremblay et al, PRL 129, 050504 (2022)] is also optimal in terms of effective distance.</jats:p>","DOI":"10.22331/q-2025-01-30-1618","page":"1618","source":"Crossref","title":"Distance-preserving stabilizer measurements in hypergraph product codes","volume":"9","author":[{"given":"Argyris Giannisis","family":"Manes","sequence":"first","affiliation":[{"name":"Departments of Physics and Applied Physics, Yale University, New Haven, CT 06520, USA Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA"}]},{"given":"Jahan","family":"Claes","sequence":"additional","affiliation":[{"name":"Departments of Physics and Applied Physics, Yale University, New Haven, CT 06520, USA Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,30]]},"URL":"http://dx.doi.org/10.22331/q-2025-01-30-1618","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2308.15520"},{"id":"arxiv:2310.07868","type":"article-journal","author":[{"given":"Anirudh","family":"Krishna"},{"given":"Inbal Livni","family":"Navon"},{"given":"Mary","family":"Wootters"}],"title":"Viderman's algorithm for quantum LDPC codes","issued":{"date-parts":[[2023,10,11]]},"note":"arxivid:2310.07868\narxiv_version_number:1"},{"id":"arxiv:2504.01728","type":"article-journal","author":[{"given":"Asit Kumar","family":"Pradhan"},{"given":"Nithin","family":"Raveendran"},{"given":"Narayanan","family":"Rengaswamy"},{"given":"Bane","family":"Vasić"}],"title":"Linear Time Iterative Decoders for Hypergraph-Product and Lifted-Product Codes","issued":{"date-parts":[[2025,4,2]]},"note":"arxivid:2504.01728\narxiv_version_number:1"},{"id":"arxiv:2309.11719","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.110.022607","source":"Crossref","title":"Long-range-enhanced surface codes","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0001-5236-8964","authenticated-orcid":true,"given":"Yifan","family":"Hong","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"},{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"}]},{"ORCID":"https://orcid.org/0000-0002-1981-8182","authenticated-orcid":true,"given":"Matteo","family":"Marinelli","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"},{"id":[{"id":"https://ror.org/008hybe55","id-type":"ROR","asserted-by":"publisher"}],"name":"JILA"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology, Boulder"}]},{"given":"Adam M.","family":"Kaufman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"},{"id":[{"id":"https://ror.org/008hybe55","id-type":"ROR","asserted-by":"publisher"}],"name":"JILA"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"National Institute of Standards and Technology, Boulder"}]},{"given":"Andrew","family":"Lucas","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"},{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado, Boulder"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,8,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.022607","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022607","note":"arxivid:2309.11719"},{"id":"arxiv:2508.04794","type":"article-journal","author":[{"given":"Noah","family":"Berthusen"},{"given":"Michael J.","family":"Gullans"},{"given":"Yifan","family":"Hong"},{"given":"Maryam","family":"Mudassar"},{"given":"Shi Jie Samuel","family":"Tan"}],"title":"Automorphism gadgets in homological product codes","issued":{"date-parts":[[2025,8,6]]},"note":"arxivid:2508.04794\narxiv_version_number:1"},{"id":"arxiv:2409.05818","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Quantum low-density parity-check (qLDPC) codes with high-performance and nonlocal stabilizers are an attractive alternative to the surface code because of their relatively high code rate and distance. However, unlike the surface code that has simple, geometrically local, stabilizer checks, qLDPC codes with nonlocal stabilizers are challenging to measure. Recent advances have shown how to deterministically perform high-fidelity, cavity-mediated many-body gates, enabling the encoding and decoding of nonlocal GHZ states. We show how to use this resource for constant-depth fault-tolerant stabilizer measurements in hypergraph product and lifted product codes. Using circuit-level noise simulations, including leakage and collective error from the noise-optimized cavity-mediated gate, we find promising thresholds of <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\"><a:mrow><a:mn>0.84</a:mn><a:mo>%</a:mo><a:mo>–</a:mo><a:mn>0.60</a:mn><a:mo>%</a:mo></a:mrow></a:math> for the hypergraph product code and pseudothreshold of <b:math xmlns:b=\"http://www.w3.org/1998/Math/MathML\"><b:mrow><b:mn>0.3</b:mn><b:mo>%</b:mo><b:mo>–</b:mo><b:mn>0.4</b:mn><b:mo>%</b:mo></b:mrow></b:math> for the lifted product codes, with cavity cooperativities in the range <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\"><c:mrow><c:mi>C</c:mi><c:mo>∼</c:mo><c:msup><c:mn>10</c:mn><c:mn>4</c:mn></c:msup><c:mo>–</c:mo><c:msup><c:mn>10</c:mn><c:mn>6</c:mn></c:msup></c:mrow></c:math>. We propose a trilayer architecture, compatible with neutral atom platforms, which enhances circuit parallelizability.</jats:p>","DOI":"10.1103/x1rk-yg29","source":"Crossref","title":"Nonlocal resources for error correction in quantum low-density parity-check codes","volume":"7","author":[{"ORCID":"https://orcid.org/0009-0008-7754-3608","authenticated-orcid":true,"given":"Omprakash","family":"Chandra","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01sf06y89","id-type":"ROR","asserted-by":"publisher"}],"name":"Macquarie University"}]},{"ORCID":"https://orcid.org/0000-0002-1454-0353","authenticated-orcid":true,"given":"Gopikrishnan","family":"Muraleedharan","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01sf06y89","id-type":"ROR","asserted-by":"publisher"}],"name":"Macquarie University"}]},{"ORCID":"https://orcid.org/0000-0002-6019-966X","authenticated-orcid":true,"given":"Gavin K.","family":"Brennen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01sf06y89","id-type":"ROR","asserted-by":"publisher"}],"name":"Macquarie University"},{"name":"BTQ Technologies"},{"name":"V7X 1M8"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,12]]},"URL":"http://dx.doi.org/10.1103/x1rk-yg29","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033247","note":"arxivid:2409.05818"},{"id":"arxiv:1804.01950","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.97.062320","source":"Crossref","title":"Numerical and analytical bounds on threshold error rates for hypergraph-product codes","volume":"97","author":[{"given":"Alexey A.","family":"Kovalev","sequence":"first","affiliation":[]},{"given":"Sanjay","family":"Prabhakar","sequence":"additional","affiliation":[]},{"given":"Ilya","family":"Dumer","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,6,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.062320","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"062320","note":"arxivid:1804.01950"},{"id":"arxiv:2305.00137","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Spatially-coupled (SC) codes is a class of convolutional LDPC codes that has been well investigated in classical coding theory thanks to their high performance and compatibility with low-latency decoders. We describe toric codes as quantum counterparts of classical two-dimensional spatially-coupled (2D-SC) codes, and introduce spatially-coupled quantum LDPC (SC-QLDPC) codes as a generalization. We use the convolutional structure to represent the parity check matrix of a 2D-SC code as a polynomial in two indeterminates, and derive an algebraic condition that is both necessary and sufficient for a 2D-SC code to be a stabilizer code. This algebraic framework facilitates the construction of new code families. While not the focus of this paper, we note that small memory facilitates physical connectivity of qubits, and it enables local encoding and low-latency windowed decoding. In this paper, we use the algebraic framework to optimize short cycles in the Tanner graph of 2D-SC hypergraph product (HGP) codes that arise from short cycles in either component code. While prior work focuses on QLDPC codes with rate less than 1/10, we construct 2D-SC HGP codes with small memories, higher rates (about 1/3), and superior thresholds.</jats:p>","DOI":"10.22331/q-2025-04-07-1693","page":"1693","source":"Crossref","title":"Spatially-Coupled QLDPC Codes","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0001-7512-1913","authenticated-orcid":false,"given":"Siyi","family":"Yang","sequence":"first","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27708, USA"}]},{"ORCID":"https://orcid.org/0000-0003-2084-9717","authenticated-orcid":false,"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27708, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,4,7]]},"URL":"http://dx.doi.org/10.22331/q-2025-04-07-1693","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2305.00137"},{"id":"arxiv:2002.06257","type":"article-journal","author":[{"given":"Muyuan","family":"Li"},{"given":"Theodore J.","family":"Yoder"}],"title":"A Numerical Study of Bravyi-Bacon-Shor and Subsystem Hypergraph Product Codes","issued":{"date-parts":[[2020,2,14]]},"note":"arxivid:2002.06257\narxiv_version_number:1"},{"id":"arxiv:2404.13010","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41467-025-56255-5","source":"Crossref","title":"High-rate quantum LDPC codes for long-range-connected neutral atom registers","volume":"16","author":[{"ORCID":"https://orcid.org/0009-0008-8710-2195","authenticated-orcid":false,"given":"Laura","family":"Pecorari","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0282-7637","authenticated-orcid":false,"given":"Sven","family":"Jandura","sequence":"additional","affiliation":[]},{"given":"Gavin K.","family":"Brennen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1549-0386","authenticated-orcid":false,"given":"Guido","family":"Pupillo","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,28]]},"URL":"http://dx.doi.org/10.1038/s41467-025-56255-5","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"1111","note":"alternative-id:56255\narxivid:2404.13010"},{"id":"arxiv:2406.02666","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.133.180601","source":"Crossref","title":"Entangling Four Logical Qubits beyond Break-Even in a Nonlocal Code","volume":"133","author":[{"ORCID":"https://orcid.org/0000-0001-5236-8964","authenticated-orcid":true,"given":"Yifan","family":"Hong","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado"}]},{"ORCID":"https://orcid.org/0009-0000-8721-9360","authenticated-orcid":true,"given":"Elijah","family":"Durso-Sabina","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"given":"David","family":"Hayes","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03ssvsv78","id-type":"ROR","asserted-by":"publisher"}],"name":"Quantinuum"}]},{"given":"Andrew","family":"Lucas","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02ttsq026","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Colorado"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.180601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180601","note":"arxivid:2406.02666"},{"id":"arxiv:1504.00822","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/focs.2015.55","page":"810-824","source":"Crossref","title":"Quantum Expander Codes","author":[{"given":"Anthony","family":"Leverrier","sequence":"first","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]},{"given":"Gilles","family":"Zemor","sequence":"additional","affiliation":[]}],"event":"2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS)","container-title":"2015 IEEE 56th Annual Symposium on Foundations of Computer Science","original-title":[],"issued":{"date-parts":[[2015,10]]},"URL":"http://dx.doi.org/10.1109/FOCS.2015.55","note":"arxivid:1504.00822"},{"id":"manual:-A.-Grospellier.-Constant-tim","type":"thesis","author":[{"family":"Grospellier","given":"A."}],"title":"Constant time decoding of quantum expander codes and application to fault-tolerant quantum computation","genre":"PhD thesis,","publisher":"Inria Paris","issued":"2019"},{"id":"arxiv:1711.08351","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3188745.3188886","page":"521-534","source":"Crossref","title":"Efficient decoding of random errors for quantum expander codes","author":[{"given":"Omar","family":"Fawzi","sequence":"first","affiliation":[{"name":"ENS Lyon, France"}]},{"given":"Antoine","family":"Grospellier","sequence":"additional","affiliation":[{"name":"Inria, France"}]},{"given":"Anthony","family":"Leverrier","sequence":"additional","affiliation":[{"name":"Inria, France"}]}],"event":"STOC '18: Symposium on Theory of Computing","container-title":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[2018,6,20]]},"URL":"http://dx.doi.org/10.1145/3188745.3188886","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3188745.3188886\n10.1145/3188745\narxivid:1711.08351"},{"id":"doi:10.1109/ISIT.2006.262128","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2006.262128","page":"1549-1553","source":"Crossref","title":"On the minimum distance of structured LDPC codes with two variable nodes of degree 2 per parity-check equation","author":[{"given":"Jean-pierre","family":"Tillich","sequence":"first","affiliation":[]},{"given":"Gilles","family":"Zemor","sequence":"additional","affiliation":[]}],"event":"2006 IEEE International Symposium on Information Theory","container-title":"2006 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2006,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2006.262128"},{"id":"arxiv:2501.19125","type":"article-journal","author":[{"given":"François","family":"Arnault"},{"given":"Philippe","family":"Gaborit"},{"given":"Wouter","family":"Rozendaal"},{"given":"Nicolas","family":"Saussay"},{"given":"Gilles","family":"Zémor"}],"title":"Upper Bounds on the Minimum Distance of Structured LDPC Codes","issued":{"date-parts":[[2025,1,31]]},"note":"arxivid:2501.19125\narxiv_version_number:1"},{"id":"arxiv:1409.1397","type":"article-journal","author":[{"given":"Tali","family":"Kaufman"},{"given":"David","family":"Kazhdan"},{"given":"Alexander","family":"Lubotzky"}],"title":"Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders","issued":{"date-parts":[[2014,9,4]]},"note":"arxivid:1409.1397\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9809054","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6732","DOI":"10.1038/20127","page":"124-126","source":"Crossref","title":"Efficient fault-tolerant quantum computing","volume":"399","author":[{"given":"Andrew M.","family":"Steane","sequence":"first","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[1999,5]]},"URL":"http://dx.doi.org/10.1038/20127","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BF20127\narxivid:quant-ph/9809054"},{"id":"arxiv:2403.09978","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.110.032411","source":"Crossref","title":"Concatenated Steane code with single-flag syndrome checks","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0001-9502-3368","authenticated-orcid":true,"given":"Balint","family":"Pato","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"}]},{"ORCID":"https://orcid.org/0000-0002-2865-0705","authenticated-orcid":true,"given":"Theerapat","family":"Tansuwannont","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"}]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":true,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.032411","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032411","note":"arxivid:2403.09978"},{"id":"arxiv:2303.06846","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.5.033049","source":"Crossref","title":"Improved quantum error correction with randomized compiling","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-7653-3204","authenticated-orcid":true,"given":"Aditya","family":"Jain","sequence":"first","affiliation":[]},{"given":"Pavithran","family":"Iyer","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Emerson","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.5.033049","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033049","note":"arxivid:2303.06846"},{"id":"arxiv:quant-ph/0702201","type":"article-journal","author":[{"given":"A. M.","family":"Stephens"},{"given":"A. G.","family":"Fowler"},{"given":"L. C. L.","family":"Hollenberg"}],"title":"Universal fault tolerant quantum computation on bilinear nearest neighbor arrays","issued":{"date-parts":[[2008,5,9]]},"note":"arxivid:quant-ph/0702201\narxiv_version_number:2"},{"id":"doi:10.1145/97444.97707","type":"proceedings-article","publisher":"ACM Press","DOI":"10.1145/97444.97707","page":"398-406","source":"Crossref","title":"A unified approach to off-line permutation routing on parallel networks","author":[{"given":"F.","family":"Annexstein","sequence":"first","affiliation":[]},{"given":"M.","family":"Baumslag","sequence":"additional","affiliation":[]}],"event":"the second annual ACM symposium","container-title":"Proceedings of the second annual ACM symposium on Parallel algorithms and architectures  - SPAA '90","original-title":[],"issued":{"date-parts":[[1990]]},"URL":"http://dx.doi.org/10.1145/97444.97707","note":"publisher-location:New York, New York, USA\nproceedings-subject:Parallel algorithms and architectures"},{"id":"arxiv:2405.07433","type":"article-journal","author":[{"given":"Nadine","family":"Meister"},{"given":"Christopher A.","family":"Pattison"},{"given":"John","family":"Preskill"}],"title":"Efficient soft-output decoders for the surface code","issued":{"date-parts":[[2024,6,1]]},"note":"arxivid:2405.07433\narxiv_version_number:2"},{"id":"arxiv:2306.06056","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2024.3402091","page":"5692-5709","source":"Crossref","title":"Quantum LDPC Codes From Intersecting Subsets","volume":"70","author":[{"ORCID":"https://orcid.org/0000-0002-4098-0969","authenticated-orcid":false,"given":"Dimiter","family":"Ostrev","sequence":"first","affiliation":[{"name":"German Aerospace Center (DLR), Institute of Communications and Navigation, We&#x00DF;ling, Germany"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2024,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2024.3402091","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2306.06056"},{"id":"doi:10.1007/s11128-018-2095-3","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"12","DOI":"10.1007/s11128-018-2095-3","source":"Crossref","title":"On quantum SPC product codes","volume":"17","author":[{"ORCID":"https://orcid.org/0000-0003-4259-1534","authenticated-orcid":false,"given":"M.","family":"Hivadi","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2018,10,19]]},"URL":"http://dx.doi.org/10.1007/s11128-018-2095-3","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","page":"324","note":"alternative-id:2095"},{"id":"arxiv:2405.07980","type":"article-journal","author":[{"given":"Olai Å.","family":"Mostad"},{"given":"Eirik","family":"Rosnes"},{"given":"Hsuan-Yin","family":"Lin"}],"title":"Generalizing Quantum Tanner Codes","issued":{"date-parts":[[2024,8,16]]},"note":"arxivid:2405.07980\narxiv_version_number:2"},{"id":"arxiv:2508.05095","type":"article-journal","author":[{"given":"Rebecca Katharina","family":"Radebold"},{"given":"Stephen D.","family":"Bartlett"},{"given":"Andrew C.","family":"Doherty"}],"title":"Explicit Instances of Quantum Tanner Codes","issued":{"date-parts":[[2025,11,14]]},"note":"arxivid:2508.05095\narxiv_version_number:3"},{"id":"arxiv:2206.07571","type":"article-journal","author":[{"given":"Anthony","family":"Leverrier"},{"given":"Gilles","family":"Zémor"}],"title":"Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes","issued":{"date-parts":[[2022,10,25]]},"note":"arxivid:2206.07571\narxiv_version_number:2"},{"id":"arxiv:2204.11469","type":"article-journal","author":[{"given":"Max","family":"Hopkins"},{"given":"Ting-Chun","family":"Lin"}],"title":"Explicit Lower Bounds Against $Ω(n)$-Rounds of Sum-of-Squares","issued":{"date-parts":[[2022,4,25]]},"note":"arxivid:2204.11469\narxiv_version_number:1"},{"id":"arxiv:2206.13228","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3564246.3585114","page":"1090-1096","source":"Crossref","title":"NLTS Hamiltonians from Good Quantum Codes","author":[{"given":"Anurag","family":"Anshu","sequence":"first","affiliation":[{"name":"Harvard University, USA"}]},{"given":"Nikolas P.","family":"Breuckmann","sequence":"additional","affiliation":[{"name":"University of Bristol, UK"}]},{"given":"Chinmay","family":"Nirkhe","sequence":"additional","affiliation":[{"name":"MIT-IBM Watson AI Lab, USA"}]}],"event":"STOC '23: 55th Annual ACM Symposium on Theory of Computing","container-title":"Proceedings of the 55th Annual ACM Symposium on Theory of Computing","original-title":[],"issued":{"date-parts":[[2023,6,2]]},"URL":"http://dx.doi.org/10.1145/3564246.3585114","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3564246.3585114\n10.1145/3564246\narxivid:2206.13228"},{"id":"manual:-N.-P.-Breuckmann-private-com","type":"document","author":[{"family":"Breuckmann","given":"N.P."}],"genre":"private communication,","issued":"2022"},{"id":"manual:-A.-Leverrier-href-https://gi","type":"document","author":[{"family":"Leverrier","given":"A."}],"title":"\\href{https://github.com/errorcorrectionzoo/eczoo_data/files/9210173/rotated.pdf}{Mapping the toric code to the rotated toric code}","issued":"2022"},{"id":"arxiv:2106.00765","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>D</mml:mi></mml:math>-dimensional hyperbolic space.</jats:p>","DOI":"10.22331/q-2022-05-13-711","page":"711","source":"Crossref","title":"Connectivity constrains quantum codes","volume":"6","author":[{"given":"Nouédyn","family":"Baspin","sequence":"first","affiliation":[{"name":"Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1"}]},{"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[{"name":"Stanford University, Stanford, CA, USA, 94305"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,13]]},"URL":"http://dx.doi.org/10.22331/q-2022-05-13-711","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2106.00765"},{"id":"arxiv:2307.03283","type":"article-journal","author":[{"given":"Nouédyn","family":"Baspin"},{"given":"Venkatesan","family":"Guruswami"},{"given":"Anirudh","family":"Krishna"},{"given":"Ray","family":"Li"}],"title":"Improved rate-distance trade-offs for quantum codes with restricted connectivity","issued":{"date-parts":[[2023,7,6]]},"note":"arxivid:2307.03283\narxiv_version_number:1"},{"id":"arxiv:2109.10982","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.129.050505","source":"Crossref","title":"Quantifying Nonlocality: How Outperforming Local Quantum Codes Is Expensive","volume":"129","author":[{"ORCID":"https://orcid.org/0000-0002-4028-3098","authenticated-orcid":true,"given":"Nouédyn","family":"Baspin","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2027-6448","authenticated-orcid":true,"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.129.050505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050505","note":"arxivid:2109.10982"},{"id":"arxiv:1205.7036","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Gilles","family":"Zémor"}],"title":"Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel","issued":{"date-parts":[[2012,5,31]]},"note":"arxivid:1205.7036\narxiv_version_number:1"},{"id":"arxiv:1703.00382","type":"article-journal","author":[{"given":"Seth","family":"Lloyd"},{"given":"Peter","family":"Shor"},{"given":"Kevin","family":"Thompson"}],"title":"polylog-LDPC Capacity Achieving Codes for the Noisy Quantum Erasure Channel","issued":{"date-parts":[[2017,7,26]]},"note":"arxivid:1703.00382\narxiv_version_number:3"},{"id":"arxiv:2602.16948","type":"article-journal","author":[{"given":"Matthias","family":"Christandl"},{"given":"Omar","family":"Fawzi"},{"given":"Ashutosh","family":"Goswami"}],"title":"Fault-tolerant interfaces for quantum LDPC codes","issued":{"date-parts":[[2026,2,18]]},"note":"arxivid:2602.16948\narxiv_version_number:1"},{"id":"arxiv:2503.05003","type":"article-journal","author":[{"given":"Alexander","family":"Cowtan"},{"given":"Zhiyang","family":"He"},{"given":"Dominic J.","family":"Williamson"},{"given":"Theodore J.","family":"Yoder"}],"title":"Parallel Logical Measurements via Quantum Code Surgery","issued":{"date-parts":[[2026,1,22]]},"note":"arxivid:2503.05003\narxiv_version_number:3"},{"id":"arxiv:2510.14895","type":"article-journal","author":[{"given":"Alexander","family":"Cowtan"},{"given":"Zhiyang","family":"He"},{"given":"Dominic J.","family":"Williamson"},{"given":"Theodore J.","family":"Yoder"}],"title":"Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery","issued":{"date-parts":[[2025,10,16]]},"note":"arxivid:2510.14895\narxiv_version_number:1"},{"id":"arxiv:2410.03628","type":"article-journal","author":[{"given":"Esha","family":"Swaroop"},{"given":"Tomas","family":"Jochym-O'Connor"},{"given":"Theodore J.","family":"Yoder"}],"title":"Universal adapters between quantum LDPC codes","issued":{"date-parts":[[2025,12,11]]},"note":"arxivid:2410.03628\narxiv_version_number:4"},{"id":"arxiv:2503.10390","type":"article-journal","author":[{"given":"Zhiyang","family":"He"},{"given":"Alexander","family":"Cowtan"},{"given":"Dominic J.","family":"Williamson"},{"given":"Theodore J.","family":"Yoder"}],"title":"Extractors: QLDPC Architectures for Efficient Pauli-Based Computation","issued":{"date-parts":[[2025,10,25]]},"note":"arxivid:2503.10390\narxiv_version_number:2"},{"id":"arxiv:0706.4094","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevb.76.201102","source":"Crossref","title":"Quantum belief propagation: An algorithm for thermal quantum systems","volume":"76","author":[{"given":"M. B.","family":"Hastings","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2007,11,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.76.201102","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"201102","note":"arxivid:0706.4094"},{"id":"arxiv:0708.1337","type":"journal-article","publisher":"Elsevier BV","issue":"8","DOI":"10.1016/j.aop.2007.10.001","page":"1899-1946","source":"Crossref","title":"Quantum Graphical Models and Belief Propagation","volume":"323","author":[{"given":"M.S.","family":"Leifer","sequence":"first","affiliation":[]},{"given":"D.","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2008,8]]},"URL":"http://dx.doi.org/10.1016/j.aop.2007.10.001","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491607001509\narxivid:0708.1337"},{"id":"arxiv:0801.1241","type":"article-journal","author":[{"given":"David","family":"Poulin"},{"given":"Yeojin","family":"Chung"}],"title":"On the iterative decoding of sparse quantum codes","issued":{"date-parts":[[2008,7,1]]},"note":"arxivid:0801.1241\narxiv_version_number:2"},{"id":"arxiv:2012.15297","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing.</jats:p>","DOI":"10.22331/q-2021-10-14-562","page":"562","source":"Crossref","title":"Trapping Sets of Quantum LDPC Codes","volume":"5","author":[{"given":"Nithin","family":"Raveendran","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA"}]},{"given":"Bane","family":"Vasić","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,10,14]]},"URL":"http://dx.doi.org/10.22331/q-2021-10-14-562","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2012.15297"},{"id":"doi:10.1109/MILCOM58377.2023.1035628","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/milcom58377.2023.10356284","page":"115-120","source":"Crossref","title":"Low-Complexity Decoding Algorithm Utilizing Degeneracy for Quantum LDPC Codes","author":[{"given":"Jaemin","family":"Kim","sequence":"first","affiliation":[{"name":"Korea Advanced Institute of Science and Technology,School of Electrical Engineering,Daejeon,Republic of Korea"}]},{"given":"Hyunwoo","family":"Jung","sequence":"additional","affiliation":[{"name":"Korea Advanced Institute of Science and Technology,School of Electrical Engineering,Daejeon,Republic of Korea"}]},{"given":"Jeongseok","family":"Ha","sequence":"additional","affiliation":[{"name":"Korea Advanced Institute of Science and Technology,School of Electrical Engineering,Daejeon,Republic of Korea"}]}],"event":"MILCOM 2023 - 2023 IEEE Military Communications Conference (MILCOM)","container-title":"MILCOM 2023 - 2023 IEEE Military Communications Conference (MILCOM)","original-title":[],"issued":{"date-parts":[[2023,10,30]]},"URL":"http://dx.doi.org/10.1109/MILCOM58377.2023.10356284"},{"id":"doi:10.1109/ICASSP48485.2024.1044615","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/icassp48485.2024.10446153","page":"9001-9005","source":"Crossref","title":"A Binary BP Decoding Using Posterior Adjustment for Quantum LDPC Codes","author":[{"given":"Tzu-Hsuan","family":"Huang","sequence":"first","affiliation":[{"name":"National Tsing Hua University,Dept. of Electrical Engineering,Hsinchu,Taiwan"}]},{"given":"Yeong-Luh","family":"Ueng","sequence":"additional","affiliation":[{"name":"National Tsing Hua University,Dept. of Electrical Engineering,Hsinchu,Taiwan"}]}],"event":"ICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)","container-title":"ICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)","original-title":[],"issued":{"date-parts":[[2024,4,14]]},"URL":"http://dx.doi.org/10.1109/ICASSP48485.2024.10446153"},{"id":"arxiv:1811.07835","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.122.200501","source":"Crossref","title":"Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes","volume":"122","author":[{"given":"Ye-Hua","family":"Liu","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,5,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.122.200501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200501","note":"arxivid:1811.07835"},{"id":"arxiv:2212.10245","type":"article-journal","author":[{"given":"Sisi","family":"Miao"},{"given":"Alexander","family":"Schnerring"},{"given":"Haizheng","family":"Li"},{"given":"Laurent","family":"Schmalen"}],"title":"Neural Belief Propagation Decoding of Quantum LDPC Codes Using Overcomplete Check Matrices","issued":{"date-parts":[[2023,3,21]]},"note":"arxivid:2212.10245\narxiv_version_number:2"},{"id":"arxiv:2307.01241","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>To leverage the full potential of quantum error-correcting stabilizer codes it is crucial to have an efficient and accurate decoder. Accurate, maximum likelihood, decoders are computationally very expensive whereas decoders based on more efficient algorithms give sub-optimal performance. In addition, the accuracy will depend on the quality of models and estimates of error rates for idling qubits, gates, measurements, and resets, and will typically assume symmetric error channels. In this work, we explore a model-free, data-driven, approach to decoding, using a graph neural network (GNN). The decoding problem is formulated as a graph classification task in which a set of stabilizer measurements is mapped to an annotated detector graph for which the neural network predicts the most likely logical error class. We show that the GNN-based decoder can outperform a matching decoder for circuit level noise on the surface code given only the simulated data, while the matching decoder is given full information of the underlying error model. Although training is computationally demanding, inference is fast and scales approximately linearly with the space-time volume of the code. We also find that we can use large, but more limited, datasets of real experimental data for the repetition code, giving decoding accuracies that are on par with minimum weight perfect matching. The results show that a purely data-driven approach to decoding may be a viable future option for practical quantum error correction, which is competitive in terms of speed, accuracy, and versatility.</jats:p>","DOI":"10.1103/physrevresearch.7.023181","source":"Crossref","title":"Data-driven decoding of quantum error correcting codes using graph neural networks","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0001-7995-6432","authenticated-orcid":true,"given":"Moritz","family":"Lange","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01tm6cn81","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Gothenburg"}]},{"ORCID":"https://orcid.org/0009-0002-4154-0233","authenticated-orcid":true,"given":"Pontus","family":"Havström","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01tm6cn81","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Gothenburg"}]},{"ORCID":"https://orcid.org/0000-0002-4972-4216","authenticated-orcid":true,"given":"Basudha","family":"Srivastava","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01tm6cn81","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Gothenburg"},{"name":"Quantinuum"}]},{"ORCID":"https://orcid.org/0009-0008-9013-7460","authenticated-orcid":true,"given":"Isak","family":"Bengtsson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/040wg7k59","id-type":"ROR","asserted-by":"publisher"}],"name":"Chalmers University of Technology"}]},{"given":"Valdemar","family":"Bergentall","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01tm6cn81","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Gothenburg"}]},{"given":"Karl","family":"Hammar","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01tm6cn81","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Gothenburg"}]},{"given":"Olivia","family":"Heuts","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01tm6cn81","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Gothenburg"}]},{"given":"Evert","family":"van Nieuwenburg","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/027bh9e22","id-type":"ROR","asserted-by":"publisher"}],"name":"Leiden University"}]},{"ORCID":"https://orcid.org/0000-0003-3185-2014","authenticated-orcid":true,"given":"Mats","family":"Granath","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01tm6cn81","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Gothenburg"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.7.023181","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023181","note":"arxivid:2307.01241"},{"id":"arxiv:2310.17758","type":"article-journal","author":[{"given":"Anqi","family":"Gong"},{"given":"Sebastian","family":"Cammerer"},{"given":"Joseph M.","family":"Renes"}],"title":"Graph Neural Networks for Enhanced Decoding of Quantum LDPC Codes","issued":{"date-parts":[[2023,11,6]]},"note":"arxivid:2310.17758\narxiv_version_number:2"},{"id":"arxiv:2305.17505","type":"article-journal","author":[{"given":"Zhengzhong","family":"Yi"},{"given":"Zhipeng","family":"Liang"},{"given":"Kaixin","family":"Zhong"},{"given":"Yulin","family":"Wu"},{"given":"Zhou","family":"Fang"},{"given":"Xuan","family":"Wang"}],"title":"Improved belief propagation decoding algorithm based on decoupling representation of Pauli operators for quantum LDPC codes","issued":{"date-parts":[[2023,12,4]]},"note":"arxivid:2305.17505\narxiv_version_number:4"},{"id":"arxiv:2205.06125","type":"article-journal","author":[{"given":"Julien","family":"du Crest"},{"given":"Mehdi","family":"Mhalla"},{"given":"Valentin","family":"Savin"}],"title":"Stabilizer Inactivation for Message-Passing Decoding of Quantum LDPC Codes","issued":{"date-parts":[[2023,3,14]]},"note":"arxivid:2205.06125\narxiv_version_number:3"},{"id":"arxiv:2406.18655","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n          <jats:p>Quantum low-density parity-check codes are a promising candidate for fault-tolerant quantum computing with considerably reduced overhead compared to the surface code. However, the lack of a practical decoding algorithm remains a barrier to their implementation. In this work, we introduce localized statistics decoding, a reliability-guided inversion decoder that is highly parallelizable and applicable to arbitrary quantum low-density parity-check codes. Our approach employs a parallel matrix factorization strategy, which we call <jats:italic>on-the-fly elimination</jats:italic>, to identify, validate, and solve local decoding regions on the decoding graph. Through numerical simulations, we show that localized statistics decoding matches the performance of state-of-the-art decoders while reducing the runtime complexity for operation in the sub-threshold regime. Importantly, our decoder is more amenable to implementation on specialized hardware, positioning it as a promising candidate for decoding real-time syndromes from experiments.</jats:p>","DOI":"10.1038/s41467-025-63214-7","source":"Crossref","title":"Localized statistics decoding for quantum low-density parity-check codes","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0002-1476-0647","authenticated-orcid":false,"given":"Timo","family":"Hillmann","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2973-1689","authenticated-orcid":false,"given":"Lucas","family":"Berent","sequence":"additional","affiliation":[]},{"given":"Armanda O.","family":"Quintavalle","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3033-1292","authenticated-orcid":false,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]},{"given":"Robert","family":"Wille","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9202-1156","authenticated-orcid":false,"given":"Joschka","family":"Roffe","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,2]]},"URL":"http://dx.doi.org/10.1038/s41467-025-63214-7","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"8214","note":"alternative-id:63214\narxivid:2406.18655"},{"id":"arxiv:2311.18488","type":"article-journal","author":[{"given":"Sana","family":"Javed"},{"given":"Francisco","family":"Garcia-Herrero"},{"given":"Bane","family":"Vasic"},{"given":"Mark F.","family":"Flanagan"}],"title":"Low-Complexity Linear Programming Based Decoding of Quantum LDPC codes","issued":{"date-parts":[[2024,1,19]]},"note":"arxivid:2311.18488\narxiv_version_number:2"},{"id":"arxiv:2312.10950","type":"article-journal","author":[{"given":"Hanwen","family":"Yao"},{"given":"Waleed Abu","family":"Laban"},{"given":"Christian","family":"Häger"},{"given":"Alexandre Graell i","family":"Amat"},{"given":"Henry D.","family":"Pfister"}],"title":"Belief Propagation Decoding of Quantum LDPC Codes with Guided Decimation","issued":{"date-parts":[[2024,6,21]]},"note":"arxivid:2312.10950\narxiv_version_number:2"},{"id":"arxiv:2412.02885","type":"article-journal","author":[{"given":"Keyi","family":"Yin"},{"given":"Xiang","family":"Fang"},{"given":"Jixuan","family":"Ruan"},{"given":"Hezi","family":"Zhang"},{"given":"Dean","family":"Tullsen"},{"given":"Andrew","family":"Sornborger"},{"given":"Chenxu","family":"Liu"},{"given":"Ang","family":"Li"},{"given":"Travis","family":"Humble"},{"given":"Yufei","family":"Ding"}],"title":"SymBreak: Mitigating Quantum Degeneracy Issues in QLDPC Code Decoders by Breaking Symmetry","issued":{"date-parts":[[2024,12,3]]},"note":"arxivid:2412.02885\narxiv_version_number:1"},{"id":"arxiv:2109.14609","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.129.050504","source":"Crossref","title":"Constant-Overhead Quantum Error Correction with Thin Planar Connectivity","volume":"129","author":[{"ORCID":"https://orcid.org/0000-0003-0868-0584","authenticated-orcid":true,"given":"Maxime A.","family":"Tremblay","sequence":"first","affiliation":[]},{"given":"Nicolas","family":"Delfosse","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0954-4509","authenticated-orcid":true,"given":"Michael E.","family":"Beverland","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.129.050504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050504","note":"arxivid:2109.14609"},{"id":"arxiv:2205.02341","type":"article-journal","author":[{"given":"Nithin","family":"Raveendran"},{"given":"Narayanan","family":"Rengaswamy"},{"given":"Asit Kumar","family":"Pradhan"},{"given":"Bane","family":"Vasić"}],"title":"Soft Syndrome Decoding of Quantum LDPC Codes for Joint Correction of Data and Syndrome Errors","issued":{"date-parts":[[2022,5,4]]},"note":"arxivid:2205.02341\narxiv_version_number:1"},{"id":"arxiv:2207.06428","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/mbits.2023.3246025","page":"5-19","source":"Crossref","title":"Conservation Laws and Quantum Error Correction: Toward a Generalized Matching Decoder","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":false,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[{"name":"T. J. Watson Research Center, IBM Quantum, Yorktown Heights, NY, USA"}]}],"container-title":"IEEE BITS the Information Theory Magazine","original-title":[],"issued":{"date-parts":[[2022,12,1]]},"URL":"http://dx.doi.org/10.1109/MBITS.2023.3246025","ISSN":["2692-4110","2692-4080"],"container-title-short":"IEEE BITS Inform. Theory Mag.","note":"arxivid:2207.06428"},{"id":"arxiv:2103.08049","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Vivien","family":"Londe"},{"given":"Michael","family":"Beverland"}],"title":"Toward a Union-Find decoder for quantum LDPC codes","issued":{"date-parts":[[2021,3,14]]},"note":"arxivid:2103.08049\narxiv_version_number:1"},{"id":"arxiv:2209.01180","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/3566097.3567934","page":"709-714","source":"Crossref","title":"Software Tools for Decoding Quantum Low-Density Parity-Check Codes","author":[{"given":"Lucas","family":"Berent","sequence":"first","affiliation":[{"name":"Technical University of Munich, Germany"}]},{"given":"Lukas","family":"Burgholzer","sequence":"additional","affiliation":[{"name":"Johannes Kepler University Linz, Austria"}]},{"given":"Robert","family":"Wille","sequence":"additional","affiliation":[{"name":"Technical University of Munich, Germany and Hagenberg GmbH, Austria"}]}],"event":"ASPDAC '23: 28th Asia and South Pacific Design Automation Conference","container-title":"Proceedings of the 28th Asia and South Pacific Design Automation Conference","original-title":[],"issued":{"date-parts":[[2023,1,16]]},"URL":"http://dx.doi.org/10.1145/3566097.3567934","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/3566097.3567934\n10.1145/3566097\narxivid:2209.01180"},{"id":"arxiv:2407.15988","type":"article-journal","author":[{"given":"Matthias C.","family":"Löbl"},{"given":"Susan X.","family":"Chen"},{"given":"Stefano","family":"Paesani"},{"given":"Anders S.","family":"Sørensen"}],"title":"Breadth-first graph traversal union-find decoder","issued":{"date-parts":[[2024,7,22]]},"note":"arxivid:2407.15988\narxiv_version_number:1"},{"id":"arxiv:2311.03307","type":"article-journal","author":[{"given":"Shilin","family":"Huang"},{"given":"Shruti","family":"Puri"}],"title":"Improved Noisy Syndrome Decoding of Quantum LDPC Codes with Sliding Window","issued":{"date-parts":[[2023,11,6]]},"note":"arxivid:2311.03307\narxiv_version_number:1"},{"id":"arxiv:2402.01532","type":"article-journal","author":[{"given":"Antonio deMarti","family":"iOlius"},{"given":"Josu Etxezarreta","family":"Martinez"}],"title":"The closed-branch decoder for quantum LDPC codes","issued":{"date-parts":[[2024,2,14]]},"note":"arxivid:2402.01532\narxiv_version_number:2"},{"id":"arxiv:1310.2984","type":"article-journal","author":[{"given":"Daniel","family":"Gottesman"}],"title":"Fault-Tolerant Quantum Computation with Constant Overhead","issued":{"date-parts":[[2014,7,22]]},"note":"arxivid:1310.2984\narxiv_version_number:3"},{"id":"arxiv:2409.01440","type":"article-journal","author":[{"given":"Antonio deMarti","family":"iOlius"},{"given":"Imanol Etxezarreta","family":"Martinez"},{"given":"Joschka","family":"Roffe"},{"given":"Josu Etxezarreta","family":"Martinez"}],"title":"An almost-linear time decoding algorithm for quantum LDPC codes under circuit-level noise","issued":{"date-parts":[[2025,8,5]]},"note":"arxivid:2409.01440\narxiv_version_number:2"},{"id":"arxiv:2412.21118","type":"article-journal","author":[{"given":"Ching-Feng","family":"Kung"},{"given":"Kao-Yueh","family":"Kuo"},{"given":"Ching-Yi","family":"Lai"}],"title":"Efficient Approximate Degenerate Ordered Statistics Decoding for Quantum Codes via Reliable Subset Reduction","issued":{"date-parts":[[2026,2,21]]},"note":"arxivid:2412.21118\narxiv_version_number:4"},{"id":"arxiv:2503.10988","type":"article-journal","author":[{"given":"Laleh Aghababaie","family":"Beni"},{"given":"Oscar","family":"Higgott"},{"given":"Noah","family":"Shutty"}],"title":"Tesseract: A Search-Based Decoder for Quantum Error Correction","issued":{"date-parts":[[2025,8,6]]},"note":"arxivid:2503.10988\narxiv_version_number:2"},{"id":"arxiv:2506.01779","type":"article-journal","author":[{"given":"Tristan","family":"Müller"},{"given":"Thomas","family":"Alexander"},{"given":"Michael E.","family":"Beverland"},{"given":"Markus","family":"Bühler"},{"given":"Blake R.","family":"Johnson"},{"given":"Thilo","family":"Maurer"},{"given":"Drew","family":"Vandeth"}],"title":"Improved belief propagation is sufficient for real-time decoding of quantum memory","issued":{"date-parts":[[2025,8,22]]},"note":"arxivid:2506.01779\narxiv_version_number:2"},{"id":"arxiv:2508.04969","type":"article-journal","author":[{"given":"Yue","family":"Wu"},{"given":"Binghong","family":"Li"},{"given":"Kathleen","family":"Chang"},{"given":"Shruti","family":"Puri"},{"given":"Lin","family":"Zhong"}],"title":"Minimum-Weight Parity Factor Decoder for Quantum Error Correction","issued":{"date-parts":[[2025,10,2]]},"note":"arxivid:2508.04969\narxiv_version_number:2"},{"id":"arxiv:2510.05795","type":"article-journal","author":[{"given":"Seok-Hyung","family":"Lee"},{"given":"Lucas H.","family":"English"},{"given":"Stephen D.","family":"Bartlett"}],"title":"Efficient Post-Selection for General Quantum LDPC Codes","issued":{"date-parts":[[2026,1,25]]},"note":"arxivid:2510.05795\narxiv_version_number:4"},{"id":"arxiv:2510.25222","type":"article-journal","author":[{"given":"Riki","family":"Toshio"},{"given":"Kaito","family":"Kishi"},{"given":"Jun","family":"Fujisaki"},{"given":"Hirotaka","family":"Oshima"},{"given":"Shintaro","family":"Sato"},{"given":"Keisuke","family":"Fujii"}],"title":"Decoder Switching: Breaking the Speed-Accuracy Tradeoff in Real-Time Quantum Error Correction","issued":{"date-parts":[[2025,10,29]]},"note":"arxivid:2510.25222\narxiv_version_number:1"},{"id":"arxiv:2510.14060","type":"journal-article","publisher":"Springer Science and Business Media LLC","DOI":"10.1038/s41467-026-70556-3","source":"Crossref","title":"Decoding correlated errors in quantum LDPC codes","author":[{"ORCID":"https://orcid.org/0009-0005-9880-123X","authenticated-orcid":false,"given":"Arshpreet Singh","family":"Maan","sequence":"first","affiliation":[]},{"given":"Francisco Miguel","family":"Garcia Herrero","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1536-8858","authenticated-orcid":false,"given":"Alexandru","family":"Paler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9362-8769","authenticated-orcid":false,"given":"Valentin","family":"Savin","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,14]]},"URL":"http://dx.doi.org/10.1038/s41467-026-70556-3","ISSN":["2041-1723"],"container-title-short":"Nat Commun","note":"alternative-id:70556\narxivid:2510.14060"},{"id":"arxiv:2510.08523","type":"article-journal","author":[{"given":"Guo","family":"Zheng"},{"given":"Liang","family":"Jiang"},{"given":"Qian","family":"Xu"}],"title":"High-Rate Surgery: towards constant-overhead logical operations","issued":{"date-parts":[[2025,10,9]]},"note":"arxivid:2510.08523\narxiv_version_number:1"},{"id":"arxiv:1412.6172","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.115.050502","source":"Crossref","title":"Thresholds for Correcting Errors, Erasures, and Faulty Syndrome Measurements in Degenerate Quantum Codes","volume":"115","author":[{"given":"Ilya","family":"Dumer","sequence":"first","affiliation":[]},{"given":"Alexey A.","family":"Kovalev","sequence":"additional","affiliation":[]},{"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2015,7,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.115.050502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050502","note":"arxivid:1412.6172"},{"id":"manual:-L.-Pryadko-and-M.-Woolls.-Qu","type":"document","author":[{"family":"Pryadko","given":"L."},{"family":"Quantum_LDPC_Codes","given":"M.Woolls"}],"issued":"2023","URL":"https://github.com/QEC-pages/Quantum_LDPC_Codes","note":"Available at \\url{https://github.com/QEC-pages/Quantum_LDPC_Codes},"},{"id":"arxiv:2502.09542","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","abstract":"<jats:p>We present a fault-tolerant Bell-pair distillation scheme achieving constant overhead through high-rate quantum low-density parity-check (qLDPC) codes. Our approach maintains a constant distillation rate equal to the code rate while requiring no additional overhead beyond the physical qubits of the code. Full circuit-level analysis demonstrates fault-tolerance for input Bell-pair infidelities below a threshold <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mo>∼</a:mo><a:mn>10</a:mn><a:mo>%</a:mo></a:math>, readily achievable with near-term capabilities. Unlike previous proposals, our scheme keeps the output Bell pairs encoded in qLDPC codes at each node, eliminating unencoding overhead and enabling direct use in distributed quantum applications through recent advances in qLDPC computation. These results establish qLDPC-based distillation as a practical route toward resource-efficient quantum networks and distributed quantum computing.</jats:p>","DOI":"10.1103/s39k-r2kq","source":"Crossref","title":"Constant-Overhead Fault-Tolerant Bell-Pair Distillation Using High-Rate Codes","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0001-5518-7907","authenticated-orcid":true,"given":"J. Pablo","family":"Bonilla Ataides","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-2148-8856","authenticated-orcid":true,"given":"Hengyun","family":"Zhou","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01njdjw78","id-type":"ROR","asserted-by":"publisher"}],"name":"QuEra Computing Incorporated"}]},{"given":"Qian","family":"Xu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"Caltech"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"Caltech"}]},{"ORCID":"https://orcid.org/0000-0002-5920-2972","authenticated-orcid":true,"given":"Gefen","family":"Baranes","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"},{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0001-5044-3131","authenticated-orcid":true,"given":"Bikun","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Chicago"}]},{"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Chicago"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,25]]},"URL":"http://dx.doi.org/10.1103/s39k-r2kq","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130804","note":"arxivid:2502.09542"},{"id":"arxiv:2504.02673","type":"article-journal","author":[{"given":"Mingyu","family":"Kang"},{"given":"Yingjia","family":"Lin"},{"given":"Hanwen","family":"Yao"},{"given":"Mert","family":"Gökduman"},{"given":"Arianna","family":"Meinking"},{"given":"Kenneth R.","family":"Brown"}],"title":"QUITS: A modular Qldpc code circUIT Simulator","issued":{"date-parts":[[2025,12,1]]},"note":"arxivid:2504.02673\narxiv_version_number:2"},{"id":"manual:-M.-Kang.-QUITS.-url-https://","type":"document","author":[{"family":"\\url{https,//github.com/mkangquantum/quits}","given":"M.Kang Q.U.I.T.S."}],"issued":"2025"},{"id":"arxiv:1608.05089","type":"article-journal","author":[{"given":"M. B.","family":"Hastings"}],"title":"Quantum Codes from High-Dimensional Manifolds","issued":{"date-parts":[[2016,8,17]]},"note":"arxivid:1608.05089\narxiv_version_number:1"},{"id":"doi:10.1109/ITW.2012.6404686","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw.2012.6404686","page":"327-331","source":"Crossref","title":"Spatially coupled quantum LDPC codes","author":[{"given":"Iryna","family":"Andriyanova","sequence":"first","affiliation":[]},{"given":"Denise","family":"Maurice","sequence":"additional","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]}],"event":"2012 IEEE Information Theory Workshop (ITW 2012)","container-title":"2012 IEEE Information Theory Workshop","original-title":[],"issued":{"date-parts":[[2012,9]]},"URL":"http://dx.doi.org/10.1109/ITW.2012.6404686"},{"id":"doi:10.1109/ISIT.2013.6620358","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2013.6620358","page":"907-911","source":"Crossref","title":"A family of quantum codes with performances close to the hashing bound under iterative decoding","author":[{"given":"Denise","family":"Maurice","sequence":"first","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]},{"given":"Iryna","family":"Andriyanova","sequence":"additional","affiliation":[]}],"event":"2013 IEEE International Symposium on Information Theory (ISIT)","container-title":"2013 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2013,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2013.6620358"},{"id":"arxiv:quant-ph/9802061","type":"article-journal","author":[{"given":"Andrew M.","family":"Steane"}],"title":"Enlargement of Calderbank Shor Steane quantum codes","issued":{"date-parts":[[1998,3,31]]},"note":"arxivid:quant-ph/9802061\narxiv_version_number:2"},{"id":"arxiv:quant-ph/9910060","type":"article-journal","author":[{"given":"Markus","family":"Grassl"},{"given":"Thomas","family":"Beth"}],"title":"Quantum BCH Codes","issued":{"date-parts":[[1999,10,14]]},"note":"arxivid:quant-ph/9910060\narxiv_version_number:1"},{"id":"arxiv:2601.20927","type":"article-journal","author":[{"given":"Jin Ming","family":"Koh"},{"given":"Anqi","family":"Gong"},{"given":"Andrei C.","family":"Diaconu"},{"given":"Daniel Bochen","family":"Tan"},{"given":"Alexandra A.","family":"Geim"},{"given":"Michael J.","family":"Gullans"},{"given":"Norman Y.","family":"Yao"},{"given":"Mikhail D.","family":"Lukin"},{"given":"Shayan","family":"Majidy"}],"title":"Entangling logical qubits without physical operations","issued":{"date-parts":[[2026,1,28]]},"note":"arxivid:2601.20927\narxiv_version_number:1"},{"id":"arxiv:1802.01520","type":"article-journal","author":[{"given":"Nikolas P.","family":"Breuckmann"}],"title":"PhD thesis: Homological Quantum Codes Beyond the Toric Code","issued":{"date-parts":[[2018,2,5]]},"note":"arxivid:1802.01520\narxiv_version_number:1"},{"id":"arxiv:2505.06082","type":"article-journal","author":[{"given":"Xiang","family":"Zou"},{"given":"Hoi-Kwong","family":"Lo"}],"title":"Algebraic Topology Principles behind Topological Quantum Error Correction","issued":{"date-parts":[[2026,1,20]]},"note":"arxivid:2505.06082\narxiv_version_number:2"},{"id":"doi:10.1007/BF01608825","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/bf01608825","page":"129-132","source":"Crossref","title":"Uncertainty relations for information entropy in wave mechanics","volume":"44","author":[{"given":"Iwo","family":"Białynicki-Birula","sequence":"first","affiliation":[]},{"given":"Jerzy","family":"Mycielski","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1975,6]]},"URL":"http://dx.doi.org/10.1007/BF01608825","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun.Math. Phys.","note":"alternative-id:BF01608825"},{"id":"doi:10.1103/PhysRevLett.50.631","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.50.631","page":"631-633","source":"Crossref","title":"Uncertainty in Quantum Measurements","volume":"50","author":[{"given":"David","family":"Deutsch","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1983,2,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.50.631","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"doi:10.1103/PhysRevLett.60.1103","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/physrevlett.60.1103","page":"1103-1106","source":"Crossref","title":"Generalized entropic uncertainty relations","volume":"60","author":[{"given":"Hans","family":"Maassen","sequence":"first","affiliation":[]},{"given":"J. B. M.","family":"Uffink","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1988,3,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.60.1103","ISSN":["0031-9007"],"container-title-short":"Phys. Rev. Lett."},{"id":"arxiv:quant-ph/0202036","type":"article-journal","author":[{"given":"Andrew M.","family":"Steane"}],"title":"Fast fault-tolerant filtering of quantum codewords","issued":{"date-parts":[[2004,4,2]]},"note":"arxivid:quant-ph/0202036\narxiv_version_number:2"},{"id":"arxiv:1410.5124","type":"article-journal","author":[{"given":"Adam","family":"Paetznick"}],"title":"Resource optimization for fault-tolerant quantum computing","issued":{"date-parts":[[2014,10,19]]},"note":"arxivid:1410.5124\narxiv_version_number:1"},{"id":"arxiv:1404.2495","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>We present a scheme for encoding and decoding an unknown state for CSS codes, based on syndrome measurements. We illustrate our method by means of Kitaev toric code, defected-lattice code, topological subsystem code and 3D Haah code. The protocol is local whenever in a given code the crossings between the logical operators consist of next neighbour pairs, which holds for the above codes. For subsystem code we also present scheme in a noisy case, where we allow for bit and phase-flip errors on qubits as well as state preparation and syndrome measurement errors. Similar scheme can be built for two other codes. We show that the fidelity of the protected qubit in the noisy scenario in a large code size limit is of \"Equation missing\"<!-- image only, no MathML or LaTex -->, where <jats:italic>p</jats:italic> is a probability of error on a single qubit per time step. Regarding Haah code we provide noiseless scheme, leaving the noisy case as an open problem.</jats:p>","DOI":"10.1038/srep08975","source":"Crossref","title":"Simple scheme for encoding and decoding a qubit in unknown state for various topological codes","volume":"5","author":[{"given":"Justyna","family":"Łodyga","sequence":"first","affiliation":[]},{"given":"Paweł","family":"Mazurek","sequence":"additional","affiliation":[]},{"given":"Andrzej","family":"Grudka","sequence":"additional","affiliation":[]},{"given":"Michał","family":"Horodecki","sequence":"additional","affiliation":[]}],"container-title":"Scientific Reports","original-title":[],"language":"en","issued":{"date-parts":[[2015,3,10]]},"URL":"http://dx.doi.org/10.1038/srep08975","ISSN":["2045-2322"],"container-title-short":"Sci Rep","page":"8975","note":"alternative-id:BFsrep08975\narxivid:1404.2495"},{"id":"arxiv:quant-ph/0210069","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevlett.90.067901","source":"Crossref","title":"Entanglement Purification for Quantum Computation","volume":"90","author":[{"given":"W.","family":"Dür","sequence":"first","affiliation":[]},{"given":"H.-J.","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2003,2,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.90.067901","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"067901","note":"arxivid:quant-ph/0210069"},{"id":"arxiv:2204.14038","type":"article-journal","author":[{"given":"Aleks","family":"Kissinger"}],"title":"Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code)","issued":{"date-parts":[[2022,4,29]]},"note":"arxivid:2204.14038\narxiv_version_number:1"},{"id":"arxiv:2406.12083","type":"article-journal","author":[{"given":"Andrey Boris","family":"Khesin"},{"given":"Alexander","family":"Li"}],"title":"Equivalence Classes of Quantum Error-Correcting Codes","issued":{"date-parts":[[2024,6,17]]},"note":"arxivid:2406.12083\narxiv_version_number:1"},{"id":"arxiv:2408.11894","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>A central ingredient in fault-tolerant quantum algorithms is the initialization of a logical state for a given quantum error-correcting code from a set of noisy qubits. A scheme that has demonstrated promising results for small code instances that are realizable on currently available hardware composes a non-fault-tolerant state preparation circuit with a verification circuit that checks for spreading errors. Known circuit constructions of this scheme are mostly obtained manually, and no algorithmic techniques for constructing depth- or gate-optimal circuits exist. As a consequence, the current state-of-the-art exploits this scheme only for specific code instances and mostly for the special case of distance <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>d</a:mi><a:mo>=</a:mo><a:mn>3</a:mn></a:math> codes only. In this work, we propose an automated approach for synthesizing fault-tolerant state preparation circuits for arbitrary CSS codes. We utilize methods based on satisfiability solving (SAT) to construct fault-tolerant state preparation circuits consisting of depth- and gate-optimal preparation and verification circuits. We also provide heuristics that can synthesize fault-tolerant state preparation circuits for code instances where no optimal solution can be obtained in an adequate time. Moreover, we give a general construction for nondeterministic state preparation circuits for codes beyond distance 3. Numerical evaluations using <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>d</c:mi><c:mo>=</c:mo><c:mn>3</c:mn></c:math>, <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>d</e:mi><e:mo>=</e:mo><e:mn>5</e:mn></e:math>, and <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:mi>d</g:mi><g:mo>=</g:mo><g:mn>7</g:mn></g:math> codes confirm that the generated circuits exhibit the desired scaling of the logical error rates. The resulting methods are publicly available as part of the  (MQT) at . Such methods are an important step in providing fault-tolerant circuit constructions that can aid in near-term demonstrations of fault-tolerant quantum computing.</jats:p>","DOI":"10.1103/prxquantum.6.020330","source":"Crossref","title":"Automated Synthesis of Fault-Tolerant State Preparation Circuits for Quantum Error-Correction Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-3434-7881","authenticated-orcid":true,"given":"Tom","family":"Peham","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"}]},{"ORCID":"https://orcid.org/0000-0002-4246-8125","authenticated-orcid":true,"given":"Ludwig","family":"Schmid","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"}]},{"ORCID":"https://orcid.org/0000-0002-2973-1689","authenticated-orcid":true,"given":"Lucas","family":"Berent","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"}]},{"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"},{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"}]},{"ORCID":"https://orcid.org/0000-0002-4993-7860","authenticated-orcid":true,"given":"Robert","family":"Wille","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"Technical University of Munich"},{"id":[{"id":"https://ror.org/02ks3nr96","id-type":"ROR","asserted-by":"publisher"}],"name":"Software Competence Center Hagenberg"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,14]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.020330","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020330","note":"update-to:{\"updated\":{\"date-parts\":[[2025,6,13]],\"date-time\":\"2025-06-13T00:00:00Z\",\"timestamp\":1749772800000},\"DOI\":\"10.1103/prxquantum.6.020330\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\nupdated-by:{\"updated\":{\"date-parts\":[[2025,6,13]],\"date-time\":\"2025-06-13T00:00:00Z\",\"timestamp\":1749772800000},\"DOI\":\"10.1103/prxquantum.6.020330\",\"type\":\"correction\",\"source\":\"publisher\",\"label\":\"Correction\"}\narxivid:2408.11894"},{"id":"arxiv:2301.13738","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We define code maps between Calderbank-Shor-Steane (CSS) codes using maps between chain complexes, and describe code surgery between such codes using a specific colimit in the category of chain complexes. As well as describing a surgery operation, this gives a general recipe for new codes. As an application we describe how to `merge' and `split' along a shared <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mover><mml:mi>X</mml:mi><mml:mo accent=\"false\">&amp;#x00AF;</mml:mo></mml:mover></mml:math> or <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mover><mml:mi>Z</mml:mi><mml:mo accent=\"false\">&amp;#x00AF;</mml:mo></mml:mover></mml:math> operator between arbitrary CSS codes in a fault-tolerant manner, so long as certain technical conditions concerning gauge fixing and code distance are satisfied. We prove that such merges and splits on LDPC codes yield codes which are themselves LDPC.</jats:p>","DOI":"10.22331/q-2024-05-14-1344","page":"1344","source":"Crossref","title":"CSS code surgery as a universal construction","volume":"8","author":[{"given":"Alexander","family":"Cowtan","sequence":"first","affiliation":[{"name":"Dept. of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, UK"},{"name":"Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, UK"}]},{"given":"Simon","family":"Burton","sequence":"additional","affiliation":[{"name":"Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, UK"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,14]]},"URL":"http://dx.doi.org/10.22331/q-2024-05-14-1344","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2301.13738"},{"id":"arxiv:2410.14631","type":"article-journal","author":[{"given":"Ting-Chun","family":"Lin"}],"title":"Transversal non-Clifford gates for quantum LDPC codes on sheaves","issued":{"date-parts":[[2024,10,18]]},"note":"arxivid:2410.14631\narxiv_version_number:1"},{"id":"arxiv:2411.15848","type":"article-journal","author":[{"given":"Po-Shen","family":"Hsin"},{"given":"Ryohei","family":"Kobayashi"},{"given":"Guanyu","family":"Zhu"}],"title":"Classifying Logical Gates in Quantum Codes via Cohomology Operations and Symmetry","issued":{"date-parts":[[2025,6,27]]},"note":"arxivid:2411.15848\narxiv_version_number:3"},{"id":"arxiv:2505.01370","type":"article-journal","author":[{"given":"Clément","family":"Poirson"},{"given":"Joschka","family":"Roffe"},{"given":"Robert I.","family":"Booth"}],"title":"Engineering CSS surgery: compiling any CNOT in any code","issued":{"date-parts":[[2025,6,27]]},"note":"arxivid:2505.01370\narxiv_version_number:3"},{"id":"arxiv:2109.00086","type":"article-journal","author":[{"given":"Toshiaki","family":"Inada"},{"given":"Wonho","family":"Jang"},{"given":"Yutaro","family":"Iiyama"},{"given":"Koji","family":"Terashi"},{"given":"Ryu","family":"Sawada"},{"given":"Junichi","family":"Tanaka"},{"given":"Shoji","family":"Asai"}],"title":"Measurement-Free Ultrafast Quantum Error Correction by Using Multi-Controlled Gates in Higher-Dimensional State Space","issued":{"date-parts":[[2021,9,6]]},"note":"arxivid:2109.00086\narxiv_version_number:2"},{"id":"arxiv:1002.1536","type":"journal-article","publisher":"American Physical Society (APS)","issue":"10","DOI":"10.1103/physrevlett.105.100501","source":"Crossref","title":"Fault Tolerance with Noisy and Slow Measurements and Preparation","volume":"105","author":[{"given":"Gerardo A.","family":"Paz-Silva","sequence":"first","affiliation":[]},{"given":"Gavin K.","family":"Brennen","sequence":"additional","affiliation":[]},{"given":"Jason","family":"Twamley","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,8,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.105.100501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"100501","note":"arxivid:1002.1536"},{"id":"arxiv:2002.11733","type":"article-journal","author":[{"given":"Albert T.","family":"Schmitz"}],"title":"Thermal Stability of Dynamical Phase Transitions in Higher Dimensional Stabilizer Codes","issued":{"date-parts":[[2020,4,12]]},"note":"arxivid:2002.11733\narxiv_version_number:2"},{"id":"arxiv:2301.11930","type":"article-journal","author":[{"given":"Yoni","family":"Choukroun"},{"given":"Lior","family":"Wolf"}],"title":"Deep Quantum Error Correction","issued":{"date-parts":[[2023,12,10]]},"note":"arxivid:2301.11930\narxiv_version_number:2"},{"id":"arxiv:2410.01673","type":"article-journal","author":[{"given":"Mohammadreza","family":"Noormandipour"},{"given":"Tobias","family":"Haug"}],"title":"MaxSAT decoders for arbitrary CSS codes","issued":{"date-parts":[[2024,10,2]]},"note":"arxivid:2410.01673\narxiv_version_number:1"},{"id":"arxiv:1603.02286","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.94.042316","source":"Crossref","title":"Transversal Clifford gates on folded surface codes","volume":"94","author":[{"given":"Jonathan E.","family":"Moussa","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,10,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.94.042316","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042316","note":"arxivid:1603.02286"},{"id":"arxiv:2406.00425","type":"article-journal","author":[{"given":"Eduardo","family":"Camps-Moreno"},{"given":"Hiram H.","family":"López"},{"given":"Gretchen L.","family":"Matthews"},{"given":"Emily","family":"McMillon"}],"title":"Toward Quantum CSS-T Codes from Sparse Matrices","issued":{"date-parts":[[2024,6,1]]},"note":"arxivid:2406.00425\narxiv_version_number:1"},{"id":"arxiv:2502.13889","type":"article-journal","author":[{"given":"Jérôme","family":"Guyot"},{"given":"Samuel","family":"Jaques"}],"title":"On the Addressability Problem on CSS Codes","issued":{"date-parts":[[2025,2,25]]},"note":"arxivid:2502.13889\narxiv_version_number:3"},{"id":"arxiv:2601.21514","type":"article-journal","author":[{"given":"Eduardo","family":"Camps-Moreno"},{"given":"Hiram H.","family":"López"},{"given":"Gretchen L.","family":"Matthews"},{"given":"Narayanan","family":"Rengaswamy"},{"given":"Rodrigo","family":"San-José"}],"title":"Transversal gates for quantum CSS codes","issued":{"date-parts":[[2026,1,29]]},"note":"arxivid:2601.21514\narxiv_version_number:1"},{"id":"arxiv:quant-ph/9611027","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.78.2252","page":"2252-2255","source":"Crossref","title":"Active Stabilization, Quantum Computation, and Quantum State Synthesis","volume":"78","author":[{"given":"A. M.","family":"Steane","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1997,3,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.78.2252","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9611027"},{"id":"arxiv:1803.09758","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.101.012342","source":"Crossref","title":"Flag fault-tolerant error correction, measurement, and quantum computation for cyclic Calderbank-Shor-Steane codes","volume":"101","author":[{"ORCID":"https://orcid.org/0000-0002-2865-0705","authenticated-orcid":true,"given":"Theerapat","family":"Tansuwannont","sequence":"first","affiliation":[]},{"given":"Christopher","family":"Chamberland","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3750-2648","authenticated-orcid":true,"given":"Debbie","family":"Leung","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,1,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.101.012342","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012342","note":"arxivid:1803.09758"},{"id":"arxiv:2208.00581","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.107.022614","source":"Crossref","title":"Parallel syndrome extraction with shared flag qubits for Calderbank-Shor-Steane codes of distance three","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0001-5377-7815","authenticated-orcid":true,"given":"Pei-Hao","family":"Liou","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1970-8167","authenticated-orcid":true,"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,2,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.022614","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022614","note":"arxivid:2208.00581"},{"id":"arxiv:2306.12862","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Lookup-table decoding is fast and distance preserving, making it attractive for near-term quantum computer architectures with small-distance quantum error-correcting codes. In this work, we develop several optimization tools that can potentially reduce the space and time overhead required for flag fault-tolerant quantum error correction (FTQEC) with lookup-table decoding on Calderbank-Shor-Steane (CSS) codes. Our techniques include the compact lookup-table construction, the meet-in-the-middle technique, the adaptive time decoding for flag FTQEC, the classical processing technique for flag information, and the separate <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mi>X</a:mi></a:math>- and <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><d:mi>Z</d:mi></d:math>-counting technique. We evaluate the performance of our tools using numerical simulation of hexagonal color codes of distances 3, 5, 7, and 9 under circuit-level noise. Combining all tools can result in an increase of more than an order of magnitude in the pseudothreshold for the hexagonal color code of distance 9, from <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><g:mo stretchy=\"false\">(</g:mo><g:mn>1.34</g:mn><g:mo>±</g:mo><g:mn>0.01</g:mn><g:mo stretchy=\"false\">)</g:mo><g:mo>×</g:mo><g:msup><g:mn>10</g:mn><g:mrow><g:mo>−</g:mo><g:mn>4</g:mn></g:mrow></g:msup></g:math> to <l:math xmlns:l=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><l:mo stretchy=\"false\">(</l:mo><l:mn>1.43</l:mn><l:mo>±</l:mo><l:mn>0.07</l:mn><l:mo stretchy=\"false\">)</l:mo><l:mo>×</l:mo><l:msup><l:mn>10</l:mn><l:mrow><l:mo>−</l:mo><l:mn>3</l:mn></l:mrow></l:msup></l:math>.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020336","source":"Crossref","title":"Optimization Tools for Distance-Preserving Flag Fault-Tolerant Error Correction","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0001-9502-3368","authenticated-orcid":true,"given":"Balint","family":"Pato","sequence":"first","affiliation":[{"name":"Duke Quantum Center, Duke University"},{"name":"Department of Electrical and Computer Engineering, Duke University"}]},{"ORCID":"https://orcid.org/0000-0002-2865-0705","authenticated-orcid":true,"given":"Theerapat","family":"Tansuwannont","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University"},{"name":"Department of Electrical and Computer Engineering, Duke University"}]},{"ORCID":"https://orcid.org/0000-0001-6731-8601","authenticated-orcid":true,"given":"Shilin","family":"Huang","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University"},{"name":"Department of Electrical and Computer Engineering, Duke University"}]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":true,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University"},{"name":"Department of Electrical and Computer Engineering, Duke University"},{"name":"Department of Physics, Duke University"},{"name":"Department of Chemistry, Duke University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,5,16]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020336","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020336","note":"arxivid:2306.12862"},{"id":"arxiv:2211.03625","type":"article-journal","author":[{"given":"Shilin","family":"Huang"},{"given":"Tomas","family":"Jochym-O'Connor"},{"given":"Theodore J.","family":"Yoder"}],"title":"Homomorphic Logical Measurements","issued":{"date-parts":[[2022,11,8]]},"note":"arxivid:2211.03625\narxiv_version_number:2"},{"id":"arxiv:2410.02753","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>We introduce homological measurement, a framework for measuring the logical Pauli operators encoded in Calderbank-Shor-Steane stabilizer codes. The framework is based on the algebraic description of such codes as chain complexes. Protocols such as lattice surgery and some of its recent generalizations are shown to be special cases of homological measurement. Using this framework, we develop a specific protocol called edge expanded homological measurement for fault-tolerant measurement of arbitrary logical Pauli operators of general quantum low density parity-check codes, requiring a number of ancillary qubits growing only linearly with the weight of the logical operator measured, and guarantee that the distance of the code is preserved. We further benchmark our protocol numerically in a photonic architecture based on Gottesman-Kitaev-Preskill qubits, showing that the logical error rates of various codes are on par with other methods requiring more ancilla qubits.</jats:p>","DOI":"10.1103/physrevx.15.021088","source":"Crossref","title":"Fault-Tolerant Logical Measurements via Homological Measurement","volume":"15","author":[{"ORCID":"https://orcid.org/0009-0008-6795-1552","authenticated-orcid":true,"given":"Benjamin","family":"Ide","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"ORCID":"https://orcid.org/0000-0002-6298-7037","authenticated-orcid":true,"given":"Manoj G.","family":"Gowda","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"ORCID":"https://orcid.org/0000-0002-1351-2959","authenticated-orcid":true,"given":"Priya J.","family":"Nadkarni","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]},{"ORCID":"https://orcid.org/0009-0004-5704-2436","authenticated-orcid":true,"given":"Guillaume","family":"Dauphinais","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qj0n416","id-type":"ROR","asserted-by":"publisher"}],"name":"Xanadu"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.15.021088","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021088","note":"arxivid:2410.02753"},{"id":"arxiv:1708.09156","type":"book-chapter","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-70694-8_16","page":"438-467","source":"Crossref","title":"Quantum Fully Homomorphic Encryption with Verification","author":[{"given":"Gorjan","family":"Alagic","sequence":"first","affiliation":[]},{"given":"Yfke","family":"Dulek","sequence":"additional","affiliation":[]},{"given":"Christian","family":"Schaffner","sequence":"additional","affiliation":[]},{"given":"Florian","family":"Speelman","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2017]]},"ISBN":["9783319706931","9783319706948"],"URL":"http://dx.doi.org/10.1007/978-3-319-70694-8_16","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Cham\narxivid:1708.09156"},{"id":"arxiv:2107.05692","type":"article-journal","author":[{"given":"Andrea","family":"Coladangelo"},{"given":"Jiahui","family":"Liu"},{"given":"Qipeng","family":"Liu"},{"given":"Mark","family":"Zhandry"}],"title":"Hidden Cosets and Applications to Unclonable Cryptography","issued":{"date-parts":[[2022,7,14]]},"note":"arxivid:2107.05692\narxiv_version_number:3"},{"id":"arxiv:1310.5376","type":"article-journal","author":[{"given":"Martin","family":"Leslie"}],"title":"Hypermap-Homology Quantum Codes (Ph.D. thesis)","issued":{"date-parts":[[2013,10,20]]},"note":"arxivid:1310.5376\narxiv_version_number:1"},{"id":"preset:BrowneNotes","type":"document","author":[{"family":"Browne","given":"D."}],"title":"\\href{https://web.archive.org/web/20240324204356/https://sites.google.com/site/danbrowneucl/teaching/lectures-on-topological-codes-and-quantum-computation}{Lecture notes}"},{"id":"arxiv:quant-ph/0003004","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.85.441","page":"441-444","source":"Crossref","title":"Simple Proof of Security of the BB84 Quantum Key Distribution Protocol","volume":"85","author":[{"given":"Peter W.","family":"Shor","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2000,7,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.85.441","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/0003004"},{"id":"arxiv:0809.3629","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.79.032325","source":"Crossref","title":"Quantum repeater with encoding","volume":"79","author":[{"given":"Liang","family":"Jiang","sequence":"first","affiliation":[]},{"given":"J. M.","family":"Taylor","sequence":"additional","affiliation":[]},{"given":"Kae","family":"Nemoto","sequence":"additional","affiliation":[]},{"given":"W. J.","family":"Munro","sequence":"additional","affiliation":[]},{"given":"Rodney","family":"Van Meter","sequence":"additional","affiliation":[]},{"given":"M. D.","family":"Lukin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.032325","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032325","note":"arxivid:0809.3629"},{"id":"doi:10.1007/s11128-006-0047-9","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s11128-006-0047-9","page":"143-158","source":"Crossref","title":"Nonbinary Quantum Goppa Codes Exceeding the Quantum Gilbert-Varshamov Bound","volume":"6","author":[{"given":"Annika","family":"Niehage","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2006,12,12]]},"URL":"http://dx.doi.org/10.1007/s11128-006-0047-9","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","note":"alternative-id:47"},{"id":"arxiv:2404.18302","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce60285.2024.00022","page":"109-116","source":"Crossref","title":"GNarsil: Splitting Stabilizers into Gauges","author":[{"given":"Oskar","family":"Novak","sequence":"first","affiliation":[{"name":"Wyant College of Optical Sciences, University of Arizona,Tucson,USA,85721"}]},{"given":"Narayanan","family":"Rengaswamy","sequence":"additional","affiliation":[{"name":"University of Arizona,Department of Electrical and Computer Engineering,Tucson,USA,85721"}]}],"event":"2024 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2024 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2024,9,15]]},"URL":"http://dx.doi.org/10.1109/QCE60285.2024.00022","note":"arxivid:2404.18302"},{"id":"arxiv:0707.4000","type":"article-journal","author":[{"given":"D.","family":"Gross"},{"given":"M. Van den","family":"Nest"}],"title":"The LU-LC conjecture, diagonal local operations and quadratic forms over GF(2)","issued":{"date-parts":[[2007,7,31]]},"note":"arxivid:0707.4000\narxiv_version_number:2"},{"id":"arxiv:quant-ph/0408190","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.71.042315","source":"Crossref","title":"Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic","volume":"71","author":[{"given":"Erik","family":"Hostens","sequence":"first","affiliation":[]},{"given":"Jeroen","family":"Dehaene","sequence":"additional","affiliation":[]},{"given":"Bart","family":"De Moor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,4,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.71.042315","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042315","note":"arxivid:quant-ph/0408190"},{"id":"arxiv:1711.07848","type":"article-journal","author":[{"given":"Héctor J.","family":"García"},{"given":"Igor L.","family":"Markov"},{"given":"Andrew W.","family":"Cross"}],"title":"On the Geometry of Stabilizer States","issued":{"date-parts":[[2017,11,20]]},"note":"arxivid:1711.07848\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0410165","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.72.014307","source":"Crossref","title":"Finite set of invariants to characterize local Clifford equivalence of stabilizer states","volume":"72","author":[{"given":"Maarten","family":"Van den Nest","sequence":"first","affiliation":[]},{"given":"Jeroen","family":"Dehaene","sequence":"additional","affiliation":[]},{"given":"Bart","family":"De Moor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,7,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.72.014307","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"014307","note":"arxivid:quant-ph/0410165"},{"id":"arxiv:quant-ph/0404106","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.70.032323","source":"Crossref","title":"Local invariants of stabilizer codes","volume":"70","author":[{"given":"Maarten","family":"Van den Nest","sequence":"first","affiliation":[]},{"given":"Jeroen","family":"Dehaene","sequence":"additional","affiliation":[]},{"given":"Bart","family":"De Moor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,9,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.70.032323","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032323","note":"arxivid:quant-ph/0404106"},{"id":"arxiv:1307.7025","type":"journal-article","publisher":"IOP Publishing","issue":"9","DOI":"10.1088/1367-2630/16/9/093021","page":"093021","source":"Crossref","title":"The ZX-calculus is complete for stabilizer quantum mechanics","volume":"16","author":[{"given":"Miriam","family":"Backens","sequence":"first","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2014,9,17]]},"URL":"http://dx.doi.org/10.1088/1367-2630/16/9/093021","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1307.7025"},{"id":"arxiv:2205.02009","type":"journal-article","publisher":"Open Publishing Association","DOI":"10.4204/eptcs.394.5","page":"66-82","source":"Crossref","title":"Complete Flow-Preserving Rewrite Rules for MBQC Patterns with Pauli Measurements","volume":"394","author":[{"given":"Tommy","family":"McElvanney","sequence":"first","affiliation":[{"name":"University of Birmingham"}]},{"given":"Miriam","family":"Backens","sequence":"additional","affiliation":[{"name":"University of Birmingham"}]}],"container-title":"Electronic Proceedings in Theoretical Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,16]]},"URL":"http://dx.doi.org/10.4204/EPTCS.394.5","ISSN":["2075-2180"],"container-title-short":"Electron. Proc. Theor. Comput. Sci.","note":"arxivid:2205.02009"},{"id":"arxiv:2109.10210","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.105.022432","source":"Crossref","title":"Improved graph formalism for quantum circuit simulation","volume":"105","author":[{"given":"Alexander Tianlin","family":"Hu","sequence":"first","affiliation":[{"name":"The Harker School, San Jose, California 95129, USA"}]},{"given":"Andrey Boris","family":"Khesin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.022432","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022432","note":"arxivid:2109.10210"},{"id":"arxiv:2308.15140","type":"journal-article","publisher":"The Open Journal","issue":"71","DOI":"10.21105/joss.04120","page":"4120","source":"Crossref","title":"QDistRnd: A GAP package for computing the distance of\nquantum error-correcting codes","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-4990-0259","authenticated-orcid":false,"given":"Leonid P.","family":"Pryadko","sequence":"first","affiliation":[]},{"given":"Vadim A.","family":"Shabashov","sequence":"additional","affiliation":[]},{"given":"Valerii K.","family":"Kozin","sequence":"additional","affiliation":[]}],"container-title":"Journal of Open Source Software","original-title":[],"issued":{"date-parts":[[2022,3,22]]},"URL":"http://dx.doi.org/10.21105/joss.04120","ISSN":["2475-9066"],"container-title-short":"JOSS","note":"alternative-id:10.21105/joss.04120\narxivid:2308.15140"},{"id":"arxiv:2408.10743","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"2","abstract":"<jats:p>The distance of a stabilizer quantum code is a very important feature since it determines the number of errors that can be detected and corrected. We present three new fast algorithms and implementations for computing the symplectic distance of the associated classical code. Our new algorithms are based on the Brouwer–Zimmermann algorithm. Our experimental study shows that these new implementations are much faster than current state-of-the-art licensed implementations on single-core processors, multicore processors, and shared-memory multiprocessors. In the most computationally-demanding cases, the performance gain in the computational time can be larger than one order of magnitude. The experimental study also shows a good scalability on shared-memory parallel architectures.</jats:p>","DOI":"10.1145/3795877","page":"1-19","source":"Crossref","title":"Fast Algorithms and Implementations for Computing the Minimum Distance of Quantum Codes","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-9758-2152","authenticated-orcid":false,"given":"Fernando","family":"Hernando","sequence":"first","affiliation":[{"name":"Dept. of Mathematics, Universitat Jaume I","place":["Castelló de la Plana, Spain"]}]},{"ORCID":"https://orcid.org/0000-0002-7912-7826","authenticated-orcid":false,"given":"Gregorio","family":"Quintana-Ortí","sequence":"additional","affiliation":[{"name":"Depto. de Ingeniería y Ciencia de Computadores, Universitat Jaume I","place":["Castelló de la Plana, Spain"]}]},{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk","place":["Gdansk, Poland"]}]}],"container-title":"ACM Transactions on Quantum Computing","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,2]]},"URL":"http://dx.doi.org/10.1145/3795877","ISSN":["2643-6809","2643-6817"],"container-title-short":"ACM Trans. Quantum Comput.","note":"alternative-id:10.1145/3795877\narxivid:2408.10743"},{"id":"arxiv:2409.13017","type":"article-journal","author":[{"given":"Mark","family":"Webster"},{"given":"Dan","family":"Browne"}],"title":"Engineering Quantum Error Correction Codes Using Evolutionary Algorithms","issued":{"date-parts":[[2024,9,19]]},"note":"arxivid:2409.13017\narxiv_version_number:1"},{"id":"arxiv:2203.04262","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2023.3286870","page":"6293-6302","source":"Crossref","title":"On the Hardness of the Minimum Distance Problem of Quantum Codes","volume":"69","author":[{"given":"Upendra","family":"Kapshikar","sequence":"first","affiliation":[{"name":"Centre of Quantum Technologies, National University of Singapore, Queenstown, Singapore"}]},{"ORCID":"https://orcid.org/0000-0002-8630-0113","authenticated-orcid":false,"given":"Srijita","family":"Kundu","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Canada"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2023,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2023.3286870","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2203.04262"},{"id":"arxiv:quant-ph/0604107","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.98.030501","source":"Crossref","title":"Degenerate Quantum Codes for Pauli Channels","volume":"98","author":[{"given":"Graeme","family":"Smith","sequence":"first","affiliation":[]},{"given":"John A.","family":"Smolin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,1,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.98.030501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030501","note":"arxivid:quant-ph/0604107"},{"id":"arxiv:2204.04699","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.laa.2022.05.002","page":"96-121","source":"Crossref","title":"A linear-algebraic and lattice-theoretical look at the Cleaning Lemma of quantum coding theory","volume":"649","author":[{"ORCID":"https://orcid.org/0000-0003-2695-3179","authenticated-orcid":false,"given":"Gleb","family":"Kalachev","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9069-6894","authenticated-orcid":false,"given":"Sergey","family":"Sadov","sequence":"additional","affiliation":[]}],"container-title":"Linear Algebra and its Applications","original-title":[],"language":"en","issued":{"date-parts":[[2022,9]]},"URL":"http://dx.doi.org/10.1016/j.laa.2022.05.002","ISSN":["0024-3795"],"container-title-short":"Linear Algebra and its Applications","note":"special_numbering:C\nalternative-id:S0024379522001896\narxivid:2204.04699"},{"id":"arxiv:2509.16887","type":"article-journal","author":[{"given":"Alex","family":"Kwiatkowski"},{"given":"Aaron J.","family":"Friedman"},{"given":"Shawn","family":"Geller"},{"given":"Jalan A.","family":"Ziyad"},{"given":"Scott","family":"Glancy"},{"given":"Emanuel","family":"Knill"}],"title":"Constructing an approximate logical Markovian model of consecutive QEC cycles of a stabilizer code","issued":{"date-parts":[[2025,9,21]]},"note":"arxivid:2509.16887\narxiv_version_number:1"},{"id":"arxiv:2601.15505","type":"article-journal","author":[{"given":"Tyler","family":"Kann"},{"given":"Matthieu R.","family":"Bloch"},{"given":"Shrinivas","family":"Kudekar"},{"given":"Ruediger","family":"Urbanke"}],"title":"Stabilizer-Code Channel Transforms Beyond Repetition Codes for Improved Hashing Bounds","issued":{"date-parts":[[2026,1,24]]},"note":"arxivid:2601.15505\narxiv_version_number:2"},{"id":"arxiv:2201.05215","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/tqe.2022.3180900","page":"1-8","source":"Crossref","title":"Depth Optimization of CZ, CNOT, and Clifford Circuits","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0001-7381-4556","authenticated-orcid":false,"given":"Dmitri","family":"Maslov","sequence":"first","affiliation":[{"name":"IBM Quantum, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA"}]},{"ORCID":"https://orcid.org/0000-0001-8630-3501","authenticated-orcid":false,"given":"Ben","family":"Zindorf","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, USA"}]}],"container-title":"IEEE Transactions on Quantum Engineering","original-title":[],"issued":{"date-parts":[[2022]]},"URL":"http://dx.doi.org/10.1109/TQE.2022.3180900","ISSN":["2689-1808"],"container-title-short":"IEEE Trans. Quantum Eng.","note":"arxivid:2201.05215"},{"id":"manual:-T.-J.-Yoder-A-generalization","type":"article-journal","author":[{"family":"Yoder","given":"T.J."}],"title":"A generalization of the stabilizer formalism for simulating arbitrary quantum circuits","volume":"l","container-title":"See","issue":"ttps://www.scottaaronson.com/showcase2/report/ted-yoder.pdf","issued":"2012"},{"id":"arxiv:0803.3232","type":"journal-article","publisher":"AIP Publishing","issue":"4","abstract":"<jats:p>The codeword stabilized (CWS) quantum code formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes [IEEE Trans. Inf. Theory 55, 433 (2009)]. This formalism reduces the problem of constructing such quantum codes to finding a binary classical code correcting an error pattern induced by a graph state. Finding such a classical code can be very difficult. Here, we consider an algorithm which maps the search for CWS codes to a problem of identifying maximum cliques in a graph. While solving this problem is in general very hard, we provide three structure theorems which reduce the search space, specifying certain admissible and optimal ((n,K,d)) additive codes. In particular, we find that the re does not exist any ((7,3,3)) CWS code though the linear programming bound does not rule it out. The complexity of the CWS-search algorithm is compared with the contrasting method introduced by Aggarwal and Calderbank [IEEE Trans. Inf. Theory 54, 1700 (2008)].</jats:p>","DOI":"10.1063/1.3086833","source":"Crossref","title":"Codeword stabilized quantum codes: Algorithm and structure","volume":"50","author":[{"given":"Isaac","family":"Chuang","sequence":"first","affiliation":[{"name":"Massachusetts Institute of Technology 1 Department of Electric Engineering and Computer Science, , Cambridge, Massachusetts 02139, USA"},{"name":"Massachusetts Institute of Technology 2 Department of Physics, , Cambridge, Massachusetts 02139, USA"}]},{"given":"Andrew","family":"Cross","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology 1 Department of Electric Engineering and Computer Science, , Cambridge, Massachusetts 02139, USA"},{"name":"IBM T.J. Watson Research Center 3 , Yorktown Heights, New York 10598, USA"}]},{"given":"Graeme","family":"Smith","sequence":"additional","affiliation":[{"name":"IBM T.J. Watson Research Center 3 , Yorktown Heights, New York 10598, USA"}]},{"given":"John","family":"Smolin","sequence":"additional","affiliation":[{"name":"IBM T.J. Watson Research Center 3 , Yorktown Heights, New York 10598, USA"}]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[{"name":"Massachusetts Institute of Technology 2 Department of Physics, , Cambridge, Massachusetts 02139, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2009,4,1]]},"URL":"http://dx.doi.org/10.1063/1.3086833","ISSN":["0022-2488","1089-7658"],"page":"042109","note":"arxivid:0803.3232"},{"id":"arxiv:quant-ph/9707049","type":"article-journal","author":[{"given":"Juan Pablo","family":"Paz"},{"given":"Wojciech Hubert","family":"Zurek"}],"title":"Continuous Error Correction","issued":{"date-parts":[[1997,7,25]]},"note":"arxivid:quant-ph/9707049\narxiv_version_number:1"},{"id":"arxiv:1310.1036","type":"journal-article","publisher":"IOP Publishing","issue":"1","DOI":"10.1088/1367-2630/16/1/013023","page":"013023","source":"Crossref","title":"An optimal dissipative encoder for the toric code","volume":"16","author":[{"given":"John","family":"Dengis","sequence":"first","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]},{"given":"Fernando","family":"Pastawski","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2014,1,16]]},"URL":"http://dx.doi.org/10.1088/1367-2630/16/1/013023","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1310.1036"},{"id":"arxiv:1310.1037","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevb.90.045101","source":"Crossref","title":"Generating topological order: No speedup by dissipation","volume":"90","author":[{"given":"Robert","family":"König","sequence":"first","affiliation":[]},{"given":"Fernando","family":"Pastawski","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2014,7,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.90.045101","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"045101","note":"arxivid:1310.1037"},{"id":"arxiv:2102.04531","type":"journal-article","publisher":"Rinton Press","issue":"9-10","abstract":"<jats:p>We formalize the problem of dissipative quantum encoding, and explore the advantages of using Markovian evolution to prepare a quantum code in the desired logical space, with emphasis on discrete-time dynamics and the possibility of exact finite-time convergence. In particular, we investigate robustness of the encoding dynamics and their ability to tolerate initialization errors, thanks to the existence of non-trivial basins of attraction.  As a key application, we show that for stabilizer quantum codes on qubits, a finite-time dissipative encoder may always be constructed, by using at most a number of quantum maps determined by the number of stabilizer generators. We find that even in situations where the target code lacks gauge degrees of freedom in its subsystem form, dissipative encoders afford nontrivial robustness against initialization errors, thus overcoming a limitation of purely unitary encoding procedures. Our general results are illustrated in a number of relevant examples, including Kitaev's toric code.</jats:p>","DOI":"10.26421/qic21.9-10-2","page":"737-770","source":"Crossref","title":"Dissipative encoding of quantum information","volume":"21","author":[{"given":"Giacomo","family":"Baggio","sequence":"first","affiliation":[]},{"given":"Francesco","family":"Ticozzi","sequence":"additional","affiliation":[]},{"given":"Peter D.","family":"Johnson","sequence":"additional","affiliation":[]},{"given":"Lorenza","family":"Viola","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2021,8]]},"URL":"http://dx.doi.org/10.26421/QIC21.9-10-2","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC","note":"arxivid:2102.04531"},{"id":"arxiv:2501.10291","type":"article-journal","author":[{"given":"Nicholas","family":"Fazio"},{"given":"Mark","family":"Webster"},{"given":"Zhenyu","family":"Cai"}],"title":"Low-overhead Magic State Circuits with Transversal CNOTs","issued":{"date-parts":[[2025,10,28]]},"note":"arxivid:2501.10291\narxiv_version_number:2"},{"id":"arxiv:2502.01393","type":"article-journal","author":[{"given":"Chandrima B.","family":"Pushpan"},{"given":"Tanoy Kanti","family":"Konar"},{"given":"Aditi Sen","family":"De"},{"given":"Amit Kumar","family":"Pal"}],"title":"Developing universal logical state-purification strategy for quantum error correcting codes","issued":{"date-parts":[[2025,2,3]]},"note":"arxivid:2502.01393\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0002039","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.62.052316","source":"Crossref","title":"Methodology for quantum logic gate construction","volume":"62","author":[{"given":"Xinlan","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Debbie W.","family":"Leung","sequence":"additional","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,10,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.62.052316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052316","note":"arxivid:quant-ph/0002039"},{"id":"arxiv:quant-ph/9806094","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.59.141","page":"141-155","source":"Crossref","title":"Robustness of entanglement","volume":"59","author":[{"given":"Guifré","family":"Vidal","sequence":"first","affiliation":[]},{"given":"Rolf","family":"Tarrach","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1999,1,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.59.141","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9806094"},{"id":"arxiv:1807.10296","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We give a new algorithm for computing the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">robustness of magic</mml:mtext></mml:mrow></mml:math>- a measure of the utility of quantum states as a computational resource. Our work is motivated by the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">magic state model</mml:mtext></mml:mrow></mml:math>of fault-tolerant quantum computation. In this model, all unitaries belong to the Clifford group. Non-Clifford operations are effected by injecting non-stabiliser states, which are referred to as<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">magic states</mml:mtext></mml:mrow></mml:math>in this context. The<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">robustness of magic</mml:mtext></mml:mrow></mml:math>measures the complexity of simulating such a circuit using a classical Monte Carlo algorithm. It is closely related to the degree negativity that slows down Monte Carlo simulations through the infamous<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">sign problem</mml:mtext></mml:mrow></mml:math>. Surprisingly, the robustness of magic is<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">sub</mml:mtext></mml:mrow></mml:math>- multiplicative. This implies that the classical simulation overhead scales subexponentially with the number of injected magic states - better than a naive analysis would suggest. However, determining the robustness of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">n</mml:mtext></mml:mrow></mml:math>copies of a magic state is difficult, as its definition involves a convex optimisation problem in a 4<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:msup><mml:mi/><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:math>-dimensional space. In this paper, we make use of inherent symmetries to reduce the problem to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">n</mml:mtext></mml:mrow></mml:math>dimensions. The total run-time of our algorithm, while still exponential in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">n</mml:mtext></mml:mrow></mml:math>, is super-polynomially faster than previously published methods. We provide a computer implementation and give the robustness of up to 10 copies of the most commonly used magic states. Guided by the exact results, we find a finite hierarchy of approximate solutions where each level can be evaluated in polynomial time and yields rigorous upper bounds to the robustness. Technically, we use symmetries of the stabiliser polytope to connect the robustness of magic to the geometry of a low-dimensional convex polytope generated by certain<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">signed quantum weight enumerators</mml:mtext></mml:mrow></mml:math>. As a by-product, we characterised the automorphism group of the stabiliser polytope, and, more generally, of projections onto complex projective 3-designs.</jats:p>","DOI":"10.22331/q-2019-04-08-132","page":"132","source":"Crossref","title":"Robustness of Magic and Symmetries of the Stabiliser Polytope","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-1334-7885","authenticated-orcid":false,"given":"Markus","family":"Heinrich","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany"}]},{"given":"David","family":"Gross","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,4,8]]},"URL":"http://dx.doi.org/10.22331/q-2019-04-08-132","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1807.10296"},{"id":"arxiv:1901.03322","type":"journal-article","publisher":"The Royal Society","issue":"2227","abstract":"<jats:p>The development of a framework for quantifying ‘non-stabilizerness’ of quantum operations is motivated by the magic state model of fault-tolerant quantum computation and by the need to estimate classical simulation cost for noisy intermediate-scale quantum (NISQ) devices. The robustness of magic was recently proposed as a well-behaved magic monotone for multi-qubit states and quantifies the simulation overhead of circuits composed of Clifford + <jats:italic>T</jats:italic>gates, or circuits using other gates from the Clifford hierarchy. Here we present a general theory of the ‘non-stabilizerness’ of quantum operations rather than states, which are useful for classical simulation of more general circuits. We introduce two magic monotones, called channel robustness and magic capacity, which are well-defined for general<jats:italic>n</jats:italic>-qubit channels and treat all stabilizer-preserving CPTP maps as free operations. We present two complementary Monte Carlo-type classical simulation algorithms with sample complexity given by these quantities and provide examples of channels where the complexity of our algorithms is exponentially better than previously known simulators. We present additional techniques that ease the difficulty of calculating our monotones for special classes of channels.</jats:p>","DOI":"10.1098/rspa.2019.0251","page":"20190251","source":"Crossref","title":"Quantifying magic for multi-qubit operations","volume":"475","author":[{"ORCID":"https://orcid.org/0000-0002-6059-4125","authenticated-orcid":true,"given":"James R.","family":"Seddon","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London, UK"}]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Sheffield, Sheffield, UK"}]}],"container-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2019,7]]},"URL":"http://dx.doi.org/10.1098/rspa.2019.0251","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. A.","note":"alternative-id:10.1098/rspa.2019.0251\narxivid:1901.03322"},{"id":"arxiv:quant-ph/9904108","type":"proceedings-article","publisher":"ACM","DOI":"10.1145/335305.335402","page":"688-696","source":"Crossref","title":"Self-testing of universal and fault-tolerant sets of quantum gates","author":[{"given":"Wim","family":"van Dam","sequence":"first","affiliation":[{"name":"C.W.I. Amsterdam; Centre for Quantum Computation, University of Oxford"}]},{"given":"Frédéic","family":"Magniez","sequence":"additional","affiliation":[{"name":"Université Paris Sud, LRI"}]},{"given":"Michele","family":"Mosca","sequence":"additional","affiliation":[{"name":"University of Waterloo; Centre for Quantum Computation, University of Oxford"}]},{"given":"Miklos","family":"Santha","sequence":"additional","affiliation":[{"name":"CNRS, Université Paris Sud, LRI"}]}],"event":"STOC00: The 32nd Annual ACM Symposium on Theory of Computing","container-title":"Proceedings of the thirty-second annual ACM symposium on Theory of computing","original-title":[],"issued":{"date-parts":[[2000,5]]},"URL":"http://dx.doi.org/10.1145/335305.335402","note":"publisher-location:New York, NY, USA\nalternative-id:10.1145/335305.335402\n10.1145/335305\narxivid:quant-ph/9904108"},{"id":"arxiv:2312.03515","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We consider models of quantum computation that involve operations performed on some fixed resourceful quantum state. Examples that fit this paradigm include magic state injection and measurement-based approaches. We introduce a framework that incorporates both of these cases and focus on the role of coherence (or superposition) in this context, as exemplified through the Hadamard gate. We prove that given access to incoherent unitaries (those that are unable to generate superposition from computational basis states, e.g. CNOT, diagonal gates), classical control, computational basis measurements, and any resourceful ancillary state (of arbitrary dimension), it is not possible to implement any coherent unitary (e.g. Hadamard) exactly with non-zero probability. We also consider the approximate case by providing lower bounds for the induced trace distance between the above operations and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> Hadamard gates. To demonstrate the stability of this result, this is then extended to a similar no-go result for the case of using <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math> Hadamard gates to exactly implement <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mi>k</mml:mi></mml:math> Hadamard gates.</jats:p>","DOI":"10.22331/q-2024-09-11-1470","page":"1470","source":"Crossref","title":"The Hadamard gate cannot be replaced by a resource state in universal quantum computation","volume":"8","author":[{"given":"Benjamin D. M.","family":"Jones","sequence":"first","affiliation":[{"name":"H. H. Wills Physics Laboratory, University of Bristol, Bristol, BS8 1TL, UK."},{"name":"School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK."},{"name":"Quantum Engineering Centre for Doctoral Training, University of Bristol, Bristol, BS8 1FD UK."}]},{"given":"Noah","family":"Linden","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK."}]},{"given":"Paul","family":"Skrzypczyk","sequence":"additional","affiliation":[{"name":"H. H. Wills Physics Laboratory, University of Bristol, Bristol, BS8 1TL, UK."},{"name":"CIFAR Azrieli Global Scholars Program, CIFAR, Toronto Canada."}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,11]]},"URL":"http://dx.doi.org/10.22331/q-2024-09-11-1470","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2312.03515"},{"id":"arxiv:quant-ph/9905027","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.63.052314","source":"Crossref","title":"Toward fault-tolerant quantum computation without concatenation","volume":"63","author":[{"given":"Eric","family":"Dennis","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2001,4,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.63.052314","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052314","note":"arxivid:quant-ph/9905027"},{"id":"arxiv:0908.0836","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.104.030503","source":"Crossref","title":"Bound States for Magic State Distillation in Fault-Tolerant Quantum Computation","volume":"104","author":[{"given":"Earl T.","family":"Campbell","sequence":"first","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,1,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.104.030503","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030503","note":"arxivid:0908.0836"},{"id":"arxiv:0908.0838","type":"article-journal","author":[{"given":"Earl T.","family":"Campbell"},{"given":"Dan E.","family":"Browne"}],"title":"On the Structure of Protocols for Magic State Distillation","issued":{"date-parts":[[2009,8,19]]},"note":"arxivid:0908.0838\narxiv_version_number:2"},{"id":"arxiv:0806.0875","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.102.070502","source":"Crossref","title":"Fault-Tolerant Holonomic Quantum Computation","volume":"102","author":[{"given":"Ognyan","family":"Oreshkov","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,2,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.102.070502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070502","note":"arxivid:0806.0875"},{"id":"arxiv:0812.4682","type":"article-journal","author":[{"given":"Ognyan","family":"Oreshkov"}],"title":"Topics in quantum information and the theory of open quantum systems","issued":{"date-parts":[[2008,12,27]]},"note":"arxivid:0812.4682\narxiv_version_number:1"},{"id":"arxiv:0904.2143","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.80.022325","source":"Crossref","title":"Scheme for fault-tolerant holonomic computation on stabilizer codes","volume":"80","author":[{"given":"Ognyan","family":"Oreshkov","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]},{"given":"Daniel A.","family":"Lidar","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.022325","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022325","note":"arxivid:0904.2143"},{"id":"arxiv:1312.0165","type":"article-journal","author":[{"given":"Ognyan","family":"Oreshkov"},{"given":"Todd A.","family":"Brun"},{"given":"Daniel A.","family":"Lidar"}],"title":"Fault tolerance for holonomic quantum computation","issued":{"date-parts":[[2013,12,1]]},"note":"arxivid:1312.0165\narxiv_version_number:1"},{"id":"arxiv:1403.5280","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.91.042315","source":"Crossref","title":"Reducing the quantum-computing overhead with complex gate distillation","volume":"91","author":[{"given":"Guillaume","family":"Duclos-Cianci","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.042315","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042315","note":"arxivid:1403.5280"},{"id":"arxiv:2404.19408","type":"article-journal","author":[{"given":"Brendan","family":"Reid"}],"title":"A simple method for compiling quantum stabilizer circuits","issued":{"date-parts":[[2024,4,30]]},"note":"arxivid:2404.19408\narxiv_version_number:1"},{"id":"arxiv:2408.01339","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.134.070602","source":"Crossref","title":"Time-Efficient Logical Operations on Quantum Low-Density Parity Check Codes","volume":"134","author":[{"given":"Guo","family":"Zhang","sequence":"first","affiliation":[{"name":"China Academy of Engineering Physics"}]},{"ORCID":"https://orcid.org/0000-0002-1705-2494","authenticated-orcid":true,"given":"Ying","family":"Li","sequence":"additional","affiliation":[{"name":"China Academy of Engineering Physics"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.070602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070602","note":"arxivid:2408.01339"},{"id":"arxiv:1710.03636","type":"journal-article","publisher":"IOP Publishing","issue":"12","DOI":"10.1088/1367-2630/aa916e","page":"123032","source":"Crossref","title":"Learning time-dependent noise to reduce logical errors: real time error rate estimation in quantum error correction","volume":"19","author":[{"given":"Ming-Xia","family":"Huo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1705-2494","authenticated-orcid":false,"given":"Ying","family":"Li","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2017,12,14]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aa916e","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1710.03636"},{"id":"arxiv:2409.14283","type":"article-journal","author":[{"given":"Suhas","family":"Vittal"},{"given":"Ali","family":"Javadi-Abhari"},{"given":"Andrew W.","family":"Cross"},{"given":"Lev S.","family":"Bishop"},{"given":"Moinuddin","family":"Qureshi"}],"title":"Flag Proxy Networks: Tackling the Architectural, Scheduling, and Decoding Obstacles of Quantum LDPC codes","issued":{"date-parts":[[2024,9,22]]},"note":"arxivid:2409.14283\narxiv_version_number:1"},{"id":"arxiv:2410.12527","type":"article-journal","author":[{"given":"Julio C. Magdalena","family":"de la Fuente"}],"title":"Dynamical weight reduction of Pauli measurements","issued":{"date-parts":[[2024,10,16]]},"note":"arxivid:2410.12527\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0402017","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.69.052324","source":"Crossref","title":"Practical scheme for error control using feedback","volume":"69","author":[{"given":"Mohan","family":"Sarovar","sequence":"first","affiliation":[]},{"given":"Charlene","family":"Ahn","sequence":"additional","affiliation":[]},{"given":"Kurt","family":"Jacobs","sequence":"additional","affiliation":[]},{"given":"Gerard J.","family":"Milburn","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,5,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.69.052324","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052324","note":"arxivid:quant-ph/0402017"},{"id":"arxiv:1009.1319","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.83.052331","source":"Crossref","title":"NP-hardness of decoding quantum error-correction codes","volume":"83","author":[{"given":"Min-Hsiu","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"François","family":"Le Gall","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,5,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.052331","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"052331","note":"arxivid:1009.1319"},{"id":"manual:-K.-Y.-Kuo-and-C.-C.-Lu-On-th","type":"paper-conference","author":[{"family":"Kuo","given":"K.-Y."},{"family":"Lu","given":"C.-C."}],"title":"On the hardness of decoding quantum stabilizer codes under the depolarizing channel","container-title":"2012 International Symposium on Information Theory and its Applications","publisher":"IEEE","issued":"2012"},{"id":"arxiv:1310.3235","type":"article-journal","author":[{"given":"Pavithran","family":"Iyer"},{"given":"David","family":"Poulin"}],"title":"Hardness of decoding quantum stabilizer codes","issued":{"date-parts":[[2013,10,11]]},"note":"arxivid:1310.3235\narxiv_version_number:1"},{"id":"arxiv:2410.18953","type":"article-journal","author":[{"given":"Alexander","family":"Poremba"},{"given":"Yihui","family":"Quek"},{"given":"Peter","family":"Shor"}],"title":"The Learning Stabilizers with Noise problem","issued":{"date-parts":[[2025,4,14]]},"note":"arxivid:2410.18953\narxiv_version_number:5"},{"id":"arxiv:2509.20697","type":"article-journal","author":[{"given":"Andrey Boris","family":"Khesin"},{"given":"Jonathan Z.","family":"Lu"},{"given":"Alexander","family":"Poremba"},{"given":"Akshar","family":"Ramkumar"},{"given":"Vinod","family":"Vaikuntanathan"}],"title":"Average-Case Complexity of Quantum Stabilizer Decoding","issued":{"date-parts":[[2025,9,25]]},"note":"arxivid:2509.20697\narxiv_version_number:1"},{"id":"arxiv:2309.15354","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Adam","family":"Paetznick"},{"given":"Jeongwan","family":"Haah"},{"given":"Matthew B.","family":"Hastings"}],"title":"Splitting decoders for correcting hypergraph faults","issued":{"date-parts":[[2023,9,27]]},"note":"arxivid:2309.15354\narxiv_version_number:1"},{"id":"arxiv:2208.02744","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/access.2023.3327214","page":"119446-119461","source":"Crossref","title":"Quantum Error Correction Via Noise Guessing Decoding","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0002-8678-5456","authenticated-orcid":false,"given":"Diogo","family":"Cruz","sequence":"first","affiliation":[{"name":"Instituto de Telecomunica&#x00E7;&#x00F5;es, Lisbon, Portugal"}]},{"ORCID":"https://orcid.org/0000-0003-2381-6320","authenticated-orcid":false,"given":"Francisco A.","family":"Monteiro","sequence":"additional","affiliation":[{"name":"Instituto de Telecomunica&#x00E7;&#x00F5;es, Lisbon, Portugal"}]},{"ORCID":"https://orcid.org/0000-0002-9980-1857","authenticated-orcid":false,"given":"Bruno C.","family":"Coutinho","sequence":"additional","affiliation":[{"name":"Instituto de Telecomunica&#x00E7;&#x00F5;es, Lisbon, Portugal"}]}],"container-title":"IEEE Access","original-title":[],"issued":{"date-parts":[[2023]]},"URL":"http://dx.doi.org/10.1109/ACCESS.2023.3327214","ISSN":["2169-3536"],"container-title-short":"IEEE Access","note":"arxivid:2208.02744"},{"id":"arxiv:1705.09334","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Neural networks can efficiently encode the probability distribution of errors in an error correcting code. Moreover, these distributions can be conditioned on the syndromes of the corresponding errors. This paves a path forward for a decoder that employs a neural network to calculate the conditional distribution, then sample from the distribution - the sample will be the predicted error for the given syndrome. We present an implementation of such an algorithm that can be applied to any stabilizer code. Testing it on the toric code, it has higher threshold than a number of known decoders thanks to naturally finding the most probable error and accounting for correlations between errors.</jats:p>","DOI":"10.1038/s41598-017-11266-1","source":"Crossref","title":"Deep Neural Network Probabilistic Decoder for Stabilizer Codes","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0001-5550-5258","authenticated-orcid":false,"given":"Stefan","family":"Krastanov","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":false,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[]}],"container-title":"Scientific Reports","original-title":[],"language":"en","issued":{"date-parts":[[2017,9,8]]},"URL":"http://dx.doi.org/10.1038/s41598-017-11266-1","ISSN":["2045-2322"],"container-title-short":"Sci Rep","page":"11003","note":"alternative-id:11266\narxivid:1705.09334"},{"id":"arxiv:2212.03214","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from decoders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">17%</mml:mtext></mml:mrow></mml:math> under independent bit-and phase-flip data noise (to be compared to the ideal threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">20.6%</mml:mtext></mml:mrow></mml:math>) and a threshold value of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">14%</mml:mtext></mml:mrow></mml:math> under depolarizing data noise (compared to the ideal threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">18.9%</mml:mtext></mml:mrow></mml:math>), which are on par with thresholds achieved by non-BP post-processing methods.</jats:p>","DOI":"10.22331/q-2023-06-07-1037","page":"1037","source":"Crossref","title":"Generalized Belief Propagation Algorithms for Decoding of Surface Codes","volume":"7","author":[{"given":"Josias","family":"Old","sequence":"first","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich, Germany"}]},{"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, Aachen, Germany"},{"name":"Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, Jülich, Germany"},{"name":"QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,6,7]]},"URL":"http://dx.doi.org/10.22331/q-2023-06-07-1037","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2212.03214"},{"id":"preset:Yedidia00","type":"chapter","author":[{"family":"Yedidia","given":"J.S."},{"family":"Freeman","given":"W.T."},{"family":"Weiss","given":"Y."}],"title":"Generalized belief propagation","container-title":"NIPS","volume":"13","issued":"2000","page":"689–695"},{"id":"arxiv:2308.16233","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Autonomous quantum memories are a way to passively protect quantum information using engineered dissipation that creates an “always-on&amp;apos;&amp;apos; decoder. We analyze Markovian autonomous decoders that can be implemented with a wide range of qubit and bosonic error-correcting codes, and derive several upper bounds and a lower bound on the logical error rate in terms of correction and noise rates. These bounds suggest that, in general, there is always a correction rate, possibly size-dependent, above which autonomous memories exhibit arbitrarily long coherence times. For any given autonomous memory, size dependence of this correction rate is difficult to rule out: we point to common scenarios where autonomous decoders that stochastically implement active error correction must operate at rates that grow with code size. For codes with a threshold, we show that it is possible to achieve faster-than-polynomial decay of the logical error rate with code size by using superlogarithmic scaling of the correction rate. We illustrate our results with several examples. One example is an exactly solvable global dissipative toric code model that can achieve an effective logical error rate that decreases exponentially with the linear lattice size, provided that the recovery rate grows proportionally with the linear lattice size.</jats:p>","DOI":"10.22331/q-2025-07-22-1804","page":"1804","source":"Crossref","title":"Bounds on Autonomous Quantum Error Correction","volume":"9","author":[{"given":"Oles","family":"Shtanko","sequence":"first","affiliation":[{"name":"IBM Quantum, IBM Research – Almaden, San Jose, CA, USA"}]},{"given":"Yu-Jie","family":"Liu","sequence":"additional","affiliation":[{"name":"Technical University of Munich, TUM School of Natural Sciences, Physics Department, 85748 Garching, Germany"},{"name":"Munich Center for Quantum Science and Technology (MCQST), Munich, Germany"}]},{"given":"Simon","family":"Lieu","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD, USA"}]},{"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST/University of Maryland, College Park, MD, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD, USA"}]},{"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,22]]},"URL":"http://dx.doi.org/10.22331/q-2025-07-22-1804","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2308.16233"},{"id":"arxiv:2308.14054","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.110.052617","source":"Crossref","title":"Efficient local operations and classical communication extraction of quantum information encoded in stabilizer codes","volume":"110","author":[{"ORCID":"https://orcid.org/0009-0005-6995-0578","authenticated-orcid":true,"given":"Koki","family":"Shiraishi","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]},{"ORCID":"https://orcid.org/0000-0003-3521-831X","authenticated-orcid":true,"given":"Hayata","family":"Yamasaki","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]},{"ORCID":"https://orcid.org/0000-0001-7861-1774","authenticated-orcid":true,"given":"Mio","family":"Murao","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"},{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Tokyo"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.052617","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052617","note":"arxivid:2308.14054"},{"id":"arxiv:2403.03272","type":"article-journal","author":[{"given":"Madelyn","family":"Cain"},{"given":"Chen","family":"Zhao"},{"given":"Hengyun","family":"Zhou"},{"given":"Nadine","family":"Meister"},{"given":"J. Pablo Bonilla","family":"Ataides"},{"given":"Arthur","family":"Jaffe"},{"given":"Dolev","family":"Bluvstein"},{"given":"Mikhail D.","family":"Lukin"}],"title":"Correlated decoding of logical algorithms with transversal gates","issued":{"date-parts":[[2025,4,7]]},"note":"arxivid:2403.03272\narxiv_version_number:2"},{"id":"arxiv:2303.15933","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In this work, we introduce a fast implementation of the minimum-weight perfect matching (MWPM) decoder, the most widely used decoder for several important families of quantum error correcting codes, including surface codes. Our algorithm, which we call sparse blossom, is a variant of the blossom algorithm which directly solves the decoding problem relevant to quantum error correction. Sparse blossom avoids the need for all-to-all Dijkstra searches, common amongst MWPM decoder implementations. For 0.1% circuit-level depolarising noise, sparse blossom processes syndrome data in both <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> bases of distance-17 surface code circuits in less than one microsecond per round of syndrome extraction on a single core, which matches the rate at which syndrome data is generated by superconducting quantum computers. Our implementation is open-source, and has been released in version 2 of the PyMatching library.</jats:p>","DOI":"10.22331/q-2025-01-20-1600","page":"1600","source":"Crossref","title":"Sparse Blossom: correcting a million errors per core second with minimum-weight matching","volume":"9","author":[{"given":"Oscar","family":"Higgott","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"},{"name":"Department of Physics & Astronomy, University College London, WC1E 6BT London, United Kingdom"}]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,20]]},"URL":"http://dx.doi.org/10.22331/q-2025-01-20-1600","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2303.15933"},{"id":"arxiv:2407.13826","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum error-correcting codes, such as subspace, subsystem, and Floquet codes, are typically constructed within the stabilizer formalism, which does not fully capture the idea of fault tolerance needed for practical quantum computing applications. In this work, we explore the remarkably powerful formalism of detector error models, which fully captures fault-tolerance at the circuit level. We introduce the detector error model formalism in a pedagogical manner and provide several examples. Additionally, we apply the formalism to three different levels of abstraction in the engineering cycle of fault-tolerant circuit designs: finding robust syndrome extraction circuits, identifying efficient measurement schedules, and constructing fault-tolerant procedures. We enhance the surface code&amp;apos;s resistance to measurement errors, devise short measurement schedules for color codes, and implement a more efficient fault-tolerant method for measuring logical operators.</jats:p>","DOI":"10.22331/q-2025-11-06-1905","page":"1905","source":"Crossref","title":"Designing fault-tolerant circuits using detector error models","volume":"9","author":[{"given":"Peter-Jan H.S.","family":"Derks","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Alex","family":"Townsend-Teague","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Ansgar G.","family":"Burchards","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,6]]},"URL":"http://dx.doi.org/10.22331/q-2025-11-06-1905","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2407.13826"},{"id":"arxiv:2411.13509","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>Erasures are the primary type of errors in physical systems dominated by leakage errors. While quantum error correction (QEC) using stabilizer codes can combat erasure errors, it remains unknown which constructions achieve capacity performance. If such codes exist, decoders with linear runtime in the code length are also desired. In this paper, we present erasure capacity-achieving quantum codes under maximum-likelihood decoding (MLD), though MLD requires cubic runtime in the code length. For QEC, using an accurate decoder with the shortest possible runtime will minimize the degradation of quantum information while awaiting the decoder’s decision. To address this, we propose belief propagation (BP) decoders that run in linear time and exploit error degeneracy in stabilizer codes, achieving capacity or near-capacity performance for a broad class of codes, including bicycle codes, product codes, and topological codes. We furthermore explore the potential of our BP decoders to handle mixed erasure and depolarizing errors, and also local deletion errors via concatenation with permutation invariant codes.</jats:p>","DOI":"10.1038/s41534-026-01212-3","source":"Crossref","title":"Degenerate quantum erasure decoding","volume":"12","author":[{"given":"Kao-Yueh","family":"Kuo","sequence":"first","affiliation":[]},{"given":"Yingkai","family":"Ouyang","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2026,3,13]]},"URL":"http://dx.doi.org/10.1038/s41534-026-01212-3","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"75","note":"alternative-id:1212\narxivid:2411.13509"},{"id":"arxiv:2412.13739","type":"article-journal","author":[{"given":"Fumiyoshi","family":"Kobayashi"},{"given":"Hidetaka","family":"Manabe"},{"given":"Gregory A. L.","family":"White"},{"given":"Terry","family":"Farrelly"},{"given":"Kavan","family":"Modi"},{"given":"Thomas M.","family":"Stace"}],"title":"Tensor-network decoders for process tensor descriptions of non-Markovian noise","issued":{"date-parts":[[2024,12,18]]},"note":"arxivid:2412.13739\narxiv_version_number:1"},{"id":"arxiv:2502.20558","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Qubit loss errors constitute a dominant source of noise in many quantum hardware systems, particularly in neutral-atom quantum computers. We develop a theoretical framework to effectively detect and correct loss errors in logical algorithms and leverage such loss information in decoding. Considering general quantum error correction codes and logical circuits, we introduce a delayed-erasure decoder for experimentally motivated error models which leverages information from delayed loss detection to accurately correct loss errors, even when the precise moment of the error is unknown. Using this decoder, we identify strategies for detecting and correcting loss errors based on the logical circuit structure. For deep circuits prior to logical measurement, we explore methods to integrate loss detection into syndrome extraction with minimal overhead, identifying optimal strategies depending on the qubit loss fraction in the noise and hardware capabilities. In contrast, we find that many key algorithmic subroutines involve frequent gate teleportation, shortening the circuit depth before logical measurement and naturally replacing qubits with no additional experimental overhead. We simulate this setting using a toy model algorithm for small-angle synthesis and find a significant performance improvement as the loss fraction increases. These results provide a path forward for advancing large-scale fault-tolerant quantum computation in systems with loss error detection.</jats:p>","DOI":"10.1103/ycwc-3myc","source":"Crossref","title":"Leveraging Qubit Loss Detection in Fault-Tolerant Quantum Algorithms","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0002-5920-2972","authenticated-orcid":true,"given":"Gefen","family":"Baranes","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-5298-3112","authenticated-orcid":true,"given":"Madelyn","family":"Cain","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0001-5518-7907","authenticated-orcid":true,"given":"J. Pablo Bonilla","family":"Ataides","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"given":"Dolev","family":"Bluvstein","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0002-1523-8295","authenticated-orcid":true,"given":"Josiah","family":"Sinclair","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-9786-0538","authenticated-orcid":true,"given":"Vladan","family":"Vuletić","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-2148-8856","authenticated-orcid":true,"given":"Hengyun","family":"Zhou","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"},{"id":[{"id":"https://ror.org/01njdjw78","id-type":"ROR","asserted-by":"publisher"}],"name":"QuEra Computing Inc."}]},{"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,2]]},"URL":"http://dx.doi.org/10.1103/ycwc-3myc","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011002","note":"arxivid:2502.20558"},{"id":"arxiv:2507.12534","type":"article-journal","author":[{"given":"Ivan","family":"Rojkov"},{"given":"Elias","family":"Zapusek"},{"given":"Florentin","family":"Reiter"}],"title":"Scalable dissipative quantum error correction for qubit codes","issued":{"date-parts":[[2026,3,18]]},"note":"arxivid:2507.12534\narxiv_version_number:2"},{"id":"arxiv:0903.5256","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.79.062322","source":"Crossref","title":"Logical operators of quantum codes","volume":"79","author":[{"given":"Mark M.","family":"Wilde","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,6,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.062322","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062322","note":"arxivid:0903.5256"},{"id":"arxiv:1409.8320","type":"article-journal","author":[{"given":"Jonas T.","family":"Anderson"},{"given":"Tomas","family":"Jochym-O'Connor"}],"title":"Classification of transversal gates in qubit stabilizer codes","issued":{"date-parts":[[2014,9,29]]},"note":"arxivid:1409.8320\narxiv_version_number:1"},{"id":"arxiv:0712.2084","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.77.042313","source":"Crossref","title":"Semi-Clifford operations, structure of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:msub><mml:mi mathvariant=\"script\">C</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math>hierarchy, and gate complexity for fault-tolerant quantum computation","volume":"77","author":[{"given":"Bei","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,4,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.042313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042313","note":"arxivid:0712.2084"},{"id":"arxiv:1704.07798","type":"article-journal","author":[{"given":"Michael","family":"Newman"},{"given":"Yaoyun","family":"Shi"}],"title":"Limitations on Transversal Computation through Quantum Homomorphic Encryption","issued":{"date-parts":[[2017,8,13]]},"note":"arxivid:1704.07798\narxiv_version_number:3"},{"id":"arxiv:1210.4626","type":"article-journal","author":[{"given":"Austin G.","family":"Fowler"}],"title":"Time-optimal quantum computation","issued":{"date-parts":[[2013,2,2]]},"note":"arxivid:1210.4626\narxiv_version_number:3"},{"id":"arxiv:2602.13395","type":"article-journal","author":[{"given":"Aranya","family":"Chakraborty"},{"given":"Daniel","family":"Gottesman"}],"title":"No-Go Theorem on Fault Tolerant Gadgets for Multiple Logical Qubits","issued":{"date-parts":[[2026,2,26]]},"note":"arxivid:2602.13395\narxiv_version_number:2"},{"id":"doi:10.1002/qua.24856","type":"journal-article","publisher":"Wiley","issue":"19","abstract":"<jats:p>Quantum algorithms for quantum chemistry map the dynamics of electrons in a molecule to the dynamics of a coupled spin system. To reach chemical accuracy for interesting molecules, a large number of quantum gates must be applied which implies the need for quantum error correction and fault‐tolerant quantum computation. Arbitrary fault‐tolerant operations can be constructed from a small, universal set of fault‐tolerant operations by gate compilation. Quantum chemistry algorithms are compiled by decomposing the dynamics of the coupled spin‐system using a Trotter formula, synthesizing the decomposed dynamics using Clifford operations and single‐qubit rotations, and finally approximating the single‐qubit rotations by a sequence of fault‐tolerant single‐qubit gates. Certain fault‐tolerant gates rely on the preparation of specific single‐qubit states referred to as magic states. As a result, gate compilation and magic state distillation are critical for solving quantum chemistry problems on a quantum computer. We review recent progress that has improved the efficiency of gate compilation and magic state distillation by orders of magnitude. © 2015 Wiley Periodicals, Inc.</jats:p>","DOI":"10.1002/qua.24856","page":"1296-1304","source":"Crossref","title":"Magic state distillation and gate compilation in quantum algorithms for quantum chemistry","volume":"115","author":[{"given":"Colin J.","family":"Trout","sequence":"first","affiliation":[{"name":"School of Chemistry and Biochemistry, Georgia Institute of Technology Atlanta Georgia 30332"}]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":false,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"School of Chemistry and Biochemistry, Georgia Institute of Technology Atlanta Georgia 30332"},{"name":"School of Physics, Georgia Institute of Technology Atlanta Georgia 30332"},{"name":"School of Computational Science and Engineering, Georgia Institute of Technology Atlanta Georgia 30332"}]}],"container-title":"International Journal of Quantum Chemistry","original-title":[],"language":"en","issued":{"date-parts":[[2015,1,24]]},"URL":"http://dx.doi.org/10.1002/qua.24856","ISSN":["0020-7608","1097-461X"],"container-title-short":"Int J of Quantum Chemistry","note":"alternative-id:10.1002/qua.24856"},{"id":"arxiv:2208.05601","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> that can correct up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230A;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230B;</mml:mo></mml:math> errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math> times in a row; in the worst-case scenario, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:math> rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:math> rounds and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn></mml:math> rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, our FTEC protocols with the adaptive schemes require no more than <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> rounds on average.</jats:p>","DOI":"10.22331/q-2023-08-08-1075","page":"1075","source":"Crossref","title":"Adaptive syndrome measurements for Shor-style error correction","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-2865-0705","authenticated-orcid":false,"given":"Theerapat","family":"Tansuwannont","sequence":"first","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA"}]},{"ORCID":"https://orcid.org/0000-0001-9502-3368","authenticated-orcid":false,"given":"Balint","family":"Pato","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA"}]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":false,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"name":"Duke Quantum Center, Duke University, Durham, NC 27701, USA"},{"name":"Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708, USA"},{"name":"Department of Physics, Duke University, Durham, NC 27708, USA"},{"name":"Department of Chemistry, Duke University, Durham, NC 27708, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,8]]},"URL":"http://dx.doi.org/10.22331/q-2023-08-08-1075","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2208.05601"},{"id":"preset:Yoder18","type":"thesis","author":[{"family":"Yoder","given":"T."}],"genre":"PhD thesis,","title":"DSpace@MIT Practical Fault-Tolerant Quantum Computation","issued":"2018"},{"id":"arxiv:0809.5063","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.79.012332","source":"Crossref","title":"Fibonacci scheme for fault-tolerant quantum computation","volume":"79","author":[{"given":"Panos","family":"Aliferis","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,1,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.79.012332","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012332","note":"arxivid:0809.5063"},{"id":"arxiv:1708.02246","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting from<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>v</mml:mi></mml:math>faults in the circuit have weight greater than<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>v</mml:mi></mml:math>. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.</jats:p>","DOI":"10.22331/q-2018-02-08-53","page":"53","source":"Crossref","title":"Flag fault-tolerant error correction with arbitrary distance codes","volume":"2","author":[{"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[{"name":"Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada"}]},{"given":"Michael E.","family":"Beverland","sequence":"additional","affiliation":[{"name":"Station Q Quantum Architectures and Computation Group, Microsoft ResearchRedmond, WA 98052, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2018,2,8]]},"URL":"http://dx.doi.org/10.22331/q-2018-02-08-53","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1708.02246"},{"id":"arxiv:2109.06248","type":"article-journal","author":[{"given":"Narayanan","family":"Rengaswamy"},{"given":"Ankur","family":"Raina"},{"given":"Nithin","family":"Raveendran"},{"given":"Bane","family":"Vasić"}],"title":"Distilling GHZ States using Stabilizer Codes","issued":{"date-parts":[[2022,3,24]]},"note":"arxivid:2109.06248\narxiv_version_number:2"},{"id":"arxiv:2212.10738","type":"article-journal","author":[{"given":"Benjamin","family":"Anker"},{"given":"Milad","family":"Marvian"}],"title":"Flag Gadgets based on Classical Codes","issued":{"date-parts":[[2024,2,20]]},"note":"arxivid:2212.10738\narxiv_version_number:2"},{"id":"arxiv:2303.17380","type":"article-journal","author":[{"given":"Hyeongrak","family":"Choi"},{"given":"Frederic T.","family":"Chong"},{"given":"Dirk","family":"Englund"},{"given":"Yongshan","family":"Ding"}],"title":"Fault Tolerant Non-Clifford State Preparation for Arbitrary Rotations","issued":{"date-parts":[[2023,3,30]]},"note":"arxivid:2303.17380\narxiv_version_number:1"},{"id":"arxiv:2501.05527","type":"article-journal","author":[{"given":"Ludwig","family":"Schmid"},{"given":"Tom","family":"Peham"},{"given":"Lucas","family":"Berent"},{"given":"Markus","family":"Müller"},{"given":"Robert","family":"Wille"}],"title":"Deterministic Fault-Tolerant State Preparation for Near-Term Quantum Error Correction: Automatic Synthesis Using Boolean Satisfiability","issued":{"date-parts":[[2025,1,9]]},"note":"arxivid:2501.05527\narxiv_version_number:1"},{"id":"arxiv:2501.14380","type":"book-chapter","publisher":"Springer Nature Switzerland","abstract":"<jats:title>Abstract</jats:title>\n          <jats:p>Quantum computers have advanced rapidly in qubit count and gate fidelity. However, large-scale fault-tolerant quantum computing still relies on quantum error correction code (QECC) to suppress noise. Manually or experimentally verifying the fault-tolerance property of complex QECC implementation is impractical due to the vast error combinations. This paper formalizes the fault-tolerance of QECC implementations within the language of quantum programs. By incorporating the techniques of quantum symbolic execution, we provide an automatic verification tool for quantum fault-tolerance. We evaluate and demonstrate the effectiveness of our tool on a universal set of logical operations across different QECCs.</jats:p>","DOI":"10.1007/978-3-031-98685-7_1","page":"3-27","source":"Crossref","title":"Verifying Fault-Tolerance of Quantum Error Correction Codes","author":[{"ORCID":"https://orcid.org/0000-0002-0772-6635","authenticated-orcid":false,"given":"Kean","family":"Chen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0009-0005-2822-0448","authenticated-orcid":false,"given":"Yuhao","family":"Liu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7628-1185","authenticated-orcid":false,"given":"Wang","family":"Fang","sequence":"additional","affiliation":[]},{"given":"Jennifer","family":"Paykin","sequence":"additional","affiliation":[]},{"given":"Xin-Chuan","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Albert","family":"Schmitz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3516-1512","authenticated-orcid":false,"given":"Steve","family":"Zdancewic","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6233-0334","authenticated-orcid":false,"given":"Gushu","family":"Li","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2025]]},"ISBN":["9783031986840","9783031986857"],"URL":"http://dx.doi.org/10.1007/978-3-031-98685-7_1","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Cham\narxivid:2501.14380"},{"id":"arxiv:quant-ph/0304016","type":"article-journal","author":[{"given":"A. M.","family":"Steane"}],"title":"Quantum Computing and Error Correction","issued":{"date-parts":[[2003,4,3]]},"note":"arxivid:quant-ph/0304016\narxiv_version_number:2"},{"id":"doi:10.1002/9783527618637.ch1","type":"other","publisher":"Wiley","DOI":"10.1002/9783527618637.ch1","page":"3-16","source":"Crossref","title":"Classical Information Theory and Classical Error Correction","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]}],"container-title":"Lectures on Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2006,11,6]]},"ISBN":["9783527405275","9783527618637"],"URL":"http://dx.doi.org/10.1002/9783527618637.ch1","note":"edition-number:1\nalternative-id:10.1002/9783527618637.ch1\n10.1002/9783527618637"},{"id":"arxiv:2507.07121","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/j.aop.2026.170353","page":"170353","source":"Crossref","title":"Introduction to quantum error correction with stabilizer codes","volume":"487","author":[{"ORCID":"https://orcid.org/0000-0002-3591-7594","authenticated-orcid":false,"given":"Zachary P.","family":"Bradshaw","sequence":"first","affiliation":[]},{"given":"Jeffrey J.","family":"Dale","sequence":"additional","affiliation":[]},{"given":"Ethan N.","family":"Evans","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2026,4]]},"URL":"http://dx.doi.org/10.1016/j.aop.2026.170353","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"special_numbering:C\nalternative-id:S0003491626000126\narxivid:2507.07121"},{"id":"arxiv:2106.12065","type":"article-journal","author":[{"given":"Nuh","family":"Aydin"},{"given":"Peihan","family":"Liu"},{"given":"Bryan","family":"Yoshino"}],"title":"Polycyclic Codes Associated with Trinomials: Good Codes and Open Questions","issued":{"date-parts":[[2021,6,22]]},"note":"arxivid:2106.12065\narxiv_version_number:1"},{"id":"arxiv:2108.03567","type":"article-journal","author":[{"given":"Nuh","family":"Aydin"},{"given":"Peihan","family":"Liu"},{"given":"Bryan","family":"Yoshino"}],"title":"A Database of Quantum Codes","issued":{"date-parts":[[2021,8,8]]},"note":"arxivid:2108.03567\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0209091","type":"journal-article","publisher":"IOP Publishing","issue":"29","DOI":"10.1088/0305-4470/36/29/316","page":"8113-8127","source":"Crossref","title":"Conversion of a general quantum stabilizer code to an entanglement distillation protocol","volume":"36","author":[{"given":"Ryutaroh","family":"Matsumoto","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[2003,7,9]]},"URL":"http://dx.doi.org/10.1088/0305-4470/36/29/316","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen.","note":"alternative-id:S0305-4470(03)61762-2\narxivid:quant-ph/0209091"},{"id":"arxiv:2303.11465","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/jsac.2024.3380094","page":"1830-1849","source":"Crossref","title":"Near-Term <i>n</i> to <i>k</i> Distillation Protocols Using Graph Codes","volume":"42","author":[{"ORCID":"https://orcid.org/0000-0002-1761-0038","authenticated-orcid":false,"given":"Kenneth","family":"Goodenough","sequence":"first","affiliation":[{"name":"QuTech, Delft University of Technology, Delft, The Netherlands"}]},{"given":"Sébastian","family":"de Bone","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0009-0009-6928-3567","authenticated-orcid":false,"given":"Vaishnavi","family":"Addala","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, USA"}]},{"given":"Stefan","family":"Krastanov","sequence":"additional","affiliation":[{"name":"College of Information and Computer Science, University of Massachusetts Amherst, Amherst, MA, USA"}]},{"ORCID":"https://orcid.org/0009-0005-5007-8855","authenticated-orcid":false,"given":"Sarah","family":"Jansen","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0003-0734-4511","authenticated-orcid":false,"given":"Dion","family":"Gijswijt","sequence":"additional","affiliation":[{"name":"Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands"}]},{"ORCID":"https://orcid.org/0000-0003-2023-2768","authenticated-orcid":false,"given":"David","family":"Elkouss","sequence":"additional","affiliation":[{"name":"QuTech, Delft University of Technology, Delft, The Netherlands"}]}],"container-title":"IEEE Journal on Selected Areas in Communications","original-title":[],"issued":{"date-parts":[[2024,7]]},"URL":"http://dx.doi.org/10.1109/JSAC.2024.3380094","ISSN":["0733-8716","1558-0008"],"container-title-short":"IEEE J. Select. Areas Commun.","note":"arxivid:2303.11465"},{"id":"arxiv:quant-ph/0310097","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tit.2005.862089","page":"748-753","source":"Crossref","title":"The minimum distance problem for two-way entanglement purification","volume":"52","author":[{"given":"A.","family":"Ambainis","sequence":"first","affiliation":[]},{"given":"D.","family":"Gottesman","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2006,2]]},"URL":"http://dx.doi.org/10.1109/TIT.2005.862089","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0310097"},{"id":"arxiv:2405.06795","type":"article-journal","author":[{"given":"Richard A.","family":"Wolf"}],"title":"Quantum Error Correction for Kids","issued":{"date-parts":[[2024,5,10]]},"note":"arxivid:2405.06795\narxiv_version_number:1"},{"id":"arxiv:1401.4174","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7505","DOI":"10.1038/nature13460","page":"351-355","source":"Crossref","title":"Contextuality supplies the ‘magic’ for quantum computation","volume":"510","author":[{"given":"Mark","family":"Howard","sequence":"first","affiliation":[]},{"given":"Joel","family":"Wallman","sequence":"additional","affiliation":[]},{"given":"Victor","family":"Veitch","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Emerson","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1038/nature13460","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:BFnature13460\narxivid:1401.4174"},{"id":"manual:-C.-Gidney.-Stim.-url-https:/","type":"document","author":[{"family":"Stim","given":"C.Gidney"}],"issued":"2021","URL":"https://github.com/quantumlib/Stim","note":"Available at \\url{https://github.com/quantumlib/Stim},"},{"id":"manual:-E.-Huang-and-A.-Pesah.-PanQE","type":"document","author":[{"family":"Huang","given":"E."},{"family":"PanQEC","given":"A.Pesah"}],"issued":"2023","URL":"https://github.com/panqec/panqec","note":"Available at \\url{https://github.com/panqec/panqec},"},{"id":"manual:-S.-Krastanov.-QuantumSavory.","type":"document","author":[{"family":"QuantumSavory","given":"S.Krastanov"}],"issued":"2025","URL":"https://github.com/QuantumSavory","note":"Available at \\url{https://github.com/QuantumSavory},"},{"id":"doi:10.5281/zenodo.5208167","type":"book","author":[{"family":"Stefan Krastanov"},{"family":"Feroz Ahmed Mian"},{"family":"Zhongyi Ni"},{"family":"Kanyang Ying"},{"family":"Praveen Viswanathan"},{"family":"Shayan Pardis"},{"family":"Sagnik Pal"},{"family":"brohan-byte"},{"family":"John Lapeyre"},{"family":"H.A."},{"family":"Anthony Micciche"},{"family":"Yuxuan Yan"},{"family":"Stefan Krastanov's bot"},{"family":"Tommy Hofmann"},{"family":"Quinten Preiß"},{"family":"gsommers"},{"family":"Lars Göttgens"},{"family":"IsaacP1234"},{"family":"Abhishek Bhatt"},{"family":"Adriano Meligrana"},{"family":"Benzillaist"},{"family":"Chen Zhao"},{"family":"ShuGe-MIT"},{"family":"Tim Holy"},{"family":"Trung Dang"},{"family":"adrianariton"},{"family":"ismoldayev"},{"family":"Ritesh Bhirud"}],"issued":{"date-parts":[[2026,1,1]]},"abstract":"QuantumClifford v0.11.0\n\nDiff since v0.10.1\n\n\n\nDrop support for Julia <1.12.\n\nImplementation of the specialized simulation algorithms for graph states (faster for large sparse tableaux).\n\n(breaking) moving all graph state functionality to a submodule GraphSim.\n\n(fix) canonicalize_gott! now properly supports non-UInt64 types.\n\nMixedDestabilizer now has a backtrack keyword argument that makes it possible to undo the canonicalization step and recover destabilizers for specific stabilizer operators.\n\nptrace for partial traces. We already had traceout!, which returns tableaux with the same number of qubits. ptrace removes the traceout qubits besides setting their stabilizing operators to I.\n\nAdd apply_right! that applies a Clifford operator to the right of a dense clifford operator.\n\nImplementing apply_inv! for direct application of the inverse of a given gate.\n\nAdd mul_right! methods for inplace operations between tableaus\n\nAdd a CliffordOperator constructor that builds a dense clifford from a PauliOperator\n\nAdd a phases getter for CliffordOperator\n\nA new non-Clifford operation was added as a Pauli channel, the rotation gate pcRx.\n\nAdapt.jl can now be used to convert various types to GPU-backed storage.\n\nThe phase storage type can now be parameterized, instead of hardcoded to UInt8.\n\nAdd an extension to QECCore -- QECCoreNemoExt for accurate matrix rank computation\n\nIntroduce metacheck_matrix_x, metacheck_matrix_z, and metacheck_matrix for CSS codes built using chain complexes and homology.\n\nQuantum codes (including (breaking) changes to API):\n_ Improvements to decoders:\n- TableDecoder now supports arbitrary error weights and a new CSSTableDecoder is available that separate decodes the x and z syndromes of CSS codes.\n- Google's tesseract decoder is available through PyTesseractDecoder.jl\n\n\nThe lifted product code constructor LPCode now supports non-commutative group algebras by appropriate switching left/right representations — particularly useful now that there is also an Oscar.jl extension, which provides many non-abelian group constructors.\n\nAdd BivariateBicycleViaPoly, implemented using multivariate quotient ring formalism\n\nIn a Hecke extension for QuantumClifford.ECC:\n\n\nAdd GeneralizedBicycle and ExtendedGeneralizedBicycle\n\nAdd GeneralizedHyperGraphProduct\n\nAdd [[2n², 2k², d]] and [[(n - k)² + n², k², d]] La-cross codes via univariate polynomial ring\n\nAdd convenience wrappers for code families honeycomb_color_codes_as_2bga and Haah_cubic_codes_as_2bga\n\n\n\nIn an Oscar extension for QuantumClifford.ECC:\n\n\nAdd TrivariateTricycle\n\nAdd GeneralizedToric on twisted tori in terms of Laurent polynomials\n\nAdd HomologicalProduct and DoubleHomologicalProduct\n\nAdd D-dimensional Surface and Toric codes through chain complexes and GF2 homology\n\n\n\nIn QECCore:\n\n\nAdd cyclic quantum Tanner graph product codes\n\nAdd [[n² + m²,(n - rank([C ∣ M]))² + (m − rank([C ∣ M]ᵀ))², d]] quantum Tillich-Zémor random_TillichZemor_code\n\nAdd Delfosse-Reichardt codes from classical self-orthogonal Reed-Muller seed codes\n\nAdd [[4p, 2(p − 2), 4]] Delfosse-Reichardt repetition DelfosseReichardtRep\n\nAdd [[8p, 4p − 2, 3]] Delfosse-Reichardt Generalized [[8,2,3]] DelfosseReichardt823\n\nIn an Oscar extension for QECCore\n\n\nAdd BivariateBicycleViaCirculantMat\n\n\n\n\n\n\n\nClassical codes (including (breaking) changes to API):\n\n\nIn QECCore:\n\n\nAdd classical Goppa\n\nAdd classical Gallager's LDPC code\n\n\n\n\n\n(breaking) Some codes are moved to QECCore from QuantumClifford, including Hamming, Golay, Triangular488 , Triangular666 , Gottesman, ReedMuller, RecursiveReedMuller, and QuantumReedMuller.\n\n(breaking) Some codes (types or functions) are slightly renamed for consistency.\n\n\nMerged pull requests:\n\n\n\nadd  pcRx (#552) (@Fe-r-oz)\n\nECC summer work:  polish Goppa codes, Gallager codes and introduce appropriate name changes to methods and structs (#630) (@Fe-r-oz)\n\nBump actions/upload-artifact from 5 to 6 (#642) (@dependabot[bot])\n\nadd tesseract-decoder extension (#643) (@Krastanov)\n\nabstract out the repetitive dependency errors and hints (#644) (@Krastanov)\n\nGeneral polish and renaming of ECC code implementations (#646) (@Krastanov)\n\nBump dawidd6/action-download-artifact from 11 to 12 (#648) (@dependabot[bot])\n\nprep for 0.11 release (#649) (@Krastanov)\n\nUpdate Nemo requirement from 0.45.5, 0.46, 0.47, 0.48, 0.49, 0.50, 0.51, 0.52 to 0.45.5, 0.46, 0.47, 0.48, 0.49, 0.50, 0.51, 0.52, 0.53 in /lib/QECCore (#650) (@dependabot[bot])\n\nprep for 0.11 release (#653) (@Krastanov)\n\n\nClosed issues:\n\n\n\nPolish the summer additions to the ECC API (#623)\n\nTagBot: Manual intervention needed for releases (#651)","DOI":"10.5281/ZENODO.5208167","publisher":"Zenodo","title":"QuantumSavory/QuantumClifford.jl: v0.11.0","URL":"https://zenodo.org/doi/10.5281/zenodo.5208167","version":"v0.11.0","note":"copyright:MIT License"},{"id":"arxiv:1910.08122","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.101.052343","source":"Crossref","title":"Local-dimension-invariant qudit stabilizer codes","volume":"101","author":[{"ORCID":"https://orcid.org/0000-0003-3893-9278","authenticated-orcid":true,"given":"Lane G.","family":"Gunderman","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,5,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.101.052343","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052343","note":"arxivid:1910.08122"},{"id":"arxiv:2110.11510","type":"article-journal","author":[{"given":"Arun J.","family":"Moorthy"},{"given":"Lane G.","family":"Gunderman"}],"title":"Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise","issued":{"date-parts":[[2021,10,21]]},"note":"arxivid:2110.11510\narxiv_version_number:1"},{"id":"arxiv:2110.15274","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.105.042424","source":"Crossref","title":"Degenerate local-dimension-invariant stabilizer codes and an alternative bound for the distance preservation condition","volume":"105","author":[{"ORCID":"https://orcid.org/0000-0003-3893-9278","authenticated-orcid":true,"given":"Lane G.","family":"Gunderman","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.042424","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042424","note":"arxivid:2110.15274"},{"id":"arxiv:quant-ph/0611214","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.75.032325","source":"Crossref","title":"Local unitary versus local Clifford equivalence of stabilizer and graph states","volume":"75","author":[{"given":"Bei","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Hyeyoun","family":"Chung","sequence":"additional","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,3,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.75.032325","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032325","note":"arxivid:quant-ph/0611214"},{"id":"arxiv:1709.03543","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.120.050504","source":"Crossref","title":"Distillation with Sublogarithmic Overhead","volume":"120","author":[{"given":"Matthew B.","family":"Hastings","sequence":"first","affiliation":[]},{"given":"Jeongwan","family":"Haah","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2018,1,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.120.050504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050504","note":"arxivid:1709.03543"},{"id":"arxiv:quant-ph/9608026","type":"article-journal","author":[{"given":"Andrew","family":"Steane"}],"title":"Quantum Reed-Muller Codes","issued":{"date-parts":[[1996,8,22]]},"note":"arxivid:quant-ph/9608026\narxiv_version_number:2"},{"id":"arxiv:quant-ph/9703045","type":"article-journal","author":[{"given":"Lin","family":"Zhang"},{"given":"Ian","family":"Fuss"}],"title":"Quantum Reed-Muller Codes","issued":{"date-parts":[[1997,3,24]]},"note":"arxivid:quant-ph/9703045\narxiv_version_number:1"},{"id":"doi:10.1109/ISIT.2016.7541599","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2016.7541599","page":"1750-1754","source":"Crossref","title":"Reed-muller codes achieve capacity on the quantum erasure channel","author":[{"given":"Santhosh","family":"Kumar","sequence":"first","affiliation":[]},{"given":"Robert","family":"Calderbank","sequence":"additional","affiliation":[]},{"given":"Henry D.","family":"Pfister","sequence":"additional","affiliation":[]}],"event":"2016 IEEE International Symposium on Information Theory (ISIT)","container-title":"2016 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2016,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2016.7541599"},{"id":"arxiv:2602.09788","type":"article-journal","author":[{"given":"Theerapat","family":"Tansuwannont"},{"given":"Tim","family":"Chan"},{"given":"Ryuji","family":"Takagi"}],"title":"Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates","issued":{"date-parts":[[2026,2,17]]},"note":"arxivid:2602.09788\narxiv_version_number:2"},{"id":"manual:-J.-Harrington-and-B.-W.-Reic","type":"article-journal","author":[{"family":"Harrington","given":"J."},{"family":"Reichardt","given":"B.W."}],"title":"Addressable multi-qubit logic via permutations","volume":"SQuInT","container-title":"Talk at Southwest Quantum Information and Technology","issued":"2011"},{"id":"arxiv:1410.0069","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.91.032330","source":"Crossref","title":"Universal transversal gates with color codes: A simplified approach","volume":"91","author":[{"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"given":"Michael E.","family":"Beverland","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,3,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.91.032330","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032330","note":"arxivid:1410.0069"},{"id":"doi:10.1109/TCOMM.2022.3231879","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","DOI":"10.1109/tcomm.2022.3231879","page":"1071-1082","source":"Crossref","title":"Construction of Multiple-Rate Quantum LDPC Codes Sharing One Scalable Stabilizer Circuit","volume":"71","author":[{"ORCID":"https://orcid.org/0000-0003-2257-1616","authenticated-orcid":false,"given":"Yun-Jiang","family":"Wang","sequence":"first","affiliation":[{"name":"State Key Laboratory of ISN and the School of Telecommunication Engineering, Xidian University, Xi&#x2019;an, China"}]},{"given":"Zhuo-Yan","family":"Xiao","sequence":"additional","affiliation":[{"name":"AVIC Optronics, Luoyang, China"}]},{"ORCID":"https://orcid.org/0000-0002-9312-2861","authenticated-orcid":false,"given":"Yi","family":"Zhang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of ISN and the School of Telecommunication Engineering, Xidian University, Xi&#x2019;an, China"}]},{"given":"Xing-Yu","family":"Xiong","sequence":"additional","affiliation":[{"name":"State Key Laboratory of ISN and the School of Telecommunication Engineering, Xidian University, Xi&#x2019;an, China"}]},{"ORCID":"https://orcid.org/0000-0002-0591-3461","authenticated-orcid":false,"given":"Sha","family":"Shi","sequence":"additional","affiliation":[{"name":"State Key Laboratory of ISN and the School of Telecommunication Engineering, Xidian University, Xi&#x2019;an, China"}]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2023,2]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2022.3231879","ISSN":["0090-6778","1558-0857"],"container-title-short":"IEEE Trans. Commun."},{"id":"arxiv:2503.19790","type":"article-journal","author":[{"given":"Theerapat","family":"Tansuwannont"},{"given":"Yugo","family":"Takada"},{"given":"Keisuke","family":"Fujii"}],"title":"Clifford gates with logical transversality for self-dual CSS codes","issued":{"date-parts":[[2025,3,25]]},"note":"arxivid:2503.19790\narxiv_version_number:1"},{"id":"arxiv:1108.5738","type":"article-journal","author":[{"given":"Andrew J.","family":"Landahl"},{"given":"Jonas T.","family":"Anderson"},{"given":"Patrick R.","family":"Rice"}],"title":"Fault-tolerant quantum computing with color codes","issued":{"date-parts":[[2011,8,29]]},"note":"arxivid:1108.5738\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0605138","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.97.180501","source":"Crossref","title":"Topological Quantum Distillation","volume":"97","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2006,10,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.97.180501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180501","note":"arxivid:quant-ph/0605138"},{"id":"arxiv:1301.6588","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2013.6620360","page":"917-921","source":"Crossref","title":"Tradeoffs for reliable quantum information storage in surface codes and color codes","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[]}],"event":"2013 IEEE International Symposium on Information Theory (ISIT)","container-title":"2013 IEEE International Symposium on Information Theory","original-title":[],"issued":{"date-parts":[[2013,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2013.6620360","note":"arxivid:1301.6588"},{"id":"doi:10.7907/059V-MG69","type":"thesis","categories":["topological quantum codes","quantum many-body physics","quantum information","Physics","quantum computation","quantum error correction"],"language":"en","author":[{"family":"Kubica","given":"Aleksander Marek"}],"contributor":[{"literal":"Institute for Quantum Information and Matter"},{"literal":"IQIM"}],"issued":{"date-parts":[[2018]]},"abstract":"This thesis is devoted to studying a class of quantum error-correcting codes — topological quantum codes. We explore the question of how one can achieve fault- tolerant quantum computation with topological codes. We treat quantum error-correcting codes not only as a compelling ingredient needed to build a quantum computer, but also as a useful theoretical tool in other areas of physics. In particular, we explore what insights topological codes can provide into challenging questions, such as the classification of quantum phases of matter.\n\nIn this thesis, we focus on a family of topological codes — color codes, which are particularly intriguing due to the rich physics they display and their computational power. We start by introducing color codes and explaining their basic properties. Then, we show how to perform fault-tolerant universal quantum computation with three-dimensional color codes by transverse gates and code switching. We later compare the resource overhead of the code-switching approach with that of a state distillation scheme. We discuss how to perform error correction with the toric and color codes, as well as introduce local decoders for those two families of codes. By exploiting a connection between error correction and statistical mechanics we estimate the storage threshold error rates for bit-flip and phase-flip noise in the three-dimensional color code. We finish by showing that the color and toric code families in d dimensions are equivalent in a sense of local unitary transformations and explore implications of this equivalence.","DOI":"10.7907/059V-MG69","publisher":"California Institute of Technology","title":"The ABCs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-Tolerant Quantum Computation and Quantum Phases Of Matter","URL":"https://resolver.caltech.edu/CaltechTHESIS:05282018-173928314","version":"Final","note":"copyright:No commercial reproduction, distribution, display or performance rights in this work are provided."},{"id":"arxiv:1308.6207","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.89.012317","source":"Crossref","title":"Decoding color codes by projection onto surface codes","volume":"89","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,1,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.89.012317","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012317","note":"arxivid:1308.6207"},{"id":"arxiv:1402.3037","type":"article-journal","author":[{"given":"Ashley M.","family":"Stephens"}],"title":"Efficient fault-tolerant decoding of topological color codes","issued":{"date-parts":[[2014,2,13]]},"note":"arxivid:1402.3037\narxiv_version_number:1"},{"id":"arxiv:2603.04234","type":"article-journal","author":[{"given":"Mark","family":"Walters"},{"given":"Mark L.","family":"Turner"}],"title":"Minimum Weight Decoding in the Colour Code is NP-hard","issued":{"date-parts":[[2026,3,4]]},"note":"arxivid:2603.04234\narxiv_version_number:1"},{"id":"arxiv:2603.22064","type":"article-journal","author":[{"given":"Shouzhen","family":"Gu"},{"given":"Lily","family":"Wang"},{"given":"Aleksander","family":"Kubica"}],"title":"The color code, the surface code, and the transversal CNOT: NP-hardness of minimum-weight decoding","issued":{"date-parts":[[2026,3,23]]},"note":"arxivid:2603.22064\narxiv_version_number:1"},{"id":"arxiv:2404.07482","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Two-dimensional color codes are a promising candidate for fault-tolerant quantum computing, as they have high encoding rates, transversal implementation of logical Clifford gates, and resource-efficient magic state preparation schemes. However, decoding color codes presents a significant challenge due to their structure, where elementary errors violate three checks instead of just two (a key feature in surface code decoding), and the complexity of extracting syndrome is greater. We introduce an efficient color-code decoder that tackles these issues by combining two matching decoders for each color, generalized to handle circuit-level noise by employing detector error models. We provide comprehensive analyses of the decoder, covering its threshold and sub-threshold scaling both for bit-flip noise with ideal measurements and for circuit-level noise. Our simulations reveal that this decoding strategy nearly reaches the best possible scaling of logical failure (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>p</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"normal\">f</mml:mi><mml:mi mathvariant=\"normal\">a</mml:mi><mml:mi mathvariant=\"normal\">i</mml:mi><mml:mi mathvariant=\"normal\">l</mml:mi></mml:mrow></mml:msub><mml:mo>&amp;#x223C;</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>) for both noise models, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>p</mml:mi></mml:math> is the noise strength, in the regime of interest for fault-tolerant quantum computing. While its noise thresholds are comparable with other matching-based decoders for color codes (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>8.2</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> for bit-flip noise and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0.46</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> for circuit-level noise), the scaling of logical failure rates below threshold significantly outperforms the best matching-based decoders.</jats:p>","DOI":"10.22331/q-2025-01-27-1609","page":"1609","source":"Crossref","title":"Color code decoder with improved scaling for correcting circuit-level noise","volume":"9","author":[{"given":"Seok-Hyung","family":"Lee","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"Andrew","family":"Li","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,27]]},"URL":"http://dx.doi.org/10.22331/q-2025-01-27-1609","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2404.07482"},{"id":"arxiv:2112.13617","type":"article-journal","author":[{"given":"Pramod","family":"Padmanabhan"},{"given":"Abhishek","family":"Chowdhury"},{"given":"Fumihiko","family":"Sugino"},{"given":"Mrittunjoy Guha","family":"Majumdar"},{"given":"Krishna Kumar","family":"Sabapathy"}],"title":"Non-CSS color codes on 2D lattices : Models and Topological Properties","issued":{"date-parts":[[2022,1,9]]},"note":"arxivid:2112.13617\narxiv_version_number:2"},{"id":"arxiv:0809.4276","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.78.062312","source":"Crossref","title":"Entanglement properties of topological color codes","volume":"78","author":[{"given":"Mehdi","family":"Kargarian","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,12,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.062312","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062312","note":"arxivid:0809.4276"},{"id":"arxiv:1107.3502","type":"article-journal","author":[{"given":"Jonas T.","family":"Anderson"}],"title":"Homological Stabilizer Codes","issued":{"date-parts":[[2011,8,10]]},"note":"arxivid:1107.3502\narxiv_version_number:2"},{"id":"arxiv:1412.5079","type":"article-journal","author":[{"given":"H.","family":"Bombin"}],"title":"Dimensional Jump in Quantum Error Correction","issued":{"date-parts":[[2016,5,12]]},"note":"arxivid:1412.5079\narxiv_version_number:3"},{"id":"arxiv:2410.13568","type":"article-journal","author":[{"given":"Friederike","family":"Butt"},{"given":"David F.","family":"Locher"},{"given":"Katharina","family":"Brechtelsbauer"},{"given":"Hans Peter","family":"Büchler"},{"given":"Markus","family":"Müller"}],"title":"Measurement-free, scalable and fault-tolerant universal quantum computing","issued":{"date-parts":[[2024,10,17]]},"note":"arxivid:2410.13568\narxiv_version_number:1"},{"id":"arxiv:1012.0425","type":"journal-article","publisher":"IOP Publishing","issue":"15","DOI":"10.1088/1751-8113/44/15/155301","page":"155301","source":"Crossref","title":"Constructions and noise threshold of topological subsystem codes","volume":"44","author":[{"given":"Martin","family":"Suchara","sequence":"first","affiliation":[]},{"given":"Sergey","family":"Bravyi","sequence":"additional","affiliation":[]},{"given":"Barbara","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2011,3,15]]},"URL":"http://dx.doi.org/10.1088/1751-8113/44/15/155301","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(11)79402-1\narxivid:1012.0425"},{"id":"manual:-Z.-Wang-private-communicatio","type":"document","author":[{"family":"Wang","given":"Z."}],"genre":"private communication,","issued":"2017"},{"id":"arxiv:0910.0573","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.81.012319","source":"Crossref","title":"Topological color codes on Union Jack lattices: a stable implementation of the whole Clifford group","volume":"81","author":[{"given":"Helmut G.","family":"Katzgraber","sequence":"first","affiliation":[]},{"given":"H.","family":"Bombin","sequence":"additional","affiliation":[]},{"given":"Ruben S.","family":"Andrist","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,1,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.012319","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012319","note":"arxivid:0910.0573"},{"id":"doi:10.1088/2399-6528/aad062","type":"journal-article","publisher":"IOP Publishing","issue":"9","DOI":"10.1088/2399-6528/aad062","page":"095011","source":"Crossref","title":"Construction of color codes from polygons","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0003-2687-5174","authenticated-orcid":false,"given":"Eduardo Brandani","family":"da Silva","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6216-8691","authenticated-orcid":false,"suffix":"Junior","given":"Waldir S","family":"Soares","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics Communications","original-title":[],"issued":{"date-parts":[[2018,9,12]]},"URL":"http://dx.doi.org/10.1088/2399-6528/aad062","ISSN":["2399-6528"],"container-title-short":"J. Phys. Commun."},{"id":"arxiv:1407.5103","type":"article-journal","author":[{"given":"Andrew J.","family":"Landahl"},{"given":"Ciaran","family":"Ryan-Anderson"}],"title":"Quantum computing by color-code lattice surgery","issued":{"date-parts":[[2014,7,18]]},"note":"arxivid:1407.5103\narxiv_version_number:1"},{"id":"arxiv:2201.07806","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Fault-tolerant quantum computation demands significant resources: large numbers of physical qubits must be checked for errors repeatedly to protect quantum data as logic gates are implemented in the presence of noise. We demonstrate that an approach based on the color code can lead to considerable reductions in the resource overheads compared with conventional methods, while remaining compatible with a two-dimensional layout. We propose a lattice surgery scheme that exploits the rich structure of the color-code phase to perform arbitrary pairs of commuting logical Pauli measurements in parallel while keeping the space cost low. Compared to lattice surgery schemes based on the surface code with the same code distance, and assuming the same amount of time is needed to complete a round of syndrome measurements, our approach yields about a <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\"><a:mrow><a:mn>3</a:mn><a:mo>×</a:mo></a:mrow></a:math> improvement in the space-time overhead, obtained from a combination of a <b:math xmlns:b=\"http://www.w3.org/1998/Math/MathML\"><b:mrow><b:mn>1.5</b:mn><b:mo>×</b:mo></b:mrow></b:math> improvement in spatial overhead together with a <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\"><c:mrow><c:mn>2</c:mn><c:mo>×</c:mo></c:mrow></c:math> speedup due to the parallelization of commuting logical measurements. Even when taking into account the color code's lower error threshold using current decoders, the overhead is reduced by 10% at a physical error rate of <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\"><d:msup><d:mn>10</d:mn><d:mrow><d:mo>−</d:mo><d:mn>3</d:mn></d:mrow></d:msup></d:math> and by 50% at <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\"><e:msup><e:mn>10</e:mn><e:mrow><e:mo>−</e:mo><e:mn>4</e:mn></e:mrow></e:msup></e:math>.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevresearch.6.043125","source":"Crossref","title":"Low-overhead quantum computing with the color code","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-7422-1517","authenticated-orcid":true,"given":"Felix","family":"Thomsen","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"given":"Markus S.","family":"Kesselring","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"}]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.6.043125","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"043125","note":"arxivid:2201.07806"},{"id":"arxiv:0907.1708","type":"article-journal","author":[{"given":"D. S.","family":"Wang"},{"given":"A. G.","family":"Fowler"},{"given":"C. D.","family":"Hill"},{"given":"L. C. L.","family":"Hollenberg"}],"title":"Graphical algorithms and threshold error rates for the 2d colour code","issued":{"date-parts":[[2009,7,10]]},"note":"arxivid:0907.1708\narxiv_version_number:1"},{"id":"manual:-G.-Duclos-Cianci-H.-Bomb-n-a","type":"personal_communication","author":[{"family":"Duclos-Cianci","given":"G."},{"family":"Bombın","given":"H."},{"family":"Poulin","given":"D."}],"title":"Fast decoding algorithm for subspace and subsystem color codes and local equivalence of topological phases","genre":"Personal communication","issued":"2011"},{"id":"arxiv:0806.4827","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.83.042310","source":"Crossref","title":"Two-dimensional color-code quantum computation","volume":"83","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,4,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.042310","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042310","note":"arxivid:0806.4827"},{"id":"doi:10.1143/JPSJ.55.3305","type":"journal-article","publisher":"Physical Society of Japan","issue":"10","DOI":"10.1143/jpsj.55.3305","page":"3305-3307","source":"Crossref","title":"Geometry-Induced Phase Transition in the ±<i>J</i>Ising Model","volume":"55","author":[{"given":"Hidetoshi","family":"Nishimori","sequence":"first","affiliation":[{"name":"Department of Physics, Tokyo Institute of Technology"}]}],"container-title":"Journal of the Physical Society of Japan","original-title":[],"language":"en","issued":{"date-parts":[[1986,10,15]]},"URL":"http://dx.doi.org/10.1143/JPSJ.55.3305","ISSN":["0031-9015","1347-4073"],"container-title-short":"J. Phys. Soc. Jpn.","note":"alternative-id:10.1143/JPSJ.55.3305"},{"id":"arxiv:0902.4845","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.103.090501","source":"Crossref","title":"Error Threshold for Color Codes and Random Three-Body Ising Models","volume":"103","author":[{"given":"Helmut G.","family":"Katzgraber","sequence":"first","affiliation":[]},{"given":"H.","family":"Bombin","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.103.090501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090501","note":"arxivid:0902.4845"},{"id":"arxiv:0903.2102","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreve.80.011141","source":"Crossref","title":"Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices","volume":"80","author":[{"given":"Masayuki","family":"Ohzeki","sequence":"first","affiliation":[]}],"container-title":"Physical Review E","original-title":[],"language":"en","issued":{"date-parts":[[2009,7,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevE.80.011141","ISSN":["1539-3755","1550-2376"],"container-title-short":"Phys. Rev. E","page":"011141","note":"arxivid:0903.2102"},{"id":"arxiv:2412.15165","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8081","DOI":"10.1038/s41586-025-09367-3","page":"620-625","source":"Crossref","title":"Experimental demonstration of logical magic state distillation","volume":"645","author":[{"ORCID":"https://orcid.org/0009-0002-8337-0762","authenticated-orcid":false,"given":"Pedro","family":"Sales Rodriguez","sequence":"first","affiliation":[]},{"given":"John M.","family":"Robinson","sequence":"additional","affiliation":[]},{"given":"Paul Niklas","family":"Jepsen","sequence":"additional","affiliation":[]},{"given":"Zhiyang","family":"He","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4656-9644","authenticated-orcid":false,"given":"Casey","family":"Duckering","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-4386-1334","authenticated-orcid":false,"given":"Chen","family":"Zhao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9126-5300","authenticated-orcid":false,"given":"Kai-Hsin","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Campo","sequence":"additional","affiliation":[]},{"given":"Kevin","family":"Bagnall","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8874-8456","authenticated-orcid":false,"given":"Minho","family":"Kwon","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Karolyshyn","sequence":"additional","affiliation":[]},{"given":"Phillip","family":"Weinberg","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5298-3112","authenticated-orcid":false,"given":"Madelyn","family":"Cain","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8986-1103","authenticated-orcid":false,"given":"Simon J.","family":"Evered","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5294-4941","authenticated-orcid":false,"given":"Alexandra A.","family":"Geim","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0605-8791","authenticated-orcid":false,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[]},{"given":"Sophie H.","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3470-1369","authenticated-orcid":false,"given":"Tom","family":"Manovitz","sequence":"additional","affiliation":[]},{"given":"Jesse","family":"Amato-Grill","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6786-0211","authenticated-orcid":false,"given":"James I.","family":"Basham","sequence":"additional","affiliation":[]},{"given":"Liane","family":"Bernstein","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5193-2711","authenticated-orcid":false,"given":"Boris","family":"Braverman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8276-8256","authenticated-orcid":false,"given":"Alexei","family":"Bylinskii","sequence":"additional","affiliation":[]},{"given":"Adam","family":"Choukri","sequence":"additional","affiliation":[]},{"given":"Robert J.","family":"DeAngelo","sequence":"additional","affiliation":[]},{"given":"Fang","family":"Fang","sequence":"additional","affiliation":[]},{"given":"Connor","family":"Fieweger","sequence":"additional","affiliation":[]},{"given":"Paige","family":"Frederick","sequence":"additional","affiliation":[]},{"given":"David","family":"Haines","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0003-0446-4013","authenticated-orcid":false,"given":"Majd","family":"Hamdan","sequence":"additional","affiliation":[]},{"given":"Julian","family":"Hammett","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0316-5102","authenticated-orcid":false,"given":"Ning","family":"Hsu","sequence":"additional","affiliation":[]},{"given":"Ming-Guang","family":"Hu","sequence":"additional","affiliation":[]},{"given":"Florian","family":"Huber","sequence":"additional","affiliation":[]},{"given":"Ningyuan","family":"Jia","sequence":"additional","affiliation":[]},{"given":"Dhruv","family":"Kedar","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3828-6377","authenticated-orcid":false,"given":"Milan","family":"Kornjača","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5690-4995","authenticated-orcid":false,"given":"Fangli","family":"Liu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0005-7557-197X","authenticated-orcid":false,"given":"John","family":"Long","sequence":"additional","affiliation":[]},{"given":"Jonathan","family":"Lopatin","sequence":"additional","affiliation":[]},{"given":"Pedro L. S.","family":"Lopes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9012-9024","authenticated-orcid":false,"given":"Xiu-Zhe","family":"Luo","sequence":"additional","affiliation":[]},{"given":"Tommaso","family":"Macrì","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7094-0124","authenticated-orcid":false,"given":"Ognjen","family":"Marković","sequence":"additional","affiliation":[]},{"given":"Luis A.","family":"Martínez-Martínez","sequence":"additional","affiliation":[]},{"given":"Xianmei","family":"Meng","sequence":"additional","affiliation":[]},{"given":"Stefan","family":"Ostermann","sequence":"additional","affiliation":[]},{"given":"Evgeny","family":"Ostroumov","sequence":"additional","affiliation":[]},{"given":"David","family":"Paquette","sequence":"additional","affiliation":[]},{"given":"Zexuan","family":"Qiang","sequence":"additional","affiliation":[]},{"given":"Vadim","family":"Shofman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0793-2682","authenticated-orcid":false,"given":"Anshuman","family":"Singh","sequence":"additional","affiliation":[]},{"given":"Manuj","family":"Singh","sequence":"additional","affiliation":[]},{"given":"Nandan","family":"Sinha","sequence":"additional","affiliation":[]},{"given":"Henry","family":"Thoreen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3320-5291","authenticated-orcid":false,"given":"Noel","family":"Wan","sequence":"additional","affiliation":[]},{"given":"Yiping","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Waxman-Lenz","sequence":"additional","affiliation":[]},{"given":"Tak","family":"Wong","sequence":"additional","affiliation":[]},{"given":"Jonathan","family":"Wurtz","sequence":"additional","affiliation":[]},{"given":"Andrii","family":"Zhdanov","sequence":"additional","affiliation":[]},{"given":"Laurent","family":"Zheng","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Greiner","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Keesling","sequence":"additional","affiliation":[]},{"given":"Nathan","family":"Gemelke","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9786-0538","authenticated-orcid":false,"given":"Vladan","family":"Vuletić","sequence":"additional","affiliation":[]},{"given":"Takuya","family":"Kitagawa","sequence":"additional","affiliation":[]},{"given":"Sheng-Tao","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9934-9530","authenticated-orcid":false,"given":"Dolev","family":"Bluvstein","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8658-1007","authenticated-orcid":false,"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Lukin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2148-8856","authenticated-orcid":false,"given":"Hengyun","family":"Zhou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9530-2559","authenticated-orcid":false,"given":"Sergio H.","family":"Cantú","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,14]]},"URL":"http://dx.doi.org/10.1038/s41586-025-09367-3","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:9367\narxivid:2412.15165"},{"id":"arxiv:1911.00355","type":"journal-article","publisher":"IOP Publishing","issue":"2","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>The color code is a topological quantum error-correcting code supporting a variety of valuable fault-tolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. To guide this experimental effort, we study the storage threshold of the triangular color code against circuit-level depolarizing noise. First, we adapt the Restriction Decoder to the setting of the triangular color code and to phenomenological noise. Then, we propose a fault-tolerant implementation of the stabilizer measurement circuits, which incorporates flag qubits. We show how information from flag qubits can be used in an efficient and scalable way with the Restriction Decoder to maintain the effective distance of the code. We numerically estimate the threshold of the triangular color code to be 0.2%, which is competitive with the thresholds of other topological quantum codes. We also prove that 1-flag stabilizer measurement circuits are sufficient to preserve the full code distance, which may be used to find simpler syndrome extraction circuits of the color code.</jats:p>","DOI":"10.1088/1367-2630/ab68fd","page":"023019","source":"Crossref","title":"Triangular color codes on trivalent graphs with flag qubits","volume":"22","author":[{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":false,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Theodore J","family":"Yoder","sequence":"additional","affiliation":[]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2020,2,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ab68fd","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1911.00355"},{"id":"arxiv:2003.03049","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Fault-tolerant quantum computing promises significant computational speedup over classical computing for a variety of important problems. One of the biggest challenges for realizing fault-tolerant quantum computing is preparing magic states with sufficiently low error rates. Magic state distillation is one of the most efficient schemes for preparing high-quality magic states. However, since magic state distillation circuits are not fault-tolerant, all the operations in the distillation circuits must be encoded in a large distance error-correcting code, resulting in a significant resource overhead. Here, we propose a fault-tolerant scheme for directly preparing high-quality magic states, which makes magic state distillation unnecessary. In particular, we introduce a concept that we call redundant ancilla encoding. The latter combined with flag qubits allows for circuits to both measure stabilizer generators of some code, while also being able to measure global operators to fault-tolerantly prepare magic states, all using nearest neighbor interactions. We apply such schemes to a planar architecture of the triangular color code family and demonstrate that our scheme requires at least an order of magnitude fewer qubits and space–time overhead compared to the most competitive magic state distillation schemes. Since our scheme requires only nearest-neighbor interactions in a planar architecture, it is suitable for various quantum computing platforms currently under development.</jats:p>","DOI":"10.1038/s41534-020-00319-5","source":"Crossref","title":"Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":false,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6318-8472","authenticated-orcid":false,"given":"Kyungjoo","family":"Noh","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2020,10,27]]},"URL":"http://dx.doi.org/10.1038/s41534-020-00319-5","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"91","note":"alternative-id:319\narxivid:2003.03049"},{"id":"arxiv:2409.07707","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>T</a:mi></a:math> measurement, achieving infidelities of <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>O</c:mi><c:mo stretchy=\"false\">(</c:mo><c:msup><c:mi>p</c:mi><c:mn>3</c:mn></c:msup><c:mo stretchy=\"false\">)</c:mo></c:math> for physical noise strength <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:mi>p</g:mi></g:math>. To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:mn>2</i:mn><i:mo>×</i:mo><i:msup><i:mn>10</i:mn><i:mrow><i:mo>−</i:mo><i:mn>16</i:mn></i:mrow></i:msup></i:math> at <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:mi>p</k:mi><k:mo>=</k:mo><k:msup><k:mn>10</k:mn><k:mrow><k:mo>−</k:mo><k:mn>3</k:mn></k:mrow></k:msup></k:math>), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.</jats:p>","DOI":"10.1103/ch5r-cnfq","source":"Crossref","title":"Low-Overhead Magic State Distillation with Color Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-1207-2752","authenticated-orcid":true,"given":"Seok-Hyung","family":"Lee","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Sydney"}]},{"given":"Felix","family":"Thomsen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Sydney"}]},{"ORCID":"https://orcid.org/0009-0008-0007-1016","authenticated-orcid":true,"given":"Nicholas","family":"Fazio","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM Quantum"},{"id":[{"id":"https://ror.org/043ncwv78","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM Denmark"}]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"The University of Sydney"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,30]]},"URL":"http://dx.doi.org/10.1103/ch5r-cnfq","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030317","note":"arxivid:2409.07707"},{"id":"arxiv:1111.0831","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.85.022317","source":"Crossref","title":"Efficient decoding of topological color codes","volume":"85","author":[{"given":"Pradeep","family":"Sarvepalli","sequence":"first","affiliation":[]},{"given":"Robert","family":"Raussendorf","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,2,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.85.022317","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022317","note":"arxivid:1111.0831"},{"id":"arxiv:1802.08680","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.99.052351","source":"Crossref","title":"Advantages of versatile neural-network decoding for topological codes","volume":"99","author":[{"given":"Nishad","family":"Maskara","sequence":"first","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Tomas","family":"Jochym-O'Connor","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,5,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.052351","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052351","note":"arxivid:1802.08680"},{"id":"arxiv:2108.11395","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.3.010310","source":"Crossref","title":"Decoder for the Triangular Color Code by Matching on a Möbius Strip","volume":"3","author":[{"given":"Kaavya","family":"Sahay","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,1,18]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.010310","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010310","note":"arxivid:2108.11395"},{"id":"arxiv:2202.06612","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit50566.2022.9834489","page":"186-191","source":"Crossref","title":"Comparison of 2D topological codes and their decoding performances","author":[{"given":"Kao-Yueh","family":"Kuo","sequence":"first","affiliation":[{"name":"National Yang Ming Chiao Tung University,Institute of Communications Engineering,Hsinchu,Taiwan,300093"}]},{"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[{"name":"National Yang Ming Chiao Tung University,Institute of Communications Engineering,Hsinchu,Taiwan,300093"}]}],"event":"2022 IEEE International Symposium on Information Theory (ISIT)","container-title":"2022 IEEE International Symposium on Information Theory (ISIT)","original-title":[],"issued":{"date-parts":[[2022,6,26]]},"URL":"http://dx.doi.org/10.1109/ISIT50566.2022.9834489","note":"arxivid:2202.06612"},{"id":"arxiv:2104.13659","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum information needs to be protected by quantum error-correcting codes due to imperfect physical devices and operations. One would like to have an efficient and high-performance decoding procedure for the class of quantum stabilizer codes. A potential candidate is Gallager’s sum-product algorithm, also known as Pearl’s belief propagation (BP), but its performance suffers from the many short cycles inherent in a quantum stabilizer code, especially highly-degenerate codes. A general impression exists that BP is not effective for topological codes. In this paper, we propose a decoding algorithm for quantum codes based on quaternary BP with additional memory effects (called MBP). This MBP is like a recursive neural network with inhibitions between neurons (edges with negative weights), which enhance the perception capability of a network. Moreover, MBP exploits the degeneracy of a quantum code so that the most probable error or its degenerate errors can be found with high probability. The decoding performance is significantly improved over the conventional BP for various quantum codes, including quantum bicycle, hypergraph-product, surface and toric codes. For MBP on the surface and toric codes over depolarizing errors, we observe error thresholds of 16% and 17.5%, respectively.</jats:p>","DOI":"10.1038/s41534-022-00623-2","source":"Crossref","title":"Exploiting degeneracy in belief propagation decoding of quantum codes","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-1390-5197","authenticated-orcid":false,"given":"Kao-Yueh","family":"Kuo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1970-8167","authenticated-orcid":false,"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,14]]},"URL":"http://dx.doi.org/10.1038/s41534-022-00623-2","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"111","note":"alternative-id:623\narxivid:2104.13659"},{"id":"arxiv:2303.14237","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In classical computing, error-correcting codes are well established and are ubiquitous both in theory and practical applications. For quantum computing, error-correction is essential as well, but harder to realize, coming along with substantial resource overheads and being concomitant with needs for substantial classical computing. Quantum error-correcting codes play a central role on the avenue towards fault-tolerant quantum computation beyond presumed near-term applications. Among those, color codes constitute a particularly important class of quantum codes that have gained interest in recent years due to favourable properties over other codes. As in classical computing, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>d</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>g</mml:mi></mml:math> is the problem of inferring an operation to restore an uncorrupted state from a corrupted one and is central in the development of fault-tolerant quantum devices. In this work, we show how the decoding problem for color codes can be reduced to a slight variation of the well-known <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext mathvariant=\"monospace\">LightsOut</mml:mtext></mml:mrow></mml:math> puzzle. We propose a novel decoder for quantum color codes using a formulation as a MaxSAT problem based on this analogy. Furthermore, we optimize the MaxSAT construction and show numerically that the decoding performance of the proposed decoder achieves state-of-the-art decoding performance on color codes. The implementation of the decoder as well as tools to automatically conduct numerical experiments are publicly available as part of the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">Munich Quantum Toolkit</mml:mtext></mml:mrow></mml:math> (MQT) on GitHub.</jats:p>","DOI":"10.22331/q-2024-10-23-1506","page":"1506","source":"Crossref","title":"Decoding quantum color codes with MaxSAT","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0002-2973-1689","authenticated-orcid":false,"given":"Lucas","family":"Berent","sequence":"first","affiliation":[{"name":"Technical University of Munich, Germany"}]},{"ORCID":"https://orcid.org/0000-0003-4699-1316","authenticated-orcid":false,"given":"Lukas","family":"Burgholzer","sequence":"additional","affiliation":[{"name":"Johannes Kepler University Linz, Austria"}]},{"ORCID":"https://orcid.org/0000-0002-9197-1309","authenticated-orcid":false,"given":"Peter-Jan H.S.","family":"Derks","sequence":"additional","affiliation":[{"name":"Freie Universität Berlin, Germany"}]},{"ORCID":"https://orcid.org/0000-0003-3033-1292","authenticated-orcid":false,"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"name":"Freie Universität Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie, Germany"}]},{"ORCID":"https://orcid.org/0000-0002-4993-7860","authenticated-orcid":false,"given":"Robert","family":"Wille","sequence":"additional","affiliation":[{"name":"Technical University of Munich, Germany"},{"name":"Software Competence Center Hagenberg GmbH (SCCH), Austria"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,23]]},"URL":"http://dx.doi.org/10.22331/q-2024-10-23-1506","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2303.14237"},{"id":"arxiv:2412.14256","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8081","DOI":"10.1038/s41586-025-09061-4","page":"614-619","source":"Crossref","title":"Scaling and logic in the colour code on a superconducting quantum processor","volume":"645","author":[{"ORCID":"https://orcid.org/0000-0003-2002-9495","authenticated-orcid":false,"given":"N.","family":"Lacroix","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8517-7475","authenticated-orcid":false,"given":"A.","family":"Bourassa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8124-2359","authenticated-orcid":false,"given":"F. J. H.","family":"Heras","sequence":"additional","affiliation":[]},{"given":"L. M.","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3189-9162","authenticated-orcid":false,"given":"J.","family":"Bausch","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2401-5691","authenticated-orcid":false,"given":"A. W.","family":"Senior","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-0473-3727","authenticated-orcid":false,"given":"T.","family":"Edlich","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6035-2812","authenticated-orcid":false,"given":"N.","family":"Shutty","sequence":"additional","affiliation":[]},{"given":"V.","family":"Sivak","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5264-4025","authenticated-orcid":false,"given":"A.","family":"Bengtsson","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9544-141X","authenticated-orcid":false,"given":"M.","family":"McEwen","sequence":"additional","affiliation":[]},{"given":"O.","family":"Higgott","sequence":"additional","affiliation":[]},{"given":"D.","family":"Kafri","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6711-4585","authenticated-orcid":false,"given":"J.","family":"Claes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9515-6184","authenticated-orcid":false,"given":"A.","family":"Morvan","sequence":"additional","affiliation":[]},{"given":"Z.","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2585-2424","authenticated-orcid":false,"given":"A.","family":"Zalcman","sequence":"additional","affiliation":[]},{"given":"S.","family":"Madhuk","sequence":"additional","affiliation":[]},{"given":"R.","family":"Acharya","sequence":"additional","affiliation":[]},{"given":"L.","family":"Aghababaie Beni","sequence":"additional","affiliation":[]},{"given":"G.","family":"Aigeldinger","sequence":"additional","affiliation":[]},{"given":"R.","family":"Alcaraz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7806-1626","authenticated-orcid":false,"given":"T. I.","family":"Andersen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8765-5621","authenticated-orcid":false,"given":"M.","family":"Ansmann","sequence":"additional","affiliation":[]},{"given":"F.","family":"Arute","sequence":"additional","affiliation":[]},{"given":"K.","family":"Arya","sequence":"additional","affiliation":[]},{"given":"A.","family":"Asfaw","sequence":"additional","affiliation":[]},{"given":"J.","family":"Atalaya","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6979-9533","authenticated-orcid":false,"given":"R.","family":"Babbush","sequence":"additional","affiliation":[]},{"given":"B.","family":"Ballard","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6523-6730","authenticated-orcid":false,"given":"J. C.","family":"Bardin","sequence":"additional","affiliation":[]},{"given":"A.","family":"Bilmes","sequence":"additional","affiliation":[]},{"given":"S.","family":"Blackwell","sequence":"additional","affiliation":[]},{"given":"J.","family":"Bovaird","sequence":"additional","affiliation":[]},{"given":"D.","family":"Bowers","sequence":"additional","affiliation":[]},{"given":"L.","family":"Brill","sequence":"additional","affiliation":[]},{"given":"M.","family":"Broughton","sequence":"additional","affiliation":[]},{"given":"D. A.","family":"Browne","sequence":"additional","affiliation":[]},{"given":"B.","family":"Buchea","sequence":"additional","affiliation":[]},{"given":"B. B.","family":"Buckley","sequence":"additional","affiliation":[]},{"given":"T.","family":"Burger","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8474-6317","authenticated-orcid":false,"given":"B.","family":"Burkett","sequence":"additional","affiliation":[]},{"given":"N.","family":"Bushnell","sequence":"additional","affiliation":[]},{"given":"A.","family":"Cabrera","sequence":"additional","affiliation":[]},{"given":"J.","family":"Campero","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9681-6108","authenticated-orcid":false,"given":"H.-S.","family":"Chang","sequence":"additional","affiliation":[]},{"given":"B.","family":"Chiaro","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9324-8168","authenticated-orcid":false,"given":"L.-Y.","family":"Chih","sequence":"additional","affiliation":[]},{"given":"A. Y.","family":"Cleland","sequence":"additional","affiliation":[]},{"given":"J.","family":"Cogan","sequence":"additional","affiliation":[]},{"given":"R.","family":"Collins","sequence":"additional","affiliation":[]},{"given":"P.","family":"Conner","sequence":"additional","affiliation":[]},{"given":"W.","family":"Courtney","sequence":"additional","affiliation":[]},{"given":"A. L.","family":"Crook","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3856-0615","authenticated-orcid":false,"given":"B.","family":"Curtin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-4853-1051","authenticated-orcid":false,"given":"S.","family":"Das","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3727-7380","authenticated-orcid":false,"given":"S.","family":"Demura","sequence":"additional","affiliation":[]},{"given":"L.","family":"De Lorenzo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5174-7310","authenticated-orcid":false,"given":"A.","family":"Di Paolo","sequence":"additional","affiliation":[]},{"given":"P.","family":"Donohoe","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8185-1194","authenticated-orcid":false,"given":"I.","family":"Drozdov","sequence":"additional","affiliation":[]},{"given":"A.","family":"Dunsworth","sequence":"additional","affiliation":[]},{"given":"A.","family":"Eickbusch","sequence":"additional","affiliation":[]},{"given":"A. Moshe","family":"Elbag","sequence":"additional","affiliation":[]},{"given":"M.","family":"Elzouka","sequence":"additional","affiliation":[]},{"given":"C.","family":"Erickson","sequence":"additional","affiliation":[]},{"given":"V. S.","family":"Ferreira","sequence":"additional","affiliation":[]},{"given":"L.","family":"Flores Burgos","sequence":"additional","affiliation":[]},{"given":"E.","family":"Forati","sequence":"additional","affiliation":[]},{"given":"A. G.","family":"Fowler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3732-168X","authenticated-orcid":false,"given":"B.","family":"Foxen","sequence":"additional","affiliation":[]},{"given":"S.","family":"Ganjam","sequence":"additional","affiliation":[]},{"given":"G.","family":"Garcia","sequence":"additional","affiliation":[]},{"given":"R.","family":"Gasca","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2226-1813","authenticated-orcid":false,"given":"É.","family":"Genois","sequence":"additional","affiliation":[]},{"given":"W.","family":"Giang","sequence":"additional","affiliation":[]},{"given":"D.","family":"Gilboa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3025-7312","authenticated-orcid":false,"given":"R.","family":"Gosula","sequence":"additional","affiliation":[]},{"given":"A.","family":"Grajales Dau","sequence":"additional","affiliation":[]},{"given":"D.","family":"Graumann","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4034-8744","authenticated-orcid":false,"given":"A.","family":"Greene","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":false,"given":"J. A.","family":"Gross","sequence":"additional","affiliation":[]},{"given":"T.","family":"Ha","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9867-2371","authenticated-orcid":false,"given":"S.","family":"Habegger","sequence":"additional","affiliation":[]},{"given":"M.","family":"Hansen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9412-0553","authenticated-orcid":false,"given":"M. P.","family":"Harrigan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0521-8378","authenticated-orcid":false,"given":"S. D.","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"S.","family":"Heslin","sequence":"additional","affiliation":[]},{"given":"P.","family":"Heu","sequence":"additional","affiliation":[]},{"given":"R.","family":"Hiltermann","sequence":"additional","affiliation":[]},{"given":"J.","family":"Hilton","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7421-5635","authenticated-orcid":false,"given":"S.","family":"Hong","sequence":"additional","affiliation":[]},{"given":"H.-Y.","family":"Huang","sequence":"additional","affiliation":[]},{"given":"A.","family":"Huff","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2735-1380","authenticated-orcid":false,"given":"W. J.","family":"Huggins","sequence":"additional","affiliation":[]},{"given":"E.","family":"Jeffrey","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0435-655X","authenticated-orcid":false,"given":"Z.","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"X.","family":"Jin","sequence":"additional","affiliation":[]},{"given":"C.","family":"Joshi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8751-4458","authenticated-orcid":false,"given":"P.","family":"Juhas","sequence":"additional","affiliation":[]},{"given":"A.","family":"Kabel","sequence":"additional","affiliation":[]},{"given":"H.","family":"Kang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7400-9604","authenticated-orcid":false,"given":"A. H.","family":"Karamlou","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0136-1428","authenticated-orcid":false,"given":"K.","family":"Kechedzhi","sequence":"additional","affiliation":[]},{"given":"T.","family":"Khaire","sequence":"additional","affiliation":[]},{"given":"T.","family":"Khattar","sequence":"additional","affiliation":[]},{"given":"M.","family":"Khezri","sequence":"additional","affiliation":[]},{"given":"S.","family":"Kim","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9481-4456","authenticated-orcid":false,"given":"P. V.","family":"Klimov","sequence":"additional","affiliation":[]},{"given":"B.","family":"Kobrin","sequence":"additional","affiliation":[]},{"given":"A. N.","family":"Korotkov","sequence":"additional","affiliation":[]},{"given":"F.","family":"Kostritsa","sequence":"additional","affiliation":[]},{"given":"J. Mark","family":"Kreikebaum","sequence":"additional","affiliation":[]},{"given":"V. D.","family":"Kurilovich","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9804-2185","authenticated-orcid":false,"given":"D.","family":"Landhuis","sequence":"additional","affiliation":[]},{"given":"T.","family":"Lange-Dei","sequence":"additional","affiliation":[]},{"given":"B. W.","family":"Langley","sequence":"additional","affiliation":[]},{"given":"P.","family":"Laptev","sequence":"additional","affiliation":[]},{"given":"K.-M.","family":"Lau","sequence":"additional","affiliation":[]},{"given":"J.","family":"Ledford","sequence":"additional","affiliation":[]},{"given":"K.","family":"Lee","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4297-2508","authenticated-orcid":false,"given":"B. J.","family":"Lester","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3110-3371","authenticated-orcid":false,"given":"L.","family":"Le Guevel","sequence":"additional","affiliation":[]},{"given":"W. Yan","family":"Li","sequence":"additional","affiliation":[]},{"given":"Y.","family":"Li","sequence":"additional","affiliation":[]},{"given":"A. T.","family":"Lill","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8399-0975","authenticated-orcid":false,"given":"W. P.","family":"Livingston","sequence":"additional","affiliation":[]},{"given":"A.","family":"Locharla","sequence":"additional","affiliation":[]},{"given":"E.","family":"Lucero","sequence":"additional","affiliation":[]},{"given":"D.","family":"Lundahl","sequence":"additional","affiliation":[]},{"given":"A.","family":"Lunt","sequence":"additional","affiliation":[]},{"given":"A.","family":"Maloney","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3908-2694","authenticated-orcid":false,"given":"S.","family":"Mandrà","sequence":"additional","affiliation":[]},{"given":"L. S.","family":"Martin","sequence":"additional","affiliation":[]},{"given":"O.","family":"Martin","sequence":"additional","affiliation":[]},{"given":"C.","family":"Maxfield","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2809-0509","authenticated-orcid":false,"given":"J. R.","family":"McClean","sequence":"additional","affiliation":[]},{"given":"S.","family":"Meeks","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6371-6140","authenticated-orcid":false,"given":"A.","family":"Megrant","sequence":"additional","affiliation":[]},{"given":"K. C.","family":"Miao","sequence":"additional","affiliation":[]},{"given":"R.","family":"Molavi","sequence":"additional","affiliation":[]},{"given":"S.","family":"Molina","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7124-4585","authenticated-orcid":false,"given":"S.","family":"Montazeri","sequence":"additional","affiliation":[]},{"given":"R.","family":"Movassagh","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8509-9836","authenticated-orcid":false,"given":"C.","family":"Neill","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6640-1072","authenticated-orcid":false,"given":"M.","family":"Newman","sequence":"additional","affiliation":[]},{"given":"A.","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"M.","family":"Nguyen","sequence":"additional","affiliation":[]},{"given":"C.-H.","family":"Ni","sequence":"additional","affiliation":[]},{"given":"M. Y.","family":"Niu","sequence":"additional","affiliation":[]},{"given":"L.","family":"Oas","sequence":"additional","affiliation":[]},{"given":"W. D.","family":"Oliver","sequence":"additional","affiliation":[]},{"given":"R.","family":"Orosco","sequence":"additional","affiliation":[]},{"given":"K.","family":"Ottosson","sequence":"additional","affiliation":[]},{"given":"A.","family":"Pizzuto","sequence":"additional","affiliation":[]},{"given":"R.","family":"Potter","sequence":"additional","affiliation":[]},{"given":"O.","family":"Pritchard","sequence":"additional","affiliation":[]},{"given":"C.","family":"Quintana","sequence":"additional","affiliation":[]},{"given":"G.","family":"Ramachandran","sequence":"additional","affiliation":[]},{"given":"M. J.","family":"Reagor","sequence":"additional","affiliation":[]},{"given":"R.","family":"Resnick","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3788-7947","authenticated-orcid":false,"given":"D. M.","family":"Rhodes","sequence":"additional","affiliation":[]},{"given":"G.","family":"Roberts","sequence":"additional","affiliation":[]},{"given":"E.","family":"Rosenberg","sequence":"additional","affiliation":[]},{"given":"E.","family":"Rosenfeld","sequence":"additional","affiliation":[]},{"given":"E.","family":"Rossi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1917-3879","authenticated-orcid":false,"given":"P.","family":"Roushan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8930-7081","authenticated-orcid":false,"given":"K.","family":"Sankaragomathi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3210-5929","authenticated-orcid":false,"given":"H. F.","family":"Schurkus","sequence":"additional","affiliation":[]},{"given":"M. J.","family":"Shearn","sequence":"additional","affiliation":[]},{"given":"A.","family":"Shorter","sequence":"additional","affiliation":[]},{"given":"V.","family":"Shvarts","sequence":"additional","affiliation":[]},{"given":"S.","family":"Small","sequence":"additional","affiliation":[]},{"given":"W. Clarke","family":"Smith","sequence":"additional","affiliation":[]},{"given":"S.","family":"Springer","sequence":"additional","affiliation":[]},{"given":"G.","family":"Sterling","sequence":"additional","affiliation":[]},{"given":"J.","family":"Suchard","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1127-2111","authenticated-orcid":false,"given":"A.","family":"Szasz","sequence":"additional","affiliation":[]},{"given":"A.","family":"Sztein","sequence":"additional","affiliation":[]},{"given":"D.","family":"Thor","sequence":"additional","affiliation":[]},{"given":"E.","family":"Tomita","sequence":"additional","affiliation":[]},{"given":"A.","family":"Torres","sequence":"additional","affiliation":[]},{"given":"M. Mert","family":"Torunbalci","sequence":"additional","affiliation":[]},{"given":"A.","family":"Vaishnav","sequence":"additional","affiliation":[]},{"given":"J.","family":"Vargas","sequence":"additional","affiliation":[]},{"given":"S.","family":"Vdovichev","sequence":"additional","affiliation":[]},{"given":"G.","family":"Vidal","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6513-9885","authenticated-orcid":false,"given":"C.","family":"Vollgraff Heidweiller","sequence":"additional","affiliation":[]},{"given":"S.","family":"Waltman","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0001-3202-5152","authenticated-orcid":false,"given":"J.","family":"Waltz","sequence":"additional","affiliation":[]},{"given":"S. X.","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3321-3198","authenticated-orcid":false,"given":"B.","family":"Ware","sequence":"additional","affiliation":[]},{"given":"T.","family":"Weidel","sequence":"additional","affiliation":[]},{"given":"T.","family":"White","sequence":"additional","affiliation":[]},{"given":"K.","family":"Wong","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7233-1576","authenticated-orcid":false,"given":"B. W. K.","family":"Woo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7754-0601","authenticated-orcid":false,"given":"M.","family":"Woodson","sequence":"additional","affiliation":[]},{"given":"C.","family":"Xing","sequence":"additional","affiliation":[]},{"given":"Z. Jamie","family":"Yao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0837-1028","authenticated-orcid":false,"given":"P.","family":"Yeh","sequence":"additional","affiliation":[]},{"given":"B.","family":"Ying","sequence":"additional","affiliation":[]},{"given":"J.","family":"Yoo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4169-5745","authenticated-orcid":false,"given":"N.","family":"Yosri","sequence":"additional","affiliation":[]},{"given":"G.","family":"Young","sequence":"additional","affiliation":[]},{"given":"Y.","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3037-2003","authenticated-orcid":false,"given":"N.","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"N.","family":"Zobrist","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9681-6746","authenticated-orcid":false,"given":"H.","family":"Neven","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7466-7997","authenticated-orcid":false,"given":"P.","family":"Kohli","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4917-5234","authenticated-orcid":false,"given":"A.","family":"Davies","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1090-7584","authenticated-orcid":false,"given":"S.","family":"Boixo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2596-2121","authenticated-orcid":false,"given":"J.","family":"Kelly","sequence":"additional","affiliation":[]},{"given":"C.","family":"Jones","sequence":"additional","affiliation":[]},{"given":"C.","family":"Gidney","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5865-0813","authenticated-orcid":false,"given":"K. J.","family":"Satzinger","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,26]]},"URL":"http://dx.doi.org/10.1038/s41586-025-09061-4","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:9061\narxivid:2412.14256"},{"id":"arxiv:1005.0777","type":"journal-article","publisher":"IOP Publishing","issue":"8","DOI":"10.1088/1367-2630/13/8/083006","page":"083006","source":"Crossref","title":"Tricolored lattice gauge theory with randomness: fault tolerance in topological color codes","volume":"13","author":[{"given":"Ruben S","family":"Andrist","sequence":"first","affiliation":[]},{"given":"Helmut G","family":"Katzgraber","sequence":"additional","affiliation":[]},{"given":"H","family":"Bombin","sequence":"additional","affiliation":[]},{"given":"M A","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2011,8,8]]},"URL":"http://dx.doi.org/10.1088/1367-2630/13/8/083006","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1005.0777"},{"id":"arxiv:2112.09584","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.106.032431","source":"Crossref","title":"Rescaling decoder for two-dimensional topological quantum color codes on 4.8.8 lattices","volume":"106","author":[{"ORCID":"https://orcid.org/0000-0002-6201-2723","authenticated-orcid":true,"given":"Pedro","family":"Parrado-Rodríguez","sequence":"first","affiliation":[{"name":"Department of Physics, College of Science, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom"}]},{"given":"Manuel","family":"Rispler","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":true,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany"},{"name":"Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,9,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.106.032431","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032431","note":"arxivid:2112.09584"},{"id":"arxiv:2512.13908","type":"article-journal","author":[{"given":"Emma","family":"Rosenfeld"},{"given":"Craig","family":"Gidney"},{"given":"Gabrielle","family":"Roberts"},{"given":"Alexis","family":"Morvan"},{"given":"Nathan","family":"Lacroix"},{"given":"Dvir","family":"Kafri"},{"given":"Jeffrey","family":"Marshall"},{"given":"Ming","family":"Li"},{"given":"Volodymyr","family":"Sivak"},{"given":"Dmitry","family":"Abanin"},{"given":"Amira","family":"Abbas"},{"given":"Rajeev","family":"Acharya"},{"given":"Laleh Aghababaie","family":"Beni"},{"given":"Georg","family":"Aigeldinger"},{"given":"Ross","family":"Alcaraz"},{"given":"Sayra","family":"Alcaraz"},{"given":"Trond I.","family":"Andersen"},{"given":"Markus","family":"Ansmann"},{"given":"Frank","family":"Arute"},{"given":"Kunal","family":"Arya"},{"given":"Walt","family":"Askew"},{"given":"Nikita","family":"Astrakhantsev"},{"given":"Juan","family":"Atalaya"},{"given":"Ryan","family":"Babbush"},{"given":"Brian","family":"Ballard"},{"given":"Joseph C.","family":"Bardin"},{"given":"Hector","family":"Bates"},{"given":"Andreas","family":"Bengtsson"},{"given":"Majid Bigdeli","family":"Karimi"},{"given":"Alexander","family":"Bilmes"},{"given":"Simon","family":"Bilodeau"},{"given":"Felix","family":"Borjans"},{"given":"Jenna","family":"Bovaird"},{"given":"Dylan","family":"Bowers"},{"given":"Leon","family":"Brill"},{"given":"Peter","family":"Brooks"},{"given":"Michael","family":"Broughton"},{"given":"David A.","family":"Browne"},{"given":"Brett","family":"Buchea"},{"given":"Bob B.","family":"Buckley"},{"given":"Tim","family":"Burger"},{"given":"Brian","family":"Burkett"},{"given":"Nicholas","family":"Bushnell"},{"given":"Jamal","family":"Busnaina"},{"given":"Anthony","family":"Cabrera"},{"given":"Juan","family":"Campero"},{"given":"Hung-Shen","family":"Chang"},{"given":"Silas","family":"Chen"},{"given":"Zijun","family":"Chen"},{"given":"Ben","family":"Chiaro"},{"given":"Liang-Ying","family":"Chih"},{"given":"Agnetta Y.","family":"Cleland"},{"given":"Bryan","family":"Cochrane"},{"given":"Matt","family":"Cockrell"},{"given":"Josh","family":"Cogan"},{"given":"Paul","family":"Conner"},{"given":"Harold","family":"Cook"},{"given":"Rodrigo G.","family":"Cortiñas"},{"given":"William","family":"Courtney"},{"given":"Alexander L.","family":"Crook"},{"given":"Ben","family":"Curtin"},{"given":"Martin","family":"Damyanov"},{"given":"Sayan","family":"Das"},{"given":"Dripto M.","family":"Debroy"},{"given":"Sean","family":"Demura"},{"given":"Paul","family":"Donohoe"},{"given":"Ilya","family":"Drozdov"},{"given":"Andrew","family":"Dunsworth"},{"given":"Valerie","family":"Ehimhen"},{"given":"Alec","family":"Eickbusch"},{"given":"Aviv Moshe","family":"Elbag"},{"given":"Lior","family":"Ella"},{"given":"Mahmoud","family":"Elzouka"},{"given":"David","family":"Enriquez"},{"given":"Catherine","family":"Erickson"},{"given":"Lara","family":"Faoro"},{"given":"Vinicius S.","family":"Ferreira"},{"given":"Marcos","family":"Flores"},{"given":"Leslie Flores","family":"Burgos"},{"given":"Sam","family":"Fontes"},{"given":"Ebrahim","family":"Forati"},{"given":"Jeremiah","family":"Ford"},{"given":"Brooks","family":"Foxen"},{"given":"Masaya","family":"Fukami"},{"given":"Alan Wing Lun","family":"Fung"},{"given":"Lenny","family":"Fuste"},{"given":"Suhas","family":"Ganjam"},{"given":"Gonzalo","family":"Garcia"},{"given":"Christopher","family":"Garrick"},{"given":"Robert","family":"Gasca"},{"given":"Helge","family":"Gehring"},{"given":"Robert","family":"Geiger"},{"given":"Élie","family":"Genois"},{"given":"William","family":"Giang"},{"given":"Dar","family":"Gilboa"},{"given":"James E.","family":"Goeders"},{"given":"Edward C.","family":"Gonzales"},{"given":"Raja","family":"Gosula"},{"given":"Stijn J.","family":"de Graaf"},{"given":"Alejandro Grajales","family":"Dau"},{"given":"Dietrich","family":"Graumann"},{"given":"Joel","family":"Grebel"},{"given":"Alex","family":"Greene"},{"given":"Jonathan A.","family":"Gross"},{"given":"Jose","family":"Guerrero"},{"given":"Loïck","family":"Le Guevel"},{"given":"Tan","family":"Ha"},{"given":"Steve","family":"Habegger"},{"given":"Tanner","family":"Hadick"},{"given":"Ali","family":"Hadjikhani"},{"given":"Michael C.","family":"Hamilton"},{"given":"Monica","family":"Hansen"},{"given":"Matthew P.","family":"Harrigan"},{"given":"Sean D.","family":"Harrington"},{"given":"Jeanne","family":"Hartshorn"},{"given":"Stephen","family":"Heslin"},{"given":"Paula","family":"Heu"},{"given":"Oscar","family":"Higgott"},{"given":"Reno","family":"Hiltermann"},{"given":"Jeremy","family":"Hilton"},{"given":"Hsin-Yuan","family":"Huang"},{"given":"Mike","family":"Hucka"},{"given":"Christopher","family":"Hudspeth"},{"given":"Ashley","family":"Huff"},{"given":"William J.","family":"Huggins"},{"given":"Lev B.","family":"Ioffe"},{"given":"Evan","family":"Jeffrey"},{"given":"Shaun","family":"Jevons"},{"given":"Zhang","family":"Jiang"},{"given":"Xiaoxuan","family":"Jin"},{"given":"Chaitali","family":"Joshi"},{"given":"Pavol","family":"Juhas"},{"given":"Andreas","family":"Kabel"},{"given":"Hui","family":"Kang"},{"given":"Kiseo","family":"Kang"},{"given":"Amir H.","family":"Karamlou"},{"given":"Ryan","family":"Kaufman"},{"given":"Kostyantyn","family":"Kechedzhi"},{"given":"Tanuj","family":"Khattar"},{"given":"Mostafa","family":"Khezri"},{"given":"Seon","family":"Kim"},{"given":"Paul V.","family":"Klimov"},{"given":"Can M.","family":"Knaut"},{"given":"Bryce","family":"Kobrin"},{"given":"Alexander N.","family":"Korotkov"},{"given":"Fedor","family":"Kostritsa"},{"given":"John Mark","family":"Kreikebaum"},{"given":"Ryuho","family":"Kudo"},{"given":"Ben","family":"Kueffler"},{"given":"Arun","family":"Kumar"},{"given":"Vladislav D.","family":"Kurilovich"},{"given":"Vitali","family":"Kutsko"},{"given":"Tiano","family":"Lange-Dei"},{"given":"Brandon W.","family":"Langley"},{"given":"Pavel","family":"Laptev"},{"given":"Kim-Ming","family":"Lau"},{"given":"Emma","family":"Leavell"},{"given":"Justin","family":"Ledford"},{"given":"Joy","family":"Lee"},{"given":"Kenny","family":"Lee"},{"given":"Brian J.","family":"Lester"},{"given":"Wendy","family":"Leung"},{"given":"Lily","family":"Li"},{"given":"Wing Yan","family":"Li"},{"given":"Alexander T.","family":"Lill"},{"given":"William P.","family":"Livingston"},{"given":"Matthew T.","family":"Lloyd"},{"given":"Aditya","family":"Locharla"},{"given":"Laura","family":"De Lorenzo"},{"given":"Erik","family":"Lucero"},{"given":"Daniel","family":"Lundahl"},{"given":"Aaron","family":"Lunt"},{"given":"Sid","family":"Madhuk"},{"given":"Aniket","family":"Maiti"},{"given":"Ashley","family":"Maloney"},{"given":"Salvatore","family":"Mandrà"},{"given":"Leigh S.","family":"Martin"},{"given":"Orion","family":"Martin"},{"given":"Eric","family":"Mascot"},{"given":"Paul Masih","family":"Das"},{"given":"Dmitri","family":"Maslov"},{"given":"Melvin","family":"Mathews"},{"given":"Cameron","family":"Maxfield"},{"given":"Jarrod R.","family":"McClean"},{"given":"Matt","family":"McEwen"},{"given":"Seneca","family":"Meeks"},{"given":"Anthony","family":"Megrant"},{"given":"Kevin C.","family":"Miao"},{"given":"Zlatko K.","family":"Minev"},{"given":"Reza","family":"Molavi"},{"given":"Sebastian","family":"Molina"},{"given":"Shirin","family":"Montazeri"},{"given":"Charles","family":"Neill"},{"given":"Michael","family":"Newman"},{"given":"Anthony","family":"Nguyen"},{"given":"Murray","family":"Nguyen"},{"given":"Chia-Hung","family":"Ni"},{"given":"Murphy Yuezhen","family":"Niu"},{"given":"Nicholas","family":"Noll"},{"given":"Logan","family":"Oas"},{"given":"William D.","family":"Oliver"},{"given":"Raymond","family":"Orosco"},{"given":"Kristoffer","family":"Ottosson"},{"given":"Alice","family":"Pagano"},{"given":"Agustin","family":"Di Paolo"},{"given":"Sherman","family":"Peek"},{"given":"David","family":"Peterson"},{"given":"Alex","family":"Pizzuto"},{"given":"Elias","family":"Portoles"},{"given":"Rebecca","family":"Potter"},{"given":"Orion","family":"Pritchard"},{"given":"Michael","family":"Qian"},{"given":"Chris","family":"Quintana"},{"given":"Ganesh","family":"Ramachandran"},{"given":"Arpit","family":"Ranadive"},{"given":"Matthew J.","family":"Reagor"},{"given":"Rachel","family":"Resnick"},{"given":"David M.","family":"Rhodes"},{"given":"Daniel","family":"Riley"},{"given":"Roberto","family":"Rodriguez"},{"given":"Emma","family":"Ropes"},{"given":"Lucia B.","family":"De Rose"},{"given":"Eliott","family":"Rosenberg"},{"given":"Dario","family":"Rosenstock"},{"given":"Elizabeth","family":"Rossi"},{"given":"Pedram","family":"Roushan"},{"given":"David A.","family":"Rower"},{"given":"Robert","family":"Salazar"},{"given":"Kannan","family":"Sankaragomathi"},{"given":"Murat Can","family":"Sarihan"},{"given":"Max","family":"Schaefer"},{"given":"Sebastian","family":"Schroeder"},{"given":"Henry F.","family":"Schurkus"},{"given":"Aria","family":"Shahingohar"},{"given":"Michael J.","family":"Shearn"},{"given":"Aaron","family":"Shorter"},{"given":"Noah","family":"Shutty"},{"given":"Vladimir","family":"Shvarts"},{"given":"Spencer","family":"Small"},{"given":"W. Clarke","family":"Smith"},{"given":"David A.","family":"Sobel"},{"given":"Barrett","family":"Spells"},{"given":"Sofia","family":"Springer"},{"given":"George","family":"Sterling"},{"given":"Jordan","family":"Suchard"},{"given":"Aaron","family":"Szasz"},{"given":"Alexander","family":"Sztein"},{"given":"Madeline","family":"Taylor"},{"given":"Jothi Priyanka","family":"Thiruraman"},{"given":"Douglas","family":"Thor"},{"given":"Dogan","family":"Timucin"},{"given":"Eifu","family":"Tomita"},{"given":"Alfredo","family":"Torres"},{"given":"M. Mert","family":"Torunbalci"},{"given":"Hao","family":"Tran"},{"given":"Abeer","family":"Vaishnav"},{"given":"Justin","family":"Vargas"},{"given":"Sergey","family":"Vdovichev"},{"given":"Guifre","family":"Vidal"},{"given":"Benjamin","family":"Villalonga"},{"given":"Catherine Vollgraff","family":"Heidweiller"},{"given":"Meghan","family":"Voorhees"},{"given":"Steven","family":"Waltman"},{"given":"Jonathan","family":"Waltz"},{"given":"Shannon X.","family":"Wang"},{"given":"Danni","family":"Wang"},{"given":"Brayden","family":"Ware"},{"given":"James D.","family":"Watson"},{"given":"Yonghua","family":"Wei"},{"given":"Travis","family":"Weidel"},{"given":"Theodore","family":"White"},{"given":"Kristi","family":"Wong"},{"given":"Bryan W. K.","family":"Woo"},{"given":"Christopher J.","family":"Wood"},{"given":"Maddy","family":"Woodson"},{"given":"Cheng","family":"Xing"},{"given":"Z. Jamie","family":"Yao"},{"given":"Ping","family":"Yeh"},{"given":"Bicheng","family":"Ying"},{"given":"Juhwan","family":"Yoo"},{"given":"Noureldin","family":"Yosri"},{"given":"Elliot","family":"Young"},{"given":"Grayson","family":"Young"},{"given":"Adam","family":"Zalcman"},{"given":"Ran","family":"Zhang"},{"given":"Yaxing","family":"Zhang"},{"given":"Ningfeng","family":"Zhu"},{"given":"Nicholas","family":"Zobrist"},{"given":"Zhenjie","family":"Zou"},{"given":"Hartmut","family":"Neven"},{"given":"Sergio","family":"Boixo"},{"given":"Cody","family":"Jones"},{"given":"Julian","family":"Kelly"},{"given":"Alexandre","family":"Bourassa"},{"given":"Kevin J.","family":"Satzinger"}],"title":"Magic state cultivation on a superconducting quantum processor","issued":{"date-parts":[[2025,12,15]]},"note":"arxivid:2512.13908\narxiv_version_number:1"},{"id":"arxiv:2404.05033","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We investigate boundaries of 3D color codes and provide a systematic classification into 101 distinct boundary types, including two novel classes. The first class consists of 1 boundary and is generated by sweeping the codimension-1 (2D) <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-domain wall across the system and attaching it to the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math>-boundary that condenses only magnetic fluxes. Since the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>-domain wall cannot condense on the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math>-boundary, a new <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">magic boundary</mml:mtext></mml:mrow></mml:math> is produced, where the boundary stabilizers contain <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>S</mml:mi></mml:math>-stabilizers going beyond the conventional Pauli stabilizer formalism, and hence contains 'magic'. Neither electric nor magnetic excitations can condense on such a magic boundary, and only the composite of the magnetic flux and codimension-2 (1D) <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>S</mml:mi></mml:math>-domain wall can condense on it, which makes the magic boundary going beyond the classification of the Lagrangian subgroup. The second class consists of 70 boundaries and is generated by sweeping the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>S</mml:mi></mml:math>-domain wall across a codimension-1 submanifold and attaching it to the boundary. This generates a codimension-2 (1D) <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">nested boundary</mml:mtext></mml:mrow></mml:math> at the intersection. We also connect these novel boundaries to their previously discovered counterpart in the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msubsup><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:msubsup></mml:math> gauge theory, equivalent to three copies of 3D toric codes, where the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>S</mml:mi></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math> domain walls correspond to gauged symmetry-protected topological (SPT) defects. New boundaries are produced whenever the corresponding symmetry of the SPT defect remains unbroken on the boundary. Applications of the magic boundaries include implementing fault-tolerant non-Clifford logical gates, e.g., in the context of fractal topological codes.</jats:p>","DOI":"10.22331/q-2025-08-27-1831","page":"1831","source":"Crossref","title":"Magic Boundaries of 3D Color Codes","volume":"9","author":[{"given":"Zijian","family":"Song","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University of California, Davis, California 95616, USA"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"IBM Quantum, T.J. Watson Research Center, Yorktown Heights, NY 10598 USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,27]]},"URL":"http://dx.doi.org/10.22331/q-2025-08-27-1831","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2404.05033"},{"id":"arxiv:1606.00960","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.98.012302","source":"Crossref","title":"Projecting three-dimensional color codes onto three-dimensional toric codes","volume":"98","author":[{"given":"Arun B.","family":"Aloshious","sequence":"first","affiliation":[]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,7,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.98.012302","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012302","note":"arxivid:1606.00960"},{"id":"arxiv:1006.5260","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1088/1367-2630/13/4/043005","page":"043005","source":"Crossref","title":"Clifford gates by code deformation","volume":"13","author":[{"given":"H","family":"Bombin","sequence":"first","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2011,4,7]]},"URL":"http://dx.doi.org/10.1088/1367-2630/13/4/043005","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1006.5260"},{"id":"arxiv:2411.15035","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/lwxd-fdlb","source":"Crossref","title":"Color Code with a Logical Control-\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <mml:mi>S</mml:mi>\n                    </mml:math>\n                    Gate Using Transversal\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <mml:mi>T</mml:mi>\n                    </mml:math>\n                    Rotations","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0265w5591","id-type":"ROR","asserted-by":"publisher"}],"name":"T. J. Watson Research Center"},{"id":[{"id":"https://ror.org/043ncwv78","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM Denmark"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,14]]},"URL":"http://dx.doi.org/10.1103/lwxd-fdlb","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070602","note":"arxivid:2411.15035"},{"id":"arxiv:quant-ph/9610011","type":"article-journal","author":[{"given":"E.","family":"Knill"},{"given":"R.","family":"Laflamme"},{"given":"W.","family":"Zurek"}],"title":"Threshold Accuracy for Quantum Computation","issued":{"date-parts":[[1996,10,15]]},"note":"arxivid:quant-ph/9610011\narxiv_version_number:3"},{"id":"arxiv:1708.07131","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.120.180501","source":"Crossref","title":"Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping","volume":"120","author":[{"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"given":"Michael E.","family":"Beverland","sequence":"additional","affiliation":[]},{"given":"Fernando","family":"Brandão","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]},{"given":"Krysta M.","family":"Svore","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.120.180501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180501","note":"arxivid:1708.07131"},{"id":"arxiv:1810.09571","type":"article-journal","author":[{"given":"Hector","family":"Bombin"}],"title":"2D quantum computation with 3D topological codes","issued":{"date-parts":[[2018,10,22]]},"note":"arxivid:1810.09571\narxiv_version_number:1"},{"id":"arxiv:1311.0277","type":"article-journal","author":[{"given":"H.","family":"Bombin"}],"title":"An Introduction to Topological Quantum Codes","issued":{"date-parts":[[2013,11,1]]},"note":"arxivid:1311.0277\narxiv_version_number:1"},{"id":"arxiv:2201.12450","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevapplied.18.014072","source":"Crossref","title":"Decoding Merged Color-Surface Codes and Finding Fault-Tolerant Clifford Circuits Using Solvers for Satisfiability Modulo Theories","volume":"18","author":[{"given":"Noah","family":"Shutty","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2022,7,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.18.014072","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"014072","note":"arxivid:2201.12450"},{"id":"arxiv:1207.0479","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.86.042336","source":"Crossref","title":"Topological subsystem codes from graphs and hypergraphs","volume":"86","author":[{"given":"Pradeep","family":"Sarvepalli","sequence":"first","affiliation":[]},{"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,10,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.042336","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042336","note":"arxivid:1207.0479"},{"id":"arxiv:1804.06382","type":"article-journal","author":[{"given":"Eduardo Brandani","family":"da Silva"},{"given":"Waldir Silva","family":"Soares"}],"title":"Hyperbolic quantum color codes","issued":{"date-parts":[[2018,4,17]]},"note":"arxivid:1804.06382\narxiv_version_number:1"},{"id":"arxiv:1306.1538","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevb.90.115118","source":"Crossref","title":"Unconventional fusion and braiding of topological defects in a lattice model","volume":"90","author":[{"given":"Jeffrey C. Y.","family":"Teo","sequence":"first","affiliation":[]},{"given":"Abhishek","family":"Roy","sequence":"additional","affiliation":[]},{"given":"Xiao","family":"Chen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.90.115118","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"115118","note":"arxivid:1306.1538"},{"id":"arxiv:1709.02318","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present a planar surface-code-based scheme for fault-tolerant quantum computation which eliminates the time overhead of single-qubit Clifford gates, and implements long-range multi-target CNOT gates with a time overhead that scales only logarithmically with the control-target separation. This is done by replacing hardware operations for single-qubit Clifford gates with a classical tracking protocol. Inter-qubit communication is added via a modified lattice surgery protocol that employs twist defects of the surface code. The long-range multi-target CNOT gates facilitate magic state distillation, which renders our scheme fault-tolerant and universal.</jats:p>","DOI":"10.22331/q-2018-05-04-62","page":"62","source":"Crossref","title":"Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes","volume":"2","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]},{"given":"Felix von","family":"Oppen","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,4]]},"URL":"http://dx.doi.org/10.22331/q-2018-05-04-62","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1709.02318"},{"id":"arxiv:2104.03669","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.104.012603","source":"Crossref","title":"Color codes with twists: Construction and universal-gate-set implementation","volume":"104","author":[{"ORCID":"https://orcid.org/0000-0002-6298-7037","authenticated-orcid":true,"given":"Manoj G.","family":"Gowda","sequence":"first","affiliation":[]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,7,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.104.012603","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012603","note":"arxivid:2104.03669"},{"id":"arxiv:2411.05402","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s11128-025-04703-y","source":"Crossref","title":"Color codes with domino twists: construction, logical measurements, and computation","volume":"24","author":[{"given":"Manoj G.","family":"Gowda","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,12]]},"URL":"http://dx.doi.org/10.1007/s11128-025-04703-y","ISSN":["1573-1332"],"container-title-short":"Quantum Inf Process","page":"87","note":"alternative-id:4703\narxivid:2411.05402"},{"id":"arxiv:2307.00054","type":"journal-article","publisher":"American Physical Society (APS)","issue":"11","DOI":"10.1103/physrevlett.133.110601","source":"Crossref","title":"Domain Wall Color Code","volume":"133","author":[{"ORCID":"https://orcid.org/0000-0002-9216-3696","authenticated-orcid":true,"given":"Konstantin","family":"Tiurev","sequence":"first","affiliation":[{"name":"HQS Quantum Simulations GmbH"}]},{"ORCID":"https://orcid.org/0000-0002-5759-6314","authenticated-orcid":true,"given":"Arthur","family":"Pesah","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02jx3x895","id-type":"ROR","asserted-by":"publisher"}],"name":"University College London"}]},{"ORCID":"https://orcid.org/0000-0002-9197-1309","authenticated-orcid":true,"given":"Peter-Jan H. S.","family":"Derks","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"}]},{"given":"Joschka","family":"Roffe","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"},{"id":[{"id":"https://ror.org/01nrxwf90","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Edinburgh"}]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"},{"id":[{"id":"https://ror.org/02aj13c28","id-type":"ROR","asserted-by":"publisher"}],"name":"Helmholtz-Zentrum Berlin für Materialien und Energie"}]},{"given":"Markus S.","family":"Kesselring","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046ak2485","id-type":"ROR","asserted-by":"publisher"}],"name":"Freie Universität Berlin"}]},{"given":"Jan-Michael","family":"Reiner","sequence":"additional","affiliation":[{"name":"HQS Quantum Simulations GmbH"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,9,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.110601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"110601","note":"arxivid:2307.00054"},{"id":"arxiv:1501.07779","type":"journal-article","publisher":"IOP Publishing","issue":"21","DOI":"10.1088/1751-8113/48/21/215302","page":"215302","source":"Crossref","title":"A family of stabilizer codes for $D({{\\mathbb{Z}}_{2}})$ anyons and Majorana modes","volume":"48","author":[{"given":"James R","family":"Wootton","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2015,5,6]]},"URL":"http://dx.doi.org/10.1088/1751-8113/48/21/215302","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1501.07779"},{"id":"arxiv:1609.07774","type":"journal-article","publisher":"IOP Publishing","issue":"1","DOI":"10.1088/2058-9565/aa5c73","page":"015006","source":"Crossref","title":"Demonstrating non-Abelian braiding of surface code defects in a five qubit experiment","volume":"2","author":[{"given":"James R","family":"Wootton","sequence":"first","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2017,3,1]]},"URL":"http://dx.doi.org/10.1088/2058-9565/aa5c73","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1609.07774"},{"id":"arxiv:2109.13308","type":"journal-article","publisher":"IOP Publishing","issue":"29","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>Matching codes are stabilizer codes based on Kitaev’s honeycomb lattice model. The hexagonal form of these codes are particularly well-suited to the heavy-hexagon device layouts currently pursued in the hardware of IBM quantum. Here we show how the stabilizers of the code can be measured solely through two-body measurements that are native to the architecture. Though the subsystem code formed by these measurements has a trivial code space, the sequence in which they are measured allows the desired logical subspace to be preserved. This therefore achieves a result similar to the recently introduced Floquet codes, but via a completely different method. The process is then run on 27 and 65 qubit devices, to compare results with simulations for a standard error model. It is found that the results correspond well to simulations where the noise strength is similar to that found in the benchmarking of the devices. The best devices show results consistent with a noise model with an error probability of around 1.5%–2%.</jats:p>","DOI":"10.1088/1751-8121/ac7a75","page":"295302","source":"Crossref","title":"Hexagonal matching codes with two-body measurements","volume":"55","author":[{"ORCID":"https://orcid.org/0000-0003-1943-5306","authenticated-orcid":true,"given":"James R","family":"Wootton","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2022,7,4]]},"URL":"http://dx.doi.org/10.1088/1751-8121/ac7a75","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:2109.13308"},{"id":"arxiv:2112.06036","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We consider a topological stabilizer code on a honeycomb grid, the \"XYZ<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi/><mml:mn>2</mml:mn></mml:msup></mml:math>\" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a \"matching code\" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mi>Z</mml:mi><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mi>Z</mml:mi></mml:math>) plaquette stabilizers and weight-two (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi><mml:mi>X</mml:mi></mml:math>) link stabilizers on a planar hexagonal grid composed of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> qubits for code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math>, or <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> only for pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math> noise, here the code distance is <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> for both pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> and pure <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Y</mml:mi></mml:math> noise. Thresholds for noise with finite <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.</jats:p>","DOI":"10.22331/q-2022-04-27-698","page":"698","source":"Crossref","title":"The XYZ<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi/><mml:mn>2</mml:mn></mml:msup></mml:math> hexagonal stabilizer code","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-4972-4216","authenticated-orcid":false,"given":"Basudha","family":"Srivastava","sequence":"first","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]},{"ORCID":"https://orcid.org/0000-0002-2534-3021","authenticated-orcid":false,"given":"Anton","family":"Frisk Kockum","sequence":"additional","affiliation":[{"name":"Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Gothenburg, Sweden"}]},{"ORCID":"https://orcid.org/0000-0003-3185-2014","authenticated-orcid":false,"given":"Mats","family":"Granath","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,27]]},"URL":"http://dx.doi.org/10.22331/q-2022-04-27-698","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2112.06036"},{"id":"arxiv:2112.01977","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.105.042616","source":"Crossref","title":"Error-rate-agnostic decoding of topological stabilizer codes","volume":"105","author":[{"ORCID":"https://orcid.org/0000-0001-5289-0752","authenticated-orcid":true,"given":"Karl","family":"Hammar","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4963-3768","authenticated-orcid":true,"given":"Alexei","family":"Orekhov","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2561-4549","authenticated-orcid":true,"given":"Patrik Wallin","family":"Hybelius","sequence":"additional","affiliation":[]},{"given":"Anna Katariina","family":"Wisakanto","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-4972-4216","authenticated-orcid":true,"given":"Basudha","family":"Srivastava","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2534-3021","authenticated-orcid":true,"given":"Anton Frisk","family":"Kockum","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3185-2014","authenticated-orcid":true,"given":"Mats","family":"Granath","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.042616","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042616","note":"arxivid:2112.01977"},{"id":"arxiv:quant-ph/9811052","type":"article-journal","author":[{"given":"S. B.","family":"Bravyi"},{"given":"A. Yu.","family":"Kitaev"}],"title":"Quantum codes on a lattice with boundary","issued":{"date-parts":[[1998,11,20]]},"note":"arxivid:quant-ph/9811052\narxiv_version_number:1"},{"id":"arxiv:1606.07116","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"},{"given":"Pavithran","family":"Iyer"},{"given":"David","family":"Poulin"}],"title":"Generalized surface codes and packing of logical qubits","issued":{"date-parts":[[2016,6,22]]},"note":"arxivid:1606.07116\narxiv_version_number:1"},{"id":"doi:10.1007/978-1-4615-5923-8_19","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-1-4615-5923-8_19","page":"181-188","source":"Crossref","title":"Quantum Error Correction with Imperfect Gates","author":[{"given":"A. Yu.","family":"Kitaev","sequence":"first","affiliation":[]}],"container-title":"Quantum Communication, Computing, and Measurement","original-title":[],"language":"en","issued":{"date-parts":[[1997]]},"ISBN":["9781461377160","9781461559238"],"URL":"http://dx.doi.org/10.1007/978-1-4615-5923-8_19","note":"publisher-location:Boston, MA"},{"id":"arxiv:2307.04418","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"6","DOI":"10.1140/epjs/s11734-023-00955-w","page":"1341-1348","source":"Crossref","title":"Quantum error correction beyond the toric code: dynamical systems meet encoding","volume":"233","author":[{"given":"Garima","family":"Rajpoot","sequence":"first","affiliation":[]},{"given":"Komal","family":"Kumari","sequence":"additional","affiliation":[]},{"given":"Sudhir Ranjan","family":"Jain","sequence":"additional","affiliation":[]}],"container-title":"The European Physical Journal Special Topics","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,2]]},"URL":"http://dx.doi.org/10.1140/epjs/s11734-023-00955-w","ISSN":["1951-6355","1951-6401"],"container-title-short":"Eur. Phys. J. Spec. Top.","note":"alternative-id:955\narxivid:2307.04418"},{"id":"arxiv:2406.18897","type":"article-journal","author":[{"given":"Shi Jie Samuel","family":"Tan"},{"given":"Christopher A.","family":"Pattison"},{"given":"Matt","family":"McEwen"},{"given":"John","family":"Preskill"}],"title":"Resilience of the surface code to error bursts","issued":{"date-parts":[[2024,6,27]]},"note":"arxivid:2406.18897\narxiv_version_number:1"},{"id":"doi:10.1063/1.1665530","type":"journal-article","publisher":"AIP Publishing","issue":"10","abstract":"<jats:p>It is shown that any Ising model with positive coupling constants is related to another Ising model by a duality transformation. We define a class of Ising models Mdn on d-dimensional lattices characterized by a number n = 1, 2, … , d (n = 1 corresponds to the Ising model with two-spin interaction). These models are related by two duality transformations. The models with 1 &amp;lt; n &amp;lt; d exhibit a phase transition without local order parameter. A nonanalyticity in the specific heat and a different qualitative behavior of certain spin correlation functions in the low and the high temperature phases indicate the existence of a phase transition. The Hamiltonian of the simple cubic dual model contains products of four Ising spin operators. Applying a star square transformation, one obtains an Ising model with competing interactions exhibiting a singularity in the specific heat but no long-range order of the spins in the low temperature phase.</jats:p>","DOI":"10.1063/1.1665530","page":"2259-2272","source":"Crossref","title":"Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters","volume":"12","author":[{"given":"Franz J.","family":"Wegner","sequence":"first","affiliation":[{"name":"Department of Physics, Brown University, Providence, Rhode Island 02912"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[1971,10,1]]},"URL":"http://dx.doi.org/10.1063/1.1665530","ISSN":["0022-2488","1089-7658"]},{"id":"arxiv:2310.08639","type":"article-journal","author":[{"given":"Shengqi","family":"Sang"},{"given":"Yijian","family":"Zou"},{"given":"Timothy H.","family":"Hsieh"}],"title":"Mixed-state Quantum Phases: Renormalization and Quantum Error Correction","issued":{"date-parts":[[2026,2,26]]},"note":"arxivid:2310.08639\narxiv_version_number:2"},{"id":"arxiv:cond-mat/0609048","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.98.070602","source":"Crossref","title":"Breakdown of a Topological Phase: Quantum Phase Transition in a Loop Gas Model with Tension","volume":"98","author":[{"given":"Simon","family":"Trebst","sequence":"first","affiliation":[]},{"given":"Philipp","family":"Werner","sequence":"additional","affiliation":[]},{"given":"Matthias","family":"Troyer","sequence":"additional","affiliation":[]},{"given":"Kirill","family":"Shtengel","sequence":"additional","affiliation":[]},{"given":"Chetan","family":"Nayak","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,2,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.98.070602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070602","note":"arxivid:cond-mat/0609048"},{"id":"arxiv:0804.3175","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevb.82.085114","source":"Crossref","title":"Topological multicritical point in the phase diagram of the toric code model and three-dimensional lattice gauge Higgs model","volume":"82","author":[{"given":"I. S.","family":"Tupitsyn","sequence":"first","affiliation":[]},{"given":"A.","family":"Kitaev","sequence":"additional","affiliation":[]},{"given":"N. V.","family":"Prokof’ev","sequence":"additional","affiliation":[]},{"given":"P. C. E.","family":"Stamp","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2010,8,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.82.085114","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"085114","note":"arxivid:0804.3175"},{"id":"arxiv:0807.0487","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevb.79.033109","source":"Crossref","title":"Low-energy effective theory of the toric code model in a parallel magnetic field","volume":"79","author":[{"given":"Julien","family":"Vidal","sequence":"first","affiliation":[]},{"given":"Sébastien","family":"Dusuel","sequence":"additional","affiliation":[]},{"given":"Kai Phillip","family":"Schmidt","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,1,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.79.033109","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"033109","note":"arxivid:0807.0487"},{"id":"arxiv:1201.6409","type":"journal-article","publisher":"American Physical Society (APS)","issue":"19","DOI":"10.1103/physrevb.85.195104","source":"Crossref","title":"Phase diagram of the toric code model in a parallel magnetic field","volume":"85","author":[{"given":"Fengcheng","family":"Wu","sequence":"first","affiliation":[]},{"given":"Youjin","family":"Deng","sequence":"additional","affiliation":[]},{"given":"Nikolay","family":"Prokof'ev","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2012,5,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.85.195104","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"195104","note":"arxivid:1201.6409"},{"id":"arxiv:1411.5815","type":"article-journal","author":[{"given":"Franz J.","family":"Wegner"}],"title":"Duality in generalized Ising models","issued":{"date-parts":[[2014,11,21]]},"note":"arxivid:1411.5815\narxiv_version_number:1"},{"id":"arxiv:1202.1016","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.90.062311","source":"Crossref","title":"Long-distance quantum communication over noisy networks without long-time quantum memory","volume":"90","author":[{"given":"Paweł","family":"Mazurek","sequence":"first","affiliation":[]},{"given":"Andrzej","family":"Grudka","sequence":"additional","affiliation":[]},{"given":"Michał","family":"Horodecki","sequence":"additional","affiliation":[]},{"given":"Paweł","family":"Horodecki","sequence":"additional","affiliation":[]},{"given":"Justyna","family":"Łodyga","sequence":"additional","affiliation":[]},{"given":"Łukasz","family":"Pankowski","sequence":"additional","affiliation":[]},{"given":"Anna","family":"Przysiężna","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,12,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.062311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062311","note":"arxivid:1202.1016"},{"id":"arxiv:1105.2111","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/13/6/065010","page":"065010","source":"Crossref","title":"Generating topological order from a two-dimensional cluster state using a duality mapping","volume":"13","author":[{"given":"Benjamin J","family":"Brown","sequence":"first","affiliation":[]},{"given":"Wonmin","family":"Son","sequence":"additional","affiliation":[]},{"given":"Christina V","family":"Kraus","sequence":"additional","affiliation":[]},{"given":"Rosario","family":"Fazio","sequence":"additional","affiliation":[]},{"given":"Vlatko","family":"Vedral","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2011,6,20]]},"URL":"http://dx.doi.org/10.1088/1367-2630/13/6/065010","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1105.2111"},{"id":"arxiv:2002.00362","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The surface code is a leading candidate quantum error correcting code, owing to its high threshold, and compatibility with existing experimental architectures. Bravyi et al. (2006) showed that encoding a state in the surface code using local unitary operations requires time at least linear in the lattice size <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi></mml:math>, however the most efficient known method for encoding an unknown state, introduced by Dennis et al. (2002), has <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> time complexity. Here, we present an optimal local unitary encoding circuit for the planar surface code that uses exactly <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mi>L</mml:mi></mml:math> time steps to encode an unknown state in a distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>L</mml:mi></mml:math> planar code. We further show how an <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> complexity local unitary encoder for the toric code can be found by enforcing locality in the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math>-depth non-local renormalisation encoder. We relate these techniques by providing an <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> local unitary circuit to convert between a toric code and a planar code, and also provide optimal encoders for the rectangular, rotated and 3D surface codes. Furthermore, we show how our encoding circuit for the planar code can be used to prepare fermionic states in the compact mapping, a recently introduced fermion to qubit mapping that has a stabiliser structure similar to that of the surface code and is particularly efficient for simulating the Fermi-Hubbard model.</jats:p>","DOI":"10.22331/q-2021-08-05-517","page":"517","source":"Crossref","title":"Optimal local unitary encoding circuits for the surface code","volume":"5","author":[{"given":"Oscar","family":"Higgott","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Matthew","family":"Wilson","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom"}]},{"given":"James","family":"Hefford","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"Department of Computer Science, University of Oxford, Oxford OX1 3QD, United Kingdom"}]},{"given":"James","family":"Dborin","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"},{"name":"London Centre for Nanotechnology, University College London, Gordon St., London WC1H 0AH, United Kingdom"}]},{"given":"Farhan","family":"Hanif","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Simon","family":"Burton","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,5]]},"URL":"http://dx.doi.org/10.22331/q-2021-08-05-517","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2002.00362"},{"id":"arxiv:1207.0253","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.88.012328","source":"Crossref","title":"Generating and verifying graph states for fault-tolerant topological measurement-based quantum computing in two-dimensional optical lattices","volume":"88","author":[{"given":"Jaewoo","family":"Joo","sequence":"first","affiliation":[]},{"given":"Emilio","family":"Alba","sequence":"additional","affiliation":[]},{"given":"Juan José","family":"García-Ripoll","sequence":"additional","affiliation":[]},{"given":"Timothy P.","family":"Spiller","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,7,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.88.012328","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012328","note":"arxivid:1207.0253"},{"id":"arxiv:1810.03912","type":"article-journal","author":[{"given":"Dorit","family":"Aharonov"},{"given":"Yonathan","family":"Touati"}],"title":"Quantum Circuit Depth Lower Bounds For Homological Codes","issued":{"date-parts":[[2018,10,9]]},"note":"arxivid:1810.03912\narxiv_version_number:1"},{"id":"arxiv:2210.06796","type":"article-journal","author":[{"given":"Arkin","family":"Tikku"},{"given":"Isaac H.","family":"Kim"}],"title":"Circuit depth versus energy in topologically ordered systems","issued":{"date-parts":[[2022,10,13]]},"note":"arxivid:2210.06796\narxiv_version_number:1"},{"id":"arxiv:1904.01502","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"10","DOI":"10.1038/s41567-020-0948-z","page":"1040-1045","source":"Crossref","title":"Quantum advantage with noisy shallow circuits","volume":"16","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"David","family":"Gosset","sequence":"additional","affiliation":[]},{"given":"Robert","family":"König","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5410-3329","authenticated-orcid":false,"given":"Marco","family":"Tomamichel","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2020,7,6]]},"URL":"http://dx.doi.org/10.1038/s41567-020-0948-z","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:948\narxivid:1904.01502"},{"id":"arxiv:2209.09774","type":"article-journal","author":[{"given":"Shin Ho","family":"Choe"},{"given":"Robert","family":"Koenig"}],"title":"Long-range data transmission in a fault-tolerant quantum bus architecture","issued":{"date-parts":[[2022,9,20]]},"note":"arxivid:2209.09774\narxiv_version_number:1"},{"id":"arxiv:2111.13729","type":"article-journal","author":[{"given":"Anbang","family":"Wu"},{"given":"Gushu","family":"Li"},{"given":"Hezi","family":"Zhang"},{"given":"Gian Giacomo","family":"Guerreschi"},{"given":"Yufei","family":"Ding"},{"given":"Yuan","family":"Xie"}],"title":"Mapping Surface Code to Superconducting Quantum Processors","issued":{"date-parts":[[2021,11,26]]},"note":"arxivid:2111.13729\narxiv_version_number:1"},{"id":"arxiv:2211.08468","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Whether it is at the fabrication stage or during the course of the quantum computation, e.g. because of high-energy events like cosmic rays, the qubits constituting an error correcting code may be rendered inoperable. Such defects may correspond to individual qubits or to clusters and could potentially disrupt the code sufficiently to generate logical errors. In this paper, we explore a novel <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>a</mml:mi><mml:mi>d</mml:mi><mml:mi>a</mml:mi><mml:mi>p</mml:mi><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:math> approach for surface code quantum error correction on a defective lattice. We show that combining an appropriate defect detection algorithm and a quarantine of the identified zone allows one to preserve the advantage of quantum error correction at finite code sizes, at the cost of a qubit overhead that scales with the size of the defect. Our numerics indicate that the code's threshold need not be significantly affected; for example, for a certain scenario where small defects repeatedly arise in each logical qubit, the noise threshold is <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2.7</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> (versus the defect-free case of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2.9</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>). These results pave the way to the experimental implementation of large-scale quantum computers where defects will be inevitable.</jats:p>","DOI":"10.22331/q-2023-07-25-1065","page":"1065","source":"Crossref","title":"Adaptive surface code for quantum error correction in the presence of temporary or permanent defects","volume":"7","author":[{"given":"Adam","family":"Siegel","sequence":"first","affiliation":[{"name":"Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom"},{"name":"Quantum Motion, 9 Sterling Way, London N7 9HJ, United Kingdom"}]},{"given":"Armands","family":"Strikis","sequence":"additional","affiliation":[{"name":"Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom"}]},{"given":"Thomas","family":"Flatters","sequence":"additional","affiliation":[{"name":"Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom"}]},{"given":"Simon","family":"Benjamin","sequence":"additional","affiliation":[{"name":"Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom"},{"name":"Quantum Motion, 9 Sterling Way, London N7 9HJ, United Kingdom"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,25]]},"URL":"http://dx.doi.org/10.22331/q-2023-07-25-1065","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2211.08468"},{"id":"arxiv:2405.06941","type":"article-journal","author":[{"given":"Keyi","family":"Yin"},{"given":"Xiang","family":"Fang"},{"given":"Yunong","family":"Shi"},{"given":"Travis","family":"Humble"},{"given":"Ang","family":"Li"},{"given":"Yufei","family":"Ding"}],"title":"Surf-Deformer: Mitigating Dynamic Defects on Surface Code via Adaptive Deformation","issued":{"date-parts":[[2024,9,16]]},"note":"arxivid:2405.06941\narxiv_version_number:3"},{"id":"arxiv:1111.4022","type":"journal-article","publisher":"IOP Publishing","issue":"12","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>In recent years, surface codes have become a leading method for quantum error correction in theoretical large-scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural two-dimensional nearest-neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar- and defect-based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. Those resources are further reduced by the use of a rotated lattice for the planar encoding. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded <jats:sc>CNOT</jats:sc> between two distance-3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D.</jats:p>","DOI":"10.1088/1367-2630/14/12/123011","page":"123011","source":"Crossref","title":"Surface code quantum computing by lattice surgery","volume":"14","author":[{"given":"Dominic","family":"Horsman","sequence":"first","affiliation":[]},{"given":"Austin G","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"Simon","family":"Devitt","sequence":"additional","affiliation":[]},{"given":"Rodney Van","family":"Meter","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2012,12,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/14/12/123011","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1111.4022"},{"id":"arxiv:1808.02892","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Given a quantum gate circuit, how does one execute it in a fault-tolerant architecture with as little overhead as possible? In this paper, we discuss strategies for surface-code quantum computing on small, intermediate and large scales. They are strategies for space-time trade-offs, going from slow computations using few qubits to fast computations using many qubits. Our schemes are based on surface-code patches, which not only feature a low space cost compared to other surface-code schemes, but are also conceptually simple~--~simple enough that they can be described as a tile-based game with a small set of rules. Therefore, no knowledge of quantum error correction is necessary to understand the schemes in this paper, but only the concepts of qubits and measurements.</jats:p>","DOI":"10.22331/q-2019-03-05-128","page":"128","source":"Crossref","title":"A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery","volume":"3","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,3,5]]},"URL":"http://dx.doi.org/10.22331/q-2019-03-05-128","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1808.02892"},{"id":"arxiv:2109.02746","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/prxquantum.3.010331","source":"Crossref","title":"Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,25]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.010331","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010331","note":"arxivid:2109.02746"},{"id":"arxiv:1202.2639","type":"article-journal","author":[{"given":"Austin G.","family":"Fowler"}],"title":"Low-overhead surface code logical Hadamard","issued":{"date-parts":[[2012,9,8]]},"note":"arxivid:1202.2639\narxiv_version_number:2"},{"id":"arxiv:1905.06903","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Despite significant overhead reductions since its first proposal, magic state distillation is often considered to be a very costly procedure that dominates the resource cost of fault-tolerant quantum computers. The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases, it can cost less to distill a magic state than to perform a logical Clifford gate on full-distance logical qubits.</jats:p>","DOI":"10.22331/q-2019-12-02-205","page":"205","source":"Crossref","title":"Magic State Distillation: Not as Costly as You Think","volume":"3","author":[{"given":"Daniel","family":"Litinski","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,2]]},"URL":"http://dx.doi.org/10.22331/q-2019-12-02-205","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1905.06903"},{"id":"arxiv:1703.00590","type":"journal-article","publisher":"IOP Publishing","issue":"3","DOI":"10.1088/2058-9565/aa7d3b","page":"035007","source":"Crossref","title":"Hyperbolic and semi-hyperbolic surface codes for quantum storage","volume":"2","author":[{"given":"Nikolas P","family":"Breuckmann","sequence":"first","affiliation":[]},{"given":"Christophe","family":"Vuillot","sequence":"additional","affiliation":[]},{"given":"Earl","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Anirudh","family":"Krishna","sequence":"additional","affiliation":[]},{"given":"Barbara M","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2017,8,2]]},"URL":"http://dx.doi.org/10.1088/2058-9565/aa7d3b","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1703.00590"},{"id":"arxiv:1903.11634","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"21","abstract":"<jats:p>We complete a universal set of fault-tolerant quantum logic gates for a two-dimensional surface code architecture.</jats:p>","DOI":"10.1126/sciadv.aay4929","source":"Crossref","title":"A fault-tolerant non-Clifford gate for the surface code in two dimensions","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"first","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia."}]}],"container-title":"Science Advances","original-title":[],"language":"en","issued":{"date-parts":[[2020,5,22]]},"URL":"http://dx.doi.org/10.1126/sciadv.aay4929","ISSN":["2375-2548"],"container-title-short":"Sci. Adv.","page":"eaay4929","note":"alternative-id:10.1126/sciadv.aay4929\narxivid:1903.11634"},{"id":"arxiv:2412.12529","type":"article-journal","author":[{"given":"Thomas R.","family":"Scruby"},{"given":"Kae","family":"Nemoto"},{"given":"Zhenyu","family":"Cai"}],"title":"Fault-tolerant Quantum Computation without Distillation on a 2D Device","issued":{"date-parts":[[2025,2,14]]},"note":"arxivid:2412.12529\narxiv_version_number:2"},{"id":"arxiv:1704.08670","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>A leading choice of error correction for scalable quantum computing is the surface code with lattice surgery. The basic lattice surgery operations, the merging and splitting of logical qubits, act non-unitarily on the logical states and are not easily captured by standard circuit notation. This raises the question of how best to design, verify, and optimise protocols that use lattice surgery, in particular in architectures with complex resource management issues. In this paper we demonstrate that the operations of the ZX calculus --- a form of quantum diagrammatic reasoning based on bialgebras --- match exactly the operations of lattice surgery. Red and green ``spider'' nodes match rough and smooth merges and splits, and follow the axioms of a dagger special associative Frobenius algebra. Some lattice surgery operations require non-trivial correction operations, which are captured natively in the use of the ZX calculus in the form of ensembles of diagrams. We give a first taste of the power of the calculus as a language for lattice surgery by considering two operations (T gates and producing a CNOT) and show how ZX diagram re-write rules give lattice surgery procedures for these operations that are novel, efficient, and highly configurable.</jats:p>","DOI":"10.22331/q-2020-01-09-218","page":"218","source":"Crossref","title":"The ZX calculus is a language for surface code lattice surgery","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0001-9549-5146","authenticated-orcid":false,"given":"Niel","family":"de Beaudrap","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD"}]},{"ORCID":"https://orcid.org/0000-0003-4965-0584","authenticated-orcid":false,"given":"Dominic","family":"Horsman","sequence":"additional","affiliation":[{"name":"Department of Physics, Durham University, South Road, Durham, DH1 1LE Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2020,1,9]]},"URL":"http://dx.doi.org/10.22331/q-2020-01-09-218","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1704.08670"},{"id":"arxiv:1812.01238","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present magic state factory constructions for producing<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. For the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory we apply the surface code lattice surgery construction techniques described in \\cite{fowler2018} to the fault-tolerant Toffoli \\cite{jones2013, eastin2013distilling}. The resulting factory has a footprint of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>12</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>6</mml:mn><mml:mi>d</mml:mi></mml:math>(where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>is the code distance) and produces one<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>every<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>5.5</mml:mn><mml:mi>d</mml:mi></mml:math>surface code cycles. Our<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state factory uses the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory's output and a catalyst<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state to exactly transform one<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>state into two<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. It has a footprint<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>25</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math>smaller than the factory in \\cite{fowler2018} but outputs<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states twice as quickly. We show how to generalize the catalyzed transformation to arbitrary phase angles, and note that the case<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mn>22.5</mml:mn><mml:mo>∘</mml:mo></mml:msup></mml:math>produces a particularly efficient circuit for producing<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msqrt><mml:mi>T</mml:mi></mml:msqrt><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>states. Compared to using the<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>12</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>8</mml:mn><mml:mi>d</mml:mi><mml:mo>×</mml:mo><mml:mn>6.5</mml:mn><mml:mi>d</mml:mi></mml:math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory of \\cite{fowler2018}, our<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>factory can quintuple the speed of algorithms that are dominated by the cost of applying Toffoli gates, including Shor's algorithm \\cite{shor1994} and the chemistry algorithm of Babbush et al. \\cite{babbush2018}. Assuming a physical gate error rate of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>, our CCZ factory can produce<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>10</mml:mn></mml:mrow></mml:msup></mml:math>states on average before an error occurs. This is sufficient for classically intractable instantiations of the chemistry algorithm, but for more demanding algorithms such as Shor's algorithm the mean number of states until failure can be increased to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:math>by increasing the factory footprint<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>∼</mml:mo><mml:mn>20</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math>.</jats:p>","DOI":"10.22331/q-2019-04-30-135","page":"135","source":"Crossref","title":"Efficient magic state factories with a catalyzed<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>T</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>transformation","volume":"3","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Inc., Santa Barbara, California 93117, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,4,30]]},"URL":"http://dx.doi.org/10.22331/q-2019-04-30-135","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1812.01238"},{"id":"arxiv:1905.08916","type":"article-journal","author":[{"given":"Craig","family":"Gidney"},{"given":"Austin G.","family":"Fowler"}],"title":"Flexible layout of surface code computations using AutoCCZ states","issued":{"date-parts":[[2019,5,22]]},"note":"arxivid:1905.08916\narxiv_version_number:1"},{"id":"arxiv:2211.10046","type":"article-journal","author":[{"given":"Jason","family":"Gavriel"},{"family":"Daniel","given":""},{"given":"Alexis","family":"Shaw"},{"given":"Michael J.","family":"Bremner"},{"given":"Alexandru","family":"Paler"},{"given":"Simon J.","family":"Devitt"}],"title":"Transversal Injection: A method for direct encoding of ancilla states for non-Clifford gates using stabiliser codes","issued":{"date-parts":[[2022,11,22]]},"note":"arxivid:2211.10046\narxiv_version_number:2"},{"id":"arxiv:1610.03485","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/physrevb.95.155126","source":"Crossref","title":"Dynamically enriched topological orders in driven two-dimensional systems","volume":"95","author":[{"given":"Andrew C.","family":"Potter","sequence":"first","affiliation":[]},{"given":"Takahiro","family":"Morimoto","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,4,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.95.155126","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155126","note":"arxivid:1610.03485"},{"id":"arxiv:1708.02360","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.97.022335","source":"Crossref","title":"Holonomic surface codes for fault-tolerant quantum computation","volume":"97","author":[{"given":"Jiang","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Simon J.","family":"Devitt","sequence":"additional","affiliation":[]},{"given":"J. Q.","family":"You","sequence":"additional","affiliation":[]},{"given":"Franco","family":"Nori","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,2,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.022335","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022335","note":"arxivid:1708.02360"},{"id":"arxiv:2411.18287","type":"article-journal","author":[{"given":"Alexandre","family":"Guernut"},{"given":"Christophe","family":"Vuillot"}],"title":"Fault-Tolerant Constant-Depth Clifford Gates on Toric Codes","issued":{"date-parts":[[2024,11,27]]},"note":"arxivid:2411.18287\narxiv_version_number:1"},{"id":"arxiv:2502.01743","type":"article-journal","author":[{"given":"Yotam","family":"Vaknin"},{"given":"Shoham","family":"Jacoby"},{"given":"Arne","family":"Grimsmo"},{"given":"Alex","family":"Retzker"}],"title":"Efficient Magic State Cultivation on the Surface Code","issued":{"date-parts":[[2026,1,30]]},"note":"arxivid:2502.01743\narxiv_version_number:4"},{"id":"arxiv:2302.02192","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The typical time-independent view of quantum error correction (QEC) codes hides significant freedom in the decomposition into circuits that are executable on hardware. Using the concept of detecting regions, we design time-dynamic QEC circuits directly instead of designing static QEC codes to decompose into circuits. In particular, we improve on the standard circuit constructions for the surface code, presenting new circuits that can embed on a hexagonal grid instead of a square grid, that can use ISWAP gates instead of CNOT or CZ gates, that can exchange qubit data and measure roles, and that move logical patches around the physical qubit grid while executing. All these constructions use no additional entangling gate layers and display essentially the same logical performance, having teraquop footprints within 25% of the standard surface code circuit. We expect these circuits to be of great interest to quantum hardware engineers, because they achieve essentially the same logical performance as standard surface code circuits while relaxing demands on hardware.</jats:p>","DOI":"10.22331/q-2023-11-07-1172","page":"1172","source":"Crossref","title":"Relaxing Hardware Requirements for Surface Code Circuits using Time-dynamics","volume":"7","author":[{"given":"Matt","family":"McEwen","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Seattle, Washington 98103, USA"}]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,7]]},"URL":"http://dx.doi.org/10.22331/q-2023-11-07-1172","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2302.02192"},{"id":"arxiv:2506.09029","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","abstract":"<jats:p>\n                    Stabilizer quantum error correction (QEC) codes, in particular topological surface codes, are prime candidates to enable practical quantum computing. While it is widely believed that strictly fault-tolerant protocols can only be implemented using single- and two-qubit gates, several quantum computing platforms, including trapped ions, neutral atoms, and superconducting qubits, support native multi-qubit operations. In this Letter, we show that stabilizer measurement circuits for unrotated surface codes can be fault tolerant using single auxiliary qubits and  gates. These gates enable lower-depth circuits with fewer fault locations and potentially shorter QEC cycle times. We find that in an optimistic parameter regime where fidelities of three-qubit gates are the same as those of two-qubit gates, the logical error rate can be up to one order of magnitude lower and the threshold significantly higher, increasing from\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mo>≈</a:mo>\n                      <a:mn>0.63</a:mn>\n                      <a:mo>%</a:mo>\n                    </a:math>\n                    to\n                    <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <c:mo>≈</c:mo>\n                      <c:mn>0.83</c:mn>\n                      <c:mo>%</c:mo>\n                    </c:math>\n                    . Our results, applicable to a wide range of platforms, motivate further investigation into multi-qubit gates for fault-tolerant QEC as they can offer substantial time and physical qubit resource advantages to reach a given target logical error rate.\n                  </jats:p>","DOI":"10.1103/sblg-fbq4","source":"Crossref","title":"Fault-Tolerant Stabilizer Measurements in Surface Codes with Three-Qubit Gates","volume":"135","author":[{"ORCID":"https://orcid.org/0009-0002-8811-1401","authenticated-orcid":true,"given":"Josias","family":"Old","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"},{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"}]},{"ORCID":"https://orcid.org/0009-0000-4765-5037","authenticated-orcid":true,"given":"Stephan","family":"Tasler","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f7hpc57","id-type":"ROR","asserted-by":"publisher"}],"name":"Friedrich-Alexander-Universität Erlangen-Nürnberg"}]},{"ORCID":"https://orcid.org/0000-0002-8207-3806","authenticated-orcid":true,"given":"Michael J.","family":"Hartmann","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00f7hpc57","id-type":"ROR","asserted-by":"publisher"}],"name":"Friedrich-Alexander-Universität Erlangen-Nürnberg"}]},{"ORCID":"https://orcid.org/0000-0002-2813-3097","authenticated-orcid":true,"given":"Markus","family":"Müller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02nv7yv05","id-type":"ROR","asserted-by":"publisher"}],"name":"Forschungszentrum Jülich"},{"id":[{"id":"https://ror.org/04xfq0f34","id-type":"ROR","asserted-by":"publisher"}],"name":"RWTH Aachen University"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,9]]},"URL":"http://dx.doi.org/10.1103/sblg-fbq4","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"240601","note":"arxivid:2506.09029"},{"id":"arxiv:2506.09028","type":"article-journal","author":[{"given":"Stephan","family":"Tasler"},{"given":"Josias","family":"Old"},{"given":"Lukas","family":"Heunisch"},{"given":"Verena","family":"Feulner"},{"given":"Timo","family":"Eckstein"},{"given":"Markus","family":"Müller"},{"given":"Michael J.","family":"Hartmann"}],"title":"Optimizing Superconducting Three-Qubit Gates for Surface-Code Error Correction","issued":{"date-parts":[[2025,6,10]]},"note":"arxivid:2506.09028\narxiv_version_number:1"},{"id":"manual:-A.-Landahl-private-communica","type":"document","author":[{"family":"Landahl","given":"A."}],"genre":"private communication,","issued":"2023"},{"id":"arxiv:1110.5133","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.108.180501","source":"Crossref","title":"Towards Practical Classical Processing for the Surface Code","volume":"108","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Adam C.","family":"Whiteside","sequence":"additional","affiliation":[]},{"given":"Lloyd C. L.","family":"Hollenberg","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,5,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.108.180501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180501","note":"arxivid:1110.5133"},{"id":"arxiv:1202.5602","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.86.042313","source":"Crossref","title":"Towards practical classical processing for the surface code: Timing analysis","volume":"86","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Adam C.","family":"Whiteside","sequence":"additional","affiliation":[]},{"given":"Lloyd C. L.","family":"Hollenberg","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,10,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.042313","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042313","note":"arxivid:1202.5602"},{"id":"doi:10.1088/0305-4470/15/2/033","type":"journal-article","publisher":"IOP Publishing","issue":"2","DOI":"10.1088/0305-4470/15/2/033","page":"673-699","source":"Crossref","title":"Morphology of ground states of two-dimensional frustration model","volume":"15","author":[{"given":"F","family":"Barahona","sequence":"first","affiliation":[]},{"given":"R","family":"Maynard","sequence":"additional","affiliation":[]},{"given":"R","family":"Rammal","sequence":"additional","affiliation":[]},{"given":"J P","family":"Uhry","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[1982,2,1]]},"URL":"http://dx.doi.org/10.1088/0305-4470/15/2/033","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen."},{"id":"arxiv:2309.10331","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit. In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>Z</mml:mi></mml:math> noise.</jats:p>","DOI":"10.22331/q-2024-10-28-1511","page":"1511","source":"Crossref","title":"Hardness results for decoding the surface code with Pauli noise","volume":"8","author":[{"given":"Alex","family":"Fischer","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87106, USA"}]},{"given":"Akimasa","family":"Miyake","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87106, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,28]]},"URL":"http://dx.doi.org/10.22331/q-2024-10-28-1511","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2309.10331"},{"id":"arxiv:2105.13082","type":"article-journal","author":[{"given":"Oscar","family":"Higgott"}],"title":"PyMatching: A Python package for decoding quantum codes with minimum-weight perfect matching","issued":{"date-parts":[[2021,7,12]]},"note":"arxivid:2105.13082\narxiv_version_number:2"},{"id":"arxiv:1405.4883","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.90.032326","source":"Crossref","title":"Efficient algorithms for maximum likelihood decoding in the surface code","volume":"90","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[]},{"given":"Martin","family":"Suchara","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Vargo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,9,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.90.032326","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032326","note":"arxivid:1405.4883"},{"id":"doi:10.1103/PhysRev.88.1332","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrev.88.1332","page":"1332-1337","source":"Crossref","title":"A Combinatorial Solution of the Two-Dimensional Ising Model","volume":"88","author":[{"given":"M.","family":"Kac","sequence":"first","affiliation":[]},{"given":"J. C.","family":"Ward","sequence":"additional","affiliation":[]}],"container-title":"Physical Review","original-title":[],"language":"en","issued":{"date-parts":[[1952,12,15]]},"URL":"http://dx.doi.org/10.1103/PhysRev.88.1332","ISSN":["0031-899X"],"container-title-short":"Phys. Rev."},{"id":"arxiv:1709.06218","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In order to build a large scale quantum computer, one must be able to correct errors extremely fast. We design a fast decoding algorithm for topological codes to correct for Pauli errors and erasure and combination of both errors and erasure. Our algorithm has a worst case complexity of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>n</mml:mi></mml:math> is the number of physical qubits and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>α</mml:mi></mml:math> is the inverse of Ackermann's function, which is very slowly growing. For all practical purposes, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>3</mml:mn></mml:math>. We prove that our algorithm performs optimally for errors of weight up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math> and for loss of up to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math> qubits, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is the minimum distance of the code. Numerically, we obtain a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>9.9</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math> for the 2d-toric code with perfect syndrome measurements and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2.6</mml:mn><mml:mi mathvariant=\"normal\">%</mml:mi></mml:math> with faulty measurements.</jats:p>","DOI":"10.22331/q-2021-12-02-595","page":"595","source":"Crossref","title":"Almost-linear time decoding algorithm for topological codes","volume":"5","author":[{"given":"Nicolas","family":"Delfosse","sequence":"first","affiliation":[{"name":"IQIM, California Institute of Technology, Pasadena, CA, USA"},{"name":"Department of Physics and Astronomy, University of California, Riverside, CA, USA"},{"name":"Station Q Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA"}]},{"given":"Naomi H.","family":"Nickerson","sequence":"additional","affiliation":[{"name":"Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,2]]},"URL":"http://dx.doi.org/10.22331/q-2021-12-02-595","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1709.06218"},{"id":"doi:10.1145/364099.364331","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"5","abstract":"<jats:p>An algorithm for assigning storage on the basis of EQUIVALENCE, DIMENSION and COMMON declarations is presented. The algorithm is based on a tree structure, and has reduced computation time by 40 percent over a previously published algorithm by identifying all equivalence classes with one scan of the EQUIVALENCE declarations. The method is applicable in any problem in which it is necessary to identify equivalence classes, given the element pairs defining the equivalence relation.</jats:p>","DOI":"10.1145/364099.364331","page":"301-303","source":"Crossref","title":"An improved equivalence algorithm","volume":"7","author":[{"given":"Bernard A.","family":"Galler","sequence":"first","affiliation":[{"name":"Univ. of Michigan, Ann Arbor"}]},{"given":"Michael J.","family":"Fisher","sequence":"additional","affiliation":[{"name":"Univ. of Michigan, Ann Arbor"}]}],"container-title":"Communications of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[1964,5]]},"URL":"http://dx.doi.org/10.1145/364099.364331","ISSN":["0001-0782","1557-7317"],"container-title-short":"Commun. ACM","note":"alternative-id:10.1145/364099.364331"},{"id":"doi:10.1137/0202024","type":"journal-article","publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","DOI":"10.1137/0202024","page":"294-303","source":"Crossref","title":"Set Merging Algorithms","volume":"2","author":[{"given":"J. E.","family":"Hopcroft","sequence":"first","affiliation":[]},{"given":"J. D.","family":"Ullman","sequence":"additional","affiliation":[]}],"container-title":"SIAM Journal on Computing","original-title":[],"language":"en","issued":{"date-parts":[[1973,12]]},"URL":"http://dx.doi.org/10.1137/0202024","ISSN":["0097-5397","1095-7111"],"container-title-short":"SIAM J. Comput.","note":"alternative-id:10.1137/0202024"},{"id":"doi:10.1145/62.2160","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"2","DOI":"10.1145/62.2160","page":"245-281","source":"Crossref","title":"Worst-case Analysis of Set Union Algorithms","volume":"31","author":[{"given":"Robert E.","family":"Tarjan","sequence":"first","affiliation":[{"name":"AT&amp;T Bell Laboratories, Murray Hill, New Jersey"}]},{"given":"Jan","family":"van Leeuwen","sequence":"additional","affiliation":[{"name":"University of Utrecht, Utrecht, The Netherlands"}]}],"container-title":"Journal of the ACM","original-title":[],"language":"en","issued":{"date-parts":[[1984,3,30]]},"URL":"http://dx.doi.org/10.1145/62.2160","ISSN":["0004-5411","1557-735X"],"container-title-short":"J. ACM","note":"alternative-id:10.1145/62.2160"},{"id":"arxiv:2107.13589","type":"article-journal","author":[{"given":"Christopher A.","family":"Pattison"},{"given":"Michael E.","family":"Beverland"},{"given":"Marcus P.","family":"da Silva"},{"given":"Nicolas","family":"Delfosse"}],"title":"Improved quantum error correction using soft information","issued":{"date-parts":[[2021,7,28]]},"note":"arxivid:2107.13589\narxiv_version_number:1"},{"id":"arxiv:2203.04948","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.13.031007","source":"Crossref","title":"Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes","volume":"13","author":[{"ORCID":"https://orcid.org/0000-0001-9880-5218","authenticated-orcid":true,"given":"Oscar","family":"Higgott","sequence":"first","affiliation":[]},{"given":"Thomas C.","family":"Bohdanowicz","sequence":"additional","affiliation":[]},{"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.13.031007","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031007","note":"arxivid:2203.04948"},{"id":"arxiv:2305.18534","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Fault-tolerant quantum computing requires classical hardware to perform the decoding necessary for error correction. The Union–Find decoder is one of the best candidates for this. It has remarkably organic characteristics, involving the growth and merger of data structures through nearest-neighbour steps; this naturally suggests the possibility of its realisation using a lattice of simple processors with nearest-neighbour links. In this way the computational load can be distributed with near-ideal parallelism. Here we show for the first time that this strict (rather than partial) locality is practical, with a worst-case runtime <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> and mean runtime subquadratic in the surface code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>. A novel parity-calculation scheme is employed which can simplify previously proposed architectures, and our approach is optimised for circuit-level noise. We compare our local realisation with one augmented by long-range links; while the latter is of course faster, we note that local asynchronous logic could negate the difference.</jats:p>","DOI":"10.22331/q-2023-11-14-1183","page":"1183","source":"Crossref","title":"Actis: A Strictly Local Union–Find Decoder","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0001-6187-7402","authenticated-orcid":false,"given":"Tim","family":"Chan","sequence":"first","affiliation":[{"name":"Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom"}]},{"ORCID":"https://orcid.org/0000-0002-7766-5348","authenticated-orcid":false,"given":"Simon C.","family":"Benjamin","sequence":"additional","affiliation":[{"name":"Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom"},{"name":"Quantum Motion, 9 Sterling Way, London N7 9HJ, United Kingdom"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,14]]},"URL":"http://dx.doi.org/10.22331/q-2023-11-14-1183","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2305.18534"},{"id":"arxiv:1202.6111","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.2.041003","source":"Crossref","title":"Topological Code Autotune","volume":"2","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]},{"given":"Adam C.","family":"Whiteside","sequence":"additional","affiliation":[]},{"given":"Angus L.","family":"McInnes","sequence":"additional","affiliation":[]},{"given":"Alimohammad","family":"Rabbani","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2012,10,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.2.041003","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041003","note":"arxivid:1202.6111"},{"id":"arxiv:1310.0863","type":"article-journal","author":[{"given":"Austin G.","family":"Fowler"}],"title":"Optimal complexity correction of correlated errors in the surface code","issued":{"date-parts":[[2013,10,2]]},"note":"arxivid:1310.0863\narxiv_version_number:1"},{"id":"arxiv:2205.09828","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We describe a pipeline approach to decoding the surface code using minimum weight perfect matching, including taking into account correlations between detection events. An independent no-communication parallelizable processing stage reweights the graph according to likely correlations, followed by another no-communication parallelizable stage for high confidence matching. A later general stage finishes the matching. This is a simplification of previous correlated matching techniques which required a complex interaction between general matching and re-weighting the graph. Despite this simplification, which gives correlated matching a better chance of achieving real-time processing, we find the logical error rate practically unchanged. We validate the new algorithm on the fully fault-tolerant toric, unrotated, and rotated surface codes, all with standard depolarizing noise. We expect these techniques to be applicable to a wide range of other decoders.</jats:p>","DOI":"10.22331/q-2023-12-12-1205","page":"1205","source":"Crossref","title":"Pipelined correlated minimum weight perfect matching of the surface code","volume":"7","author":[{"given":"Alexandru","family":"Paler","sequence":"first","affiliation":[{"name":"Aalto University, Espoo 02150, Finland"},{"name":"University of Texas at Dallas, Richardson, TX 75080, USA"}]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Inc., Santa Barbara, 93117 CA, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,12,12]]},"URL":"http://dx.doi.org/10.22331/q-2023-12-12-1205","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2205.09828"},{"id":"arxiv:1812.01505","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevapplied.12.064006","source":"Crossref","title":"High-Threshold Code for Modular Hardware With Asymmetric Noise","volume":"12","author":[{"ORCID":"https://orcid.org/0000-0002-4894-8322","authenticated-orcid":true,"given":"Xiaosi","family":"Xu","sequence":"first","affiliation":[]},{"given":"Qi","family":"Zhao","sequence":"additional","affiliation":[]},{"given":"Xiao","family":"Yuan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7766-5348","authenticated-orcid":true,"given":"Simon C.","family":"Benjamin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2019,12,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.12.064006","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"064006","note":"arxivid:1812.01505"},{"id":"arxiv:2305.08307","type":"article-journal","author":[{"given":"Yue","family":"Wu"},{"given":"Lin","family":"Zhong"}],"title":"Fusion Blossom: Fast MWPM Decoders for QEC","issued":{"date-parts":[[2023,5,15]]},"note":"arxivid:2305.08307\narxiv_version_number:1"},{"id":"arxiv:2405.01151","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/lcomm.2024.3396882","page":"1509-1513","source":"Crossref","title":"Spanning Tree Matching Decoder for Quantum Surface Codes","volume":"28","author":[{"given":"Diego","family":"Forlivesi","sequence":"first","affiliation":[{"name":"Department of Electrical, Electronic, and Information Engineering &#x201C;Guglielmo Marconi&#x201D; and the CNIT/WiLab, University of Bologna, Bologna, Italy"}]},{"ORCID":"https://orcid.org/0000-0002-8417-6454","authenticated-orcid":false,"given":"Lorenzo","family":"Valentini","sequence":"additional","affiliation":[{"name":"Department of Electrical, Electronic, and Information Engineering &#x201C;Guglielmo Marconi&#x201D; and the CNIT/WiLab, University of Bologna, Bologna, Italy"}]},{"ORCID":"https://orcid.org/0000-0001-8782-8318","authenticated-orcid":false,"given":"Marco","family":"Chiani","sequence":"additional","affiliation":[{"name":"Department of Electrical, Electronic, and Information Engineering &#x201C;Guglielmo Marconi&#x201D; and the CNIT/WiLab, University of Bologna, Bologna, Italy"}]}],"container-title":"IEEE Communications Letters","original-title":[],"issued":{"date-parts":[[2024,7]]},"URL":"http://dx.doi.org/10.1109/LCOMM.2024.3396882","ISSN":["1089-7798","1558-2558","2373-7891"],"container-title-short":"IEEE Commun. Lett.","note":"arxivid:2405.01151"},{"id":"arxiv:2408.01393","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Recent experimental advances have made it possible to implement logical multiqubit transversal gates on surface codes in a multitude of platforms. A transversal controlled- (t) gate on two surface codes introduces correlated errors across the code blocks and thus requires modified decoding compared to established methods of decoding surface-code quantum memory (SCQM) or lattice-surgery operations. In this work, we examine and benchmark the performance of three different decoding strategies for the t for scalable fault-tolerant quantum computation. In particular, we present a low-complexity decoder based on minimum-weight perfect matching (MWPM) that achieves the same threshold as the SCQM MWPM decoder. We extend our analysis with a study of tailored decoding of a transversal-teleportation circuit, along with a comparison between the performance of lattice-surgery and transversal operations under Pauli- and erasure-noise models. Our investigation builds toward systematic estimation of the cost of implementing large-scale quantum algorithms based on transversal gates in the surface code.</jats:p>","DOI":"10.1103/prxquantum.6.020326","source":"Crossref","title":"Error Correction of Transversal cnot Gates for Scalable Surface-Code Computation","volume":"6","author":[{"ORCID":"https://orcid.org/0009-0002-4587-5620","authenticated-orcid":true,"given":"Kaavya","family":"Sahay","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-3561-6628","authenticated-orcid":true,"given":"Yingjia","family":"Lin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"}]},{"ORCID":"https://orcid.org/0000-0001-6731-8601","authenticated-orcid":true,"given":"Shilin","family":"Huang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0001-7716-1425","authenticated-orcid":true,"given":"Kenneth R.","family":"Brown","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"},{"id":[{"id":"https://ror.org/00py81415","id-type":"ROR","asserted-by":"publisher"}],"name":"Duke University"}]},{"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,5,7]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.6.020326","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020326","note":"arxivid:2408.01393"},{"id":"arxiv:2502.14787","type":"article-journal","author":[{"given":"Yue","family":"Wu"},{"given":"Namitha","family":"Liyanage"},{"given":"Lin","family":"Zhong"}],"title":"Micro Blossom: Accelerated Minimum-Weight Perfect Matching Decoding for Quantum Error Correction","issued":{"date-parts":[[2025,2,20]]},"note":"arxivid:2502.14787\narxiv_version_number:1"},{"id":"arxiv:2505.13599","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>\n                    Transversal logical gates offer the opportunity for fast and low-noise logic, particularly when interspersed by a single round of parity check measurements of the underlying code. Using such circuits for the surface code requires decoding across logical gates, complicating the decoding task. We show how one can decode across an arbitrary sequence of transversal gates for the unrotated surface code, using a fast “logical observable” minimum-weight perfect matching-based decoder, and benchmark its performance in Clifford circuits under circuit-level noise. We propose  logical observable matching decoders to address the problem of fully efficient decoding: our basic windowed decoder is computationally efficient under the restriction of quiescent (slow) resets. Our “advanced” two-step windowed decoder can be computationally inefficient but allows fast resets. For both windowed decoders we identify errors which scale sublinearly in\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>d</a:mi>\n                    </a:math>\n                    —depending on the structure of the circuit—which can lead to logical failure, and we propose methods to adapt the decoding to remove such failures. Our work highlights the complexity and interest in efficient decoding of fast logic for the surface code.\n                  </jats:p>","DOI":"10.1103/sk5y-25b1","source":"Crossref","title":"Decoding across Transversal Clifford Gates in the Surface Code","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-8000-8701","authenticated-orcid":true,"given":"Marc","family":"Serra-Peralta","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04wf30j82","id-type":"ROR","asserted-by":"publisher"}],"name":"QuTech"},{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"},{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-0776-886X","authenticated-orcid":true,"given":"Mackenzie H.","family":"Shaw","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04wf30j82","id-type":"ROR","asserted-by":"publisher"}],"name":"QuTech"},{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"},{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"}]},{"ORCID":"https://orcid.org/0000-0003-0218-6614","authenticated-orcid":true,"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04wf30j82","id-type":"ROR","asserted-by":"publisher"}],"name":"QuTech"},{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"},{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,2,19]]},"URL":"http://dx.doi.org/10.1103/sk5y-25b1","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010335","note":"arxivid:2505.13599"},{"id":"arxiv:2401.12434","type":"article-journal","author":[{"given":"Noah","family":"Shutty"},{"given":"Michael","family":"Newman"},{"given":"Benjamin","family":"Villalonga"}],"title":"Efficient near-optimal decoding of the surface code through ensembling","issued":{"date-parts":[[2024,3,15]]},"note":"arxivid:2401.12434\narxiv_version_number:3"},{"id":"arxiv:2408.12135","type":"article-journal","author":[{"given":"Cody","family":"Jones"}],"title":"Improved accuracy for decoding surface codes with matching synthesis","issued":{"date-parts":[[2024,8,22]]},"note":"arxivid:2408.12135\narxiv_version_number:1"},{"id":"arxiv:0911.0581","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.104.050504","source":"Crossref","title":"Fast Decoders for Topological Quantum Codes","volume":"104","author":[{"given":"Guillaume","family":"Duclos-Cianci","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2010,2,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.104.050504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050504","note":"arxivid:0911.0581"},{"id":"arxiv:1304.6100","type":"article-journal","author":[{"given":"Guillaume","family":"Duclos-Cianci"},{"given":"David","family":"Poulin"}],"title":"Fault-Tolerant Renormalization Group Decoder for Abelian Topological Codes","issued":{"date-parts":[[2013,4,22]]},"note":"arxivid:1304.6100\narxiv_version_number:1"},{"id":"arxiv:1411.3028","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.92.032309","source":"Crossref","title":"Fast fault-tolerant decoder for qubit and qudit surface codes","volume":"92","author":[{"given":"Fern H. E.","family":"Watson","sequence":"first","affiliation":[]},{"given":"Hussain","family":"Anwar","sequence":"additional","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2015,9,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.92.032309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032309","note":"arxivid:1411.3028"},{"id":"arxiv:1310.2393","type":"journal-article","publisher":"MDPI AG","issue":"4","abstract":"<jats:p>Here we study an efficient algorithm for decoding topological codes. It is a simple form of HDRG decoder, which could be straightforwardly generalized to complex decoding problems. Specific results are obtained for the planar code with both i.i.d. and spatially correlated errors. The method is shown to compare well with existing ones, despite its simplicity.</jats:p>","DOI":"10.3390/e17041946","page":"1946-1957","source":"Crossref","title":"A Simple Decoder for Topological Codes","volume":"17","author":[{"given":"James","family":"Wootton","sequence":"first","affiliation":[{"name":"Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland"}]}],"container-title":"Entropy","original-title":[],"language":"en","issued":{"date-parts":[[2015,4,1]]},"URL":"http://dx.doi.org/10.3390/e17041946","ISSN":["1099-4300"],"container-title-short":"Entropy","note":"alternative-id:e17041946\narxivid:1310.2393"},{"id":"arxiv:1801.01879","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreve.97.051302","source":"Crossref","title":"Linear-time general decoding algorithm for the surface code","volume":"97","author":[{"given":"Andrew S.","family":"Darmawan","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review E","original-title":[],"language":"en","issued":{"date-parts":[[2018,5,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevE.97.051302","ISSN":["2470-0045","2470-0053"],"container-title-short":"Phys. Rev. E","page":"051302","note":"arxivid:1801.01879"},{"id":"arxiv:1302.2669","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.89.022326","source":"Crossref","title":"Efficient Markov chain Monte Carlo algorithm for the surface code","volume":"89","author":[{"given":"Adrian","family":"Hutter","sequence":"first","affiliation":[]},{"given":"James R.","family":"Wootton","sequence":"additional","affiliation":[]},{"given":"Daniel","family":"Loss","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.89.022326","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022326","note":"arxivid:1302.2669"},{"id":"arxiv:1610.04238","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.119.030501","source":"Crossref","title":"Neural Decoder for Topological Codes","volume":"119","author":[{"given":"Giacomo","family":"Torlai","sequence":"first","affiliation":[]},{"given":"Roger G.","family":"Melko","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,7,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.119.030501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"030501","note":"arxivid:1610.04238"},{"id":"arxiv:1802.06441","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1088/2058-9565/aad1f7","page":"044002","source":"Crossref","title":"Deep neural decoders for near term fault-tolerant experiments","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":false,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9591-9727","authenticated-orcid":false,"given":"Pooya","family":"Ronagh","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2018,7,31]]},"URL":"http://dx.doi.org/10.1088/2058-9565/aad1f7","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:1802.06441"},{"id":"arxiv:2208.01178","type":"journal-article","publisher":"IOP Publishing","issue":"4","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Implementing algorithms on a fault-tolerant quantum computer will require fast decoding throughput and latency times to prevent an exponential increase in buffer times between the applications of gates. In this work we begin by quantifying these requirements. We then introduce the construction of local neural network (NN) decoders using three-dimensional convolutions. These local decoders are adapted to circuit-level noise and can be applied to surface code volumes of arbitrary size. Their application removes errors arising from a certain number of faults, which serves to substantially reduce the syndrome density. Remaining errors can then be corrected by a global decoder, such as Blossom or union find, with their implementation significantly accelerated due to the reduced syndrome density. However, in the circuit-level setting, the corrections applied by the local decoder introduce many vertical pairs of highlighted vertices. To obtain a low syndrome density in the presence of vertical pairs, we consider a strategy of performing a syndrome collapse which removes many vertical pairs and reduces the size of the decoding graph used by the global decoder. We also consider a strategy of performing a vertical cleanup, which consists of removing all local vertical pairs prior to implementing the global decoder. By applying our local NN decoder and the vertical cleanup strategy to a <jats:italic>d</jats:italic> = 17 surface code volume, we show a <jats:inline-formula>\n                     <jats:tex-math/>\n                     <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                           <mml:mn>10</mml:mn>\n                           <mml:mn>6</mml:mn>\n                        </mml:msup>\n                        <mml:mo>×</mml:mo>\n                     </mml:math>\n                     <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"qstace64dieqn1.gif\" xlink:type=\"simple\"/>\n                  </jats:inline-formula> speedup of the minimum-weight perfect matching decoder. Lastly, we estimate the cost of implementing our local decoders on field programmable gate arrays.</jats:p>","DOI":"10.1088/2058-9565/ace64d","page":"045011","source":"Crossref","title":"Techniques for combining fast local decoders with global decoders under circuit-level noise","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Luis","family":"Goncalves","sequence":"additional","affiliation":[]},{"given":"Prasahnt","family":"Sivarajah","sequence":"additional","affiliation":[]},{"given":"Eric","family":"Peterson","sequence":"additional","affiliation":[]},{"given":"Sebastian","family":"Grimberg","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2023,7,27]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ace64d","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2208.01178"},{"id":"arxiv:2208.05758","type":"article-journal","author":[{"given":"Yosuke","family":"Ueno"},{"given":"Masaaki","family":"Kondo"},{"given":"Masamitsu","family":"Tanaka"},{"given":"Yasunari","family":"Suzuki"},{"given":"Yutaka","family":"Tabuchi"}],"title":"NEO-QEC: Neural Network Enhanced Online Superconducting Decoder for Surface Codes","issued":{"date-parts":[[2022,9,1]]},"note":"arxivid:2208.05758\narxiv_version_number:2"},{"id":"arxiv:2307.03280","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Neural network decoders can achieve a lower logical error rate compared to conventional decoders, like minimum-weight perfect matching, when decoding the surface code. Furthermore, these decoders require no prior information about the physical error rates, making them highly adaptable. In this study, we investigate the performance of such a decoder using both simulated and experimental data obtained from a transmon-qubit processor, focusing on small-distance surface codes. We first show that the neural network typically outperforms the matching decoder due to better handling of errors leading to multiple correlated syndrome defects, such as <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\"><a:mi>Y</a:mi></a:math> errors. When applied to the experimental data of Google Quantum AI [R. Acharya  , ], the neural network decoder achieves logical error rates approximately <b:math xmlns:b=\"http://www.w3.org/1998/Math/MathML\"><b:mrow><b:mn>25</b:mn><b:mo>%</b:mo></b:mrow></b:math> lower than minimum-weight perfect matching, approaching the performance of a maximum-likelihood decoder. To demonstrate the flexibility of this decoder, we incorporate the soft information available in the analog readout of transmon qubits and evaluate the performance of this decoder in simulation using a symmetric Gaussian-noise model. Considering the soft information leads to an approximately <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\"><c:mrow><c:mn>10</c:mn><c:mo>%</c:mo></c:mrow></c:math> lower logical error rate, depending on the probability of a measurement error. The good logical performance, flexibility, and computational efficiency make neural network decoders well-suited for near-term demonstrations of quantum memories.</jats:p>","DOI":"10.1103/physrevresearch.7.013029","source":"Crossref","title":"Neural network decoder for near-term surface-code experiments","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0001-7124-8933","authenticated-orcid":true,"given":"Boris M.","family":"Varbanov","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"}]},{"ORCID":"https://orcid.org/0000-0002-8000-8701","authenticated-orcid":true,"given":"Marc","family":"Serra-Peralta","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"},{"name":"Technische Universiteit Delft"}]},{"ORCID":"https://orcid.org/0009-0004-4535-4504","authenticated-orcid":true,"given":"David","family":"Byfield","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/001377z89","id-type":"ROR","asserted-by":"publisher"}],"name":"Riverlane"}]},{"ORCID":"https://orcid.org/0000-0003-0218-6614","authenticated-orcid":true,"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02e2c7k09","id-type":"ROR","asserted-by":"publisher"}],"name":"Delft University of Technology"},{"name":"Technische Universiteit Delft"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.7.013029","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"013029","note":"arxivid:2307.03280"},{"id":"arxiv:2501.14525","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce60285.2024.00149","page":"1258-1263","source":"Crossref","title":"Towards a Cryogenic CMOS-Memristor Neural Decoder for Quantum Error Correction","author":[{"given":"Pierre-Antoine","family":"Mouny","sequence":"first","affiliation":[{"name":"Irr&#x00E9;versible Inc.,Sherbrooke, Qu&#x00E9;bec,Canada"}]},{"given":"Maher","family":"Benhouria","sequence":"additional","affiliation":[{"name":"Irr&#x00E9;versible Inc.,Sherbrooke, Qu&#x00E9;bec,Canada"}]},{"given":"Victor","family":"Yon","sequence":"additional","affiliation":[{"name":"Irr&#x00E9;versible Inc.,Sherbrooke, Qu&#x00E9;bec,Canada"}]},{"given":"Patrick","family":"Dufour","sequence":"additional","affiliation":[{"name":"Institut Interdisplinaire d&#x0027;Innovation Technologique (3IT), Universit&#x00E9; de Sherbrooke,Sherbrooke,Québec,Canada"}]},{"given":"Linxiang","family":"Huang","sequence":"additional","affiliation":[{"name":"Institut Interdisplinaire d&#x0027;Innovation Technologique (3IT), Universit&#x00E9; de Sherbrooke,Sherbrooke,Québec,Canada"}]},{"given":"Yann","family":"Beilliard","sequence":"additional","affiliation":[{"name":"Irr&#x00E9;versible Inc.,Sherbrooke, Qu&#x00E9;bec,Canada"}]},{"given":"Sophie","family":"Rochette","sequence":"additional","affiliation":[{"name":"Irr&#x00E9;versible Inc.,Sherbrooke, Qu&#x00E9;bec,Canada"}]},{"given":"Dominique","family":"Drouin","sequence":"additional","affiliation":[{"name":"Institut Interdisplinaire d&#x0027;Innovation Technologique (3IT), Universit&#x00E9; de Sherbrooke,Sherbrooke,Québec,Canada"}]},{"given":"Pooya","family":"Ronagh","sequence":"additional","affiliation":[{"name":"Irr&#x00E9;versible Inc.,Sherbrooke, Qu&#x00E9;bec,Canada"}]}],"event":"2024 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2024 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2024,9,15]]},"URL":"http://dx.doi.org/10.1109/QCE60285.2024.00149","note":"arxivid:2501.14525"},{"id":"arxiv:2506.16113","type":"journal-article","publisher":"IOP Publishing","issue":"2","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    Artificial neural networks (ANNs) are a promising approach to the decoding problem of quantum error correction (QEC), but have observed consistent difficulty when generalising performance to larger QEC codes. Recent scalability-focused approaches have split the decoding workload by using local ANNs to perform initial syndrome processing and leaving final processing to a global residual decoder. We investigated ANN surface code decoding under a scheme exploiting the spatiotemporal structure of syndrome data. In particular, we present a vectorised method for surface code data simulation and benchmark decoding performance when such data defines a multi-label classification problem and generative modelling problem for rotated surface codes with circuit noise after each gate and idle timestep. Performance was found to generalise to rotated surface codes of sizes up to\n                    <jats:italic>d</jats:italic>\n                     = 97, with depolarisation parameter thresholds of up to 0.7% achieved, competitive with minimum weight perfect matching (MWPM). Improved timings, compared with MWPM alone, were found starting at code distances of\n                    <jats:italic>d</jats:italic>\n                     = 33 and\n                    <jats:italic>d</jats:italic>\n                     = 89 under noise models above and below threshold respectively. These results suggest promising prospects for ANN-based frameworks for surface code decoding with performance sufficient to support the demands expected from fault-tolerant resource estimates.\n                  </jats:p>","DOI":"10.1088/2058-9565/ae5fc9","page":"025048","source":"Crossref","title":"Fully convolutional 3D neural network decoders for surface codes with syndrome circuit noise","volume":"11","author":[{"ORCID":"https://orcid.org/0000-0003-4757-6851","authenticated-orcid":true,"given":"Spiro","family":"Gicev","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7672-6965","authenticated-orcid":true,"given":"Lloyd C L","family":"Hollenberg","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3476-2348","authenticated-orcid":false,"given":"Muhammad","family":"Usman","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2026,5,7]]},"URL":"http://dx.doi.org/10.1088/2058-9565/ae5fc9","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2506.16113"},{"id":"arxiv:1810.07207","type":"journal-article","publisher":"IOP Publishing","issue":"2","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Topological error correcting codes, and particularly the surface code, currently provide the most feasible road-map towards large-scale fault-tolerant quantum computation. As such, obtaining fast and flexible decoding algorithms for these codes, within the experimentally realistic and challenging context of faulty syndrome measurements, without requiring any final read-out of the physical qubits, is of critical importance. In this work, we show that the problem of decoding such codes can be naturally reformulated as a process of repeated interactions between a decoding agent and a code environment, to which the machinery of reinforcement learning can be applied to obtain decoding agents. While in principle this framework can be instantiated with environments modelling circuit level noise, we take a first step towards this goal by using deepQ learning to obtain decoding agents for a variety of simplified phenomenological noise models, which yield faulty syndrome measurements without including the propagation of errors which arise in full circuit level noise models.</jats:p>","DOI":"10.1088/2632-2153/abc609","page":"025005","source":"Crossref","title":"Reinforcement learning decoders for fault-tolerant quantum computation","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-6202-8864","authenticated-orcid":false,"given":"Ryan","family":"Sweke","sequence":"first","affiliation":[]},{"given":"Markus S","family":"Kesselring","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0323-0031","authenticated-orcid":false,"given":"Evert P L","family":"van Nieuwenburg","sequence":"additional","affiliation":[]},{"given":"Jens","family":"Eisert","sequence":"additional","affiliation":[]}],"container-title":"Machine Learning: Science and Technology","original-title":[],"issued":{"date-parts":[[2020,12,28]]},"URL":"http://dx.doi.org/10.1088/2632-2153/abc609","ISSN":["2632-2153"],"container-title-short":"Mach. Learn.: Sci. Technol.","note":"arxivid:1810.07207"},{"id":"arxiv:1811.12338","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We implement a quantum error correction algorithm for bit-flip errors on the topological toric code using deep reinforcement learning. An action-value Q-function encodes the discounted value of moving a defect to a neighboring site on the square grid (the action) depending on the full set of defects on the torus (the syndrome or state). The Q-function is represented by a deep convolutional neural network. Using the translational invariance on the torus allows for viewing each defect from a central perspective which significantly simplifies the state space representation independently of the number of defect pairs. The training is done using experience replay, where data from the algorithm being played out is stored and used for mini-batch upgrade of the Q-network. We find performance which is close to, and for small error rates asymptotically equivalent to, that achieved by the Minimum Weight Perfect Matching algorithm for code distances up to<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:math>. Our results show that it is possible for a self-trained agent without supervision or support algorithms to find a decoding scheme that performs on par with hand-made algorithms, opening up for future machine engineered decoders for more general error models and error correcting codes.</jats:p>","DOI":"10.22331/q-2019-09-02-183","page":"183","source":"Crossref","title":"Quantum error correction for the toric code using deep reinforcement learning","volume":"3","author":[{"given":"Philip","family":"Andreasson","sequence":"first","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]},{"given":"Joel","family":"Johansson","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]},{"given":"Simon","family":"Liljestrand","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]},{"given":"Mats","family":"Granath","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,9,2]]},"URL":"http://dx.doi.org/10.22331/q-2019-09-02-183","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1811.12338"},{"id":"arxiv:2212.11890","type":"article-journal","author":[{"given":"Elisha Siddiqui","family":"Matekole"},{"given":"Esther","family":"Ye"},{"given":"Ramya","family":"Iyer"},{"given":"Samuel Yen-Chi","family":"Chen"}],"title":"Decoding surface codes with deep reinforcement learning and probabilistic policy reuse","issued":{"date-parts":[[2022,12,22]]},"note":"arxivid:2212.11890\narxiv_version_number:1"},{"id":"arxiv:2101.07285","type":"journal-article","publisher":"American Physical Society (APS)","issue":"8","DOI":"10.1103/physrevlett.128.080505","source":"Crossref","title":"Scalable Neural Decoder for Topological Surface Codes","volume":"128","author":[{"ORCID":"https://orcid.org/0000-0002-9141-7113","authenticated-orcid":true,"given":"Kai","family":"Meinerz","sequence":"first","affiliation":[]},{"given":"Chae-Yeun","family":"Park","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1479-9736","authenticated-orcid":true,"given":"Simon","family":"Trebst","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.128.080505","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"080505","note":"arxivid:2101.07285"},{"id":"arxiv:2311.16082","type":"article-journal","author":[{"given":"Hanrui","family":"Wang"},{"given":"Pengyu","family":"Liu"},{"given":"Kevin","family":"Shao"},{"given":"Dantong","family":"Li"},{"given":"Jiaqi","family":"Gu"},{"given":"David Z.","family":"Pan"},{"given":"Yongshan","family":"Ding"},{"given":"Song","family":"Han"}],"title":"Transformer-QEC: Quantum Error Correction Code Decoding with Transferable Transformers","issued":{"date-parts":[[2023,11,27]]},"note":"arxivid:2311.16082\narxiv_version_number:1"},{"id":"arxiv:2506.02734","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/xmp9-bh84","source":"Crossref","title":"Self-attention U-Net decoder for toric codes","volume":"23","author":[{"ORCID":"https://orcid.org/0000-0002-8164-9527","authenticated-orcid":true,"given":"Wei-Wei","family":"Zhang","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01y0j0j86","id-type":"ROR","asserted-by":"publisher"}],"name":"Northwestern Polytechnical University"}]},{"given":"Zhuo","family":"Xia","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0220qvk04","id-type":"ROR","asserted-by":"publisher"}],"name":"Shanghai Jiao Tong University"},{"id":[{"id":"https://ror.org/01y0j0j86","id-type":"ROR","asserted-by":"publisher"}],"name":"Northwestern Polytechnical University"}]},{"given":"Wei","family":"Zhao","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Security Communication"}]},{"given":"Wei","family":"Pan","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01y0j0j86","id-type":"ROR","asserted-by":"publisher"}],"name":"Northwestern Polytechnical University"}]},{"given":"Haobin","family":"Shi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01y0j0j86","id-type":"ROR","asserted-by":"publisher"}],"name":"Northwestern Polytechnical University"}]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,23]]},"URL":"http://dx.doi.org/10.1103/xmp9-bh84","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"064052","note":"arxivid:2506.02734"},{"id":"arxiv:2310.05900","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8040","abstract":"<jats:title>Abstract</jats:title><jats:p>Building a large-scale quantum computer requires effective strategies to correct errors that inevitably arise in physical quantum systems<jats:sup>1</jats:sup>. Quantum error-correction codes<jats:sup>2</jats:sup> present a way to reach this goal by encoding logical information redundantly into many physical qubits. A key challenge in implementing such codes is accurately decoding noisy syndrome information extracted from redundancy checks to obtain the correct encoded logical information. Here we develop a recurrent, transformer-based neural network that learns to decode the surface code, the leading quantum error-correction code<jats:sup>3</jats:sup>. Our decoder outperforms other state-of-the-art decoders on real-world data from Google’s Sycamore quantum processor for distance-3 and distance-5 surface codes<jats:sup>4</jats:sup>. On distances up to 11, the decoder maintains its advantage on simulated data with realistic noise including cross-talk and leakage, utilizing soft readouts and leakage information. After training on approximate synthetic data, the decoder adapts to the more complex, but unknown, underlying error distribution by training on a limited budget of experimental samples. Our work illustrates the ability of machine learning to go beyond human-designed algorithms by learning from data directly, highlighting machine learning as a strong contender for decoding in quantum computers.</jats:p>","DOI":"10.1038/s41586-024-08148-8","page":"834-840","source":"Crossref","title":"Learning high-accuracy error decoding for quantum processors","volume":"635","author":[{"ORCID":"https://orcid.org/0000-0003-3189-9162","authenticated-orcid":false,"given":"Johannes","family":"Bausch","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2401-5691","authenticated-orcid":false,"given":"Andrew W.","family":"Senior","sequence":"additional","affiliation":[]},{"given":"Francisco J. H.","family":"Heras","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-0473-3727","authenticated-orcid":false,"given":"Thomas","family":"Edlich","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4917-5234","authenticated-orcid":false,"given":"Alex","family":"Davies","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"given":"Cody","family":"Jones","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5865-0813","authenticated-orcid":false,"given":"Kevin","family":"Satzinger","sequence":"additional","affiliation":[]},{"given":"Murphy Yuezhen","family":"Niu","sequence":"additional","affiliation":[]},{"given":"Sam","family":"Blackwell","sequence":"additional","affiliation":[]},{"given":"George","family":"Holland","sequence":"additional","affiliation":[]},{"given":"Dvir","family":"Kafri","sequence":"additional","affiliation":[]},{"given":"Juan","family":"Atalaya","sequence":"additional","affiliation":[]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2812-9917","authenticated-orcid":false,"given":"Demis","family":"Hassabis","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1090-7584","authenticated-orcid":false,"given":"Sergio","family":"Boixo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9681-6746","authenticated-orcid":false,"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7466-7997","authenticated-orcid":false,"given":"Pushmeet","family":"Kohli","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,20]]},"URL":"http://dx.doi.org/10.1038/s41586-024-08148-8","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:8148\narxivid:2310.05900"},{"id":"arxiv:2512.07737","type":"article-journal","author":[{"given":"Andrew W.","family":"Senior"},{"given":"Thomas","family":"Edlich"},{"given":"Francisco J. H.","family":"Heras"},{"given":"Lei M.","family":"Zhang"},{"given":"Oscar","family":"Higgott"},{"given":"James S.","family":"Spencer"},{"given":"Taylor","family":"Applebaum"},{"given":"Sam","family":"Blackwell"},{"given":"Justin","family":"Ledford"},{"given":"Akvilė","family":"Žemgulytė"},{"given":"Augustin","family":"Žídek"},{"given":"Noah","family":"Shutty"},{"given":"Andrew","family":"Cowie"},{"given":"Yin","family":"Li"},{"given":"George","family":"Holland"},{"given":"Peter","family":"Brooks"},{"given":"Charlie","family":"Beattie"},{"given":"Michael","family":"Newman"},{"given":"Alex","family":"Davies"},{"given":"Cody","family":"Jones"},{"given":"Sergio","family":"Boixo"},{"given":"Hartmut","family":"Neven"},{"given":"Pushmeet","family":"Kohli"},{"given":"Johannes","family":"Bausch"}],"title":"A scalable and real-time neural decoder for topological quantum codes","issued":{"date-parts":[[2026,3,11]]},"note":"arxivid:2512.07737\narxiv_version_number:2"},{"id":"arxiv:2108.06569","type":"article-journal","author":[{"given":"Poulami","family":"Das"},{"given":"Aditya","family":"Locharla"},{"given":"Cody","family":"Jones"}],"title":"LILLIPUT: A Lightweight Low-Latency Lookup-Table Based Decoder for Near-term Quantum Error Correction","issued":{"date-parts":[[2021,8,14]]},"note":"arxivid:2108.06569\narxiv_version_number:1"},{"id":"arxiv:2001.11427","type":"article-journal","author":[{"given":"Nicolas","family":"Delfosse"}],"title":"Hierarchical decoding to reduce hardware requirements for quantum computing","issued":{"date-parts":[[2020,1,30]]},"note":"arxivid:2001.11427\narxiv_version_number:1"},{"id":"arxiv:2208.04660","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevapplied.19.034050","source":"Crossref","title":"Local Predecoder to Reduce the Bandwidth and Latency of Quantum Error Correction","volume":"19","author":[{"ORCID":"https://orcid.org/0000-0002-4865-7015","authenticated-orcid":true,"given":"Samuel C.","family":"Smith","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Applied","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevApplied.19.034050","ISSN":["2331-7019"],"container-title-short":"Phys. Rev. Applied","page":"034050","note":"arxivid:2208.04660"},{"id":"arxiv:2208.08547","type":"article-journal","author":[{"given":"Gokul Subramanian","family":"Ravi"},{"given":"Jonathan M.","family":"Baker"},{"given":"Arash","family":"Fayyazi"},{"given":"Sophia Fuhui","family":"Lin"},{"given":"Ali","family":"Javadi-Abhari"},{"given":"Massoud","family":"Pedram"},{"given":"Frederic T.","family":"Chong"}],"title":"Better Than Worst-Case Decoding for Quantum Error Correction","issued":{"date-parts":[[2022,10,25]]},"note":"arxivid:2208.08547\narxiv_version_number:2"},{"id":"arxiv:2209.09219","type":"article-journal","author":[{"given":"Xinyu","family":"Tan"},{"given":"Fang","family":"Zhang"},{"given":"Rui","family":"Chao"},{"given":"Yaoyun","family":"Shi"},{"given":"Jianxin","family":"Chen"}],"title":"Scalable surface code decoders with parallelization in time","issued":{"date-parts":[[2022,9,29]]},"note":"arxivid:2209.09219\narxiv_version_number:2"},{"id":"arxiv:2209.08552","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Large-scale quantum computers have the potential to hold computational capabilities beyond conventional computers. However, the physical qubits are prone to noise which must be corrected in order to perform fault-tolerant quantum computations. Quantum Error Correction (QEC) provides the path for realizing such computations. QEC generates a continuous stream of data that decoders must process at the rate it is received, which can be as fast as 1<jats:italic>μ</jats:italic>s per QEC round in superconducting quantum computers. If the decoder infrastructure cannot keep up, a data backlog problem is encountered and the computation runs exponentially slower. Today’s leading approaches to quantum error correction are not scalable as existing decoders typically run slower as the problem size is increased, inevitably hitting the backlog problem. Here, we show how to parallelize decoding to achieve almost arbitrary speed, removing this roadblock to scalability. Our parallelization requires some classical feed forward decisions to be delayed, slowing-down the logical clock speed. However, the slow-down is now only polynomial in the size of the QEC code, averting the exponential slowdown. We numerically demonstrate our parallel decoder for the surface code, showing no noticeable reduction in logical fidelity compared to previous decoders and demonstrating the predicted speedup.</jats:p>","DOI":"10.1038/s41467-023-42482-1","source":"Crossref","title":"Parallel window decoding enables scalable fault tolerant quantum computation","volume":"14","author":[{"given":"Luka","family":"Skoric","sequence":"first","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]},{"given":"Kenton M.","family":"Barnes","sequence":"additional","affiliation":[]},{"given":"Neil I.","family":"Gillespie","sequence":"additional","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2023,11,3]]},"URL":"http://dx.doi.org/10.1038/s41467-023-42482-1","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"7040","note":"alternative-id:42482\narxivid:2209.08552"},{"id":"arxiv:2412.05115","type":"article-journal","author":[{"given":"Joshua","family":"Viszlai"},{"given":"Jason D.","family":"Chadwick"},{"given":"Sarang","family":"Joshi"},{"given":"Gokul Subramanian","family":"Ravi"},{"given":"Yanjing","family":"Li"},{"given":"Frederic T.","family":"Chong"}],"title":"Predictive Window Decoding for Fault-Tolerant Quantum Programs","issued":{"date-parts":[[2024,12,6]]},"note":"arxivid:2412.05115\narxiv_version_number:1"},{"id":"arxiv:2402.04834","type":"article-journal","author":[{"given":"Aviad","family":"Kaufmann"},{"given":"Itai","family":"Arad"}],"title":"A blockBP decoder for the surface code","issued":{"date-parts":[[2024,4,16]]},"note":"arxivid:2402.04834\narxiv_version_number:2"},{"id":"arxiv:2407.11523","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/tqe.2025.3577769","page":"1-16","source":"Crossref","title":"Improved Belief Propagation Decoding Algorithms for Surface Codes","volume":"6","author":[{"ORCID":"https://orcid.org/0009-0002-9717-3647","authenticated-orcid":false,"given":"Jiahan","family":"Chen","sequence":"first","affiliation":[{"name":"Harbin Institute of Technology, Shenzhen, China"}]},{"ORCID":"https://orcid.org/0000-0001-7386-1846","authenticated-orcid":false,"given":"Zhengzhong","family":"Yi","sequence":"additional","affiliation":[{"name":"Harbin Institute of Technology, Shenzhen, China"}]},{"given":"Zhipeng","family":"Liang","sequence":"additional","affiliation":[{"name":"Harbin Institute of Technology, Shenzhen, China"}]},{"ORCID":"https://orcid.org/0000-0002-3512-0649","authenticated-orcid":false,"given":"Xuan","family":"Wang","sequence":"additional","affiliation":[{"name":"Harbin Institute of Technology, Shenzhen, China"}]}],"container-title":"IEEE Transactions on Quantum Engineering","original-title":[],"issued":{"date-parts":[[2025]]},"URL":"http://dx.doi.org/10.1109/TQE.2025.3577769","ISSN":["2689-1808"],"container-title-short":"IEEE Trans. Quantum Eng.","note":"arxivid:2407.11523"},{"id":"arxiv:2406.14968","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/jsait.2025.3571313","page":"138-149","source":"Crossref","title":"A Blindness Property of the Min-Sum Decoding for the Toric Code","volume":"6","author":[{"ORCID":"https://orcid.org/0009-0006-3538-590X","authenticated-orcid":false,"given":"Julien Du","family":"Crest","sequence":"first","affiliation":[{"name":"Universit&#x00E9; Grenoble Alpes, Grenoble INP, LIG, Grenoble, France"}]},{"ORCID":"https://orcid.org/0000-0003-4178-5396","authenticated-orcid":false,"given":"Mehdi","family":"Mhalla","sequence":"additional","affiliation":[{"name":"Universit&#x00E9; Grenoble Alpes, CNRS, Grenoble INP, LIG, Grenoble, France"}]},{"ORCID":"https://orcid.org/0000-0001-9362-8769","authenticated-orcid":false,"given":"Valentin","family":"Savin","sequence":"additional","affiliation":[{"name":"Universit&#x00E9; Grenoble Alpes, CEA-L&#x00E9;ti, Grenoble, France"}]}],"container-title":"IEEE Journal on Selected Areas in Information Theory","original-title":[],"issued":{"date-parts":[[2025]]},"URL":"http://dx.doi.org/10.1109/jsait.2025.3571313","ISSN":["2641-8770"],"container-title-short":"IEEE J. Sel. Areas Inf. Theory","note":"arxivid:2406.14968"},{"id":"arxiv:2306.16476","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The development of practical, high-performance decoding algorithms reduces the resource cost of fault-tolerant quantum computing. Here we propose a decoder for the surface code that finds low-weight correction operators for errors produced by the depolarising noise model. The decoder is obtained by mapping the syndrome of the surface code onto that of the color code, thereby allowing us to adopt more sophisticated color-code decoding algorithms. Analytical arguments and exhaustive testing show that the resulting decoder can find a least-weight correction for all weight <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:math> depolarising errors for even code distance <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>. This improves the logical error rate by an exponential factor <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> compared with decoders that treat bit-flip and dephasing errors separately. We demonstrate this improvement with analytical arguments and supporting numerical simulations at low error rates. Of independent interest, we also demonstrate an exponential improvement in logical error rate for our decoder used to correct independent and identically distributed bit-flip errors affecting the color code compared with more conventional color-code decoding algorithms.</jats:p>","DOI":"10.22331/q-2025-02-17-1632","page":"1632","source":"Crossref","title":"Minimising surface-code failures using a color-code decoder","volume":"9","author":[{"given":"Asmae","family":"Benhemou","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom"}]},{"given":"Kaavya","family":"Sahay","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA"}]},{"given":"Lingling","family":"Lao","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom"}]},{"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[{"name":"Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,17]]},"URL":"http://dx.doi.org/10.22331/q-2025-02-17-1632","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2306.16476"},{"id":"arxiv:2402.15924","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tcomm.2024.3454029","page":"1419-1430","source":"Crossref","title":"Progressive-Proximity Bit-Flipping for Decoding Surface Codes","volume":"73","author":[{"ORCID":"https://orcid.org/0009-0002-9648-9041","authenticated-orcid":false,"given":"Michele","family":"Pacenti","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, AZ, USA"}]},{"ORCID":"https://orcid.org/0000-0001-6552-7020","authenticated-orcid":false,"given":"Mark F.","family":"Flanagan","sequence":"additional","affiliation":[{"name":"School of Electrical and Electronic Engineering, University College Dublin, Dublin 4, Ireland"}]},{"ORCID":"https://orcid.org/0009-0006-7260-9263","authenticated-orcid":false,"given":"Dimitris","family":"Chytas","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, AZ, USA"}]},{"ORCID":"https://orcid.org/0000-0003-2365-4106","authenticated-orcid":false,"given":"Bane","family":"Vasić","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, The University of Arizona, Tucson, AZ, USA"}]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2025,3]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2024.3454029","ISSN":["0090-6778","1558-0857"],"container-title-short":"IEEE Trans. Commun.","note":"arxivid:2402.15924"},{"id":"arxiv:2404.07251","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.134.070403","source":"Crossref","title":"Stability of Mixed-State Quantum Phases via Finite Markov Length","volume":"134","author":[{"given":"Shengqi","family":"Sang","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"},{"id":[{"id":"https://ror.org/01aff2v68","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Waterloo"},{"id":[{"id":"https://ror.org/02t274463","id-type":"ROR","asserted-by":"publisher"}],"name":"Kavli Institute for Theoretical Physics, University of California"}]},{"given":"Timothy H.","family":"Hsieh","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013m0ej23","id-type":"ROR","asserted-by":"publisher"}],"name":"Perimeter Institute for Theoretical Physics"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.070403","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070403","note":"arxivid:2404.07251"},{"id":"arxiv:2405.03766","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s42005-024-01883-4","source":"Crossref","title":"Mitigating errors in logical qubits","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0002-4865-7015","authenticated-orcid":false,"given":"Samuel C.","family":"Smith","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":false,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":false,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"Communications Physics","original-title":[],"language":"en","issued":{"date-parts":[[2024,11,28]]},"URL":"http://dx.doi.org/10.1038/s42005-024-01883-4","ISSN":["2399-3650"],"container-title-short":"Commun Phys","page":"386","note":"alternative-id:1883\narxivid:2405.03766"},{"id":"arxiv:2504.01654","type":"article-journal","author":[{"given":"Diego","family":"Forlivesi"},{"given":"Lorenzo","family":"Valentini"},{"given":"Marco","family":"Chiani"}],"title":"Bubble Clustering Decoder for Quantum Topological Codes","issued":{"date-parts":[[2025,4,2]]},"note":"arxivid:2504.01654\narxiv_version_number:1"},{"id":"arxiv:2506.14745","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/jsait.2025.3581810","page":"163-175","source":"Crossref","title":"Union-Intersection Union-Find for Decoding Depolarizing Errors in Topological Codes","volume":"6","author":[{"given":"Tzu-Hao","family":"Lin","sequence":"first","affiliation":[{"name":"Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu, Taiwan"}]},{"ORCID":"https://orcid.org/0000-0003-1970-8167","authenticated-orcid":false,"given":"Ching-Yi","family":"Lai","sequence":"additional","affiliation":[{"name":"Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu, Taiwan"}]}],"container-title":"IEEE Journal on Selected Areas in Information Theory","original-title":[],"issued":{"date-parts":[[2025]]},"URL":"http://dx.doi.org/10.1109/JSAIT.2025.3581810","ISSN":["2641-8770"],"container-title-short":"IEEE J. Sel. Areas Inf. Theory","note":"arxivid:2506.14745"},{"id":"arxiv:2302.07395","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In this paper, I cut the cost of Y basis measurement and initialization in the surface code by nearly an order of magnitude. Fusing twist defects diagonally across the surface code patch reaches the Y basis in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230A;</mml:mo><mml:mi>d</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x230B;</mml:mo><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:math> rounds, without leaving the bounding box of the patch and without reducing the code distance. I use Monte Carlo sampling to benchmark the performance of the construction under circuit noise, and to analyze the distribution of logical errors. Cheap inplace Y basis measurement reduces the cost of S gates and magic state factories, and unlocks Pauli measurement tomography of surface code qubits on space-limited hardware.</jats:p>","DOI":"10.22331/q-2024-04-08-1310","page":"1310","source":"Crossref","title":"Inplace Access to the Surface Code Y Basis","volume":"8","author":[{"given":"Craig","family":"Gidney","sequence":"first","affiliation":[{"name":"Google Quantum AI, Santa Barbara, California 93117, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,8]]},"URL":"http://dx.doi.org/10.22331/q-2024-04-08-1310","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2302.07395"},{"id":"arxiv:2302.12292","type":"article-journal","author":[{"given":"Craig","family":"Gidney"}],"title":"Cleaner magic states with hook injection","issued":{"date-parts":[[2023,2,23]]},"note":"arxivid:2302.12292\narxiv_version_number:1"},{"id":"arxiv:2502.00957","type":"article-journal","author":[{"given":"Kwok Ho","family":"Wan"},{"given":"Zhenghao","family":"Zhong"}],"title":"Pauli webs spun by transversal $|Y\\rangle$ state initialisation","issued":{"date-parts":[[2025,2,2]]},"note":"arxivid:2502.00957\narxiv_version_number:1"},{"id":"arxiv:2012.08536","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop and simulate an explicit recipe for a just-in-time decoding scheme in three 3D surface codes, which can be used to implement a transversal (non-Clifford) <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mover><mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:mrow><mml:mo accent=\"false\">&amp;#x00AF;</mml:mo></mml:mover></mml:math> between three 2D surface codes in time linear in the code distance. We present a fully detailed set of bounded-height lattice slices through the 3D codes which retain the code distance and measurement-error detecting properties of the full 3D code and admit a dimension-jumping process which expands from/collapses to 2D surface codes supported on the boundaries of each slice. At each timestep of the procedure the slices agree on a common set of overlapping qubits on which <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:math> should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>X</mml:mi></mml:math> and measurement errors and find evidence for a threshold <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>&amp;#x223C;</mml:mo><mml:mn>0.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> in all three codes. We expect that this threshold could be improved by optimisation of the decoder.</jats:p>","DOI":"10.22331/q-2022-05-24-721","page":"721","source":"Crossref","title":"Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code","volume":"6","author":[{"given":"T. R.","family":"Scruby","sequence":"first","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"},{"name":"Dept. of Physics and Astronomy, University College London, London, WC1E 6BT, UK"}]},{"given":"D. E.","family":"Browne","sequence":"additional","affiliation":[{"name":"Dept. of Physics and Astronomy, University College London, London, WC1E 6BT, UK"}]},{"given":"P.","family":"Webster","sequence":"additional","affiliation":[{"name":"Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia"}]},{"given":"M.","family":"Vasmer","sequence":"additional","affiliation":[{"name":"Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,24]]},"URL":"http://dx.doi.org/10.22331/q-2022-05-24-721","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2012.08536"},{"id":"arxiv:2211.15465","type":"article-journal","author":[{"given":"Daniel","family":"Litinski"},{"given":"Naomi","family":"Nickerson"}],"title":"Active volume: An architecture for efficient fault-tolerant quantum computers with limited non-local connections","issued":{"date-parts":[[2022,11,28]]},"note":"arxivid:2211.15465\narxiv_version_number:1"},{"id":"arxiv:1209.0510","type":"article-journal","author":[{"given":"Austin G.","family":"Fowler"},{"given":"Simon J.","family":"Devitt"}],"title":"A bridge to lower overhead quantum computation","issued":{"date-parts":[[2013,4,9]]},"note":"arxivid:1209.0510\narxiv_version_number:4"},{"id":"arxiv:2212.00813","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>We introduce a framework for fault-tolerant postselection (FTPS) of fault-tolerant codes and channels—such as those based on surface codes—using soft-information metrics based on visible syndrome and erasure information. We introduce several metrics for ranking configurations of syndromes and erasures. In particular, we introduce the  (and variants thereof) as a powerful soft-information metric for predicting logical error rates of fault-tolerant channels based on topological error-correcting codes. The logical gap is roughly the unsigned weight difference between inequivalent logical corrections and is adaptable to any tailored noise model or decoder. We deploy this framework to prepare high-quality surface-code magic states with low overhead under a model of independent and identically distributed (IID) Pauli and erasure errors. Postselection strategies based on the logical gap can suppress the encoding error rate (EER) of a magic state preparation channel to the level of the physical error rate with low overhead. For example, when operating at <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mn>60</a:mn><a:mi mathvariant=\"normal\">%</a:mi></a:math> of the bulk threshold of the corresponding surface code, an overall reduction of the EER by a factor of <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><e:mn>15</e:mn></e:math> is achievable with a relative overhead factor of <h:math xmlns:h=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><h:mo>&lt;</h:mo><h:mn>2</h:mn></h:math> (approximately <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><k:mn>23</k:mn></k:math> times less than that of simple syndrome-counting rules). We analyze a schematic buffer architecture for implementing postselection rules on magic state factories in the context of magic state distillation. The FTPS framework can be utilized for mitigating errors in more general fault-tolerant logical channels.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.010302","source":"Crossref","title":"Fault-Tolerant Postselection for Low-Overhead Magic State Preparation","volume":"5","author":[{"given":"Héctor","family":"Bombín","sequence":"first","affiliation":[{"name":"PsiQuantum, Palo Alto, California 94304, USA"}]},{"given":"Mihir","family":"Pant","sequence":"additional","affiliation":[{"name":"PsiQuantum, Palo Alto, California 94304, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4652-389X","authenticated-orcid":true,"given":"Sam","family":"Roberts","sequence":"additional","affiliation":[{"name":"PsiQuantum, Palo Alto, California 94304, USA"}]},{"ORCID":"https://orcid.org/0000-0003-1928-8019","authenticated-orcid":true,"given":"Karthik I.","family":"Seetharam","sequence":"additional","affiliation":[{"name":"PsiQuantum, Palo Alto, California 94304, USA"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,1,4]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.010302","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010302","note":"arxivid:2212.00813"},{"id":"arxiv:2403.03991","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely , using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill -type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mn>100</a:mn><a:mo>×</a:mo><a:msup><a:mi>p</a:mi><a:mn>2</a:mn></a:msup></a:math> in terms of the physical error rate <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>p</c:mi></c:math>. For example, with a physical error rate of <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>p</e:mi><e:mo>=</e:mo><e:msup><e:mn>10</e:mn><e:mrow><e:mo>−</e:mo><e:mn>4</e:mn></e:mrow></e:msup></e:math> (<g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:msup><g:mn>10</g:mn><g:mrow><g:mo>−</g:mo><g:mn>3</g:mn></g:mrow></g:msup></g:math>), the logical error rate is reduced to <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:msub><i:mi>p</i:mi><i:mi>L</i:mi></i:msub><i:mo>=</i:mo><i:msup><i:mn>10</i:mn><i:mrow><i:mo>−</i:mo><i:mn>6</i:mn></i:mrow></i:msup></i:math> (<k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:msup><k:mn>10</k:mn><k:mrow><k:mo>−</k:mo><k:mn>4</k:mn></k:mrow></k:msup></k:math>), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.</jats:p>","DOI":"10.1103/thxx-njr6","source":"Crossref","title":"Efficient Magic State Distillation by Zero-Level Distillation","volume":"6","author":[{"ORCID":"https://orcid.org/0009-0001-4184-7748","authenticated-orcid":true,"given":"Tomohiro","family":"Itogawa","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/035t8zc32","id-type":"ROR","asserted-by":"publisher"}],"name":"Osaka University"}]},{"ORCID":"https://orcid.org/0009-0008-4589-3419","authenticated-orcid":true,"given":"Yugo","family":"Takada","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/035t8zc32","id-type":"ROR","asserted-by":"publisher"}],"name":"Osaka University"}]},{"ORCID":"https://orcid.org/0009-0003-5123-9513","authenticated-orcid":true,"given":"Yutaka","family":"Hirano","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/035t8zc32","id-type":"ROR","asserted-by":"publisher"}],"name":"Osaka University"}]},{"given":"Keisuke","family":"Fujii","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/035t8zc32","id-type":"ROR","asserted-by":"publisher"}],"name":"Osaka University"},{"id":[{"id":"https://ror.org/035t8zc32","id-type":"ROR","asserted-by":"publisher"}],"name":"Osaka University"},{"id":[{"id":"https://ror.org/02tt21044","id-type":"ROR","asserted-by":"publisher"}],"name":"RIKEN Center for Quantum Computing (RQC)"},{"id":[{"id":"https://ror.org/035t8zc32","id-type":"ROR","asserted-by":"publisher"}],"name":"Osaka University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,20]]},"URL":"http://dx.doi.org/10.1103/thxx-njr6","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020356","note":"arxivid:2403.03991"},{"id":"arxiv:2409.17595","type":"article-journal","author":[{"given":"Craig","family":"Gidney"},{"given":"Noah","family":"Shutty"},{"given":"Cody","family":"Jones"}],"title":"Magic state cultivation: growing T states as cheap as CNOT gates","issued":{"date-parts":[[2024,9,26]]},"note":"arxivid:2409.17595\narxiv_version_number:1"},{"id":"arxiv:2303.08829","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>There are several models of quantum computation which exhibit shared fundamental fault-tolerance properties. This article makes commonalities explicit by presenting these different models in a unifying framework based on the ZX calculus. We focus on models of topological fault tolerance – specifically surface codes – including circuit-based, measurement-based and fusion-based quantum computation, as well as the recently introduced model of Floquet codes. We find that all of these models can be viewed as different flavors of the same underlying stabilizer fault-tolerance structure, and sustain this through a set of local equivalence transformations which allow mapping between flavors. We anticipate that this unifying perspective will pave the way to transferring progress among the different views of stabilizer fault-tolerance and help researchers familiar with one model easily understand others.</jats:p>","DOI":"10.22331/q-2024-06-18-1379","page":"1379","source":"Crossref","title":"Unifying flavors of fault tolerance with the ZX calculus","volume":"8","author":[{"given":"Hector","family":"Bombin","sequence":"first","affiliation":[{"name":"PsiQuantum, Palo Alto"}]},{"given":"Daniel","family":"Litinski","sequence":"additional","affiliation":[{"name":"PsiQuantum, Palo Alto"}]},{"given":"Naomi","family":"Nickerson","sequence":"additional","affiliation":[{"name":"PsiQuantum, Palo Alto"}]},{"given":"Fernando","family":"Pastawski","sequence":"additional","affiliation":[{"name":"PsiQuantum, Palo Alto"}]},{"given":"Sam","family":"Roberts","sequence":"additional","affiliation":[{"name":"PsiQuantum, Palo Alto"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,18]]},"URL":"http://dx.doi.org/10.22331/q-2024-06-18-1379","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2303.08829"},{"id":"arxiv:2410.14891","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Recently, usage of detecting regions facilitated the discovery of new circuits for fault-tolerantly implementing the surface code. Building on these ideas, we present LUCI, a framework for constructing fault-tolerant circuits flexible enough to construct aperiodic and anisotropic circuits, making it a clear step towards quantum error correction beyond static codes. We show that LUCI can be used to adapt surface code circuits to lattices with imperfect qubit and coupler yield, a key challenge for fault-tolerant quantum computers using solid-state architectures. These circuits preserve spacelike distance for isolated broken couplers or isolated broken measure qubits in exchange for halving timelike distance, substantially reducing the penalty for dropout compared to the state of the art and creating opportunities in device architecture design. For qubit and coupler dropout rates of 1% and a patch diameter of 15, LUCI achieves an average spacelike distance of 13.1, compared to 9.1 for the best method in the literature. For a SI1000(0.001) circuit noise model, this translates to a 36x improvement in median logical error rate per round, a factor which increases with device performance. At these dropout and error rates, LUCI requires roughly 25% fewer physical qubits to reach algorithmically relevant one-in-a-trillion logical codeblock error rates.</jats:p>","DOI":"10.22331/q-2025-12-11-1936","page":"1936","source":"Crossref","title":"LUCI in the Surface Code with Dropouts","volume":"9","author":[{"given":"Dripto M.","family":"Debroy","sequence":"first","affiliation":[{"name":"Google Quantum AI, Venice, CA 90291, USA"}]},{"given":"Matt","family":"McEwen","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, CA 93117, USA"}]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Santa Barbara, CA 93117, USA"}]},{"given":"Noah","family":"Shutty","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Venice, CA 90291, USA"}]},{"given":"Adam","family":"Zalcman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Venice, CA 90291, USA"},{"name":"Google Quantum AI, Tokyo, Japan"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,11]]},"URL":"http://dx.doi.org/10.22331/q-2025-12-11-1936","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2410.14891"},{"id":"arxiv:2502.10355","type":"article-journal","author":[{"given":"Dripto M.","family":"Debroy"}],"title":"Diamond Circuits for Surface Codes","issued":{"date-parts":[[2025,2,14]]},"note":"arxivid:2502.10355\narxiv_version_number:1"},{"id":"arxiv:0905.0531","type":"article-journal","author":[{"given":"D. S.","family":"Wang"},{"given":"A. G.","family":"Fowler"},{"given":"A. M.","family":"Stephens"},{"given":"L. C. L.","family":"Hollenberg"}],"title":"Threshold error rates for the toric and surface codes","issued":{"date-parts":[[2009,5,5]]},"note":"arxivid:0905.0531\narxiv_version_number:1"},{"id":"arxiv:1609.06373","type":"article-journal","author":[{"given":"Bettina","family":"Heim"},{"given":"Krysta M.","family":"Svore"},{"given":"Matthew B.","family":"Hastings"}],"title":"Optimal Circuit-Level Decoding for Surface Codes","issued":{"date-parts":[[2016,9,20]]},"note":"arxivid:1609.06373\narxiv_version_number:1"},{"id":"arxiv:1206.0800","type":"journal-article","publisher":"American Physical Society (APS)","issue":"18","DOI":"10.1103/physrevlett.109.180502","source":"Crossref","title":"Proof of Finite Surface Code Threshold for Matching","volume":"109","author":[{"given":"Austin G.","family":"Fowler","sequence":"first","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,11,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.109.180502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"180502","note":"arxivid:1206.0800"},{"id":"arxiv:quant-ph/0207088","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/s0003-4916(02)00019-2","page":"31-58","source":"Crossref","title":"Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory","volume":"303","author":[{"given":"Chenyang","family":"Wang","sequence":"first","affiliation":[]},{"given":"Jim","family":"Harrington","sequence":"additional","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Annals of Physics","original-title":[],"language":"en","issued":{"date-parts":[[2003,1]]},"URL":"http://dx.doi.org/10.1016/S0003-4916(02)00019-2","ISSN":["0003-4916"],"container-title-short":"Annals of Physics","note":"alternative-id:S0003491602000192\narxivid:quant-ph/0207088"},{"id":"arxiv:0811.0464","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreve.79.021129","source":"Crossref","title":"Locations of multicritical points for spin glasses on regular lattices","volume":"79","author":[{"given":"Masayuki","family":"Ohzeki","sequence":"first","affiliation":[]}],"container-title":"Physical Review E","original-title":[],"language":"en","issued":{"date-parts":[[2009,2,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevE.79.021129","ISSN":["1539-3755","1550-2376"],"container-title-short":"Phys. Rev. E","page":"021129","note":"arxivid:0811.0464"},{"id":"arxiv:1009.3686","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.83.020302","source":"Crossref","title":"Surface code quantum computing with error rates over 1%","volume":"83","author":[{"given":"David S.","family":"Wang","sequence":"first","affiliation":[]},{"given":"Austin G.","family":"Fowler","sequence":"additional","affiliation":[]},{"given":"Lloyd C. L.","family":"Hollenberg","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,2,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.020302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"020302","note":"arxivid:1009.3686"},{"id":"arxiv:1311.5003","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.89.022321","source":"Crossref","title":"Fault-tolerant thresholds for quantum error correction with the surface code","volume":"89","author":[{"given":"Ashley M.","family":"Stephens","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.89.022321","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022321","note":"arxivid:1311.5003"},{"id":"arxiv:2602.20238","type":"article-journal","author":[{"given":"Satoshi","family":"Yoshida"},{"given":"Ethan","family":"Lake"},{"given":"Hayata","family":"Yamasaki"}],"title":"Proof of a finite threshold for the union-find decoder","issued":{"date-parts":[[2026,2,23]]},"note":"arxivid:2602.20238\narxiv_version_number:1"},{"id":"arxiv:1706.04912","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.96.042316","source":"Crossref","title":"Fault-tolerance thresholds for the surface code with fabrication errors","volume":"96","author":[{"given":"James M.","family":"Auger","sequence":"first","affiliation":[]},{"given":"Hussain","family":"Anwar","sequence":"additional","affiliation":[]},{"given":"Mercedes","family":"Gimeno-Segovia","sequence":"additional","affiliation":[]},{"given":"Thomas M.","family":"Stace","sequence":"additional","affiliation":[]},{"given":"Dan E.","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2017,10,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.96.042316","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042316","note":"arxivid:1706.04912"},{"id":"arxiv:2401.04530","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.110.012417","source":"Crossref","title":"Coherent errors in stabilizer codes caused by quasistatic phase damping","volume":"110","author":[{"ORCID":"https://orcid.org/0000-0002-8818-3723","authenticated-orcid":true,"given":"Dávid","family":"Pataki","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02w42ss30","id-type":"ROR","asserted-by":"publisher"}],"name":"Budapest University of Technology and Economics"}]},{"given":"Áron","family":"Márton","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02w42ss30","id-type":"ROR","asserted-by":"publisher"}],"name":"Budapest University of Technology and Economics"}]},{"ORCID":"https://orcid.org/0000-0003-0962-7523","authenticated-orcid":true,"given":"János K.","family":"Asbóth","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02w42ss30","id-type":"ROR","asserted-by":"publisher"}],"name":"Budapest University of Technology and Economics"},{"id":[{"id":"https://ror.org/035dsb084","id-type":"ROR","asserted-by":"publisher"}],"name":"Wigner Research Centre for Physics"}]},{"given":"András","family":"Pályi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02w42ss30","id-type":"ROR","asserted-by":"publisher"}],"name":"Budapest University of Technology and Economics"},{"name":"HUN-REN–BME Quantum Dynamics and Correlations Research Group"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.110.012417","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012417","note":"arxivid:2401.04530"},{"id":"arxiv:2408.02082","type":"article-journal","author":[{"given":"Ian","family":"Hesner"},{"given":"Bence","family":"Hetényi"},{"given":"James R.","family":"Wootton"}],"title":"Using Detector Likelihood for Benchmarking Quantum Error Correction","issued":{"date-parts":[[2024,8,4]]},"note":"arxivid:2408.02082\narxiv_version_number:1"},{"id":"arxiv:2104.01180","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6572","abstract":"<jats:title>Synthesizing topological order</jats:title>\n          <jats:p>\n            Topologically ordered matter exhibits long-range quantum entanglement. However, measuring this entanglement in real materials is extremely tricky. Now, two groups take a different approach and turn to synthetic systems to engineer the topological order of the so-called toric code type (see the Perspective by Bartlett). Satzinger\n            <jats:italic>et al</jats:italic>\n            . used a quantum processor to study the ground state and excitations of the toric code. Semeghini\n            <jats:italic>et al</jats:italic>\n            . detected signatures of a toric code–type quantum spin liquid in a two-dimensional array of Rydberg atoms held in optical tweezers. —JS\n          </jats:p>","DOI":"10.1126/science.abi8378","page":"1237-1241","source":"Crossref","title":"Realizing topologically ordered states on a quantum processor","volume":"374","author":[{"ORCID":"https://orcid.org/0000-0001-5865-0813","authenticated-orcid":true,"given":"K. J.","family":"Satzinger","sequence":"first","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-7657-9464","authenticated-orcid":true,"given":"Y.-J","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Physics, Technical University of Munich, 85748 Garching, Germany."},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany."}]},{"ORCID":"https://orcid.org/0000-0002-6744-4592","authenticated-orcid":true,"given":"A.","family":"Smith","sequence":"additional","affiliation":[{"name":"Department of Physics, Technical University of Munich, 85748 Garching, Germany."},{"name":"School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK."},{"name":"Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, UK."}]},{"ORCID":"https://orcid.org/0000-0002-5982-8107","authenticated-orcid":true,"given":"C.","family":"Knapp","sequence":"additional","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA."},{"name":"Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, USA."}]},{"given":"M.","family":"Newman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"C.","family":"Jones","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"Z.","family":"Chen","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"C.","family":"Quintana","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-0507-0211","authenticated-orcid":true,"given":"X.","family":"Mi","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"A.","family":"Dunsworth","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"C.","family":"Gidney","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"I.","family":"Aleiner","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"F.","family":"Arute","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6486-7100","authenticated-orcid":true,"given":"K.","family":"Arya","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"J.","family":"Atalaya","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-6979-9533","authenticated-orcid":true,"given":"R.","family":"Babbush","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6523-6730","authenticated-orcid":true,"given":"J. C.","family":"Bardin","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."},{"name":"Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA, USA."}]},{"given":"R.","family":"Barends","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5547-691X","authenticated-orcid":true,"given":"J.","family":"Basso","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5264-4025","authenticated-orcid":true,"given":"A.","family":"Bengtsson","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"A.","family":"Bilmes","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"M.","family":"Broughton","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5139-7931","authenticated-orcid":true,"given":"B. B.","family":"Buckley","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"D. A.","family":"Buell","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-8474-6317","authenticated-orcid":true,"given":"B.","family":"Burkett","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-7445-1654","authenticated-orcid":true,"given":"N.","family":"Bushnell","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"B.","family":"Chiaro","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6164-0007","authenticated-orcid":true,"given":"R.","family":"Collins","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"W.","family":"Courtney","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-3727-7380","authenticated-orcid":true,"given":"S.","family":"Demura","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"A. R.","family":"Derk","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-7134-5733","authenticated-orcid":true,"given":"D.","family":"Eppens","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"C.","family":"Erickson","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"L.","family":"Faoro","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique Theorique et Hautes Energies, Sorbonne Université, 75005 Paris, France."}]},{"given":"E.","family":"Farhi","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9048-2896","authenticated-orcid":true,"given":"A. G.","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-3732-168X","authenticated-orcid":true,"given":"B.","family":"Foxen","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"M.","family":"Giustina","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-4034-8744","authenticated-orcid":true,"given":"A.","family":"Greene","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."},{"name":"Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA."}]},{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":true,"given":"J. A.","family":"Gross","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-9412-0553","authenticated-orcid":true,"given":"M. P.","family":"Harrigan","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-0521-8378","authenticated-orcid":true,"given":"S. D.","family":"Harrington","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"J.","family":"Hilton","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"S.","family":"Hong","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"T.","family":"Huang","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"W. J.","family":"Huggins","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8777-6789","authenticated-orcid":true,"given":"L. B.","family":"Ioffe","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"S. V.","family":"Isakov","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"E.","family":"Jeffrey","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-0435-655X","authenticated-orcid":true,"given":"Z.","family":"Jiang","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-9160-5176","authenticated-orcid":true,"given":"D.","family":"Kafri","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-0136-1428","authenticated-orcid":true,"given":"K.","family":"Kechedzhi","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"T.","family":"Khattar","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-3397-0444","authenticated-orcid":true,"given":"S.","family":"Kim","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"P. V.","family":"Klimov","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"A. N.","family":"Korotkov","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."},{"name":"Department of Electrical and Computer Engineering, University of California, Riverside, CA, USA."}]},{"given":"F.","family":"Kostritsa","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-9804-2185","authenticated-orcid":true,"given":"D.","family":"Landhuis","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"P.","family":"Laptev","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"A.","family":"Locharla","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6449-2273","authenticated-orcid":true,"given":"E.","family":"Lucero","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9576-573X","authenticated-orcid":true,"given":"O.","family":"Martin","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2809-0509","authenticated-orcid":true,"given":"J. R.","family":"McClean","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9544-141X","authenticated-orcid":true,"given":"M.","family":"McEwen","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."},{"name":"Department of Physics, University of California, Santa Barbara, CA, USA."}]},{"given":"K. C.","family":"Miao","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"M.","family":"Mohseni","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"S.","family":"Montazeri","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8497-6363","authenticated-orcid":true,"given":"W.","family":"Mruczkiewicz","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"J.","family":"Mutus","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-7760-9186","authenticated-orcid":true,"given":"O.","family":"Naaman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-5548-0051","authenticated-orcid":true,"given":"M.","family":"Neeley","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6563-3568","authenticated-orcid":true,"given":"C.","family":"Neill","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"M. Y.","family":"Niu","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8406-6626","authenticated-orcid":true,"given":"T. E.","family":"O’Brien","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"A.","family":"Opremcak","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"B.","family":"Pató","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"A.","family":"Petukhov","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-3963-1830","authenticated-orcid":true,"given":"N. C.","family":"Rubin","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0001-8188-364X","authenticated-orcid":true,"given":"D.","family":"Sank","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"V.","family":"Shvarts","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"D.","family":"Strain","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-1310-9174","authenticated-orcid":true,"given":"M.","family":"Szalay","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"B.","family":"Villalonga","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9803-7471","authenticated-orcid":true,"given":"T. C.","family":"White","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-1806-5454","authenticated-orcid":true,"given":"Z.","family":"Yao","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0003-0837-1028","authenticated-orcid":true,"given":"P.","family":"Yeh","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"J.","family":"Yoo","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2585-2424","authenticated-orcid":true,"given":"A.","family":"Zalcman","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9681-6746","authenticated-orcid":true,"given":"H.","family":"Neven","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-1090-7584","authenticated-orcid":true,"given":"S.","family":"Boixo","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6371-6140","authenticated-orcid":true,"given":"A.","family":"Megrant","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-7473-6745","authenticated-orcid":true,"given":"Y.","family":"Chen","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"J.","family":"Kelly","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-3000-6732","authenticated-orcid":true,"given":"V.","family":"Smelyanskiy","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]},{"given":"A.","family":"Kitaev","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."},{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA."},{"name":"Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, USA."}]},{"ORCID":"https://orcid.org/0000-0002-7093-9502","authenticated-orcid":true,"given":"M.","family":"Knap","sequence":"additional","affiliation":[{"name":"Department of Physics, Technical University of Munich, 85748 Garching, Germany."},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany."},{"name":"Institute for Advanced Study, Technical University of Munich, 85748 Garching, Germany."}]},{"ORCID":"https://orcid.org/0000-0003-0320-9304","authenticated-orcid":true,"given":"F.","family":"Pollmann","sequence":"additional","affiliation":[{"name":"Department of Physics, Technical University of Munich, 85748 Garching, Germany."},{"name":"Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany."}]},{"ORCID":"https://orcid.org/0000-0003-1917-3879","authenticated-orcid":true,"given":"P.","family":"Roushan","sequence":"additional","affiliation":[{"name":"Google Quantum AI, Mountain View, CA, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,3]]},"URL":"http://dx.doi.org/10.1126/science.abi8378","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.abi8378\narxivid:2104.01180"},{"id":"arxiv:2104.04119","type":"journal-article","publisher":"American Association for the Advancement of Science (AAAS)","issue":"6572","abstract":"<jats:title>Synthesizing topological order</jats:title>\n          <jats:p>\n            Topologically ordered matter exhibits long-range quantum entanglement. However, measuring this entanglement in real materials is extremely tricky. Now, two groups take a different approach and turn to synthetic systems to engineer the topological order of the so-called toric code type (see the Perspective by Bartlett). Satzinger\n            <jats:italic>et al</jats:italic>\n            . used a quantum processor to study the ground state and excitations of the toric code. Semeghini\n            <jats:italic>et al</jats:italic>\n            . detected signatures of a toric code–type quantum spin liquid in a two-dimensional array of Rydberg atoms held in optical tweezers. —JS\n          </jats:p>","DOI":"10.1126/science.abi8794","page":"1242-1247","source":"Crossref","title":"Probing topological spin liquids on a programmable quantum simulator","volume":"374","author":[{"ORCID":"https://orcid.org/0000-0001-9071-2279","authenticated-orcid":true,"given":"G.","family":"Semeghini","sequence":"first","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0001-8270-3233","authenticated-orcid":true,"given":"H.","family":"Levine","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0003-3931-0949","authenticated-orcid":true,"given":"A.","family":"Keesling","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."},{"name":"QuEra Computing, Boston, MA 02135, USA."}]},{"ORCID":"https://orcid.org/0000-0003-4146-3637","authenticated-orcid":true,"given":"S.","family":"Ebadi","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0003-3107-2579","authenticated-orcid":true,"given":"T. T.","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9934-9530","authenticated-orcid":true,"given":"D.","family":"Bluvstein","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":true,"given":"R.","family":"Verresen","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0003-2144-536X","authenticated-orcid":true,"given":"H.","family":"Pichler","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria."},{"name":"Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria."}]},{"given":"M.","family":"Kalinowski","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0001-5171-7798","authenticated-orcid":true,"given":"R.","family":"Samajdar","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2253-0278","authenticated-orcid":true,"given":"A.","family":"Omran","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."},{"name":"QuEra Computing, Boston, MA 02135, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2432-7070","authenticated-orcid":true,"given":"S.","family":"Sachdev","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."},{"name":"School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA."}]},{"ORCID":"https://orcid.org/0000-0002-6306-2263","authenticated-orcid":true,"given":"A.","family":"Vishwanath","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0002-2935-2363","authenticated-orcid":true,"given":"M.","family":"Greiner","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]},{"ORCID":"https://orcid.org/0000-0002-9786-0538","authenticated-orcid":true,"given":"V.","family":"Vuletić","sequence":"additional","affiliation":[{"name":"Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA."}]},{"ORCID":"https://orcid.org/0000-0002-8658-1007","authenticated-orcid":true,"given":"M. D.","family":"Lukin","sequence":"additional","affiliation":[{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA."}]}],"container-title":"Science","original-title":[],"language":"en","issued":{"date-parts":[[2021,12,3]]},"URL":"http://dx.doi.org/10.1126/science.abi8794","ISSN":["0036-8075","1095-9203"],"container-title-short":"Science","note":"alternative-id:10.1126/science.abi8794\narxivid:2104.04119"},{"id":"arxiv:2412.14360","type":"article-journal","author":[{"given":"Alec","family":"Eickbusch"},{"given":"Matt","family":"McEwen"},{"given":"Volodymyr","family":"Sivak"},{"given":"Alexandre","family":"Bourassa"},{"given":"Juan","family":"Atalaya"},{"given":"Jahan","family":"Claes"},{"given":"Dvir","family":"Kafri"},{"given":"Craig","family":"Gidney"},{"given":"Christopher W.","family":"Warren"},{"given":"Jonathan","family":"Gross"},{"given":"Alex","family":"Opremcak"},{"given":"Nicholas","family":"Zobrist"},{"given":"Kevin C.","family":"Miao"},{"given":"Gabrielle","family":"Roberts"},{"given":"Kevin J.","family":"Satzinger"},{"given":"Andreas","family":"Bengtsson"},{"given":"Matthew","family":"Neeley"},{"given":"William P.","family":"Livingston"},{"given":"Alex","family":"Greene"},{"given":"Rajeev","family":"Acharya"},{"given":"Laleh Aghababaie","family":"Beni"},{"given":"Georg","family":"Aigeldinger"},{"given":"Ross","family":"Alcaraz"},{"given":"Trond I.","family":"Andersen"},{"given":"Markus","family":"Ansmann"},{"given":"Frank","family":"Arute"},{"given":"Kunal","family":"Arya"},{"given":"Abraham","family":"Asfaw"},{"given":"Ryan","family":"Babbush"},{"given":"Brian","family":"Ballard"},{"given":"Joseph C.","family":"Bardin"},{"given":"Alexander","family":"Bilmes"},{"given":"Jenna","family":"Bovaird"},{"given":"Dylan","family":"Bowers"},{"given":"Leon","family":"Brill"},{"given":"Michael","family":"Broughton"},{"given":"David A.","family":"Browne"},{"given":"Brett","family":"Buchea"},{"given":"Bob B.","family":"Buckley"},{"given":"Tim","family":"Burger"},{"given":"Brian","family":"Burkett"},{"given":"Nicholas","family":"Bushnell"},{"given":"Anthony","family":"Cabrera"},{"given":"Juan","family":"Campero"},{"given":"Hung-Shen","family":"Chang"},{"given":"Ben","family":"Chiaro"},{"given":"Liang-Ying","family":"Chih"},{"given":"Agnetta Y.","family":"Cleland"},{"given":"Josh","family":"Cogan"},{"given":"Roberto","family":"Collins"},{"given":"Paul","family":"Conner"},{"given":"William","family":"Courtney"},{"given":"Alexander L.","family":"Crook"},{"given":"Ben","family":"Curtin"},{"given":"Sayan","family":"Das"},{"given":"Alexander Del Toro","family":"Barba"},{"given":"Sean","family":"Demura"},{"given":"Laura","family":"De Lorenzo"},{"given":"Agustin","family":"Di Paolo"},{"given":"Paul","family":"Donohoe"},{"given":"Ilya K.","family":"Drozdov"},{"given":"Andrew","family":"Dunsworth"},{"given":"Aviv Moshe","family":"Elbag"},{"given":"Mahmoud","family":"Elzouka"},{"given":"Catherine","family":"Erickson"},{"given":"Vinicius S.","family":"Ferreira"},{"given":"Leslie Flores","family":"Burgos"},{"given":"Ebrahim","family":"Forati"},{"given":"Austin G.","family":"Fowler"},{"given":"Brooks","family":"Foxen"},{"given":"Suhas","family":"Ganjam"},{"given":"Gonzalo","family":"Garcia"},{"given":"Robert","family":"Gasca"},{"given":"Élie","family":"Genois"},{"given":"William","family":"Giang"},{"given":"Dar","family":"Gilboa"},{"given":"Raja","family":"Gosula"},{"given":"Alejandro Grajales","family":"Dau"},{"given":"Dietrich","family":"Graumann"},{"given":"Tan","family":"Ha"},{"given":"Steve","family":"Habegger"},{"given":"Michael C.","family":"Hamilton"},{"given":"Monica","family":"Hansen"},{"given":"Matthew P.","family":"Harrigan"},{"given":"Sean D.","family":"Harrington"},{"given":"Stephen","family":"Heslin"},{"given":"Paula","family":"Heu"},{"given":"Oscar","family":"Higgott"},{"given":"Reno","family":"Hiltermann"},{"given":"Jeremy","family":"Hilton"},{"given":"Hsin-Yuan","family":"Huang"},{"given":"Ashley","family":"Huff"},{"given":"William J.","family":"Huggins"},{"given":"Evan","family":"Jeffrey"},{"given":"Zhang","family":"Jiang"},{"given":"Xiaoxuan","family":"Jin"},{"given":"Cody","family":"Jones"},{"given":"Chaitali","family":"Joshi"},{"given":"Pavol","family":"Juhas"},{"given":"Andreas","family":"Kabel"},{"given":"Hui","family":"Kang"},{"given":"Amir H.","family":"Karamlou"},{"given":"Kostyantyn","family":"Kechedzhi"},{"given":"Trupti","family":"Khaire"},{"given":"Tanuj","family":"Khattar"},{"given":"Mostafa","family":"Khezri"},{"given":"Seon","family":"Kim"},{"given":"Bryce","family":"Kobrin"},{"given":"Alexander N.","family":"Korotkov"},{"given":"Fedor","family":"Kostritsa"},{"given":"John Mark","family":"Kreikebaum"},{"given":"Vladislav D.","family":"Kurilovich"},{"given":"David","family":"Landhuis"},{"given":"Tiano","family":"Lange-Dei"},{"given":"Brandon W.","family":"Langley"},{"given":"Kim-Ming","family":"Lau"},{"given":"Justin","family":"Ledford"},{"given":"Kenny","family":"Lee"},{"given":"Brian J.","family":"Lester"},{"given":"Loïck","family":"Le Guevel"},{"given":"Wing Yan","family":"Li"},{"given":"Alexander T.","family":"Lill"},{"given":"Aditya","family":"Locharla"},{"given":"Erik","family":"Lucero"},{"given":"Daniel","family":"Lundahl"},{"given":"Aaron","family":"Lunt"},{"given":"Sid","family":"Madhuk"},{"given":"Ashley","family":"Maloney"},{"given":"Salvatore","family":"Mandrà"},{"given":"Leigh S.","family":"Martin"},{"given":"Orion","family":"Martin"},{"given":"Cameron","family":"Maxfield"},{"given":"Jarrod R.","family":"McClean"},{"given":"Seneca","family":"Meeks"},{"given":"Anthony","family":"Megrant"},{"given":"Reza","family":"Molavi"},{"given":"Sebastian","family":"Molina"},{"given":"Shirin","family":"Montazeri"},{"given":"Ramis","family":"Movassagh"},{"given":"Michael","family":"Newman"},{"given":"Anthony","family":"Nguyen"},{"given":"Murray","family":"Nguyen"},{"given":"Chia-Hung","family":"Ni"},{"given":"Logan","family":"Oas"},{"given":"Raymond","family":"Orosco"},{"given":"Kristoffer","family":"Ottosson"},{"given":"Alex","family":"Pizzuto"},{"given":"Rebecca","family":"Potter"},{"given":"Orion","family":"Pritchard"},{"given":"Chris","family":"Quintana"},{"given":"Ganesh","family":"Ramachandran"},{"given":"Matthew J.","family":"Reagor"},{"given":"David M.","family":"Rhodes"},{"given":"Eliott","family":"Rosenberg"},{"given":"Elizabeth","family":"Rossi"},{"given":"Kannan","family":"Sankaragomathi"},{"given":"Henry F.","family":"Schurkus"},{"given":"Michael J.","family":"Shearn"},{"given":"Aaron","family":"Shorter"},{"given":"Noah","family":"Shutty"},{"given":"Vladimir","family":"Shvarts"},{"given":"Spencer","family":"Small"},{"given":"W. Clarke","family":"Smith"},{"given":"Sofia","family":"Springer"},{"given":"George","family":"Sterling"},{"given":"Jordan","family":"Suchard"},{"given":"Aaron","family":"Szasz"},{"given":"Alex","family":"Sztein"},{"given":"Douglas","family":"Thor"},{"given":"Eifu","family":"Tomita"},{"given":"Alfredo","family":"Torres"},{"given":"M. Mert","family":"Torunbalci"},{"given":"Abeer","family":"Vaishnav"},{"given":"Justin","family":"Vargas"},{"given":"Sergey","family":"Vdovichev"},{"given":"Guifre","family":"Vidal"},{"given":"Catherine Vollgraff","family":"Heidweiller"},{"given":"Steven","family":"Waltman"},{"given":"Jonathan","family":"Waltz"},{"given":"Shannon X.","family":"Wang"},{"given":"Brayden","family":"Ware"},{"given":"Travis","family":"Weidel"},{"given":"Theodore","family":"White"},{"given":"Kristi","family":"Wong"},{"given":"Bryan W. K.","family":"Woo"},{"given":"Maddy","family":"Woodson"},{"given":"Cheng","family":"Xing"},{"given":"Z. Jamie","family":"Yao"},{"given":"Ping","family":"Yeh"},{"given":"Bicheng","family":"Ying"},{"given":"Juhwan","family":"Yoo"},{"given":"Noureldin","family":"Yosri"},{"given":"Grayson","family":"Young"},{"given":"Adam","family":"Zalcman"},{"given":"Yaxing","family":"Zhang"},{"given":"Ningfeng","family":"Zhu"},{"given":"Sergio","family":"Boixo"},{"given":"Julian","family":"Kelly"},{"given":"Vadim","family":"Smelyanskiy"},{"given":"Hartmut","family":"Neven"},{"given":"Dave","family":"Bacon"},{"given":"Zijun","family":"Chen"},{"given":"Paul V.","family":"Klimov"},{"given":"Pedram","family":"Roushan"},{"given":"Charles","family":"Neill"},{"given":"Yu","family":"Chen"},{"given":"Alexis","family":"Morvan"}],"title":"Demonstrating dynamic surface codes","issued":{"date-parts":[[2025,6,19]]},"note":"arxivid:2412.14360\narxiv_version_number:2"},{"id":"doi:10.21468/SciPostPhysLectNotes.49","type":"journal-article","publisher":"Stichting SciPost","abstract":"<jats:p>This chapter provides an introduction to the surface code, discussing details about how physical qubits are connected and operated to create logical qubits, how errors are handled, and what performance metrics are needed to generate logical qubits with performance that exceeds their underlying physical qubit performance.</jats:p>","DOI":"10.21468/scipostphyslectnotes.49","source":"Crossref","title":"An introduction to the surface code","author":[{"given":"Andrew","family":"Cleland","sequence":"first","affiliation":[{"name":"University of Chicago"}]}],"container-title":"SciPost Physics Lecture Notes","original-title":[],"issued":{"date-parts":[[2022,5,3]]},"URL":"http://dx.doi.org/10.21468/SciPostPhysLectNotes.49","ISSN":["2590-1990"],"container-title-short":"SciPost Phys. Lect. Notes","page":"49"},{"id":"arxiv:2307.14989","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Quantum technologies have the potential to solve certain computationally hard problems with polynomial or super-polynomial speedups when compared to classical methods. Unfortunately, the unstable nature of quantum information makes it prone to errors. For this reason, quantum error correction is an invaluable tool to make quantum information reliable and enable the ultimate goal of fault-tolerant quantum computing. Surface codes currently stand as the most promising candidates to build near term error corrected qubits given their two-dimensional architecture, the requirement of only local operations, and high tolerance to quantum noise. Decoding algorithms are an integral component of any error correction scheme, as they are tasked with producing accurate estimates of the errors that affect quantum information, so that they can subsequently be corrected. A critical aspect of decoding algorithms is their speed, since the quantum state will suffer additional errors with the passage of time. This poses a connundrum, where decoding performance is improved at the expense of complexity and viceversa. In this review, a thorough discussion of state-of-the-art decoding algorithms for surface codes is provided. The target audience of this work are both readers with an introductory understanding of the field as well as those seeking to further their knowledge of the decoding paradigm of surface codes. We describe the core principles of these decoding methods as well as existing variants that show promise for improved results. In addition, both the decoding performance, in terms of error correction capability, and decoding complexity, are compared. A review of the existing software tools regarding surface codes decoding is also provided.</jats:p>","DOI":"10.22331/q-2024-10-10-1498","page":"1498","source":"Crossref","title":"Decoding algorithms for surface codes","volume":"8","author":[{"given":"Antonio","family":"deMarti iOlius","sequence":"first","affiliation":[{"name":"Department of Basic Sciences, Tecnun - University of Navarra, 20018 San Sebastian, Spain."}]},{"given":"Patricio","family":"Fuentes","sequence":"additional","affiliation":[{"name":"Photonic Inc., Vancouver, British Columbia, Canada."}]},{"given":"Román","family":"Orús","sequence":"additional","affiliation":[{"name":"Multiverse Computing, Pio Baroja 37, 20008 San Sebastián, Spain"},{"name":"Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain"},{"name":"IKERBASQUE, Basque Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain"}]},{"given":"Pedro M.","family":"Crespo","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Tecnun - University of Navarra, 20018 San Sebastian, Spain."}]},{"given":"Josu","family":"Etxezarreta Martinez","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Tecnun - University of Navarra, 20018 San Sebastian, Spain."}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,10]]},"URL":"http://dx.doi.org/10.22331/q-2024-10-10-1498","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2307.14989"},{"id":"arxiv:2412.02036","type":"article-journal","author":[{"given":"Xiang","family":"Fang"},{"given":"Keyi","family":"Yin"},{"given":"Yuchen","family":"Zhu"},{"given":"Jixuan","family":"Ruan"},{"given":"Dean","family":"Tullsen"},{"given":"Zhiding","family":"Liang"},{"given":"Andrew","family":"Sornborger"},{"given":"Ang","family":"Li"},{"given":"Travis","family":"Humble"},{"given":"Yufei","family":"Ding"},{"given":"Yunong","family":"Shi"}],"title":"CaliScalpel: In-Situ and Fine-Grained Qubit Calibration Integrated with Surface Code Quantum Error Correction","issued":{"date-parts":[[2024,12,2]]},"note":"arxivid:2412.02036\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0703272","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.76.012305","source":"Crossref","title":"Optimal resources for topological two-dimensional stabilizer codes: Comparative study","volume":"76","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2007,7,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.76.012305","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012305","note":"arxivid:quant-ph/0703272"},{"id":"arxiv:2409.14765","type":"article-journal","author":[{"given":"Anthony Ryan","family":"O'Rourke"},{"given":"Simon","family":"Devitt"}],"title":"Compare the Pair: Rotated vs. Unrotated Surface Codes at Equal Logical Error Rates","issued":{"date-parts":[[2024,10,11]]},"note":"arxivid:2409.14765\narxiv_version_number:2"},{"id":"arxiv:2402.02185","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Topological quantum error correction is a milestone in the scaling roadmap of quantum computers, which targets circuits with trillions of gates that would allow running quantum algorithms for real-world problems. The square-lattice surface code has become the workhorse to address this challenge, as it poses milder requirements on current devices both in terms of required error rates and small local connectivities. In some platforms, however, the connectivities are kept even lower in order to minimise gate errors at the hardware level, which limits the error correcting codes that can be directly implemented on them. In this work, we make a comparative study of possible strategies to overcome this limitation for the heavy-hexagonal lattice, the architecture of current IBM superconducting quantum computers. We explore two complementary strategies: the search for an efficient embedding of the surface code into the heavy-hexagonal lattice, as well as the use of codes whose connectivity requirements are naturally tailored to this architecture, such as subsystem-type and Floquet codes. Using noise models of increased complexity, we assess the performance of these strategies for IBM devices in terms of their error thresholds and qubit footprints. An optimized SWAP-based embedding of the surface code is found to be the most promising strategy towards a near-term demonstration of quantum error correction advantage.</jats:p>","DOI":"10.22331/q-2025-02-06-1623","page":"1623","source":"Crossref","title":"Comparative study of quantum error correction strategies for the heavy-hexagonal lattice","volume":"9","author":[{"given":"César","family":"Benito","sequence":"first","affiliation":[{"name":"Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain"}]},{"given":"Esperanza","family":"López","sequence":"additional","affiliation":[{"name":"Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain"}]},{"given":"Borja","family":"Peropadre","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM Research, Cambridge, MA 02142, USA"}]},{"given":"Alejandro","family":"Bermudez","sequence":"additional","affiliation":[{"name":"Currently on sabbatical at Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom"},{"name":"Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,6]]},"URL":"http://dx.doi.org/10.22331/q-2025-02-06-1623","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2402.02185"},{"id":"arxiv:cond-mat/0405313","type":"journal-article","publisher":"Physical Society of Japan","issue":"10","DOI":"10.1143/jpsj.73.2701","page":"2701-2707","source":"Crossref","title":"Error Counting in a Quantum Error-correcting Code and the Ground-State Energy of a Spin Glass","volume":"73","author":[{"given":"Hidetoshi","family":"Nishimori","sequence":"first","affiliation":[{"name":"Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551"}]},{"given":"Peter","family":"Sollich","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom"}]}],"container-title":"Journal of the Physical Society of Japan","original-title":[],"language":"en","issued":{"date-parts":[[2004,10,15]]},"URL":"http://dx.doi.org/10.1143/JPSJ.73.2701","ISSN":["0031-9015","1347-4073"],"container-title-short":"J. Phys. Soc. Jpn.","note":"alternative-id:10.1143/JPSJ.73.2701\narxivid:cond-mat/0405313"},{"id":"arxiv:2403.08706","type":"article-journal","author":[{"given":"Andrew S.","family":"Darmawan"}],"title":"Optimal adaptation of surface-code decoders to local noise","issued":{"date-parts":[[2024,3,13]]},"note":"arxivid:2403.08706\narxiv_version_number:1"},{"id":"arxiv:1609.00510","type":"article-journal","author":[{"given":"Nikolas P.","family":"Breuckmann"},{"given":"Kasper","family":"Duivenvoorden"},{"given":"Dominik","family":"Michels"},{"given":"Barbara M.","family":"Terhal"}],"title":"Local Decoders for the 2D and 4D Toric Code","issued":{"date-parts":[[2016,9,19]]},"note":"arxivid:1609.00510\narxiv_version_number:2"},{"id":"arxiv:2410.07598","type":"journal-article","publisher":"American Physical Society (APS)","issue":"12","DOI":"10.1103/nh49-52y2","source":"Crossref","title":"Thresholds for Postselected Quantum Error Correction from Statistical Mechanics","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0003-2569-8516","authenticated-orcid":true,"given":"Lucas H.","family":"English","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0002-8029-6408","authenticated-orcid":true,"given":"Dominic J.","family":"Williamson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0384j8v12","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Sydney"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,18]]},"URL":"http://dx.doi.org/10.1103/nh49-52y2","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"120603","note":"arxivid:2410.07598"},{"id":"arxiv:cond-mat/0010143","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.87.047201","source":"Crossref","title":"Universality Class of the Nishimori Point in the 2D<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mo>±</mml:mo><mml:mi mathvariant=\"italic\">J</mml:mi></mml:math>Random-Bond Ising Model","volume":"87","author":[{"given":"A.","family":"Honecker","sequence":"first","affiliation":[]},{"given":"M.","family":"Picco","sequence":"additional","affiliation":[]},{"given":"P.","family":"Pujol","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2001,7,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.87.047201","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"047201","note":"arxivid:cond-mat/0010143"},{"id":"arxiv:cond-mat/0106023","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevb.65.054425","source":"Crossref","title":"Two-dimensional random-bond Ising model, free fermions, and the network model","volume":"65","author":[{"given":"F.","family":"Merz","sequence":"first","affiliation":[]},{"given":"J. T.","family":"Chalker","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2002,1,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.65.054425","ISSN":["0163-1829","1095-3795"],"container-title-short":"Phys. Rev. B","page":"054425","note":"arxivid:cond-mat/0106023"},{"id":"arxiv:0811.2101","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5-6","DOI":"10.1007/s10955-009-9705-5","page":"1039-1061","source":"Crossref","title":"Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model","volume":"135","author":[{"given":"Francesco","family":"Parisen Toldin","sequence":"first","affiliation":[]},{"given":"Andrea","family":"Pelissetto","sequence":"additional","affiliation":[]},{"given":"Ettore","family":"Vicari","sequence":"additional","affiliation":[]}],"container-title":"Journal of Statistical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2009,3,14]]},"URL":"http://dx.doi.org/10.1007/s10955-009-9705-5","ISSN":["0022-4715","1572-9613"],"container-title-short":"J Stat Phys","note":"alternative-id:9705\narxivid:0811.2101"},{"id":"arxiv:0902.4153","type":"journal-article","publisher":"American Physical Society (APS)","issue":"17","DOI":"10.1103/physrevb.79.174408","source":"Crossref","title":"Location and properties of the multicritical point in the Gaussian and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>J</mml:mi></mml:mrow></mml:math>Ising spin glasses","volume":"79","author":[{"given":"S. L. A.","family":"de Queiroz","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,5,7]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.79.174408","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"174408","note":"arxivid:0902.4153"},{"id":"arxiv:2402.16937","type":"journal-article","publisher":"American Physical Society (APS)","issue":"25","DOI":"10.1103/hlfh-86yz","source":"Crossref","title":"Exact Calculations of Coherent Information for Toric Codes under Decoherence: Identifying the Fundamental Error Threshold","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0002-7387-3326","authenticated-orcid":true,"given":"Jong Yeon","family":"Lee","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01an7q238","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California"},{"id":[{"id":"https://ror.org/047426m28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Illinois at Urbana-Champaign"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,25]]},"URL":"http://dx.doi.org/10.1103/hlfh-86yz","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"250601","note":"arxivid:2402.16937"},{"id":"arxiv:2512.10399","type":"article-journal","author":[{"given":"Lucas H.","family":"English"},{"given":"Sam","family":"Roberts"},{"given":"Stephen D.","family":"Bartlett"},{"given":"Andrew C.","family":"Doherty"},{"given":"Dominic J.","family":"Williamson"}],"title":"Ising on the donut: Regimes of topological quantum error correction from statistical mechanics","issued":{"date-parts":[[2025,12,11]]},"note":"arxivid:2512.10399\narxiv_version_number:1"},{"id":"arxiv:1202.2743","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.86.020303","source":"Crossref","title":"Error and loss tolerances of surface codes with general lattice structures","volume":"86","author":[{"given":"Keisuke","family":"Fujii","sequence":"first","affiliation":[]},{"given":"Yuuki","family":"Tokunaga","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,8,21]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.020303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"020303","note":"arxivid:1202.2743"},{"id":"arxiv:2212.11632","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.108.022401","source":"Crossref","title":"Performance enhancement of surface codes via recursive minimum-weight perfect-match decoding","volume":"108","author":[{"ORCID":"https://orcid.org/0000-0003-0602-5535","authenticated-orcid":true,"given":"Antonio deMarti","family":"iOlius","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7058-8426","authenticated-orcid":true,"given":"Josu Etxezarreta","family":"Martinez","sequence":"additional","affiliation":[]},{"given":"Patricio","family":"Fuentes","sequence":"additional","affiliation":[]},{"given":"Pedro M.","family":"Crespo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.108.022401","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022401","note":"arxivid:2212.11632"},{"id":"arxiv:1202.1852","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevx.2.021004","source":"Crossref","title":"Strong Resilience of Topological Codes to Depolarization","volume":"2","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]},{"given":"Ruben S.","family":"Andrist","sequence":"additional","affiliation":[]},{"given":"Masayuki","family":"Ohzeki","sequence":"additional","affiliation":[]},{"given":"Helmut G.","family":"Katzgraber","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2012,4,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.2.021004","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021004","note":"arxivid:1202.1852"},{"id":"doi:10.1103/Physics.5.50","type":"journal-article","publisher":"American Physical Society (APS)","DOI":"10.1103/physics.5.50","source":"Crossref","title":"Keeping One Step Ahead of Errors","volume":"5","author":[{"given":"Daniel","family":"Gottesman","sequence":"first","affiliation":[]}],"container-title":"Physics","original-title":[],"language":"en","issued":{"date-parts":[[2012,4,30]]},"URL":"http://dx.doi.org/10.1103/Physics.5.50","ISSN":["1943-2879"],"container-title-short":"Physics","page":"50"},{"id":"arxiv:0904.3556","type":"journal-article","publisher":"American Physical Society (APS)","issue":"20","DOI":"10.1103/physrevlett.102.200501","source":"Crossref","title":"Thresholds for Topological Codes in the Presence of Loss","volume":"102","author":[{"given":"Thomas M.","family":"Stace","sequence":"first","affiliation":[]},{"given":"Sean D.","family":"Barrett","sequence":"additional","affiliation":[]},{"given":"Andrew C.","family":"Doherty","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2009,5,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.102.200501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"200501","note":"arxivid:0904.3556"},{"id":"arxiv:0912.1159","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.81.022317","source":"Crossref","title":"Error correction and degeneracy in surface codes suffering loss","volume":"81","author":[{"given":"Thomas M.","family":"Stace","sequence":"first","affiliation":[]},{"given":"Sean D.","family":"Barrett","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,2,18]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.022317","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022317","note":"arxivid:0912.1159"},{"id":"arxiv:1810.09621","type":"article-journal","author":[{"given":"Naomi","family":"Nickerson"},{"given":"Héctor","family":"Bombín"}],"title":"Measurement based fault tolerance beyond foliation","issued":{"date-parts":[[2018,10,23]]},"note":"arxivid:1810.09621\narxiv_version_number:1"},{"id":"arxiv:1607.06460","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevlett.119.040502","source":"Crossref","title":"Tensor-Network Simulations of the Surface Code under Realistic Noise","volume":"119","author":[{"given":"Andrew S.","family":"Darmawan","sequence":"first","affiliation":[]},{"given":"David","family":"Poulin","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2017,7,27]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.119.040502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"040502","note":"arxivid:1607.06460"},{"id":"arxiv:1209.2157","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevlett.110.010502","source":"Crossref","title":"Surface Code Threshold in the Presence of Correlated Errors","volume":"110","author":[{"given":"E.","family":"Novais","sequence":"first","affiliation":[]},{"given":"Eduardo R.","family":"Mucciolo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2013,1,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.110.010502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"010502","note":"arxivid:1209.2157"},{"id":"arxiv:1304.2975","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.88.012336","source":"Crossref","title":"Fidelity of the surface code in the presence of a bosonic bath","volume":"88","author":[{"given":"P.","family":"Jouzdani","sequence":"first","affiliation":[]},{"given":"E.","family":"Novais","sequence":"additional","affiliation":[]},{"given":"E. R.","family":"Mucciolo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,7,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.88.012336","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012336","note":"arxivid:1304.2975"},{"id":"arxiv:2410.22436","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>The surface code, one of the leading candidates for quantum error correction, is known to protect encoded quantum information against stochastic, i.e., incoherent errors. The protection against coherent errors, such as from unwanted gate rotations, is however understood only for special cases, such as rotations about the <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mi>X</a:mi></a:math> or <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>Z</c:mi></c:math> axes. Here we consider generic single-qubit coherent errors in the surface code, i.e., rotations by angle <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>α</e:mi></e:math> about an axis that can be chosen arbitrarily. We develop a statistical mechanical mapping for such errors and perform entanglement analysis in transfer matrix space to numerically establish the existence of an error-correcting phase, which we chart in a subspace of rotation axes to estimate the corresponding maximum-likelihood thresholds <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:msub><g:mi>α</g:mi><g:mi>th</g:mi></g:msub></g:math>. The classical statistical mechanics model we derive is a random-bond Ising model with complex couplings and four-spin interactions (i.e., a complex-coupled Ashkin-Teller model). The error-correcting phase, <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:mi>α</i:mi><i:mo>&lt;</i:mo><i:msub><i:mi>α</i:mi><i:mi>th</i:mi></i:msub></i:math>, where the logical error rate decreases exponentially with code distance, is shown to correspond in transfer matrix space to a gapped one-dimensional quantum Hamiltonian exhibiting spontaneous breaking of a <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:msub><k:mrow><k:mi mathvariant=\"double-struck\">Z</k:mi></k:mrow><k:mn>2</k:mn></k:msub></k:math> symmetry. Our numerical results rest on two key ingredients: (i) we show that the state evolution under the transfer matrix, a nonunitary <n:math xmlns:n=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><n:mo stretchy=\"false\">(</n:mo><n:mn>1</n:mn><n:mo>+</n:mo><n:mn>1</n:mn><n:mo stretchy=\"false\">)</n:mo></n:math>-dimensional quantum circuit, can be efficiently numerically simulated using matrix product states; and (ii) based on this approach, we also develop an algorithm to (approximately) sample syndromes based on their Born probability. The <r:math xmlns:r=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><r:msub><r:mi>α</r:mi><r:mi>th</r:mi></r:msub></r:math> values we find show that the maximum-likelihood thresholds for coherent errors are larger than those for the corresponding incoherent errors (from the Pauli twirl), and significantly exceed the values found using minimum weight perfect matching.</jats:p>","DOI":"10.1103/gskb-t5ql","source":"Crossref","title":"Statistical Mechanical Mapping and Maximum-Likelihood Thresholds for the Surface Code under Generic Single-Qubit Coherent Errors","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-2727-061X","authenticated-orcid":true,"given":"Jan","family":"Behrends","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"Cavendish Laboratory"},{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Cambridge"}]},{"ORCID":"https://orcid.org/0000-0001-9933-9108","authenticated-orcid":true,"given":"Benjamin","family":"Béri","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"Cavendish Laboratory"},{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Cambridge"},{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"DAMTP, University of Cambridge"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,7]]},"URL":"http://dx.doi.org/10.1103/gskb-t5ql","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040305","note":"arxivid:2410.22436"},{"id":"arxiv:2411.05785","type":"article-journal","author":[{"given":"Yimu","family":"Bao"},{"given":"Sajant","family":"Anand"}],"title":"Phases of decodability in the surface code with unitary errors","issued":{"date-parts":[[2024,11,8]]},"note":"arxivid:2411.05785\narxiv_version_number:1"},{"id":"arxiv:2412.21055","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    We consider the surface code under errors featuring both coherent and incoherent components and study the coherence of the corresponding logical noise channel and how this impacts information-theoretic measures of code performance, namely coherent information and quantum relative entropy. Using numerical simulations and developing a phenomenological field theory, focusing on the most general single-qubit\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mi>X</a:mi>\n                    </a:math>\n                    -error channel, we show that, for any nonzero incoherent noise component, the coherence of the logical noise is exponentially suppressed with the code distance. We also find that the information-theoretic measures require this suppression to detect optimal thresholds for Pauli recovery; for this they thus require increasingly large distances for increasing error coherence and ultimately break down for fully coherent errors. To obtain our results, we develop a statistical mechanics mapping and a corresponding matrix-product-state algorithm for approximate syndrome sampling. These methods enable the large-scale simulation of these non-Pauli errors, including their maximum-likelihood thresholds, away from the limits captured by previous approaches.\n                  </jats:p>","DOI":"10.1103/psf5-b6j2","source":"Crossref","title":"The Surface Code beyond Pauli Channels: Logical Noise Coherence, Information-Theoretic Measures, and Errorfield-Double Phenomenology","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-2727-061X","authenticated-orcid":true,"given":"Jan","family":"Behrends","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"T.C.M. Group, Cavendish Laboratory, University of Cambridge"}]},{"ORCID":"https://orcid.org/0000-0001-9933-9108","authenticated-orcid":true,"given":"Benjamin","family":"Béri","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"T.C.M. Group, Cavendish Laboratory, University of Cambridge"},{"id":[{"id":"https://ror.org/013meh722","id-type":"ROR","asserted-by":"publisher"}],"name":"DAMTP, University of Cambridge"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,2]]},"URL":"http://dx.doi.org/10.1103/psf5-b6j2","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040350","note":"arxivid:2412.21055"},{"id":"arxiv:2112.03923","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"7906","abstract":"<jats:title>Abstract</jats:title><jats:p>The ability to engineer parallel, programmable operations between desired qubits within a quantum processor is key for building scalable quantum information systems<jats:sup>1,2</jats:sup>. In most state-of-the-art approaches, qubits interact locally, constrained by the connectivity associated with their fixed spatial layout. Here we demonstrate a quantum processor with dynamic, non-local connectivity, in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions, between layers of single- and two-qubit operations. Our approach makes use of neutral atom arrays trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation<jats:sup>3–5</jats:sup>. We use this architecture to realize programmable generation of entangled graph states, such as cluster states and a seven-qubit Steane code state<jats:sup>6,7</jats:sup>. Furthermore, we shuttle entangled ancilla arrays to realize a surface code state with thirteen data and six ancillary qubits<jats:sup>8</jats:sup>and a toric code state on a torus with sixteen data and eight ancillary qubits<jats:sup>9</jats:sup>. Finally, we use this architecture to realize a hybrid analogue–digital evolution<jats:sup>2</jats:sup>and use it for measuring entanglement entropy in quantum simulations<jats:sup>10–12</jats:sup>, experimentally observing non-monotonic entanglement dynamics associated with quantum many-body scars<jats:sup>13,14</jats:sup>. Realizing a long-standing goal, these results provide a route towards scalable quantum processing and enable applications ranging from simulation to metrology.</jats:p>","DOI":"10.1038/s41586-022-04592-6","page":"451-456","source":"Crossref","title":"A quantum processor based on coherent transport of entangled atom arrays","volume":"604","author":[{"given":"Dolev","family":"Bluvstein","sequence":"first","affiliation":[]},{"given":"Harry","family":"Levine","sequence":"additional","affiliation":[]},{"given":"Giulia","family":"Semeghini","sequence":"additional","affiliation":[]},{"given":"Tout T.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Sepehr","family":"Ebadi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0605-8791","authenticated-orcid":false,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[]},{"given":"Alexander","family":"Keesling","sequence":"additional","affiliation":[]},{"given":"Nishad","family":"Maskara","sequence":"additional","affiliation":[]},{"given":"Hannes","family":"Pichler","sequence":"additional","affiliation":[]},{"given":"Markus","family":"Greiner","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9786-0538","authenticated-orcid":false,"given":"Vladan","family":"Vuletić","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8658-1007","authenticated-orcid":false,"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2022,4,20]]},"URL":"http://dx.doi.org/10.1038/s41586-022-04592-6","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:4592\narxivid:2112.03923"},{"id":"arxiv:0811.2036","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.79.075124","source":"Crossref","title":"Three-dimensional topological phase on the diamond lattice","volume":"79","author":[{"given":"Shinsei","family":"Ryu","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,2,26]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.79.075124","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"075124","note":"arxivid:0811.2036"},{"id":"arxiv:cond-mat/0302460","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.67.245316","source":"Crossref","title":"Fermions, strings, and gauge fields in lattice spin models","volume":"67","author":[{"given":"Michael","family":"Levin","sequence":"first","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2003,6,20]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.67.245316","ISSN":["0163-1829","1095-3795"],"container-title-short":"Phys. Rev. B","page":"245316","note":"arxivid:cond-mat/0302460"},{"id":"arxiv:2311.05674","type":"journal-article","publisher":"Stichting SciPost","issue":"5","abstract":"<jats:p>It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_2</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> gauge theory with an emergent fermion, and point out that pumping chiral <jats:inline-formula><jats:alternatives><jats:tex-math>p+ip</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> topological states gives rise to a <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{Z}_{8}</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msub><mml:mi>ℤ</mml:mi><mml:mn>8</mml:mn></mml:msub></mml:math></jats:alternatives></jats:inline-formula> 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization of <jats:inline-formula><jats:alternatives><jats:tex-math>T^3</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msup><mml:mi>T</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math></jats:alternatives></jats:inline-formula> (3-torus) and <jats:inline-formula><jats:alternatives><jats:tex-math>T^2 \\rtimes_{C_2} S^1</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mo>⋊</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:msub><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> (2-torus bundle over the circle) respectively, and pumping <jats:inline-formula><jats:alternatives><jats:tex-math>p+ip</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> states. Our considerations also imply the possibility of a logical <jats:inline-formula><jats:alternatives><jats:tex-math>T</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mi>T</mml:mi></mml:math></jats:alternatives></jats:inline-formula> gate by placing the code on <jats:inline-formula><jats:alternatives><jats:tex-math>\\mathbb{RP}^3</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi><mml:mi>ℙ</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></jats:alternatives></jats:inline-formula> and pumping a <jats:inline-formula><jats:alternatives><jats:tex-math>p+ip</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> topological state.</jats:p>","DOI":"10.21468/scipostphys.16.5.122","source":"Crossref","title":"Higher-group symmetry of (3+1)D fermionic $\\mathbb{Z}_2$ gauge theory: Logical CCZ, CS, and T gates from higher symmetry","volume":"16","author":[{"ORCID":"https://orcid.org/0000-0002-4322-9433","authenticated-orcid":false,"given":"Maissam","family":"Barkeshli","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]},{"ORCID":"https://orcid.org/0000-0002-4764-1476","authenticated-orcid":false,"given":"Po-Shen","family":"Hsin","sequence":"additional","affiliation":[{"name":"Mani L. Bhaumik Institute for Theoretical Physics"}]},{"given":"Ryohei","family":"Kobayashi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland, College Park"}]}],"container-title":"SciPost Physics","original-title":[],"issued":{"date-parts":[[2024,5,7]]},"URL":"http://dx.doi.org/10.21468/SciPostPhys.16.5.122","ISSN":["2542-4653"],"container-title-short":"SciPost Phys.","page":"122","note":"arxivid:2311.05674"},{"id":"arxiv:1012.0859","type":"article-journal","author":[{"given":"Isaac H.","family":"Kim"}],"title":"Local non-CSS quantum error correcting code on a three-dimensional lattice","issued":{"date-parts":[[2013,4,25]]},"note":"arxivid:1012.0859\narxiv_version_number:3"},{"id":"arxiv:cond-mat/0411752","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevb.72.035307","source":"Crossref","title":"String and membrane condensation on three-dimensional lattices","volume":"72","author":[{"given":"Alioscia","family":"Hamma","sequence":"first","affiliation":[]},{"given":"Paolo","family":"Zanardi","sequence":"additional","affiliation":[]},{"given":"Xiao-Gang","family":"Wen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2005,7,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.72.035307","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"035307","note":"arxivid:cond-mat/0411752"},{"id":"arxiv:2106.05274","type":"journal-article","publisher":"AIP Publishing","issue":"9","abstract":"<jats:p>In this paper, we derive the explicit formula for higher cup products on hypercubic lattices based on the recently developed geometrical interpretation on the simplicial complexes. We illustrate how this formalism can elucidate lattice constructions on hypercubic lattices for various models and derive them from spacetime actions. In particular, we demonstrate explicitly that the (3 + 1)D SPT S=12∫w22+w14 (where w1 and w2 are the first and second Stiefel-Whitney classes) is dual to the 3-fermion Walker-Wang model constructed on the cubic lattice. Other examples include the double-semion model and also the “fermionic” toric code in arbitrary dimensions on hypercubic lattices. In addition, we extend previous constructions of exact boson-fermion dualities and the Gu-Wen Grassmann integral to arbitrary dimensions. Another result that may be of independent interest is a derivation of a cochain-level action for the generalized double-semion model, reproducing a recently derived action on the cohomology level.</jats:p>","DOI":"10.1063/5.0095189","source":"Crossref","title":"Higher cup products on hypercubic lattices: Application to lattice models of topological phases","volume":"64","author":[{"ORCID":"https://orcid.org/0000-0002-8810-9355","authenticated-orcid":false,"given":"Yu-An","family":"Chen","sequence":"first","affiliation":[{"name":"International Center for Quantum Materials, School of Physics, Peking University 1 , Beijing 100871, China"}]},{"ORCID":"https://orcid.org/0000-0002-1360-4773","authenticated-orcid":false,"given":"Sri","family":"Tata","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yale University 2 , New Haven, Connecticut 06511, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,9,1]]},"URL":"http://dx.doi.org/10.1063/5.0095189","ISSN":["0022-2488","1089-7658"],"page":"091902","note":"arxivid:2106.05274"},{"id":"arxiv:2312.09111","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1038/s41534-024-00945-3","source":"Crossref","title":"Efficient fault-tolerant implementations of non-Clifford gates with reconfigurable atom arrays","volume":"10","author":[{"ORCID":"https://orcid.org/0009-0003-1126-1628","authenticated-orcid":false,"given":"Yifei","family":"Wang","sequence":"first","affiliation":[]},{"given":"Yixu","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8810-9355","authenticated-orcid":false,"given":"Yu-An","family":"Chen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9343-390X","authenticated-orcid":false,"given":"Wenjun","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Tao","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2937-4810","authenticated-orcid":false,"given":"Jiazhong","family":"Hu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1521-5365","authenticated-orcid":false,"given":"Wenlan","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Yingfei","family":"Gu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3402-9763","authenticated-orcid":false,"given":"Zi-Wen","family":"Liu","sequence":"additional","affiliation":[]}],"container-title":"npj Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,31]]},"URL":"http://dx.doi.org/10.1038/s41534-024-00945-3","ISSN":["2056-6387"],"container-title-short":"npj Quantum Inf","page":"136","note":"alternative-id:945\narxivid:2312.09111"},{"id":"arxiv:2212.06985","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> is an extremely simple and maximally local classical decoder which has been used to great effect in certain classes of classical codes. When applied to quantum codes there exist constant-weight errors (such as half of a stabiliser) which are uncorrectable for this decoder, so previous studies have considered modified versions of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math>, sometimes in conjunction with other decoders. We argue that this may not always be necessary, and present numerical evidence for the existence of a threshold for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> when applied to the looplike syndromes of a three-dimensional toric code on a cubic lattice. This result can be attributed to the fact that the lowest-weight uncorrectable errors for this decoder are closer (in terms of Hamming distance) to correctable errors than to other uncorrectable errors, and so they are likely to become correctable in future code cycles after transformation by additional noise. Introducing randomness into the decoder can allow it to correct these \"uncorrectable\" errors with finite probability, and for a decoding strategy that uses a combination of belief propagation and probabilistic <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"monospace\">f</mml:mi><mml:mi mathvariant=\"monospace\">l</mml:mi><mml:mi mathvariant=\"monospace\">i</mml:mi><mml:mi mathvariant=\"monospace\">p</mml:mi></mml:mrow></mml:math> we observe a threshold of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>5.5</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math> under phenomenological noise. This is comparable to the best known threshold for this code (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>7.1</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) which was achieved using belief propagation and ordered statistics decoding [Higgott and Breuckmann, 2022], a strategy with a runtime of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>n</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math> as opposed to the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math> when parallelised) runtime of our local decoder. We expect that this strategy could be generalised to work well in other low-density parity check codes, and hope that these results will prompt investigation of other previously overlooked decoders.</jats:p>","DOI":"10.22331/q-2023-08-29-1093","page":"1093","source":"Crossref","title":"Local Probabilistic Decoding of a Quantum Code","volume":"7","author":[{"given":"T. R.","family":"Scruby","sequence":"first","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"}]},{"given":"K.","family":"Nemoto","sequence":"additional","affiliation":[{"name":"Okinawa Institute of Science and Technology, Okinawa, 904-0495, Japan"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,29]]},"URL":"http://dx.doi.org/10.22331/q-2023-08-29-1093","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2212.06985"},{"id":"arxiv:2310.10722","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Decoding algorithms based on approximate tensor-network (TN) contraction have proven tremendously successful in decoding two-dimensional (2D) local quantum codes such as toric or surface codes and color codes, effectively achieving optimal decoding accuracy. In this work, we introduce several techniques to generalize TN decoding to higher dimensions so that it can be applied to three-dimensional (3D) codes as well as 2D codes with noisy syndrome measurements (phenomenological noise or circuit-level noise). The 3D case is significantly more challenging than 2D, as the involved approximate tensor contraction is dramatically less well behaved than its 2D counterpart. Nonetheless, we numerically demonstrate that the decoding accuracy of our approach outperforms state-of-the-art decoders on the 3D surface code, both in the point and loop sectors, as well as for depolarizing noise. Our techniques could prove useful in near-term experimental demonstrations of quantum error correction, when decoding is to be performed off line and accuracy is of the utmost importance. To this end, we show how TN decoding can be applied to circuit-level noise and demonstrate that it outperforms the matching decoder on the rotated surface code.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.040303","source":"Crossref","title":"Tensor-Network Decoding Beyond 2D","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-0591-9740","authenticated-orcid":true,"given":"Christophe","family":"Piveteau","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"}]},{"ORCID":"https://orcid.org/0000-0002-2668-1567","authenticated-orcid":true,"given":"Christopher T.","family":"Chubb","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"}]},{"ORCID":"https://orcid.org/0000-0003-2302-8025","authenticated-orcid":true,"given":"Joseph M.","family":"Renes","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05a28rw58","id-type":"ROR","asserted-by":"publisher"}],"name":"ETH Zürich"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,10,8]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.040303","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040303","note":"arxivid:2310.10722"},{"id":"arxiv:1808.03092","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.100.012311","source":"Crossref","title":"Decoding the three-dimensional toric codes and welded codes on cubic lattices","volume":"100","author":[{"given":"Abhishek","family":"Kulkarni","sequence":"first","affiliation":[]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.100.012311","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012311","note":"arxivid:1808.03092"},{"id":"arxiv:2409.04300","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qcnc64685.2025.00112","page":"672-676","source":"Crossref","title":"An Equivariant Machine Learning Decoder for 3D Toric Codes","author":[{"given":"Oliver","family":"Weißl","sequence":"first","affiliation":[{"name":"Fortiss GmbH"}]},{"given":"Evgenii","family":"Egorov","sequence":"additional","affiliation":[{"name":"University of Amsterdam"}]}],"event":"2025 International Conference on Quantum Communications, Networking, and Computing (QCNC)","container-title":"2025 International Conference on Quantum Communications, Networking, and Computing (QCNC)","original-title":[],"issued":{"date-parts":[[2025,3,31]]},"URL":"http://dx.doi.org/10.1109/QCNC64685.2025.00112","note":"arxivid:2409.04300"},{"id":"arxiv:2510.20489","type":"article-journal","author":[{"given":"Ji-Ze","family":"Xu"},{"given":"Yin","family":"Zhong"},{"given":"Miguel A.","family":"Martin-Delgado"},{"given":"Hao","family":"Song"},{"given":"Ke","family":"Liu"}],"title":"Phenomenological Noise Models and Optimal Thresholds of the 3D Toric Code","issued":{"date-parts":[[2025,10,29]]},"note":"arxivid:2510.20489\narxiv_version_number:2"},{"id":"arxiv:1010.2901","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.83.012304","source":"Crossref","title":"Quantum memories based on engineered dissipation","volume":"83","author":[{"given":"Fernando","family":"Pastawski","sequence":"first","affiliation":[]},{"given":"Lucas","family":"Clemente","sequence":"additional","affiliation":[]},{"given":"Juan Ignacio","family":"Cirac","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,1,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.012304","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012304","note":"arxivid:1010.2901"},{"id":"arxiv:hep-th/0310279","type":"journal-article","publisher":"Elsevier BV","issue":"3","DOI":"10.1016/j.nuclphysb.2004.03.006","page":"377-396","source":"Crossref","title":"Self-dual random-plaquette gauge model and the quantum toric code","volume":"686","author":[{"given":"Koujin","family":"Takeda","sequence":"first","affiliation":[]},{"given":"Hidetoshi","family":"Nishimori","sequence":"additional","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[2004,5]]},"URL":"http://dx.doi.org/10.1016/j.nuclphysb.2004.03.006","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"alternative-id:S0550321304001725\narxivid:hep-th/0310279"},{"id":"arxiv:2201.03568","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>&amp;#x03F5;</mml:mi></mml:math>, which admits a fault-tolerant non-Clifford CCZ gate \\cite{zhu2021topological}. We investigate the performance of such FSCs as fault-tolerant quantum memories. We prove that there exist decoding strategies with non-zero thresholds for bit-flip and phase-flip errors in the FSCs with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>&amp;#x03F5;</mml:mi></mml:math>. For the bit-flip errors, we adapt the sweep decoder, developed for string-like syndromes in the regular 3D surface code, to the FSCs by designing suitable modifications on the boundaries of the holes in the fractal lattice. Our adaptation of the sweep decoder for the FSCs maintains its self-correcting and single-shot nature. For the phase-flip errors, we employ the minimum-weight-perfect-matching (MWPM) decoder for the point-like syndromes. We report a sustainable fault-tolerant threshold (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>&amp;#x223C;</mml:mo><mml:mn>1.7</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) under phenomenological noise for the sweep decoder and the code capacity threshold (lower bounded by <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>2.95</mml:mn><mml:mi mathvariant=\"normal\">&amp;#x0025;</mml:mi></mml:math>) for the MWPM decoder for a particular FSC with Hausdorff dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>D</mml:mi><mml:mi>H</mml:mi></mml:msub><mml:mo>&amp;#x2248;</mml:mo><mml:mn>2.966</mml:mn></mml:math>. The latter can be mapped to a lower bound of the critical point of a  confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.</jats:p>","DOI":"10.22331/q-2023-09-26-1122","page":"1122","source":"Crossref","title":"Quantum error correction with fractal topological codes","volume":"7","author":[{"given":"Arpit","family":"Dua","sequence":"first","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125 USA"}]},{"given":"Tomas","family":"Jochym-O&apos;Connor","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 USA"},{"name":"IBM Almaden Research Center, San Jose, CA 95120 USA"}]},{"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 USA"},{"name":"IBM Almaden Research Center, San Jose, CA 95120 USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,9,26]]},"URL":"http://dx.doi.org/10.22331/q-2023-09-26-1122","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2201.03568"},{"id":"arxiv:quant-ph/0602063","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.73.062303","source":"Crossref","title":"Topological quantum error correction with optimal encoding rate","volume":"73","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2006,6,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.73.062303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062303","note":"arxivid:quant-ph/0602063"},{"id":"manual:-C.-D.-Albuquerque-et-al.-On-","type":"article-journal","author":[{"family":"Albuquerque","given":"C.D."}],"title":"On Toric Quantum Codes","volume":"50","container-title":"Int. J. Pure Appl. Math","issued":"2009","page":"221–226"},{"id":"arxiv:2205.13582","type":"article-journal","author":[{"given":"Cibele Cristina","family":"Trinca"},{"given":"J. Carmelo","family":"Interlando"},{"given":"Reginaldo","family":"Palazzo"},{"given":"Antonio Aparecido","family":"de Andrade"},{"given":"Ricardo Augusto","family":"Watanabe"}],"title":"On the Construction of New Toric Quantum Codes and Quantum Burst-Error Correcting Codes","issued":{"date-parts":[[2022,5,26]]},"note":"arxivid:2205.13582\narxiv_version_number:1"},{"id":"doi:10.1201/9781420035377-13","type":"book-chapter","publisher":"Chapman and Hall/CRC","DOI":"10.1201/9781420035377-13","page":"303-338","source":"Crossref","title":"Z2-systolic freedom and quantum codes","container-title":"Mathematics of Quantum Computation","original-title":[],"language":"en","issued":{"date-parts":[[2002,2,14]]},"ISBN":["9780429122798"],"URL":"http://dx.doi.org/10.1201/9781420035377-13","note":"edition-number:0"},{"id":"doi:10.1063/1.4726034","type":"journal-article","publisher":"AIP Publishing","issue":"6","abstract":"<jats:p>Homological quantum codes (also called topological codes) are low density parity check error correcting codes that come from surfaces and higher dimension manifolds. Homological codes from surfaces, i.e., surface codes, have also been suggested as a possible way to construct stable quantum memory and fault-tolerant computation. It has been conjectured that all homological codes have a square root bound on there distance and therefore cannot produce good codes. This claim has been disputed in dimension four using the geometric property of systolic freedom. We will show in this paper that the conjecture holds in dimension two due to the negation of systolic freedom, i.e., systolic rigidity.</jats:p>","DOI":"10.1063/1.4726034","source":"Crossref","title":"Bounding the distance of quantum surface codes","volume":"53","author":[{"given":"Ethan","family":"Fetaya","sequence":"first","affiliation":[{"name":"The Hebrew University of Jerusalem , Department of Mathematics, Jerusalem, Israel"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2012,6,1]]},"URL":"http://dx.doi.org/10.1063/1.4726034","ISSN":["0022-2488","1089-7658"],"page":"062202"},{"id":"doi:10.7907/THD5-A335","type":"thesis","categories":["quantum information","Physics","quantum control"],"language":"en","author":[{"family":"Ahn","given":"Charlene Sonja"}],"issued":{"date-parts":[[2004]]},"abstract":"Quantum mechanical applications range from quantum computers to quantum key distribution to teleportation. In these applications, quantum error correction is extremely important for protecting quantum states against decoherence. Here I present two main results regarding quantum error correction protocols.\n\nThe first main topic I address is the development of continuous-time quantum error correction protocols via combination with techniques from quantum control. These protocols rely on weak measurement and Hamiltonian feedback instead of the projective measurements and unitary gates usually assumed by canonical quantum error correction. I show that a subclass of these protocols can be understood as a quantum feedback protocol, and analytically analyze the general case using the stabilizer formalism; I show that in this case perfect feedback can perfectly protect a stabilizer subspace. I also show through numerical simulations that another subclass of these protocols does better than canonical quantum error correction when the time between corrections is limited.\n\nThe second main topic is development of improved overhead results for fault-tolerant computation. In particular, through analysis of topological quantum error correcting codes, it will be shown that the required blowup in depth of a noisy circuit performing a fault-tolerant computation can be reduced to a factor of O(log log L), an improvement over previous results. Showing this requires investigation into a local method of performing fault-tolerant correction on a topological code of arbitrary dimension.","DOI":"10.7907/THD5-A335","publisher":"California Institute of Technology","title":"Extending Quantum Error Correction: New Continuous Measurement Protocols and Improved Fault-Tolerant Overhead","URL":"https://resolver.caltech.edu/CaltechETD:etd-05192004-164713","version":"Final","note":"copyright:No commercial reproduction, distribution, display or performance rights in this work are provided."},{"id":"arxiv:1809.10145","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.123.020501","source":"Crossref","title":"Cellular-Automaton Decoders with Provable Thresholds for Topological Codes","volume":"123","author":[{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":true,"given":"Aleksander","family":"Kubica","sequence":"first","affiliation":[]},{"given":"John","family":"Preskill","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.123.020501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020501","note":"arxivid:1809.10145"},{"id":"arxiv:1709.00020","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.97.012330","source":"Crossref","title":"Locality-preserving logical operators in topological stabilizer codes","volume":"97","author":[{"given":"Paul","family":"Webster","sequence":"first","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,1,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.97.012330","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012330","note":"arxivid:1709.00020"},{"id":"manual:-J.-Q.-Broshuis-The-Small-Ste","type":"thesis","author":[{"family":"Broshuis","given":"J.Q."}],"title":"The Small Stellated Dodecahedron Code: Finding Interleaved Measurement Schedules","genre":"Bachelor's thesis.","URL":"https://resolver.tudelft.nl/uuid:4e6852c1-b18d-4b6b-8cc4-dc4587bff260"},{"id":"doi:10.1063/1.3081056","type":"journal-article","publisher":"AIP Publishing","issue":"2","abstract":"<jats:p>In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g≥2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2–5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s≡1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3–5.</jats:p>","DOI":"10.1063/1.3081056","source":"Crossref","title":"Topological quantum codes on compact surfaces with genus g≥2","volume":"50","author":[{"given":"C. D.","family":"Albuquerque","sequence":"first","affiliation":[{"name":"Universidade Estadual de Campinas 1 Departamento de Telemática, Faculdade de Engenharia Elétrica e Computação, , Campinas, Sao Paulo 13083-970, Brazil"}]},{"suffix":"Jr.","given":"R.","family":"Palazzo","sequence":"additional","affiliation":[{"name":"Universidade Estadual de Campinas 1 Departamento de Telemática, Faculdade de Engenharia Elétrica e Computação, , Campinas, Sao Paulo 13083-970, Brazil"}]},{"given":"E. B.","family":"Silva","sequence":"additional","affiliation":[{"name":"Universidade Estadual de Maringá 2 Departamento de Matemática, , Maringá, Paraná 87020-900, Brazil"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2009,2,1]]},"URL":"http://dx.doi.org/10.1063/1.3081056","ISSN":["0022-2488","1089-7658"],"page":"023513"},{"id":"doi:10.26421/QIC10.11-12-6","type":"journal-article","publisher":"Rinton Press","issue":"11&12","abstract":"<jats:p>In this paper we present six classes of topological quantum codes (TQC) on compact surfaces with genus $g\\ge 2$. These codes are derived from self-dual, quasi self-dual and denser tessellations associated with embeddings of self-dual complete graphs and complete bipartite graphs on the corresponding compact surfaces. The majority of the new classes has the self-dual tessellations as their algebraic and geometric supporting mathematical structures. Every code achieves minimum distance 3 and its encoding rate is such that $\\frac{k}{n} \\rightarrow 1$ as $n \\rightarrow \\infty$, except for the one case where $\\frac{k}{n} \\rightarrow \\frac{1}{3}$ as $n \\rightarrow \\infty$.</jats:p>","DOI":"10.26421/qic10.11-12-6","page":"956-970","source":"Crossref","title":"New classes of TQC associated with self-dual, quasi self-dual and denser tessellations","volume":"10","author":[{"given":"C. D.","family":"Albuquerque","sequence":"first","affiliation":[]},{"given":"R.","family":"Palazzo Jr.","sequence":"additional","affiliation":[]},{"given":"E. B.","family":"Silva","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2010,11]]},"URL":"http://dx.doi.org/10.26421/QIC10.11-12-6","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC"},{"id":"arxiv:1506.04029","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2016.2555700","page":"3731-3744","source":"Crossref","title":"Constructions and Noise Threshold of Hyperbolic Surface Codes","volume":"62","author":[{"given":"Nikolas P.","family":"Breuckmann","sequence":"first","affiliation":[]},{"given":"Barbara M.","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2016,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2016.2555700","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1506.04029"},{"id":"arxiv:1805.00644","type":"journal-article","publisher":"AIP Publishing","issue":"8","abstract":"<jats:p>We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) &amp;gt; 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, Tc(f)&amp;lt;Tc(w).</jats:p>","DOI":"10.1063/1.5039735","source":"Crossref","title":"Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank","volume":"60","author":[{"given":"Yi","family":"Jiang","sequence":"first","affiliation":[{"name":"Department of Physics and Astronomy, University of California 1 , Riverside, California 92521, USA"}]},{"given":"Ilya","family":"Dumer","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, University of California 2 , Riverside, California 92521, USA"}]},{"ORCID":"https://orcid.org/0000-0002-5884-7154","authenticated-orcid":false,"given":"Alexey A.","family":"Kovalev","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska 3 , Lincoln, Nebraska 68588, USA"}]},{"ORCID":"https://orcid.org/0000-0002-4990-0259","authenticated-orcid":false,"given":"Leonid P.","family":"Pryadko","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of California 1 , Riverside, California 92521, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,1]]},"URL":"http://dx.doi.org/10.1063/1.5039735","ISSN":["0022-2488","1089-7658"],"page":"083302","note":"arxivid:1805.00644"},{"id":"arxiv:1310.5555","type":"journal-article","publisher":"AIP Publishing","issue":"8","abstract":"<jats:p>Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are low density parity check codes with linear rate and distance nε. Their rate is evaluated via Euler characteristic arguments and their distance using \\documentclass[12pt]{minimal}\\begin{document}$\\mathbb {Z}_2$\\end{document}Z2-systolic geometry. This construction answers a question of Zémor [“On Cayley graphs, surface codes, and the limits of homological coding for quantum error correction,” in Proceedings of Second International Workshop on Coding and Cryptology (IWCC), Lecture Notes in Computer Science Vol. 5557 (2009), pp. 259–273], who asked whether homological codes with such parameters could exist at all.</jats:p>","DOI":"10.1063/1.4891487","source":"Crossref","title":"Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds","volume":"55","author":[{"given":"Larry","family":"Guth","sequence":"first","affiliation":[{"name":"MIT 1 Department of Mathematics, , Cambridge, Massachusetts 02139, USA"}]},{"given":"Alexander","family":"Lubotzky","sequence":"additional","affiliation":[{"name":"Hebrew University 2 Institute of Mathematics, , Jerusalem 91904, Israel"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2014,8,1]]},"URL":"http://dx.doi.org/10.1063/1.4891487","ISSN":["0022-2488","1089-7658"],"page":"082202","note":"arxivid:1310.5555"},{"id":"arxiv:1610.03870","type":"article-journal","author":[{"given":"Plinio G. P.","family":"Murillo"}],"title":"Systole of congruence coverings of arithmetic hyperbolic manifolds","issued":{"date-parts":[[2017,10,3]]},"note":"arxivid:1610.03870\narxiv_version_number:2"},{"id":"arxiv:2001.03568","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"1","DOI":"10.1109/tit.2021.3122352","page":"272-286","source":"Crossref","title":"Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0002-7211-5515","authenticated-orcid":false,"given":"Nikolas P.","family":"Breuckmann","sequence":"first","affiliation":[]},{"given":"Vivien","family":"Londe","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,1]]},"URL":"http://dx.doi.org/10.1109/TIT.2021.3122352","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2001.03568"},{"id":"doi:10.2140/gtm.1999.2.113","type":"proceedings-article","publisher":"Mathematical Sciences Publishers","DOI":"10.2140/gtm.1999.2.113","page":"113-123","source":"Crossref","title":"<b>Z</b><sub>2</sub>–Systolic-Freedom","author":[{"given":"Michael H","family":"Freedman","sequence":"first","affiliation":[]}],"event":"Low Dimensional Topology -- The Kirbyfest","container-title":"Geometry &amp; Topology Monographs","original-title":[],"issued":{"date-parts":[[1999,11,17]]},"URL":"http://dx.doi.org/10.2140/gtm.1999.2.113","ISSN":["1464-8997","1464-8989"]},{"id":"arxiv:1108.2886","type":"article-journal","author":[{"given":"Ethan","family":"Fetaya"}],"title":"Homological Error Correcting Codes and Systolic Geometry","issued":{"date-parts":[[2011,8,14]]},"note":"arxivid:1108.2886\narxiv_version_number:1"},{"id":"arxiv:1712.08578","type":"journal-issue","publisher":"Rinton Press","issue":"5&6","DOI":"10.26421/qic19.5-6","source":"Crossref","title":[],"volume":"19","container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2019,5]]},"URL":"http://dx.doi.org/10.26421/QIC19.5-6","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC","note":"arxivid:1712.08578"},{"id":"manual:-I.-H.-Kim.-Quantum-codes-on-","type":"thesis","author":[{"family":"Kim","given":"I.H."}],"title":"Quantum codes on Hurwitz surfaces","genre":"PhD thesis,","publisher":"Massachusetts Institute of Technology","issued":"2007"},{"id":"arxiv:1408.3379","type":"article-journal","author":[{"given":"M. B.","family":"Hastings"},{"given":"A.","family":"Geller"}],"title":"Reduced Space-Time and Time Costs Using Dislocation Codes and Arbitrary Ancillas","issued":{"date-parts":[[2015,4,6]]},"note":"arxivid:1408.3379\narxiv_version_number:2"},{"id":"arxiv:2201.05678","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevresearch.4.023090","source":"Crossref","title":"Circuit-level protocol and analysis for twist-based lattice surgery","volume":"4","author":[{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"given":"Earl T.","family":"Campbell","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2022,5,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.4.023090","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023090","note":"arxivid:2201.05678"},{"id":"arxiv:0704.2540","type":"journal-article","publisher":"IOP Publishing","issue":"9","DOI":"10.1088/1751-8113/42/9/095302","page":"095302","source":"Crossref","title":"Quantum measurements and gates by code deformation","volume":"42","author":[{"given":"H","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M A","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2009,2,4]]},"URL":"http://dx.doi.org/10.1088/1751-8113/42/9/095302","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(09)86826-1\narxivid:0704.2540"},{"id":"arxiv:2211.09802","type":"journal-article","publisher":"IOP Publishing","issue":"6","abstract":"<jats:p>Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter. They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons, i.e., the Ising anyons. In particular, we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type, and can be explored to encode topological logical qubits. Furthermore, we demonstrate how to implement both single- and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits. Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons, offering a new lens into the study of such peculiar quasiparticles.</jats:p>","DOI":"10.1088/0256-307x/40/6/060301","page":"060301","source":"Crossref","title":"Digital Simulation of Projective Non-Abelian Anyons with 68 Superconducting Qubits","volume":"40","author":[{"given":"Shibo","family":"Xu","sequence":"first","affiliation":[]},{"given":"Zheng-Zhi","family":"Sun","sequence":"additional","affiliation":[]},{"given":"Ke","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Liang","family":"Xiang","sequence":"additional","affiliation":[]},{"given":"Zehang","family":"Bao","sequence":"additional","affiliation":[]},{"given":"Zitian","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Fanhao","family":"Shen","sequence":"additional","affiliation":[]},{"given":"Zixuan","family":"Song","sequence":"additional","affiliation":[]},{"given":"Pengfei","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Wenhui","family":"Ren","sequence":"additional","affiliation":[]},{"given":"Xu","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Hang","family":"Dong","sequence":"additional","affiliation":[]},{"given":"Jinfeng","family":"Deng","sequence":"additional","affiliation":[]},{"given":"Jiachen","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Yaozu","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Ziqi","family":"Tan","sequence":"additional","affiliation":[]},{"given":"Yu","family":"Gao","sequence":"additional","affiliation":[]},{"given":"Feitong","family":"Jin","sequence":"additional","affiliation":[]},{"given":"Xuhao","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Chuanyu","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Ning","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Yiren","family":"Zou","sequence":"additional","affiliation":[]},{"given":"Jiarun","family":"Zhong","sequence":"additional","affiliation":[]},{"given":"Aosai","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Weikang","family":"Li","sequence":"additional","affiliation":[]},{"given":"Wenjie","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Li-Wei","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Yunyan","family":"Yao","sequence":"additional","affiliation":[]},{"given":"Zhen","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Hekang","family":"Li","sequence":"additional","affiliation":[]},{"given":"Qiujiang","family":"Guo","sequence":"additional","affiliation":[]},{"given":"Chao","family":"Song","sequence":"additional","affiliation":[]},{"given":"H.","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Dong-Ling","family":"Deng","sequence":"additional","affiliation":[]}],"container-title":"Chinese Physics Letters","original-title":[],"issued":{"date-parts":[[2023,6,1]]},"URL":"http://dx.doi.org/10.1088/0256-307X/40/6/060301","ISSN":["0256-307X","1741-3540"],"container-title-short":"Chinese Phys. Lett.","note":"arxivid:2211.09802"},{"id":"doi:10.1103/PhysRevA.102.042616","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.102.042616","source":"Crossref","title":"Quantum computation with generalized dislocation codes","volume":"102","author":[{"ORCID":"https://orcid.org/0000-0002-6298-7037","authenticated-orcid":true,"given":"Manoj G.","family":"Gowda","sequence":"first","affiliation":[]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,10,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.042616","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042616"},{"id":"arxiv:2409.18689","type":"article-journal","author":[{"given":"Kao-Yueh","family":"Kuo"},{"given":"Ching-Yi","family":"Lai"}],"title":"Fault-Tolerant Belief Propagation for Practical Quantum Memory","issued":{"date-parts":[[2024,9,27]]},"note":"arxivid:2409.18689\narxiv_version_number:1"},{"id":"arxiv:1907.02554","type":"journal-article","publisher":"American Physical Society (APS)","issue":"13","DOI":"10.1103/physrevlett.124.130501","source":"Crossref","title":"Fault-Tolerant Thresholds for the Surface Code in Excess of \n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mn>5</mml:mn><mml:mo>%</mml:mo></mml:math>\n Under Biased Noise","volume":"124","author":[{"ORCID":"https://orcid.org/0000-0002-3776-2864","authenticated-orcid":true,"given":"David K.","family":"Tuckett","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3975-0226","authenticated-orcid":true,"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8060-8109","authenticated-orcid":true,"given":"Benjamin J.","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2020,3,30]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.124.130501","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"130501","note":"arxivid:1907.02554"},{"id":"arxiv:1812.08186","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physrevx.9.041031","source":"Crossref","title":"Tailoring Surface Codes for Highly Biased Noise","volume":"9","author":[{"ORCID":"https://orcid.org/0000-0002-3776-2864","authenticated-orcid":true,"given":"David K.","family":"Tuckett","sequence":"first","affiliation":[]},{"given":"Andrew S.","family":"Darmawan","sequence":"additional","affiliation":[]},{"given":"Christopher T.","family":"Chubb","sequence":"additional","affiliation":[]},{"given":"Sergey","family":"Bravyi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4387-670X","authenticated-orcid":true,"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]},{"given":"Steven T.","family":"Flammia","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2019,11,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.9.041031","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"041031","note":"arxivid:1812.08186"},{"id":"arxiv:2211.14038","type":"article-journal","author":[{"given":"Younghun","family":"Kim"},{"given":"Jeongsoo","family":"Kang"},{"given":"Younghun","family":"Kwon"}],"title":"Design of Quantum error correcting code for biased error on heavy-hexagon structure","issued":{"date-parts":[[2022,11,25]]},"note":"arxivid:2211.14038\narxiv_version_number:1"},{"id":"arxiv:1201.3757","type":"journal-article","publisher":"American Physical Society (APS)","issue":"26","DOI":"10.1103/physrevlett.108.260504","source":"Crossref","title":"From Majorana fermions to topological order","volume":"108","author":[{"given":"Barbara M.","family":"Terhal","sequence":"first","affiliation":[]},{"given":"Fabian","family":"Hassler","sequence":"additional","affiliation":[]},{"given":"David P.","family":"DiVincenzo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2012,6,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.108.260504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"260504","note":"arxivid:1201.3757"},{"id":"arxiv:2201.03540","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Executing quantum algorithms on error-corrected logical qubits is a critical step for scalable quantum computing, but the requisite numbers of qubits and physical error rates are demanding for current experimental hardware. Recently, the development of error correcting codes tailored to particular physical noise models has helped relax these requirements. In this work, we propose a qubit encoding and gate protocol for<jats:sup>171</jats:sup>Yb neutral atom qubits that converts the dominant physical errors into erasures, that is, errors in known locations. The key idea is to encode qubits in a metastable electronic level, such that gate errors predominantly result in transitions to disjoint subspaces whose populations can be continuously monitored via fluorescence. We estimate that 98% of errors can be converted into erasures. We quantify the benefit of this approach via circuit-level simulations of the surface code, finding a threshold increase from 0.937% to 4.15%. We also observe a larger code distance near the threshold, leading to a faster decrease in the logical error rate for the same number of physical qubits, which is important for near-term implementations. Erasure conversion should benefit any error correcting code, and may also be applied to design new gates and encodings in other qubit platforms.</jats:p>","DOI":"10.1038/s41467-022-32094-6","source":"Crossref","title":"Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays","volume":"13","author":[{"given":"Yue","family":"Wu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7095-1547","authenticated-orcid":false,"given":"Shimon","family":"Kolkowitz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5194-0591","authenticated-orcid":false,"given":"Shruti","family":"Puri","sequence":"additional","affiliation":[]},{"given":"Jeff D.","family":"Thompson","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,9]]},"URL":"http://dx.doi.org/10.1038/s41467-022-32094-6","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"4657","note":"alternative-id:32094\narxivid:2201.03540"},{"id":"arxiv:2210.10255","type":"article-journal","author":[{"given":"Trond I.","family":"Andersen"},{"given":"Yuri D.","family":"Lensky"},{"given":"Kostyantyn","family":"Kechedzhi"},{"given":"Ilya","family":"Drozdov"},{"given":"Andreas","family":"Bengtsson"},{"given":"Sabrina","family":"Hong"},{"given":"Alexis","family":"Morvan"},{"given":"Xiao","family":"Mi"},{"given":"Alex","family":"Opremcak"},{"given":"Rajeev","family":"Acharya"},{"given":"Richard","family":"Allen"},{"given":"Markus","family":"Ansmann"},{"given":"Frank","family":"Arute"},{"given":"Kunal","family":"Arya"},{"given":"Abraham","family":"Asfaw"},{"given":"Juan","family":"Atalaya"},{"given":"Ryan","family":"Babbush"},{"given":"Dave","family":"Bacon"},{"given":"Joseph C.","family":"Bardin"},{"given":"Gina","family":"Bortoli"},{"given":"Alexandre","family":"Bourassa"},{"given":"Jenna","family":"Bovaird"},{"given":"Leon","family":"Brill"},{"given":"Michael","family":"Broughton"},{"given":"Bob B.","family":"Buckley"},{"given":"David A.","family":"Buell"},{"given":"Tim","family":"Burger"},{"given":"Brian","family":"Burkett"},{"given":"Nicholas","family":"Bushnell"},{"given":"Zijun","family":"Chen"},{"given":"Ben","family":"Chiaro"},{"given":"Desmond","family":"Chik"},{"given":"Charina","family":"Chou"},{"given":"Josh","family":"Cogan"},{"given":"Roberto","family":"Collins"},{"given":"Paul","family":"Conner"},{"given":"William","family":"Courtney"},{"given":"Alexander L.","family":"Crook"},{"given":"Ben","family":"Curtin"},{"given":"Dripto M.","family":"Debroy"},{"given":"Alexander Del Toro","family":"Barba"},{"given":"Sean","family":"Demura"},{"given":"Andrew","family":"Dunsworth"},{"given":"Daniel","family":"Eppens"},{"given":"Catherine","family":"Erickson"},{"given":"Lara","family":"Faoro"},{"given":"Edward","family":"Farhi"},{"given":"Reza","family":"Fatemi"},{"given":"Vinicius S.","family":"Ferreira"},{"given":"Leslie Flores","family":"Burgos"},{"given":"Ebrahim","family":"Forati"},{"given":"Austin G.","family":"Fowler"},{"given":"Brooks","family":"Foxen"},{"given":"William","family":"Giang"},{"given":"Craig","family":"Gidney"},{"given":"Dar","family":"Gilboa"},{"given":"Marissa","family":"Giustina"},{"given":"Raja","family":"Gosula"},{"given":"Alejandro Grajales","family":"Dau"},{"given":"Jonathan A.","family":"Gross"},{"given":"Steve","family":"Habegger"},{"given":"Michael C.","family":"Hamilton"},{"given":"Monica","family":"Hansen"},{"given":"Matthew P.","family":"Harrigan"},{"given":"Sean D.","family":"Harrington"},{"given":"Paula","family":"Heu"},{"given":"Jeremy","family":"Hilton"},{"given":"Markus R.","family":"Hoffmann"},{"given":"Trent","family":"Huang"},{"given":"Ashley","family":"Huff"},{"given":"William J.","family":"Huggins"},{"given":"Lev B.","family":"Ioffe"},{"given":"Sergei V.","family":"Isakov"},{"given":"Justin","family":"Iveland"},{"given":"Evan","family":"Jeffrey"},{"given":"Zhang","family":"Jiang"},{"given":"Cody","family":"Jones"},{"given":"Pavol","family":"Juhas"},{"given":"Dvir","family":"Kafri"},{"given":"Tanuj","family":"Khattar"},{"given":"Mostafa","family":"Khezri"},{"given":"Mária","family":"Kieferová"},{"given":"Seon","family":"Kim"},{"given":"Alexei","family":"Kitaev"},{"given":"Paul V.","family":"Klimov"},{"given":"Andrey R.","family":"Klots"},{"given":"Alexander N.","family":"Korotkov"},{"given":"Fedor","family":"Kostritsa"},{"given":"John Mark","family":"Kreikebaum"},{"given":"David","family":"Landhuis"},{"given":"Pavel","family":"Laptev"},{"given":"Kim-Ming","family":"Lau"},{"given":"Lily","family":"Laws"},{"given":"Joonho","family":"Lee"},{"given":"Kenny","family":"Lee"},{"given":"Brian J.","family":"Lester"},{"given":"Alexander","family":"Lill"},{"given":"Wayne","family":"Liu"},{"given":"Aditya","family":"Locharla"},{"given":"Erik","family":"Lucero"},{"given":"Fionn D.","family":"Malone"},{"given":"Orion","family":"Martin"},{"given":"Jarrod R.","family":"McClean"},{"given":"Trevor","family":"McCourt"},{"given":"Matt","family":"McEwen"},{"given":"Kevin C.","family":"Miao"},{"given":"Amanda","family":"Mieszala"},{"given":"Masoud","family":"Mohseni"},{"given":"Shirin","family":"Montazeri"},{"given":"Emily","family":"Mount"},{"given":"Ramis","family":"Movassagh"},{"given":"Wojciech","family":"Mruczkiewicz"},{"given":"Ofer","family":"Naaman"},{"given":"Matthew","family":"Neeley"},{"given":"Charles","family":"Neill"},{"given":"Ani","family":"Nersisyan"},{"given":"Michael","family":"Newman"},{"given":"Jiun How","family":"Ng"},{"given":"Anthony","family":"Nguyen"},{"given":"Murray","family":"Nguyen"},{"given":"Murphy Yuezhen","family":"Niu"},{"given":"Thomas E.","family":"O'Brien"},{"given":"Seun","family":"Omonije"},{"given":"Andre","family":"Petukhov"},{"given":"Rebecca","family":"Potter"},{"given":"Leonid P.","family":"Pryadko"},{"given":"Chris","family":"Quintana"},{"given":"Charles","family":"Rocque"},{"given":"Nicholas C.","family":"Rubin"},{"given":"Negar","family":"Saei"},{"given":"Daniel","family":"Sank"},{"given":"Kannan","family":"Sankaragomathi"},{"given":"Kevin J.","family":"Satzinger"},{"given":"Henry F.","family":"Schurkus"},{"given":"Christopher","family":"Schuster"},{"given":"Michael J.","family":"Shearn"},{"given":"Aaron","family":"Shorter"},{"given":"Noah","family":"Shutty"},{"given":"Vladimir","family":"Shvarts"},{"given":"Jindra","family":"Skruzny"},{"given":"W. Clarke","family":"Smith"},{"given":"Rolando","family":"Somma"},{"given":"George","family":"Sterling"},{"given":"Doug","family":"Strain"},{"given":"Marco","family":"Szalay"},{"given":"Alfredo","family":"Torres"},{"given":"Guifre","family":"Vidal"},{"given":"Benjamin","family":"Villalonga"},{"given":"Catherine Vollgraff","family":"Heidweiller"},{"given":"Theodore","family":"White"},{"given":"Bryan W. K.","family":"Woo"},{"given":"Cheng","family":"Xing"},{"given":"Z. Jamie","family":"Yao"},{"given":"Ping","family":"Yeh"},{"given":"Juhwan","family":"Yoo"},{"given":"Grayson","family":"Young"},{"given":"Adam","family":"Zalcman"},{"given":"Yaxing","family":"Zhang"},{"given":"Ningfeng","family":"Zhu"},{"given":"Nicholas","family":"Zobrist"},{"given":"Hartmut","family":"Neven"},{"given":"Sergio","family":"Boixo"},{"given":"Anthony","family":"Megrant"},{"given":"Julian","family":"Kelly"},{"given":"Yu","family":"Chen"},{"given":"Vadim","family":"Smelyanskiy"},{"given":"Eun-Ah","family":"Kim"},{"given":"Igor","family":"Aleiner"},{"given":"Pedram","family":"Roushan"}],"title":"Non-Abelian braiding of graph vertices in a superconducting processor","issued":{"date-parts":[[2023,5,31]]},"note":"arxivid:2210.10255\narxiv_version_number:2"},{"id":"arxiv:2312.17057","type":"article-journal","author":[{"given":"Diego","family":"Forlivesi"},{"given":"Lorenzo","family":"Valentini"},{"given":"Marco","family":"Chiani"}],"title":"Logical Error Rates of XZZX and Rotated Quantum Surface Codes","issued":{"date-parts":[[2023,12,28]]},"note":"arxivid:2312.17057\narxiv_version_number:1"},{"id":"arxiv:2506.04084","type":"article-journal","author":[{"given":"Pei-Kai","family":"Tsai"},{"given":"Shruti","family":"Puri"}],"title":"A Unitary Encoder for Surface Codes","issued":{"date-parts":[[2025,6,4]]},"note":"arxivid:2506.04084\narxiv_version_number:1"},{"id":"arxiv:2501.15566","type":"article-journal","author":[{"given":"Kwok Ho","family":"Wan"},{"given":"Zhenghao","family":"Zhong"}],"title":"Pauli web of the $|Y\\rangle$ state surface code injection","issued":{"date-parts":[[2025,3,20]]},"note":"arxivid:2501.15566\narxiv_version_number:3"},{"id":"arxiv:2407.20976","type":"article-journal","author":[{"given":"Kwok Ho","family":"Wan"},{"given":"Mark","family":"Webber"},{"given":"Austin G.","family":"Fowler"},{"given":"Winfried K.","family":"Hensinger"}],"title":"An iterative transversal CNOT decoder","issued":{"date-parts":[[2025,4,10]]},"note":"arxivid:2407.20976\narxiv_version_number:3"},{"id":"arxiv:2412.01391","type":"article-journal","author":[{"given":"Zi-Han","family":"Chen"},{"given":"Ming-Cheng","family":"Chen"},{"given":"Chao-Yang","family":"Lu"},{"given":"Jian-Wei","family":"Pan"}],"title":"Transversal Logical Clifford gates on rotated surface codes with reconfigurable neutral atom arrays","issued":{"date-parts":[[2024,12,2]]},"note":"arxivid:2412.01391\narxiv_version_number:1"},{"id":"arxiv:2408.00829","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Erasure qubits offer a promising avenue toward reducing the overhead of quantum error correction (QEC) protocols. However, they require additional operations, such as erasure checks, that may add extra noise and increase the run-time of QEC protocols. To assess the benefits provided by erasure qubits, we focus on the performance of the surface code as a quantum memory. In particular, we analyze various erasure check schedules, find the correctable regions in the phase space of error parameters, and probe the subthreshold scaling of the logical error rate. We then consider a realization of erasure qubits in the superconducting hardware architectures via dual-rail qubits. We use the standard transmon-based implementation of the surface code as the performance benchmark. Our results indicate that QEC protocols with erasure qubits can outperform the ones with state-of-the-art transmons, even in the absence of precise information about the locations of erasure errors.</jats:p>","DOI":"10.1103/985g-58gd","source":"Crossref","title":"Optimizing Quantum Error-Correction Protocols with Erasure Qubits","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0003-2560-4209","authenticated-orcid":true,"given":"Shouzhen","family":"Gu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]},{"ORCID":"https://orcid.org/0000-0002-3916-1819","authenticated-orcid":true,"given":"Yotam","family":"Vaknin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"},{"id":[{"id":"https://ror.org/03qxff017","id-type":"ROR","asserted-by":"publisher"}],"name":"The Hebrew University of Jerusalem"}]},{"given":"Alex","family":"Retzker","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"},{"id":[{"id":"https://ror.org/03qxff017","id-type":"ROR","asserted-by":"publisher"}],"name":"The Hebrew University of Jerusalem"}]},{"ORCID":"https://orcid.org/0000-0001-8213-8190","authenticated-orcid":true,"given":"Aleksander","family":"Kubica","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04mv4n011","id-type":"ROR","asserted-by":"publisher"}],"name":"AWS Center for Quantum Computing"},{"id":[{"id":"https://ror.org/05dxps055","id-type":"ROR","asserted-by":"publisher"}],"name":"California Institute of Technology"},{"id":[{"id":"https://ror.org/03v76x132","id-type":"ROR","asserted-by":"publisher"}],"name":"Yale University"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,12,4]]},"URL":"http://dx.doi.org/10.1103/985g-58gd","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040354","note":"arxivid:2408.00829"},{"id":"arxiv:2410.23779","type":"journal-article","publisher":"IOP Publishing","issue":"3","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>The realization of fault-tolerant quantum computers hinges on effective quantum error correction (QEC) protocols, whose performance significantly relies on the nature of the underlying noise. In this work, we directly study the structure of non-Markovian correlated errors and their impact on surface code memory performance. Specifically, we compare surface code performance under non-Markovian noise and independent circuit-level noise, while keeping marginal error rates constant. Our analysis shows that while not all temporally correlated structures are detrimental, certain structures, particularly multi-time ‘streaky’ correlations affecting syndrome qubits and two-qubit gates, can severely degrade logical error rate scaling. Furthermore, we discuss our results in the context of recent QEC experiments on physical devices. These findings underscore the importance of understanding and mitigating non-Markovian noise toward achieving practical, fault-tolerant quantum computing.</jats:p>","DOI":"10.1088/2058-9565/adebab","page":"035060","source":"Crossref","title":"Detrimental non-Markovian errors for surface code memory","volume":"10","author":[{"ORCID":"https://orcid.org/0009-0008-7597-1094","authenticated-orcid":true,"given":"John","family":"F Kam","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4757-6851","authenticated-orcid":false,"given":"Spiro","family":"Gicev","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2054-9901","authenticated-orcid":true,"given":"Kavan","family":"Modi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0006-4192-0366","authenticated-orcid":false,"given":"Angus","family":"Southwell","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3476-2348","authenticated-orcid":false,"given":"Muhammad","family":"Usman","sequence":"additional","affiliation":[]}],"container-title":"Quantum Science and Technology","original-title":[],"issued":{"date-parts":[[2025,7,16]]},"URL":"http://dx.doi.org/10.1088/2058-9565/adebab","ISSN":["2058-9565"],"container-title-short":"Quantum Sci. Technol.","note":"arxivid:2410.23779"},{"id":"arxiv:2602.21293","type":"article-journal","author":[{"given":"Yiren","family":"Zou"},{"given":"Hong-Kuan","family":"Xia"},{"given":"Aosai","family":"Zhang"},{"given":"Xuhao","family":"Zhu"},{"given":"Feitong","family":"Jin"},{"given":"Qingyuan","family":"Wang"},{"given":"Yu","family":"Gao"},{"given":"Chuanyu","family":"Zhang"},{"given":"Ning","family":"Wang"},{"given":"Zhengyi","family":"Cui"},{"given":"Fanhao","family":"Shen"},{"given":"Zehang","family":"Bao"},{"given":"Zitian","family":"Zhu"},{"given":"Jiarun","family":"Zhong"},{"given":"Gongyu","family":"Liu"},{"given":"Jia-Nan","family":"Yang"},{"given":"Yihang","family":"Han"},{"given":"Yiyang","family":"He"},{"given":"Jiayuan","family":"Shen"},{"given":"Han","family":"Wang"},{"given":"Yanzhe","family":"Wang"},{"given":"Jiahua","family":"Huang"},{"given":"Xinrong","family":"Zhang"},{"given":"Sailang","family":"Zhou"},{"given":"Hang","family":"Dong"},{"given":"Jinfeng","family":"Deng"},{"given":"Yaozu","family":"Wu"},{"given":"Zixuan","family":"Song"},{"given":"Hekang","family":"Li"},{"given":"Zhen","family":"Wang"},{"given":"Chao","family":"Song"},{"given":"Qiujiang","family":"Guo"},{"given":"Pengfei","family":"Zhang"},{"given":"Guo-Yi","family":"Zhu"},{"given":"H.","family":"Wang"}],"title":"Teleportation transition of surface codes on a superconducting quantum processor","issued":{"date-parts":[[2026,2,24]]},"note":"arxivid:2602.21293\narxiv_version_number:1"},{"id":"arxiv:1907.09528","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physrevx.10.011022","source":"Crossref","title":"Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits","volume":"10","author":[{"ORCID":"https://orcid.org/0000-0003-3239-5783","authenticated-orcid":true,"given":"Christopher","family":"Chamberland","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3375-4445","authenticated-orcid":true,"given":"Guanyu","family":"Zhu","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9614-2836","authenticated-orcid":true,"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]},{"given":"Jared B.","family":"Hertzberg","sequence":"additional","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2020,1,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.10.011022","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"011022","note":"arxivid:1907.09528"},{"id":"arxiv:2010.05960","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","abstract":"<jats:title>A<jats:sc>bstract</jats:sc>\n                     </jats:title><jats:p>We explicitly construct a class of holographic quantum error correction codes with non-trivial centers in the code subalgebra. Specifically, we use the Bacon-Shor codes and perfect tensors to construct a gauge code (or a stabilizer code with gauge-fixing), which we call the holographic hybrid code. This code admits a local log-depth encoding/decoding circuit, and can be represented as a holographic tensor network which satisfies an analog of the Ryu-Takayanagi formula and reproduces features of the sub-region duality. We then construct approximate versions of the holographic hybrid codes by “skewing” the code subspace, where the size of skewing is analogous to the size of the gravitational constant in holography. These approximate hybrid codes are not necessarily stabilizer codes, but they can be expressed as the superposition of holographic tensor networks that are stabilizer codes. For such constructions, different logical states, representing different bulk matter content, can “back-react” on the emergent geometry, resembling a key feature of gravity. The locality of the bulk degrees of freedom becomes subspace-dependent and approximate. Such subspace-dependence is manifest from the point of view of the “entanglement wedge” and bulk operator reconstruction from the boundary. Exact complementary error correction breaks down for certain bipartition of the boundary degrees of freedom; however, a limited, state-dependent form is preserved for particular subspaces. We also construct an example where the connected two-point correlation functions can have a power-law decay. Coupled with known constraints from holography, a weakly back-reacting bulk also forces these skewed tensor network models to the “large <jats:italic>N</jats:italic> limit” where they are built by concatenating a large <jats:italic>N</jats:italic> number of copies.</jats:p>","DOI":"10.1007/jhep05(2021)127","source":"Crossref","title":"Approximate Bacon-Shor code and holography","volume":"2021","author":[{"ORCID":"https://orcid.org/0000-0002-5761-5474","authenticated-orcid":false,"given":"ChunJun","family":"Cao","sequence":"first","affiliation":[]},{"given":"Brad","family":"Lackey","sequence":"additional","affiliation":[]}],"container-title":"Journal of High Energy Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,5]]},"URL":"http://dx.doi.org/10.1007/JHEP05(2021)127","ISSN":["1029-8479"],"container-title-short":"J. High Energ. Phys.","page":"127","note":"alternative-id:15690\narxivid:2010.05960"},{"id":"arxiv:quant-ph/0305025","type":"article-journal","author":[{"given":"D.","family":"Bacon"}],"title":"Decoherence, Control, and Symmetry in Quantum Computers","issued":{"date-parts":[[2003,5,5]]},"note":"arxivid:quant-ph/0305025\narxiv_version_number:1"},{"id":"arxiv:1609.08062","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Topological error correction codes are promising candidates to protect quantum computations from the deteriorating effects of noise. While some codes provide high noise thresholds suitable for robust quantum memories, others allow straightforward gate implementation needed for data processing. To exploit the particular advantages of different topological codes for fault-tolerant quantum computation, it is necessary to be able to switch between them. Here we propose a practical solution, subsystem lattice surgery, which requires only two-body nearest-neighbor interactions in a fixed layout in addition to the indispensable error correction. This method can be used for the fault-tolerant transfer of quantum information between arbitrary topological subsystem codes in two dimensions and beyond. In particular, it can be employed to create a simple interface, a quantum bus, between noise resilient surface code memories and flexible color code processors.</jats:p>","DOI":"10.1038/s41467-017-01418-2","source":"Crossref","title":"Fault-tolerant interface between quantum memories and quantum processors","volume":"8","author":[{"ORCID":"https://orcid.org/0000-0001-7815-7006","authenticated-orcid":false,"given":"Hendrik","family":"Poulsen Nautrup","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1950-8640","authenticated-orcid":false,"given":"Nicolai","family":"Friis","sequence":"additional","affiliation":[]},{"given":"Hans J.","family":"Briegel","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2017,11,6]]},"URL":"http://dx.doi.org/10.1038/s41467-017-01418-2","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"1321","note":"alternative-id:1418\narxivid:1609.08062"},{"id":"arxiv:2412.15187","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>\n                    We show that universal quantum computation can be concretely made fault-tolerant without mid-circuit measurements. To this end, we introduce a measurement-free deformation protocol of the Bacon-Shor code to realize a logical  gate. Combined with a fold-transversal logical Hadamard gate, this enables a universal set of fault-tolerant operations using only transversal gates and qubit permutations. For the purpose of benchmarking under circuit-level noise, we develop an efficient method to simulate non-Clifford circuits with a small number of Hadamard gates. Separately, we demonstrate that certain CSS codes can be concatenated without measurements or having to rely on a universal logical gate set. This is made possible by means of a resource-efficient gadget—termed the “disposable Toffoli gadget”—that realizes the error-correcting feedback. Then, under concatenation of the Bacon-Shor code, we observe a fault-tolerance threshold at a circuit-level depolarizing noise rate of approximately\n                    <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <a:mn>0.12</a:mn>\n                      <a:mi mathvariant=\"normal\">%</a:mi>\n                    </a:math>\n                    . Together, the deformation and concatenation protocols outline a blueprint for a fully fault-tolerant architecture without any feed-forward operation, particularly suited to state-of-the-art neutral-atom platforms.\n                  </jats:p>","DOI":"10.1103/lkk1-v6wp","source":"Crossref","title":"Universal Quantum Computation via Scalable Measurement-Free Error Correction","volume":"6","author":[{"ORCID":"https://orcid.org/0009-0004-7947-0677","authenticated-orcid":true,"given":"Stefano","family":"Veroni","sequence":"first","affiliation":[{"name":"PlanQC GmbH"}]},{"ORCID":"https://orcid.org/0000-0002-1536-8858","authenticated-orcid":true,"given":"Alexandru","family":"Paler","sequence":"additional","affiliation":[{"name":"PlanQC GmbH"},{"id":[{"id":"https://ror.org/020hwjq30","id-type":"ROR","asserted-by":"publisher"}],"name":"Aalto University"}]},{"ORCID":"https://orcid.org/0000-0001-7141-2553","authenticated-orcid":true,"given":"Giacomo","family":"Giudice","sequence":"additional","affiliation":[{"name":"PlanQC GmbH"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,14]]},"URL":"http://dx.doi.org/10.1103/lkk1-v6wp","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040337","note":"arxivid:2412.15187"},{"id":"arxiv:2403.01659","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.111.032427","source":"Crossref","title":"Improved performance of the Bacon-Shor code with Steane's syndrome extraction method","volume":"111","author":[{"ORCID":"https://orcid.org/0009-0009-7061-6824","authenticated-orcid":true,"given":"Guillermo","family":"Escobar-Arrieta","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02yzgww51","id-type":"ROR","asserted-by":"publisher"}],"name":"Universidad de Costa Rica"},{"id":[{"id":"https://ror.org/02yzgww51","id-type":"ROR","asserted-by":"publisher"}],"name":"Universidad de Costa Rica"}]},{"ORCID":"https://orcid.org/0000-0002-2148-1363","authenticated-orcid":true,"given":"Mauricio","family":"Gutiérrez","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02yzgww51","id-type":"ROR","asserted-by":"publisher"}],"name":"Universidad de Costa Rica"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,3,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.111.032427","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032427","note":"arxivid:2403.01659"},{"id":"arxiv:1212.3564","type":"journal-article","publisher":"IOP Publishing","issue":"3","DOI":"10.1088/1367-2630/15/3/035014","page":"035014","source":"Crossref","title":"Gauge subsystems, separability and robustness in autonomous quantum memories","volume":"15","author":[{"given":"Gopal","family":"Sarma","sequence":"first","affiliation":[]},{"given":"Hideo","family":"Mabuchi","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2013,3,12]]},"URL":"http://dx.doi.org/10.1088/1367-2630/15/3/035014","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1212.3564"},{"id":"arxiv:quant-ph/0610063","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/physrevlett.98.220502","source":"Crossref","title":"Subsystem Fault Tolerance with the Bacon-Shor Code","volume":"98","author":[{"given":"Panos","family":"Aliferis","sequence":"first","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2007,5,31]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.98.220502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"220502","note":"arxivid:quant-ph/0610063"},{"id":"arxiv:1705.01686","type":"article-journal","author":[{"given":"Theodore J.","family":"Yoder"}],"title":"Universal fault-tolerant quantum computation with Bacon-Shor codes","issued":{"date-parts":[[2017,5,4]]},"note":"arxivid:1705.01686\narxiv_version_number:1"},{"id":"arxiv:1903.03937","type":"journal-article","publisher":"IOP Publishing","issue":"7","abstract":"<jats:title>Abstract</jats:title>\n               <jats:p>Leakage is a particularly damaging error that occurs when a qubit state falls out of its two-level computational subspace. Compared to independent depolarizing noise, leaked qubits may produce many more configurations of harmful correlated errors during error-correction. In this work, we investigate different local codes in the low-error regime of a leakage gate error model. When restricting to bare-ancilla extraction, we observe that subsystem codes are good candidates for handling leakage, as their locality can limit damaging correlated errors. As a case study, we compare subspace surface codes to the subsystem surface codes introduced by Bravyi <jats:italic>et al</jats:italic>. In contrast to depolarizing noise, subsystem surface codes outperform same-distance subspace surface codes below error rates as high as ⪅ 7.5 × 10<jats:sup>−4</jats:sup> while offering better per-qubit distance protection. Furthermore, we show that at low to intermediate distances, Bacon–Shor codes offer better per-qubit error protection against leakage in an ion-trap motivated error model below error rates as high as ⪅ 1.2 × 10<jats:sup>−3</jats:sup>. For restricted leakage models, this advantage can be extended to higher distances by relaxing to unverified two-qubit cat state extraction in the surface code. These results highlight an intrinsic benefit of subsystem code locality to error-corrective performance.</jats:p>","DOI":"10.1088/1367-2630/ab3372","page":"073055","source":"Crossref","title":"Handling leakage with subsystem codes","volume":"21","author":[{"given":"Natalie C","family":"Brown","sequence":"first","affiliation":[]},{"given":"Michael","family":"Newman","sequence":"additional","affiliation":[]},{"given":"Kenneth R","family":"Brown","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2019,7,1]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ab3372","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1903.03937"},{"id":"arxiv:2305.12046","type":"article-journal","author":[{"given":"Craig","family":"Gidney"},{"given":"Dave","family":"Bacon"}],"title":"Less Bacon More Threshold","issued":{"date-parts":[[2023,5,20]]},"note":"arxivid:2305.12046\narxiv_version_number:1"},{"id":"arxiv:2503.03867","type":"journal-article","publisher":"American Physical Society (APS)","issue":"22","DOI":"10.1103/425n-6k9s","source":"Crossref","title":"Logical Operations with a Dynamical Qubit in Floquet-Bacon-Shor Code","volume":"135","author":[{"ORCID":"https://orcid.org/0009-0009-9220-0977","authenticated-orcid":true,"given":"Xuandong","family":"Sun","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0002-5259-9807","authenticated-orcid":true,"given":"Longcheng","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0090r4d87","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute of Computing Technology"},{"id":[{"id":"https://ror.org/05qbk4x57","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chinese Academy of Sciences"}]},{"given":"Zhiyi","family":"Wu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02v51f717","id-type":"ROR","asserted-by":"publisher"}],"name":"Peking University"},{"name":"International Quantum Academy"}]},{"given":"Zechen","family":"Guo","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Peisheng","family":"Huang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04j7b2v61","id-type":"ROR","asserted-by":"publisher"}],"name":"Ningxia University"},{"name":"International Quantum Academy"}]},{"given":"Wenhui","family":"Huang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Qixian","family":"Li","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Yongqi","family":"Liang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Yiting","family":"Liu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Daxiong","family":"Sun","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"ORCID":"https://orcid.org/0000-0003-0210-5124","authenticated-orcid":true,"given":"Zilin","family":"Wang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04j7b2v61","id-type":"ROR","asserted-by":"publisher"}],"name":"Ningxia University"},{"name":"International Quantum Academy"}]},{"given":"Changrong","family":"Xie","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"ORCID":"https://orcid.org/0009-0006-8540-3299","authenticated-orcid":true,"given":"Yuzhe","family":"Xiong","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"ORCID":"https://orcid.org/0009-0008-4555-1275","authenticated-orcid":true,"given":"Xiaohan","family":"Yang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Jiajian","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Jiawei","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Libo","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Zihao","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Weijie","family":"Guo","sequence":"additional","affiliation":[{"name":"International Quantum Academy"}]},{"given":"Ji","family":"Jiang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"}]},{"given":"Song","family":"Liu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"Shenzhen Branch, Hefei National Laboratory"}]},{"given":"Xiayu","family":"Linpeng","sequence":"additional","affiliation":[{"name":"International Quantum Academy"}]},{"given":"Jingjing","family":"Niu","sequence":"additional","affiliation":[{"name":"International Quantum Academy"},{"name":"Shenzhen Branch, Hefei National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-9475-6452","authenticated-orcid":true,"given":"Jiawei","family":"Qiu","sequence":"additional","affiliation":[{"name":"International Quantum Academy"}]},{"given":"Wenhui","family":"Ren","sequence":"additional","affiliation":[{"name":"International Quantum Academy"}]},{"given":"Ziyu","family":"Tao","sequence":"additional","affiliation":[{"name":"International Quantum Academy"}]},{"given":"Yuefeng","family":"Yuan","sequence":"additional","affiliation":[{"name":"International Quantum Academy"}]},{"given":"Yuxuan","family":"Zhou","sequence":"additional","affiliation":[{"name":"International Quantum Academy"}]},{"ORCID":"https://orcid.org/0000-0003-4193-3280","authenticated-orcid":true,"given":"Ji","family":"Chu","sequence":"additional","affiliation":[{"name":"International Quantum Academy"}]},{"ORCID":"https://orcid.org/0000-0002-8120-6259","authenticated-orcid":true,"given":"Youpeng","family":"Zhong","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/049tv2d57","id-type":"ROR","asserted-by":"publisher"}],"name":"Southern University of Science and Technology"},{"name":"International Quantum Academy"},{"name":"Shenzhen Branch, Hefei National Laboratory"}]},{"ORCID":"https://orcid.org/0000-0002-0281-1670","authenticated-orcid":true,"given":"Xiaoming","family":"Sun","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/0090r4d87","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute of Computing Technology"},{"id":[{"id":"https://ror.org/05qbk4x57","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chinese Academy of Sciences"}]},{"given":"Dapeng","family":"Yu","sequence":"additional","affiliation":[{"name":"International Quantum Academy"},{"name":"Shenzhen Branch, Hefei National Laboratory"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,25]]},"URL":"http://dx.doi.org/10.1103/425n-6k9s","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"220601","note":"arxivid:2503.03867"},{"id":"arxiv:0806.2188","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.78.062317","source":"Crossref","title":"Message passing in fault-tolerant quantum error correction","volume":"78","author":[{"given":"Zachary W. E.","family":"Evans","sequence":"first","affiliation":[]},{"given":"Ashley M.","family":"Stephens","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,12,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.062317","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062317","note":"arxivid:0806.2188"},{"id":"arxiv:0805.4213","type":"article-journal","author":[{"given":"Federico M.","family":"Spedalieri"},{"given":"Vwani P.","family":"Roychowdhury"}],"title":"Latency in local, two-dimensional, fault-tolerant quantum computing","issued":{"date-parts":[[2008,5,27]]},"note":"arxivid:0805.4213\narxiv_version_number:1"},{"id":"arxiv:2009.11482","type":"article-journal","author":[{"given":"Laird","family":"Egan"},{"given":"Dripto M.","family":"Debroy"},{"given":"Crystal","family":"Noel"},{"given":"Andrew","family":"Risinger"},{"given":"Daiwei","family":"Zhu"},{"given":"Debopriyo","family":"Biswas"},{"given":"Michael","family":"Newman"},{"given":"Muyuan","family":"Li"},{"given":"Kenneth R.","family":"Brown"},{"given":"Marko","family":"Cetina"},{"given":"Christopher","family":"Monroe"}],"title":"Fault-Tolerant Operation of a Quantum Error-Correction Code","issued":{"date-parts":[[2021,1,7]]},"note":"arxivid:2009.11482\narxiv_version_number:2"},{"id":"arxiv:1901.06319","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.99.052333","source":"Crossref","title":"Optimal quantum subsystem codes in two dimensions","volume":"99","author":[{"given":"Theodore J.","family":"Yoder","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,5,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.052333","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052333","note":"arxivid:1901.06319"},{"id":"arxiv:2405.09287","type":"article-journal","author":[{"given":"Balint","family":"Pato"},{"given":"J. Wilson","family":"Staples"},{"given":"Kenneth R.","family":"Brown"}],"title":"Logical coherence in 2D compass codes","issued":{"date-parts":[[2025,2,11]]},"note":"arxivid:2405.09287\narxiv_version_number:2"},{"id":"arxiv:1805.12542","type":"article-journal","author":[{"given":"Vinuta V.","family":"Gayatri"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Decoding Algorithms for Hypergraph Subsystem Codes and Generalized Subsystem Surface Codes","issued":{"date-parts":[[2018,5,31]]},"note":"arxivid:1805.12542\narxiv_version_number:1"},{"id":"arxiv:2311.18003","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli \\cite{AKS06}, we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko \\cite{KP13}, we show that any subsystem stabilizer code can be \"doubled\" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis \\cite{BTL10}. Using Goursat&amp;apos;s Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes&amp;apos; properties.</jats:p>","DOI":"10.22331/q-2024-07-10-1403","page":"1403","source":"Crossref","title":"Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat&amp;apos;s Lemma","volume":"8","author":[{"given":"Michael Liaofan","family":"Liu","sequence":"first","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST, and University of Maryland, College Park, Maryland 20740, USA"},{"name":"Department of Mathematics, Amherst College, Amherst, Massachusetts 01002, USA"}]},{"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[{"name":"Walter Burke Institute for Theoretical Physics and Department of Physics, California Institute of Technology, Pasadena, California 91125, USA"}]},{"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST, and University of Maryland, College Park, Maryland 20740, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,10]]},"URL":"http://dx.doi.org/10.22331/q-2024-07-10-1403","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2311.18003"},{"id":"arxiv:0811.1570","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"},{"given":"Andreas","family":"Klappenecker"}],"title":"Constructions of Subsystem Codes over Finite Fields","issued":{"date-parts":[[2008,11,10]]},"note":"arxivid:0811.1570\narxiv_version_number:1"},{"id":"manual:-P.-Lisonek-A.-Roy-and-S.-Tra","type":"document","author":[{"family":"Lisonek","given":"P."},{"family":"Roy","given":"A."},{"family":"Trandafir","given":"S."}],"genre":"private communication,","issued":"2019"},{"id":"arxiv:0806.4954","type":"article-journal","author":[{"given":"Pradeep Kiran","family":"Sarvepalli"},{"given":"Andreas","family":"Klappenecker"}],"title":"Encoding Subsystem Codes","issued":{"date-parts":[[2008,6,30]]},"note":"arxivid:0806.4954\narxiv_version_number:1"},{"id":"arxiv:1009.2203","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.83.022307","source":"Crossref","title":"Automated searching for quantum subsystem codes","volume":"83","author":[{"given":"Gregory M.","family":"Crosswhite","sequence":"first","affiliation":[]},{"given":"Dave","family":"Bacon","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2011,2,10]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.83.022307","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022307","note":"arxivid:1009.2203"},{"id":"arxiv:2502.02553","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","abstract":"<jats:p>Fault-tolerant quantum computation requires quantum error correction (QEC), which relies on entanglement to protect information from local noise. Achieving universality, however, demands overcoming the Eastin-Knill theorem. This is often accomplished through strategies like magic state distillation, a process that prepares computational resources—namely, magic states—whose power is now understood to be rooted in quantum contextuality, a fundamental nonclassical feature generalizing Bell nonlocality. Yet, the broader role of contextuality in enabling universality, including its significance as an inherent feature of QEC codes and protocols themselves, has remained largely unexplored. In this work, we develop a rigorous framework for contextuality in QEC and prove three main results. Fundamentally, we show that subsystem stabilizer codes with two or more gauge qubits are strongly contextual in their partial closure, while others are noncontextual, establishing a clear criterion for identifying contextual codes. Mathematically, we unify Abramsky-Brandenburger’s sheaf-theoretic and Kirby-Love’s tree-based definitions of contextuality, resolving a conjecture of Kim and Abramsky. Practically, we prove that many widely studied code-switching protocols which admit universal transversal gate sets, such as the doubled color codes introduced by Bravyi and Cross, are necessarily strongly contextual in their partial closure. Collectively, our results establish quantum contextuality as an intrinsic characteristic of fault-tolerant quantum codes and protocols, complementing entanglement and magic as resources for scalable quantum computation. For quantum coding theorists, this provides a new invariant: contextuality classifies which subsystem stabilizer codes can participate in universal fault-tolerant protocols. These findings position contextuality not only as a foundational concept but also as a practical guide for the design and analysis of future QEC architectures.</jats:p>","DOI":"10.1103/7zb5-vs4x","source":"Crossref","title":"Contextuality of Quantum Error-Correcting Codes","volume":"7","author":[{"ORCID":"https://orcid.org/0009-0000-0293-0664","authenticated-orcid":true,"given":"Derek","family":"Khu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/053rfa017","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Infocomm Research (I"},{"id":[{"id":"https://ror.org/036wvzt09","id-type":"ROR","asserted-by":"publisher"}],"name":"Agency for Science, Technology and Research (A*STAR)"}]},{"ORCID":"https://orcid.org/0009-0001-1655-6842","authenticated-orcid":true,"given":"Andrew","family":"Tanggara","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01mgdzc49","id-type":"ROR","asserted-by":"publisher"}],"name":"Centre for Quantum Technologies"},{"id":[{"id":"https://ror.org/02e7b5302","id-type":"ROR","asserted-by":"publisher"}],"name":"Nanyang Technological University"}]},{"given":"Chao","family":"Jin","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/053rfa017","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Infocomm Research (I"},{"id":[{"id":"https://ror.org/036wvzt09","id-type":"ROR","asserted-by":"publisher"}],"name":"Agency for Science, Technology and Research (A*STAR)"}]},{"given":"Kishor","family":"Bharti","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/036wvzt09","id-type":"ROR","asserted-by":"publisher"}],"name":"Agency for Science Technology and Research (A*STAR)"},{"id":[{"id":"https://ror.org/02n0ejh50","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute of High Performance Computing (IHPC)"},{"id":[{"id":"https://ror.org/036wvzt09","id-type":"ROR","asserted-by":"publisher"}],"name":"Agency for Science, Technology and Research (A*STAR)"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2026,1,28]]},"URL":"http://dx.doi.org/10.1103/7zb5-vs4x","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"010319","note":"arxivid:2502.02553"},{"id":"arxiv:1204.1838","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.85.050302","source":"Crossref","title":"Optimal error correction in topological subsystem codes","volume":"85","author":[{"given":"Ruben S.","family":"Andrist","sequence":"first","affiliation":[]},{"given":"H.","family":"Bombin","sequence":"additional","affiliation":[]},{"given":"Helmut G.","family":"Katzgraber","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,5,14]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.85.050302","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"050302","note":"arxivid:1204.1838"},{"id":"doi:10.1109/TCOMM.2023.3277534","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tcomm.2023.3277534","page":"4181-4192","source":"Crossref","title":"Decoding Topological Subsystem Color Codes Over the Erasure Channel Using Gauge Fixing","volume":"71","author":[{"ORCID":"https://orcid.org/0000-0002-5207-7830","authenticated-orcid":false,"given":"Hiteshvi Manish","family":"Solanki","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India"}]},{"ORCID":"https://orcid.org/0000-0001-8047-6946","authenticated-orcid":false,"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[{"name":"Qualcomm India Pvt. Ltd., Bengaluru, Karnataka, India"}]}],"container-title":"IEEE Transactions on Communications","original-title":[],"issued":{"date-parts":[[2023,7]]},"URL":"http://dx.doi.org/10.1109/TCOMM.2023.3277534","ISSN":["0090-6778","1558-0857"],"container-title-short":"IEEE Trans. Commun."},{"id":"arxiv:2111.14594","type":"article-journal","author":[{"given":"Hiteshvi Manish","family":"Solanki"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Decoding Topological Subsystem Color Codes Over the Erasure Channel using Gauge Fixing","issued":{"date-parts":[[2022,4,15]]},"note":"arxivid:2111.14594\narxiv_version_number:2"},{"id":"doi:10.1109/ITW46852.2021.9457583","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/itw46852.2021.9457583","page":"1-5","source":"Crossref","title":"Correcting Erasures with Topological Subsystem Color Codes","author":[{"given":"Hiteshvi Manish","family":"Solanki","sequence":"first","affiliation":[]},{"given":"Pradeep","family":"Kiran Sarvepalli","sequence":"additional","affiliation":[]}],"event":"2020 IEEE Information Theory Workshop (ITW)","container-title":"2020 IEEE Information Theory Workshop (ITW)","original-title":[],"issued":{"date-parts":[[2021,4,11]]},"URL":"http://dx.doi.org/10.1109/ITW46852.2021.9457583"},{"id":"arxiv:2106.02649","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/prxquantum.3.030322","source":"Crossref","title":"Achieving Fault Tolerance on Capped Color Codes with Few Ancillas","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-2865-0705","authenticated-orcid":true,"given":"Theerapat","family":"Tansuwannont","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3750-2648","authenticated-orcid":true,"given":"Debbie","family":"Leung","sequence":"additional","affiliation":[]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,8,11]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.3.030322","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"030322","note":"arxivid:2106.02649"},{"id":"arxiv:1509.04255","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.93.022323","source":"Crossref","title":"Stacked codes: Universal fault-tolerant quantum computation in a two-dimensional layout","volume":"93","author":[{"given":"Tomas","family":"Jochym-O'Connor","sequence":"first","affiliation":[]},{"given":"Stephen D.","family":"Bartlett","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,2,17]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.022323","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022323","note":"arxivid:1509.04255"},{"id":"arxiv:1512.04193","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.93.052332","source":"Crossref","title":"Gauge color codes in two dimensions","volume":"93","author":[{"given":"Cody","family":"Jones","sequence":"first","affiliation":[]},{"given":"Peter","family":"Brooks","sequence":"additional","affiliation":[]},{"given":"Jim","family":"Harrington","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,5,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.052332","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052332","note":"arxivid:1512.04193"},{"id":"arxiv:2501.18554","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8080","DOI":"10.1038/s41586-025-09475-0","page":"341-347","source":"Crossref","title":"Probing the Kitaev honeycomb model on a neutral-atom quantum computer","volume":"645","author":[{"ORCID":"https://orcid.org/0000-0001-8986-1103","authenticated-orcid":false,"given":"Simon J.","family":"Evered","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0605-8791","authenticated-orcid":false,"given":"Marcin","family":"Kalinowski","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5294-4941","authenticated-orcid":false,"given":"Alexandra A.","family":"Geim","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-3470-1369","authenticated-orcid":false,"given":"Tom","family":"Manovitz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9934-9530","authenticated-orcid":false,"given":"Dolev","family":"Bluvstein","sequence":"additional","affiliation":[]},{"given":"Sophie H.","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5775-9542","authenticated-orcid":false,"given":"Nishad","family":"Maskara","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2148-8856","authenticated-orcid":false,"given":"Hengyun","family":"Zhou","sequence":"additional","affiliation":[]},{"given":"Sepehr","family":"Ebadi","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2384-0208","authenticated-orcid":false,"given":"Muqing","family":"Xu","sequence":"additional","affiliation":[]},{"given":"Joseph","family":"Campo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5298-3112","authenticated-orcid":false,"given":"Madelyn","family":"Cain","sequence":"additional","affiliation":[]},{"given":"Stefan","family":"Ostermann","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1655-9151","authenticated-orcid":false,"given":"Susanne F.","family":"Yelin","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2432-7070","authenticated-orcid":false,"given":"Subir","family":"Sachdev","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2935-2363","authenticated-orcid":false,"given":"Markus","family":"Greiner","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9786-0538","authenticated-orcid":false,"given":"Vladan","family":"Vuletić","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8658-1007","authenticated-orcid":false,"given":"Mikhail D.","family":"Lukin","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,10]]},"URL":"http://dx.doi.org/10.1038/s41586-025-09475-0","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:9475\narxivid:2501.18554"},{"id":"arxiv:1701.01440","type":"journal-article","publisher":"American Physical Society (APS)","issue":"24","DOI":"10.1103/physrevb.96.245116","source":"Crossref","title":"Radical chiral Floquet phases in a periodically driven Kitaev model and beyond","volume":"96","author":[{"given":"Hoi Chun","family":"Po","sequence":"first","affiliation":[]},{"given":"Lukasz","family":"Fidkowski","sequence":"additional","affiliation":[]},{"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]},{"given":"Andrew C.","family":"Potter","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2017,12,12]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.96.245116","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"245116","note":"arxivid:1701.01440"},{"id":"arxiv:2501.18461","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8080","abstract":"<jats:title>Abstract</jats:title>\n          <jats:p>Out-of-equilibrium phases in many-body systems constitute a new paradigm in quantum matter—they exhibit dynamical properties that may otherwise be forbidden by equilibrium thermodynamics. Among these non-equilibrium phases are periodically driven (Floquet) systems<jats:sup>1–5</jats:sup>, which are generically difficult to simulate classically because of their high entanglement. Here we realize a Floquet topologically ordered state theoretically proposed in ref. <jats:sup>6</jats:sup>, on an array of superconducting qubits. We image the characteristic dynamics of its chiral edge modes and characterize its emergent anyonic excitations. Devising an interferometric algorithm allows us to introduce and measure a bulk topological invariant to probe the dynamical transmutation of anyons for system sizes up to 58 qubits. Our work demonstrates that quantum processors can provide key insights into the thus-far largely unexplored landscape of highly entangled non-equilibrium phases of matter.</jats:p>","DOI":"10.1038/s41586-025-09456-3","page":"348-353","source":"Crossref","title":"Probing non-equilibrium topological order on a quantum processor","volume":"645","author":[{"ORCID":"https://orcid.org/0009-0003-7057-6611","authenticated-orcid":false,"given":"M.","family":"Will","sequence":"first","affiliation":[]},{"given":"T. A.","family":"Cochran","sequence":"additional","affiliation":[]},{"given":"E.","family":"Rosenberg","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7027-3918","authenticated-orcid":false,"given":"B.","family":"Jobst","sequence":"additional","affiliation":[]},{"given":"N. M.","family":"Eassa","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1917-3879","authenticated-orcid":false,"given":"P.","family":"Roushan","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7093-9502","authenticated-orcid":false,"given":"M.","family":"Knap","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6744-4592","authenticated-orcid":false,"given":"A.","family":"Gammon-Smith","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0320-9304","authenticated-orcid":false,"given":"F.","family":"Pollmann","sequence":"additional","affiliation":[]}],"container-title":"Nature","original-title":[],"language":"en","issued":{"date-parts":[[2025,9,10]]},"URL":"http://dx.doi.org/10.1038/s41586-025-09456-3","ISSN":["0028-0836","1476-4687"],"container-title-short":"Nature","note":"alternative-id:9456\narxivid:2501.18461"},{"id":"arxiv:2405.14927","type":"article-journal","author":[{"given":"Benedikt","family":"Placke"},{"given":"S. A.","family":"Parameswaran"}],"title":"Slow measurement-only dynamics of entanglement in Pauli subsystem codes","issued":{"date-parts":[[2024,12,11]]},"note":"arxivid:2405.14927\narxiv_version_number:2"},{"id":"arxiv:2404.15989","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","abstract":"<jats:p>Designs for quantum error correction depend strongly on the connectivity of the qubits. For solid-state qubits, the most straightforward approach is to have connectivity constrained to a planar graph. Practical considerations may also further restrict the connectivity, resulting in a relatively sparse graph such as the heavy-hexagonal (“heavy-hex”) architecture of current IBM Quantum devices. In such cases, it is hard to use all qubits to their full potential. Instead, in order to emulate the denser connectivity required to implement well-known quantum error-correcting codes, many qubits remain effectively unused. In this work, we show how this bug can be turned into a feature. By using the unused qubits of one code to execute another, two codes can be implemented on top of each other, allowing easy application of fault-tolerant entangling gates and measurements. We demonstrate this by realizing a surface code and a Bacon-Shor code on a 133-qubit IBM Quantum device. Using transversal controlled-<a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:mi>X</a:mi></a:math> () gates and lattice surgery, we demonstrate entanglement between these logical qubits with code distance up to <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><d:mi>d</d:mi><d:mo>=</d:mo><d:mn>4</d:mn></d:math> and five rounds of stabilizer-measurement cycles. The nonplanar coupling between the qubits allows us to simultaneously measure the logical <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><g:mi>X</g:mi><g:mi>X</g:mi></g:math>, <j:math xmlns:j=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><j:mi>Y</j:mi><j:mi>Y</j:mi></j:math>, and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><m:mi>Z</m:mi><m:mi>Z</m:mi></m:math> observables. With this, we verify the violation of Bell’s inequality for both the <p:math xmlns:p=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><p:mi>d</p:mi><p:mo>=</p:mo><p:mn>2</p:mn></p:math> case with postselection featuring a fidelity of 94% and the <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><s:mi>d</s:mi><s:mo>=</s:mo><s:mn>3</s:mn></s:math> instance using only quantum error correction.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.040334","source":"Crossref","title":"Creating Entangled Logical Qubits in the Heavy-Hex Lattice with Topological Codes","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-8432-3178","authenticated-orcid":true,"given":"Bence","family":"Hetényi","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02js37d36","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM Research Zurich"}]},{"ORCID":"https://orcid.org/0000-0003-1943-5306","authenticated-orcid":true,"given":"James R.","family":"Wootton","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02js37d36","id-type":"ROR","asserted-by":"publisher"}],"name":"IBM Research Zurich"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,3]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.040334","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"040334","note":"arxivid:2404.15989"},{"id":"arxiv:2412.15751","type":"article-journal","author":[{"given":"Hansol","family":"Kim"},{"given":"Wonjae","family":"Choi"},{"given":"Younghun","family":"Kwon"}],"title":"Implementation of Magic State Injection within Heavy-Hexagon Architecture","issued":{"date-parts":[[2024,12,26]]},"note":"arxivid:2412.15751\narxiv_version_number:2"},{"id":"arxiv:2210.09730","type":"journal-article","publisher":"Association for Computing Machinery (ACM)","issue":"1","abstract":"<jats:p>Error syndromes for heavy hexagonal code and other topological codes such as surface code have typically been decoded by using Minimum Weight Perfect Matching– (MWPM) based methods. Recent advances have shown that topological codes can be efficiently decoded by deploying machine learning (ML) techniques, in particular with neural networks. In this work, we first propose an ML-based decoder for heavy hexagonal code and establish its efficiency in terms of the values of threshold and pseudo-threshold for various noise models. We show that the proposed ML-based decoding method achieves ~ 5 × higher values of threshold than that for MWPM. Next, exploiting the property of subsystem codes, we define gauge equivalence for heavy hexagonal code, by which two distinct errors can belong to the same error class. A linear search-based method is proposed for determining the equivalent error classes. This provides a quadratic reduction in the number of error classes to be considered for both bit flip and phase flip errors and thus a further improvement of ~ 14% in the threshold over the basic ML decoder. Last, a novel technique based on rank to determine the equivalent error classes is presented, which is empirically faster than the one based on linear search.</jats:p>","DOI":"10.1145/3636516","page":"1-27","source":"Crossref","title":"Efficient Syndrome Decoder for Heavy Hexagonal QECC via Machine Learning","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-1795-5298","authenticated-orcid":false,"given":"Debasmita","family":"Bhoumik","sequence":"first","affiliation":[{"name":"Advanced Computing &amp; Microelectronics Unit, Indian Statistical Institute, Baranagar, West Bengal"}]},{"ORCID":"https://orcid.org/0000-0003-0730-0084","authenticated-orcid":false,"given":"Ritajit","family":"Majumdar","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM India Research Lab, Thanisandra, Bengaluru, Karnataka"}]},{"ORCID":"https://orcid.org/0009-0004-0629-0018","authenticated-orcid":false,"given":"Dhiraj","family":"Madan","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM India Research Lab, Thanisandra, Bengaluru, Karnataka"}]},{"ORCID":"https://orcid.org/0000-0001-9388-7154","authenticated-orcid":false,"given":"Dhinakaran","family":"Vinayagamurthy","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM India Research Lab, Thanisandra, Bengaluru, Karnataka"}]},{"ORCID":"https://orcid.org/0000-0003-2323-8938","authenticated-orcid":false,"given":"Shesha","family":"Raghunathan","sequence":"additional","affiliation":[{"name":"IBM Quantum, IBM India Research Lab, Thanisandra, Bengaluru, Karnataka"}]},{"ORCID":"https://orcid.org/0000-0002-2052-3779","authenticated-orcid":false,"given":"Susmita","family":"Sur-Kolay","sequence":"additional","affiliation":[{"name":"Advanced Computing &amp; Microelectronics Unit, Indian Statistical Institute, Baranagar, West Bengal"}]}],"container-title":"ACM Transactions on Quantum Computing","original-title":[],"language":"en","issued":{"date-parts":[[2024,2,24]]},"URL":"http://dx.doi.org/10.1145/3636516","ISSN":["2643-6809","2643-6817"],"container-title-short":"ACM Trans. Quantum Comput.","note":"alternative-id:10.1145/3636516\narxivid:2210.09730"},{"id":"arxiv:2311.15146","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","abstract":"<jats:p>Syndrome decoding is an integral but computationally demanding step in the implementation of quantum error correction for fault-tolerant quantum computing. Here, we report the development and benchmarking of Artificial Neural Network (ANN) decoding on IBM quantum processors. We demonstrate that ANNs can efficiently decode syndrome measurement data from heavy-hexagonal code architecture and apply appropriate corrections to facilitate error protection. The current physical error rates of IBM devices are above the code's threshold and restrict the scope of our ANN decoder for logical error rate suppression. However, our work confirms the applicability of ANN decoding methods of syndrome data retrieved from experimental devices and establishes machine learning as a promising pathway for quantum error correction when quantum devices with below threshold error rates become available in the near future.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevresearch.6.l032004","source":"Crossref","title":"Artificial neural network syndrome decoding on IBM quantum processors","volume":"6","author":[{"ORCID":"https://orcid.org/0009-0007-2368-7190","authenticated-orcid":true,"given":"Brhyeton","family":"Hall","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/01ej9dk98","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Melbourne"}]},{"given":"Spiro","family":"Gicev","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01ej9dk98","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Melbourne"}]},{"ORCID":"https://orcid.org/0000-0003-3476-2348","authenticated-orcid":true,"given":"Muhammad","family":"Usman","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01ej9dk98","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Melbourne"},{"id":[{"id":"https://ror.org/03q397159","id-type":"ROR","asserted-by":"publisher"}],"name":"Data61"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2024,7,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.6.L032004","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"L032004","note":"arxivid:2311.15146"},{"id":"arxiv:2203.07205","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>Quantum error correction offers a promising path for performing high fidelity quantum computations. Although fully fault-tolerant executions of algorithms remain unrealized, recent improvements in control electronics and quantum hardware enable increasingly advanced demonstrations of the necessary operations for error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. We encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow for the correction of any single fault in the circuitry. Using real-time feedback, we reset syndrome and flag qubits conditionally after each syndrome extraction cycle. We report decoder dependent logical error, with average logical error per syndrome measurement in Z(X)-basis of ~0.040 (~0.088) and ~0.037 (~0.087) for matching and maximum likelihood decoders, respectively, on leakage post-selected data.</jats:p>","DOI":"10.1038/s41467-023-38247-5","source":"Crossref","title":"Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders","volume":"14","author":[{"ORCID":"https://orcid.org/0000-0002-0772-2853","authenticated-orcid":false,"given":"Neereja","family":"Sundaresan","sequence":"first","affiliation":[]},{"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8486-9162","authenticated-orcid":false,"given":"Youngseok","family":"Kim","sequence":"additional","affiliation":[]},{"given":"Muyuan","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7217-7137","authenticated-orcid":false,"given":"Edward H.","family":"Chen","sequence":"additional","affiliation":[]},{"given":"Grace","family":"Harper","sequence":"additional","affiliation":[]},{"given":"Ted","family":"Thorbeck","sequence":"additional","affiliation":[]},{"given":"Andrew W.","family":"Cross","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7800-0399","authenticated-orcid":false,"given":"Antonio D.","family":"Córcoles","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8871-4638","authenticated-orcid":false,"given":"Maika","family":"Takita","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2023,5,18]]},"URL":"http://dx.doi.org/10.1038/s41467-023-38247-5","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"2852","note":"alternative-id:38247\narxivid:2203.07205"},{"id":"arxiv:2412.01446","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n                  <jats:p>\n                    The surface code family is a promising approach to implementing fault-tolerant quantum computations. Universal fault-tolerance requires error-corrected non-Clifford operations, in addition to Clifford gates, and for the former, it is imperative to experimentally demonstrate additional resources known as magic states. Another challenge is to efficiently embed surface codes into quantum hardware with connectivity constraints. This work simultaneously addresses both challenges by employing a qubit-efficient rotated heavy-hexagonal surface code for IBM quantum processors () and implementing the magic state injection protocol. Our work reports error thresholds for both logical bit- and phase-flip errors, of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\approx 0.37\\%$$</jats:tex-math>\n                    </jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\approx 0.31\\%$$</jats:tex-math>\n                    </jats:inline-formula>\n                    , respectively, which are higher than the threshold values previously reported with traditional embedding. The post-selection-based preparation of logical magic states\n                    <jats:inline-formula>\n                      <jats:tex-math>$$|H_L\\rangle$$</jats:tex-math>\n                    </jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$|T_L\\rangle$$</jats:tex-math>\n                    </jats:inline-formula>\n                    achieve fidelities of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$0.8806\\pm 0.0002$$</jats:tex-math>\n                    </jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$0.8665\\pm 0.0003$$</jats:tex-math>\n                    </jats:inline-formula>\n                    , respectively, which are both above the magic state distillation threshold. The post-selection process yields an average success rate of\n                    <jats:inline-formula>\n                      <jats:tex-math>$$36.28 \\pm 0.09\\%$$</jats:tex-math>\n                    </jats:inline-formula>\n                    . Additionally, we report the minimum fidelity among injected arbitrary single logical qubit states as\n                    <jats:inline-formula>\n                      <jats:tex-math>$$0.8356\\pm 0.0003$$</jats:tex-math>\n                    </jats:inline-formula>\n                    . Our work demonstrates the potential for realising non-Clifford logical gates by producing high-fidelity logical magic states on IBM quantum devices.\n                  </jats:p>","DOI":"10.1038/s41598-026-40381-1","source":"Crossref","title":"Magic state injection on IBM quantum processors above the distillation threshold","volume":"16","author":[{"given":"Younghun","family":"Kim","sequence":"first","affiliation":[]},{"given":"Martin","family":"Sevior","sequence":"additional","affiliation":[]},{"given":"Muhammad","family":"Usman","sequence":"additional","affiliation":[]}],"container-title":"Scientific Reports","original-title":[],"language":"en","issued":{"date-parts":[[2026,2,26]]},"URL":"http://dx.doi.org/10.1038/s41598-026-40381-1","ISSN":["2045-2322"],"container-title-short":"Sci Rep","page":"11189","note":"alternative-id:40381\narxivid:2412.01446"},{"id":"arxiv:0712.1377","type":"article-journal","author":[{"given":"Eric","family":"Rowell"},{"given":"Richard","family":"Stong"},{"given":"Zhenghan","family":"Wang"}],"title":"On classification of modular tensor categories","issued":{"date-parts":[[2009,11,9]]},"note":"arxivid:0712.1377\narxiv_version_number:4"},{"id":"arxiv:0811.0911","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevb.80.075111","source":"Crossref","title":"Interacting anyonic fermions in a two-body color code model","volume":"80","author":[{"given":"H.","family":"Bombin","sequence":"first","affiliation":[]},{"given":"M.","family":"Kargarian","sequence":"additional","affiliation":[]},{"given":"M. A.","family":"Martin-Delgado","sequence":"additional","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2009,8,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.80.075111","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"075111","note":"arxivid:0811.0911"},{"id":"arxiv:1207.1443","type":"article-journal","author":[{"given":"Sergey","family":"Bravyi"},{"given":"Guillaume","family":"Duclos-Cianci"},{"given":"David","family":"Poulin"},{"given":"Martin","family":"Suchara"}],"title":"Subsystem surface codes with three-qubit check operators","issued":{"date-parts":[[2013,12,17]]},"note":"arxivid:1207.1443\narxiv_version_number:2"},{"id":"arxiv:cond-mat/0510816","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physrevb.73.064410","source":"Crossref","title":"Multicritical point of Ising spin glasses on triangular and honeycomb lattices","volume":"73","author":[{"given":"S. L. A.","family":"de Queiroz","sequence":"first","affiliation":[]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2006,2,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevB.73.064410","ISSN":["1098-0121","1550-235X"],"container-title-short":"Phys. Rev. B","page":"064410","note":"arxivid:cond-mat/0510816"},{"id":"arxiv:2410.02022","type":"article-journal","author":[{"given":"Andrew","family":"Tanggara"},{"given":"Mile","family":"Gu"},{"given":"Kishor","family":"Bharti"}],"title":"Simple Construction of Qudit Floquet Codes on a Family of Lattices","issued":{"date-parts":[[2025,1,9]]},"note":"arxivid:2410.02022\narxiv_version_number:2"},{"id":"arxiv:0808.3086","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.78.062315","source":"Crossref","title":"Nonbinary codeword-stabilized quantum codes","volume":"78","author":[{"given":"Xie","family":"Chen","sequence":"first","affiliation":[]},{"given":"Bei","family":"Zeng","sequence":"additional","affiliation":[]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,12,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.78.062315","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062315","note":"arxivid:0808.3086"},{"id":"arxiv:1303.7020","type":"document","categories":["CWS Codes","Union Stabilizer Codes","Permutation Symmetry","Toric Code"],"language":"en","author":[{"family":"Beigi","given":"Salman"},{"family":"Chen","given":"Jianxin"},{"family":"Grassl","given":"Markus"},{"family":"Ji","given":"Zhengfeng"},{"family":"Wang","given":"Qiang"},{"family":"Zeng","given":"Bei"}],"contributor":[{"family":"Severini","given":"Simone"},{"family":"Brandao","given":"Fernando"}],"editor":[{"family":"Severini","given":"Simone"},{"family":"Brandao","given":"Fernando"}],"issued":{"date-parts":[[2013]]},"abstract":"Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes.  Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code Q can be represented by a self-dual additive code S and a classical code C, i.e., Q=(S,C), however this representation is in general not unique. We show that for any CWS code Q with certain permutation symmetry, one can always find a self-dual additive code S with the same permutation symmetry as Q such that Q=(S,C). As many good CWS codes have been found by starting from a chosen S, this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of S with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes.  For CWS codes, so far mainly the standard form (G,C) has been considered, where G is a graph state. We analyze the symmetry of the corresponding graph of G, which in general cannot possess the same permutation symmetry as Q.  We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant.","container-title":"LIPIcs, Volume 22, TQC 2013","DOI":"10.4230/LIPICS.TQC.2013.192","volume":"22","number":"12","page":"192-206","page-first":"192","publisher":"Schloss Dagstuhl – Leibniz-Zentrum für Informatik","title":"Symmetries of Codeword Stabilized Quantum Codes","URL":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.192","note":"copyright:Creative Commons Attribution 3.0 Unported license\narxivid:1303.7020"},{"id":"arxiv:1505.00283","type":"article-journal","author":[{"given":"Douglas Frederico Guimarães","family":"Santiago"},{"given":"Geraldo Samuel Sena","family":"Otoni"}],"title":"A new approach to codeword stabilized quantum codes using the algebraic structure of modules","issued":{"date-parts":[[2015,5,1]]},"note":"arxivid:1505.00283\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0401155","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.70.062101","source":"Crossref","title":"Discrete phase space based on finite fields","volume":"70","author":[{"given":"Kathleen S.","family":"Gibbons","sequence":"first","affiliation":[]},{"given":"Matthew J.","family":"Hoffman","sequence":"additional","affiliation":[]},{"given":"William K.","family":"Wootters","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2004,12,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.70.062101","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062101","note":"arxivid:quant-ph/0401155"},{"id":"arxiv:quant-ph/0410117","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.72.012309","source":"Crossref","title":"Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem","volume":"72","author":[{"given":"Juan Pablo","family":"Paz","sequence":"first","affiliation":[]},{"given":"Augusto José","family":"Roncaglia","sequence":"additional","affiliation":[]},{"given":"Marcos","family":"Saraceno","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2005,7,11]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.72.012309","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012309","note":"arxivid:quant-ph/0410117"},{"id":"arxiv:2503.09353","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/s5wn-mysr","source":"Crossref","title":"Grand unification of all discrete Wigner functions on\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                      <mml:mrow>\n                        <mml:mi>d</mml:mi>\n                        <mml:mo>×</mml:mo>\n                        <mml:mi>d</mml:mi>\n                      </mml:mrow>\n                    </mml:math>\n                    phase space","volume":"112","author":[{"ORCID":"https://orcid.org/0009-0001-5792-6555","authenticated-orcid":true,"given":"Lucky K.","family":"Antonopoulos","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"ORCID":"https://orcid.org/0009-0007-9225-9560","authenticated-orcid":true,"given":"Dominic G.","family":"Lewis","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"ORCID":"https://orcid.org/0000-0003-1705-1798","authenticated-orcid":true,"given":"Jack","family":"Davis","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05a0dhs15","id-type":"ROR","asserted-by":"publisher"}],"name":"École Normale Supérieure"},{"id":[{"id":"https://ror.org/052bz7812","id-type":"ROR","asserted-by":"publisher"}],"name":"PSL University"},{"id":[{"id":"https://ror.org/00s19x989","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS"}]},{"ORCID":"https://orcid.org/0000-0002-1100-0431","authenticated-orcid":true,"given":"Nicholas","family":"Funai","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]},{"ORCID":"https://orcid.org/0000-0002-3964-233X","authenticated-orcid":true,"given":"Nicolas C.","family":"Menicucci","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/04ttjf776","id-type":"ROR","asserted-by":"publisher"}],"name":"RMIT University"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,11,20]]},"URL":"http://dx.doi.org/10.1103/s5wn-mysr","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052219","note":"arxivid:2503.09353"},{"id":"arxiv:quant-ph/9802007","type":"journal-article","publisher":"Elsevier BV","issue":"10","DOI":"10.1016/s0960-0779(98)00218-5","page":"1749-1758","source":"Crossref","title":"Full length article","volume":"10","container-title":"Chaos, Solitons &amp; Fractals","original-title":[],"language":"en","issued":{"date-parts":[[1999,9]]},"URL":"http://dx.doi.org/10.1016/S0960-0779(98)00218-5","ISSN":["0960-0779"],"container-title-short":"Chaos, Solitons &amp; Fractals","note":"alternative-id:S0960077998002185\narxivid:quant-ph/9802007"},{"id":"arxiv:2302.07966","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Qudits with local dimension <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mn>2</mml:mn></mml:math> can have unique structure and uses that qubits (<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math>) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. We study the structure of the qudit Pauli group for any, including composite, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> in several ways. To cover composite values of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>, we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.</jats:p>","DOI":"10.22331/q-2024-04-04-1307","page":"1307","source":"Crossref","title":"The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems","volume":"8","author":[{"given":"Rahul","family":"Sarkar","sequence":"first","affiliation":[{"name":"Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA 94305"}]},{"given":"Theodore J.","family":"Yoder","sequence":"additional","affiliation":[{"name":"IBM T.J. Watson Research Center, Yorktown Heights, NY"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,4]]},"URL":"http://dx.doi.org/10.22331/q-2024-04-04-1307","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2302.07966"},{"id":"doi:10.1007/BF01457956","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","DOI":"10.1007/bf01457956","page":"570-578","source":"Crossref","title":"Die Eindeutigkeit der Schrödingerschen Operatoren","volume":"104","author":[{"given":"J.","family":"v. Neumann","sequence":"first","affiliation":[]}],"container-title":"Mathematische Annalen","original-title":[],"language":"de","issued":{"date-parts":[[1931,12]]},"URL":"http://dx.doi.org/10.1007/BF01457956","ISSN":["0025-5831","1432-1807"],"container-title-short":"Math. Ann.","note":"alternative-id:BF01457956"},{"id":"doi:10.1201/9780429497933","type":"book","publisher":"CRC Press","DOI":"10.1201/9780429497933","source":"Crossref","title":"Quantum Kinematics and Dynamics","author":[{"given":"Julian","family":"Schwinger","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2018,3,5]]},"ISBN":["9780429497933"],"URL":"http://dx.doi.org/10.1201/9780429497933","note":"edition-number:1"},{"id":"manual:-J.-J.-Sylvester.-The-Collect","type":"book","author":[{"family":"Sylvester","given":"J.J."}],"title":"The Collected Mathematical Papers of James Joseph Sylvester","volume":"3","publisher":"University Press","issued":"1909"},{"id":"doi:10.1016/0550-3213(78)90153-0","type":"journal-article","publisher":"Elsevier BV","issue":"1","DOI":"10.1016/0550-3213(78)90153-0","page":"1-25","source":"Crossref","title":"On the phase transition towards permanent quark confinement","volume":"138","author":[{"given":"G.","family":"'t Hooft","sequence":"first","affiliation":[]}],"container-title":"Nuclear Physics B","original-title":[],"language":"en","issued":{"date-parts":[[1978,6]]},"URL":"http://dx.doi.org/10.1016/0550-3213(78)90153-0","ISSN":["0550-3213"],"container-title-short":"Nuclear Physics B","note":"alternative-id:0550321378901530"},{"id":"arxiv:quant-ph/0602001","type":"journal-article","publisher":"AIP Publishing","issue":"12","abstract":"<jats:p>We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.</jats:p>","DOI":"10.1063/1.2393152","source":"Crossref","title":"Hudson’s theorem for finite-dimensional quantum systems","volume":"47","author":[{"given":"D.","family":"Gross","sequence":"first","affiliation":[{"name":"Institute for Mathematical Sciences , Imperial College London, London SW7 2BW, United Kingdom and QOLS, Blackett Laboratory, , London SW7 2BW, United Kingdom"},{"name":"Imperial College London , Imperial College London, London SW7 2BW, United Kingdom and QOLS, Blackett Laboratory, , London SW7 2BW, United Kingdom"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2006,12,1]]},"URL":"http://dx.doi.org/10.1063/1.2393152","ISSN":["0022-2488","1089-7658"],"page":"122107","note":"arxivid:quant-ph/0602001"},{"id":"doi:10.1007/978-1-4684-9148-7_43","type":"book-chapter","publisher":"Springer US","DOI":"10.1007/978-1-4684-9148-7_43","page":"423-430","source":"Crossref","title":"Hamiltonian Flows, SU(∞), SO(∞), USp(∞), and Strings","author":[{"given":"Cosmas","family":"Zachos","sequence":"first","affiliation":[]}],"container-title":"NATO ASI Series","original-title":[],"language":"en","issued":{"date-parts":[[1990]]},"ISBN":["9781468491500","9781468491487"],"URL":"http://dx.doi.org/10.1007/978-1-4684-9148-7_43","ISSN":["0258-1221"],"note":"publisher-location:Boston, MA"},{"id":"arxiv:2401.12105","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physrevlett.134.050202","source":"Crossref","title":"Magic Resource Can Enhance the Quantum Capacity of Channels","volume":"134","author":[{"ORCID":"https://orcid.org/0000-0001-8419-8736","authenticated-orcid":true,"given":"Kaifeng","family":"Bu","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/00rs6vg23","id-type":"ROR","asserted-by":"publisher"}],"name":"The Ohio State University"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]},{"ORCID":"https://orcid.org/0000-0003-4236-4833","authenticated-orcid":true,"given":"Arthur","family":"Jaffe","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"},{"id":[{"id":"https://ror.org/03vek6s52","id-type":"ROR","asserted-by":"publisher"}],"name":"Harvard University"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.134.050202","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"050202","note":"arxivid:2401.12105"},{"id":"arxiv:quant-ph/0512155","type":"journal-article","publisher":"IOP Publishing","issue":"11","DOI":"10.1088/0305-4470/39/11/010","page":"2701-2721","source":"Crossref","title":"Valence bond solid formalism for<i>d</i>-level one-way quantum computation","volume":"39","author":[{"given":"Sean","family":"Clark","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and General","original-title":[],"issued":{"date-parts":[[2006,3,1]]},"URL":"http://dx.doi.org/10.1088/0305-4470/39/11/010","ISSN":["0305-4470","1361-6447"],"container-title-short":"J. Phys. A: Math. Gen.","note":"alternative-id:S0305-4470(06)16463-X\narxivid:quant-ph/0512155"},{"id":"doi:10.1088/1751-8113/43/4/042001","type":"journal-article","publisher":"IOP Publishing","issue":"4","DOI":"10.1088/1751-8113/43/4/042001","page":"042001","source":"Crossref","title":"Symplectic transformations and quantum tomography in finite quantum systems","volume":"43","author":[{"given":"A","family":"Vourdas","sequence":"first","affiliation":[]},{"given":"C","family":"Banderier","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2010,1,4]]},"URL":"http://dx.doi.org/10.1088/1751-8113/43/4/042001","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(10)38437-X"},{"id":"arxiv:1101.1519","type":"journal-article","publisher":"Elsevier BV","issue":"5-6","DOI":"10.1016/j.physleta.2013.12.009","page":"505-509","source":"Crossref","title":"Standard form of qudit stabilizer groups","volume":"378","author":[{"given":"Vlad","family":"Gheorghiu","sequence":"first","affiliation":[]}],"container-title":"Physics Letters A","original-title":[],"language":"en","issued":{"date-parts":[[2014,1]]},"URL":"http://dx.doi.org/10.1016/j.physleta.2013.12.009","ISSN":["0375-9601"],"container-title-short":"Physics Letters A","note":"alternative-id:S0375960113011080\narxivid:1101.1519"},{"id":"arxiv:1102.3354","type":"journal-article","publisher":"Rinton Press","issue":"1&2","abstract":"<jats:p>The stabilizer formalism is a scheme, generalizing well-known techniques developed by Gottesman~\\cite{GottPhD} in the case of qubits, to efficiently simulate a class of transformations (\\emph{stabilizer circuits}, which include the quantum Fourier transform and highly entangling operations) on standard basis states of $d$-dimensional qudits. To determine the state of a simulated system, existing treatments involve the computation of cumulative phase factors which involve quadratic dependencies. We present a simple formalism in which Pauli operators are represented using displacement operators in discrete phase space, expressing the evolution of the state via linear transformations modulo $D \\le 2d$. We thus obtain a simple proof that simulating stabilizer circuits on $n$ qudits, involving any constant number of measurement rounds, is complete for the complexity class \\coMod[d]\\Log\\ and may be simulated by $O(\\log(n)^2)$-depth circuits for any constant $d \\ge 2$.</jats:p>","DOI":"10.26421/qic13.1-2-6","page":"73-115","source":"Crossref","title":"A linearized stabilizer formalism for systems of finite dimension","volume":"13","author":[{"given":"Niel","family":"de Beaudrap","sequence":"first","affiliation":[]}],"container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2013,1]]},"URL":"http://dx.doi.org/10.26421/QIC13.1-2-6","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC","note":"arxivid:1102.3354"},{"id":"arxiv:2008.00959","type":"journal-article","publisher":"Frontiers Media SA","DOI":"10.3389/fphy.2020.589504","source":"Crossref","title":"Qudits and High-Dimensional Quantum Computing","volume":"8","author":[{"given":"Yuchen","family":"Wang","sequence":"first","affiliation":[]},{"given":"Zixuan","family":"Hu","sequence":"additional","affiliation":[]},{"given":"Barry C.","family":"Sanders","sequence":"additional","affiliation":[]},{"given":"Sabre","family":"Kais","sequence":"additional","affiliation":[]}],"container-title":"Frontiers in Physics","original-title":[],"issued":{"date-parts":[[2020,11,10]]},"URL":"http://dx.doi.org/10.3389/fphy.2020.589504","ISSN":["2296-424X"],"container-title-short":"Front. Phys.","page":"589504","note":"alternative-id:10.3389/fphy.2020.589504\narxivid:2008.00959"},{"id":"arxiv:1307.5087","type":"journal-article","publisher":"IOP Publishing","issue":"30","DOI":"10.1088/1751-8113/47/30/305303","page":"305303","source":"Crossref","title":"An ideal characterization of the Clifford operators","volume":"47","author":[{"given":"J M","family":"Farinholt","sequence":"first","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2014,7,15]]},"URL":"http://dx.doi.org/10.1088/1751-8113/47/30/305303","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"arxivid:1307.5087"},{"id":"arxiv:1803.03228","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.98.032304","source":"Crossref","title":"Normal form for single-qutrit Clifford+<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>T</mml:mi></mml:math>operators and synthesis of single-qutrit gates","volume":"98","author":[{"given":"Shiroman","family":"Prakash","sequence":"first","affiliation":[]},{"given":"Akalank","family":"Jain","sequence":"additional","affiliation":[]},{"given":"Bhakti","family":"Kapur","sequence":"additional","affiliation":[]},{"given":"Shubangi","family":"Seth","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2018,9,5]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.98.032304","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032304","note":"arxivid:1803.03228"},{"id":"arxiv:2011.07970","type":"article-journal","author":[{"given":"Akalank","family":"Jain"},{"given":"Amolak Ratan","family":"Kalra"},{"given":"Shiroman","family":"Prakash"}],"title":"A Normal Form for Single-Qudit Clifford+$T$ Operators","issued":{"date-parts":[[2020,11,16]]},"note":"arxivid:2011.07970\narxiv_version_number:1"},{"id":"arxiv:1504.03383","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.93.012313","source":"Crossref","title":"Efficient topological compilation for a weakly integral anyonic model","volume":"93","author":[{"given":"Alex","family":"Bocharov","sequence":"first","affiliation":[]},{"given":"Xingshan","family":"Cui","sequence":"additional","affiliation":[]},{"given":"Vadym","family":"Kliuchnikov","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2016,1,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.93.012313","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"012313","note":"arxivid:1504.03383"},{"id":"arxiv:1810.04710","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s00039-022-00593-9","page":"193-235","source":"Crossref","title":"Ramanujan Complexes and Golden Gates in PU(3)","volume":"32","author":[{"given":"Shai","family":"Evra","sequence":"first","affiliation":[]},{"given":"Ori","family":"Parzanchevski","sequence":"additional","affiliation":[]}],"container-title":"Geometric and Functional Analysis","original-title":[],"language":"en","issued":{"date-parts":[[2022,3,26]]},"URL":"http://dx.doi.org/10.1007/s00039-022-00593-9","ISSN":["1016-443X","1420-8970"],"container-title-short":"Geom. Funct. Anal.","note":"alternative-id:593\narxivid:1810.04710"},{"id":"arxiv:2311.08696","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In this paper we study single qutrit circuits consisting of words over the Clifford<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>+</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi></mml:mrow></mml:math> cyclotomic gate set, where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mtext>diag</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>&amp;#x00B1;</mml:mo><mml:msup><mml:mi>&amp;#x03BE;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>a</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>&amp;#x00B1;</mml:mo><mml:msup><mml:mi>&amp;#x03BE;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mo>&amp;#x00B1;</mml:mo><mml:msup><mml:mi>&amp;#x03BE;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>c</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03BE;</mml:mi></mml:math> is a primitive <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>9</mml:mn></mml:math>-th root of unity and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:math> are integers. We characterize classes of qutrit unit vectors <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>z</mml:mi></mml:math> with entries in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>&amp;#x03BE;</mml:mi><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&amp;#x03C7;</mml:mi></mml:mfrac><mml:mo stretchy=\"false\">]</mml:mo></mml:math> based on the possibility of reducing their smallest denominator exponent (sde) with respect to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03C7;</mml:mi><mml:mo>:=</mml:mo><mml:mn>1</mml:mn><mml:mo>&amp;#x2013;</mml:mo><mml:mi>&amp;#x03BE;</mml:mi><mml:mo>,</mml:mo></mml:math> by acting an appropriate gate in Clifford<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>+</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi></mml:mrow></mml:math>. We do this by studying the notion of `derivatives mod <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>3</mml:mn></mml:math>' of an arbitrary element of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>&amp;#x03BE;</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:math> and using it to study the smallest denominator exponent of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>H</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi></mml:mrow><mml:mi>z</mml:mi></mml:math> where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>H</mml:mi></mml:math> is the qutrit Hadamard gate and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi></mml:mrow></mml:math>. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>+</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi></mml:mrow></mml:math> gates naturally arise as gates with sde <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>0</mml:mn></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>3</mml:mn></mml:math> in the group <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>&amp;#x03BE;</mml:mi><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&amp;#x03C7;</mml:mi></mml:mfrac><mml:mo stretchy=\"false\">]</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math> of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>3</mml:mn><mml:mo>&amp;#x00D7;</mml:mo><mml:mn>3</mml:mn></mml:math> unitaries with entries in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"double-struck\">Z</mml:mi></mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>&amp;#x03BE;</mml:mi><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&amp;#x03C7;</mml:mi></mml:mfrac><mml:mo stretchy=\"false\">]</mml:mo></mml:math>. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>+</mml:mo><mml:mi>R</mml:mi></mml:math> and recover the previously known exact synthesis algorithm of Kliuchnikov, Maslov, Mosca (2012). The framework developed to formulate qutrit gate synthesis for Clifford<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo>+</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">D</mml:mi></mml:mrow></mml:math> extends to qudits of arbitrary prime power.</jats:p>","DOI":"10.22331/q-2025-02-26-1647","page":"1647","source":"Crossref","title":"Synthesis and Arithmetic of Single Qutrit Circuits","volume":"9","author":[{"given":"Amolak Ratan","family":"Kalra","sequence":"first","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada"},{"name":"David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada"},{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada"}]},{"given":"Michele","family":"Mosca","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada"},{"name":"Dept. of Combinatorics $\\&$ Optimization, University of Waterloo, Waterloo, Ontario, Canada"},{"name":"Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada"}]},{"given":"Dinesh","family":"Valluri","sequence":"additional","affiliation":[{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada"},{"name":"Dept. of Combinatorics $\\&$ Optimization, University of Waterloo, Waterloo, Ontario, Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,2,26]]},"URL":"http://dx.doi.org/10.22331/q-2025-02-26-1647","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2311.08696"},{"id":"arxiv:2401.16120","type":"article-journal","author":[{"given":"Shai","family":"Evra"},{"given":"Ori","family":"Parzanchevski"}],"title":"Arithmeticity, thinness and efficiency of qutrit Clifford+T gates","issued":{"date-parts":[[2024,11,12]]},"note":"arxivid:2401.16120\narxiv_version_number:2"},{"id":"arxiv:2405.08147","type":"article-journal","author":[{"given":"Amolak Ratan","family":"Kalra"},{"given":"Manimugdha","family":"Saikia"},{"given":"Dinesh","family":"Valluri"},{"given":"Sam","family":"Winnick"},{"given":"Jon","family":"Yard"}],"title":"Multi-qutrit exact synthesis","issued":{"date-parts":[[2024,5,13]]},"note":"arxivid:2405.08147\narxiv_version_number:1"},{"id":"arxiv:1808.02406","type":"journal-article","publisher":"American Physical Society (APS)","issue":"5","DOI":"10.1103/physreva.99.052307","source":"Crossref","title":"Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits","volume":"99","author":[{"given":"Yifei","family":"Huang","sequence":"first","affiliation":[]},{"given":"Peter","family":"Love","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2019,5,6]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.99.052307","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"052307","note":"arxivid:1808.02406"},{"id":"arxiv:1206.1598","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.86.022316","source":"Crossref","title":"Qudit versions of the qubit<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><mml:mrow><mml:mi>π</mml:mi><mml:mo>/</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:math>gate","volume":"86","author":[{"given":"Mark","family":"Howard","sequence":"first","affiliation":[]},{"given":"Jiri","family":"Vala","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2012,8,15]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.86.022316","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022316","note":"arxivid:1206.1598"},{"id":"arxiv:2209.07607","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Multipartite entanglement is one of the hallmarks of quantum mechanics and is central to quantum information processing. In this work we show that concentratable entanglement (CE), an operationally motivated entanglement measure, induces a hierarchy upon pure states from which different entanglement structures can be experimentally certified. In particular, we find that nearly all genuine multipartite entangled states can be verified through the CE. Interestingly, GHZ states prove to be far from maximally entangled according to this measure. Instead we find the exact maximal value and corresponding states for up to 18 qubits and show that these correspond to extremal quantum error correcting codes. The latter allows us to unravel a deep connection between CE and coding theory. Finally, our results also offer an alternative proof, on up to 31 qubits, that absolutely maximally entangled states do not exist.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/physrevresearch.6.023019","source":"Crossref","title":"Hierarchy of multipartite correlations based on concentratable entanglement","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-1712-9148","authenticated-orcid":true,"given":"Louis","family":"Schatzki","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA; Illinois Quantum Information Science and Technology (IQUIST) Center, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA; and Information Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA"}]},{"ORCID":"https://orcid.org/0009-0005-4883-4148","authenticated-orcid":true,"given":"Guangkuo","family":"Liu","sequence":"additional","affiliation":[{"name":"JILA, University of Colorado/NIST, Boulder, Colorado 80309, USA and Department of Physics, University of Colorado, Boulder CO 80309, USA"}]},{"ORCID":"https://orcid.org/0000-0002-2757-3170","authenticated-orcid":true,"given":"M.","family":"Cerezo","sequence":"additional","affiliation":[{"name":"Information Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and Quantum Science Center, Oak Ridge, Tennessee 37931, USA"}]},{"ORCID":"https://orcid.org/0000-0001-6990-7821","authenticated-orcid":true,"given":"Eric","family":"Chitambar","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA and Illinois Quantum Information Science and Technology (IQUIST) Center, University of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA"}]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2024,4,4]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.6.023019","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"023019","note":"arxivid:2209.07607"},{"id":"arxiv:2501.07812","type":"article-journal","author":[{"given":"Svyatoslav","family":"Kushnarev"},{"given":"Hassan Jameel","family":"Asghar"}],"title":"The Qudit Cirq Library: An Extension of Google's Cirq Library for Qudits","issued":{"date-parts":[[2025,1,16]]},"note":"arxivid:2501.07812\narxiv_version_number:2"},{"id":"doi:10.1007/978-3-319-44906-7","type":"book","publisher":"Springer International Publishing","DOI":"10.1007/978-3-319-44906-7","source":"Crossref","title":"Group Representation for Quantum Theory","author":[{"given":"Masahito","family":"Hayashi","sequence":"first","affiliation":[]}],"container-title":[],"original-title":[],"language":"en","issued":{"date-parts":[[2017]]},"ISBN":["9783319449043","9783319449067"],"URL":"http://dx.doi.org/10.1007/978-3-319-44906-7","note":"publisher-location:Cham"},{"id":"arxiv:quant-ph/9702033","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.56.r1","page":"R1-R4","source":"Crossref","title":"Five quantum register error correction code for higher spin systems","volume":"56","author":[{"given":"H. F.","family":"Chau","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,7,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.56.R1","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9702033"},{"id":"arxiv:2509.25587","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/tqe.2026.3669054","page":"1-15","source":"Crossref","title":"Encoder Circuit Optimization for Nonbinary Quantum Error Correction Codes in Prime Dimensions: An Algorithmic Framework","volume":"7","author":[{"ORCID":"https://orcid.org/0009-0003-6438-084X","authenticated-orcid":false,"given":"Aditya","family":"Sodhani","sequence":"first","affiliation":[{"name":"University of Minnesota Twin Cities, Minneapolis, MN, USA"}]},{"ORCID":"https://orcid.org/0000-0001-6543-2793","authenticated-orcid":false,"given":"Keshab K.","family":"Parhi","sequence":"additional","affiliation":[{"name":"University of Minnesota Twin Cities, Minneapolis, MN, USA"}]}],"container-title":"IEEE Transactions on Quantum Engineering","original-title":[],"issued":{"date-parts":[[2026]]},"URL":"http://dx.doi.org/10.1109/TQE.2026.3669054","ISSN":["2689-1808"],"container-title-short":"IEEE Trans. Quantum Eng.","note":"arxivid:2509.25587"},{"id":"arxiv:1202.2326","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/14/6/063006","page":"063006","source":"Crossref","title":"Qutrit magic state distillation","volume":"14","author":[{"given":"Hussain","family":"Anwar","sequence":"first","affiliation":[]},{"given":"Earl T","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Dan E","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2012,6,7]]},"URL":"http://dx.doi.org/10.1088/1367-2630/14/6/063006","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1202.2326"},{"id":"arxiv:quant-ph/9901025","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevlett.83.648","page":"648-651","source":"Crossref","title":"How to Share a Quantum Secret","volume":"83","author":[{"given":"Richard","family":"Cleve","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Gottesman","sequence":"additional","affiliation":[]},{"given":"Hoi-Kwong","family":"Lo","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[1999,7,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.83.648","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","note":"arxivid:quant-ph/9901025"},{"id":"arxiv:quant-ph/9610023","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.55.r839","page":"R839-R841","source":"Crossref","title":"Correcting quantum errors in higher spin systems","volume":"55","author":[{"given":"H. F.","family":"Chau","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[1997,2,1]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.55.R839","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","note":"arxivid:quant-ph/9610023"},{"id":"arxiv:quant-ph/9910059","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/3-540-46796-3_23","page":"231-244","source":"Crossref","title":"Quantum Reed—Solomon Codes","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]},{"given":"Willi","family":"Geiselmann","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Beth","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[1999]]},"ISBN":["9783540667230","9783540467960"],"URL":"http://dx.doi.org/10.1007/3-540-46796-3_23","ISSN":["0302-9743"],"note":"publisher-location:Berlin, Heidelberg\narxivid:quant-ph/9910059"},{"id":"arxiv:1811.08461","type":"journal-article","publisher":"American Physical Society (APS)","issue":"7","DOI":"10.1103/physrevlett.123.070507","source":"Crossref","title":"Towards Low Overhead Magic State Distillation","volume":"123","author":[{"given":"Anirudh","family":"Krishna","sequence":"first","affiliation":[]},{"given":"Jean-Pierre","family":"Tillich","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2019,8,16]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.123.070507","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"070507","note":"arxivid:1811.08461"},{"id":"arxiv:1712.05892","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevx.8.031051","source":"Crossref","title":"Fracton Models on General Three-Dimensional Manifolds","volume":"8","author":[{"given":"Wilbur","family":"Shirley","sequence":"first","affiliation":[]},{"given":"Kevin","family":"Slagle","sequence":"additional","affiliation":[]},{"given":"Zhenghan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2018,8,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.8.031051","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"031051","note":"arxivid:1712.05892"},{"id":"arxiv:2001.01722","type":"journal-article","publisher":"World Scientific Pub Co Pte Ltd","issue":"06","abstract":"<jats:p> Fractons are a new type of quasiparticle which are immobile in isolation, but can often move by forming bound states. Fractons are found in a variety of physical settings, such as spin liquids and elasticity theory, and exhibit unusual phenomenology, such as gravitational physics and localization. The past several years have seen a surge of interest in these exotic particles, which have come to the forefront of modern condensed matter theory. In this review, we provide a broad treatment of fractons, ranging from pedagogical introductory material to discussions of recent advances in the field. We begin by demonstrating how the fracton phenomenon naturally arises as a consequence of higher moment conservation laws, often accompanied by the emergence of tensor gauge theories. We then provide a survey of fracton phases in spin models, along with the various tools used to characterize them, such as the foliation framework. We discuss in detail the manifestation of fracton physics in elasticity theory, as well as the connections of fractons with localization and gravitation. Finally, we provide an overview of some recently proposed platforms for fracton physics, such as Majorana islands and hole-doped antiferromagnets. We conclude with some open questions and an outlook on the field. </jats:p>","DOI":"10.1142/s0217751x20300033","page":"2030003","source":"Crossref","title":"Fracton phases of matter","volume":"35","author":[{"given":"Michael","family":"Pretko","sequence":"first","affiliation":[{"name":"Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA"}]},{"given":"Xie","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA"}]},{"given":"Yizhi","family":"You","sequence":"additional","affiliation":[{"name":"Princeton Center for Theoretical Science, Princeton University, NJ 08544, USA"}]}],"container-title":"International Journal of Modern Physics A","original-title":[],"language":"en","issued":{"date-parts":[[2020,2,29]]},"URL":"http://dx.doi.org/10.1142/S0217751X20300033","ISSN":["0217-751X","1793-656X"],"container-title-short":"Int. J. Mod. Phys. A","note":"alternative-id:10.1142/S0217751X20300033\narxivid:2001.01722"},{"id":"arxiv:1202.0052","type":"article-journal","author":[{"given":"Isaac H.","family":"Kim"}],"title":"3D local qupit quantum code without string logical operator","issued":{"date-parts":[[2012,1,31]]},"note":"arxivid:1202.0052\narxiv_version_number:1"},{"id":"arxiv:2504.09847","type":"journal-article","publisher":"American Physical Society (APS)","issue":"15","DOI":"10.1103/qctl-hwvh","source":"Crossref","title":"<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi mathvariant=\"double-struck\">Z</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math>\n generalizations of three-dimensional stabilizer codes","volume":"112","author":[{"ORCID":"https://orcid.org/0009-0005-2103-0991","authenticated-orcid":true,"given":"Chanbeen","family":"Lee","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/04xysgw12","id-type":"ROR","asserted-by":"publisher"}],"name":"Pohang University of Science and Technology"},{"id":[{"id":"https://ror.org/00y0zf565","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Basic Science"}]},{"ORCID":"https://orcid.org/0009-0002-9009-5949","authenticated-orcid":true,"given":"Yaozong","family":"Hu","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Tokyo"}]},{"ORCID":"https://orcid.org/0000-0003-4957-3013","authenticated-orcid":true,"given":"Gil Young","family":"Cho","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05apxxy63","id-type":"ROR","asserted-by":"publisher"}],"name":"Korea Advanced Institute of Science and Technology"},{"id":[{"id":"https://ror.org/00y0zf565","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Basic Science"},{"id":[{"id":"https://ror.org/011hxwn54","id-type":"ROR","asserted-by":"publisher"}],"name":"Asia-Pacific Center for Theoretical Physics"}]},{"ORCID":"https://orcid.org/0000-0002-8112-021X","authenticated-orcid":true,"given":"Haruki","family":"Watanabe","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/057zh3y96","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Tokyo"}]}],"container-title":"Physical Review B","original-title":[],"language":"en","issued":{"date-parts":[[2025,10,16]]},"URL":"http://dx.doi.org/10.1103/qctl-hwvh","ISSN":["2469-9950","2469-9969"],"container-title-short":"Phys. Rev. B","page":"155136","note":"arxivid:2504.09847"},{"id":"doi:10.23919/ISITA.2018.8664389","type":"proceedings-article","publisher":"IEEE","DOI":"10.23919/isita.2018.8664389","page":"334-338","source":"Crossref","title":"Performance of Nonbinary Cubic Codes","author":[{"given":"Arun John","family":"Moncy","sequence":"first","affiliation":[]},{"given":"Pradeep","family":"Kiran Sarvepalli","sequence":"additional","affiliation":[]}],"event":"2018 International Symposium on Information Theory and Its Applications (ISITA)","container-title":"2018 International Symposium on Information Theory and Its Applications (ISITA)","original-title":[],"issued":{"date-parts":[[2018,10]]},"URL":"http://dx.doi.org/10.23919/ISITA.2018.8664389"},{"id":"arxiv:2403.06228","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We show that using qutrits rather than qubits leads to a substantial reduction in the overhead cost associated with an approach to fault-tolerant quantum computing known as magic state distillation. We construct a family of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>9</mml:mn><mml:mi>m</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mn>3</mml:mn></mml:msub></mml:math> triorthogonal qutrit error-correcting codes for any positive integers <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>m</mml:mi></mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi></mml:math> with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi><mml:mo>&amp;#x2264;</mml:mo><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn></mml:math> that are suitable for magic state distillation. In magic state distillation, the number of ancillae required to produce a magic state with target error rate <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi></mml:math> is <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>O</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>log</mml:mi><mml:mi>&amp;#x03B3;</mml:mi></mml:msup><mml:mo>&amp;#x2061;</mml:mo><mml:msup><mml:mi>&amp;#x03F5;</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where the yield parameter <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B3;</mml:mi></mml:math> characterizes the overhead cost. For <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn></mml:math>, our codes have <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B3;</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>log</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mn>6</mml:mn><mml:mrow><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, which tends to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn></mml:math> as <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">&amp;#x2192;</mml:mo><mml:mi mathvariant=\"normal\">&amp;#x221E;</mml:mi></mml:math>. Moreover, the <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>20</mml:mn><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mn>3</mml:mn></mml:msub></mml:math> qutrit code that arises from our construction when <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math> already has a yield parameter of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1.51</mml:mn></mml:math> which outperforms all known qubit triorthogonal codes of size less than a few hundred qubits.</jats:p>","DOI":"10.22331/q-2025-06-12-1768","page":"1768","source":"Crossref","title":"Low Overhead Qutrit Magic State Distillation","volume":"9","author":[{"given":"Shiroman","family":"Prakash","sequence":"first","affiliation":[{"name":"Department of Physics and Computer Science, Dayalbagh Educational Institute, Agra-282005, India"}]},{"given":"Tanay","family":"Saha","sequence":"additional","affiliation":[{"name":"Department of Physics and Computer Science, Dayalbagh Educational Institute, Agra-282005, India"},{"name":"Present Address: Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2025,6,12]]},"URL":"http://dx.doi.org/10.22331/q-2025-06-12-1768","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2403.06228"},{"id":"arxiv:quant-ph/0304054","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.68.062303","source":"Crossref","title":"Quantum computation based on<i>d</i>-level cluster state","volume":"68","author":[{"given":"D. L.","family":"Zhou","sequence":"first","affiliation":[]},{"given":"B.","family":"Zeng","sequence":"additional","affiliation":[]},{"given":"Z.","family":"Xu","sequence":"additional","affiliation":[]},{"given":"C. P.","family":"Sun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2003,12,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.68.062303","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062303","note":"arxivid:quant-ph/0304054"},{"id":"arxiv:0912.2017","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.81.032326","source":"Crossref","title":"Location of quantum information in additive graph codes","volume":"81","author":[{"given":"Vlad","family":"Gheorghiu","sequence":"first","affiliation":[]},{"given":"Shiang Yong","family":"Looi","sequence":"additional","affiliation":[]},{"given":"Robert B.","family":"Griffiths","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2010,3,25]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.81.032326","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032326","note":"arxivid:0912.2017"},{"id":"doi:10.1038/s41567-018-0347-x","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1038/s41567-018-0347-x","page":"148-153","source":"Crossref","title":"High-dimensional one-way quantum processing implemented on d-level cluster states","volume":"15","author":[{"given":"Christian","family":"Reimer","sequence":"first","affiliation":[]},{"given":"Stefania","family":"Sciara","sequence":"additional","affiliation":[]},{"given":"Piotr","family":"Roztocki","sequence":"additional","affiliation":[]},{"given":"Mehedi","family":"Islam","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9534-4476","authenticated-orcid":false,"given":"Luis","family":"Romero Cortés","sequence":"additional","affiliation":[]},{"given":"Yanbing","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Bennet","family":"Fischer","sequence":"additional","affiliation":[]},{"given":"Sébastien","family":"Loranger","sequence":"additional","affiliation":[]},{"given":"Raman","family":"Kashyap","sequence":"additional","affiliation":[]},{"given":"Alfonso","family":"Cino","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8263-8507","authenticated-orcid":false,"given":"Sai T.","family":"Chu","sequence":"additional","affiliation":[]},{"given":"Brent E.","family":"Little","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5195-1744","authenticated-orcid":false,"given":"David J.","family":"Moss","sequence":"additional","affiliation":[]},{"given":"Lucia","family":"Caspani","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1835-2250","authenticated-orcid":false,"given":"William J.","family":"Munro","sequence":"additional","affiliation":[]},{"given":"José","family":"Azaña","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5323-3850","authenticated-orcid":false,"given":"Michael","family":"Kues","sequence":"additional","affiliation":[]},{"given":"Roberto","family":"Morandotti","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2018,12,3]]},"URL":"http://dx.doi.org/10.1038/s41567-018-0347-x","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nature Phys","note":"alternative-id:347"},{"id":"doi:10.1142/S0219749914500208","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"04","abstract":"<jats:p> This paper considers the construction of quantum error correcting codes from linear codes over finite commutative Frobenius rings. We extend the Calderbank–Shor–Steane (CSS) construction to these rings. Further, quantum codes are extended to matrix product codes. Quantum codes over 𝔽<jats:sub>p<jats:sup>k</jats:sup></jats:sub> are also obtained from linear codes over rings using the generalized Gray map. </jats:p>","DOI":"10.1142/s0219749914500208","page":"1450020","source":"Crossref","title":"Quantum codes over rings","volume":"12","author":[{"given":"Kenza","family":"Guenda","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, USTHB, BP. 32 El Alia Bab Ezzouar, 16111 Algiers, Algeria"}]},{"given":"T. Aaron","family":"Gulliver","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Victoria, P. O. Box 1700, STN CSS, Victoria, BC, Canada V8W2Y2, Canada"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2014,6]]},"URL":"http://dx.doi.org/10.1142/S0219749914500208","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749914500208"},{"id":"arxiv:2405.03559","type":"article-journal","author":[{"given":"Samo","family":"Novák"}],"title":"Homological Quantum Error Correction with Torsion","issued":{"date-parts":[[2024,5,6]]},"note":"arxivid:2405.03559\narxiv_version_number:1"},{"id":"arxiv:2501.04888","type":"article-journal","author":[{"given":"Lane G.","family":"Gunderman"}],"title":"Beyond Integral-Domain Stabilizer Codes","issued":{"date-parts":[[2025,1,8]]},"note":"arxivid:2501.04888\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0702004","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1007/s00340-006-2510-9","page":"367-370","source":"Crossref","title":"Non-negative Wigner functions in prime dimensions","volume":"86","author":[{"given":"D.","family":"Gross","sequence":"first","affiliation":[]}],"container-title":"Applied Physics B","original-title":[],"language":"en","issued":{"date-parts":[[2006,12,5]]},"URL":"http://dx.doi.org/10.1007/s00340-006-2510-9","ISSN":["0946-2171","1432-0649"],"container-title-short":"Appl. Phys. B","note":"alternative-id:2510\narxivid:quant-ph/0702004"},{"id":"arxiv:1712.08628","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","abstract":"<jats:title>Abstract</jats:title><jats:p>Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the <jats:italic>t</jats:italic>-fold tensor powers <jats:inline-formula><jats:alternatives><jats:tex-math>$$U^{\\otimes t}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msup>\n                    <mml:mi>U</mml:mi>\n                    <mml:mrow>\n                      <mml:mo>⊗</mml:mo>\n                      <mml:mi>t</mml:mi>\n                    </mml:mrow>\n                  </mml:msup>\n                </mml:math></jats:alternatives></jats:inline-formula> of all unitaries <jats:inline-formula><jats:alternatives><jats:tex-math>$$U\\in U(d)$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>U</mml:mi>\n                    <mml:mo>∈</mml:mo>\n                    <mml:mi>U</mml:mi>\n                    <mml:mo>(</mml:mo>\n                    <mml:mi>d</mml:mi>\n                    <mml:mo>)</mml:mo>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula> is spanned by the permutations of the <jats:italic>t</jats:italic> tensor factors. In this work, we describe a similar duality theory for tensor powers of <jats:italic>Clifford unitaries</jats:italic>. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications:\n<jats:list list-type=\"bullet\">\n                <jats:list-item>\n                  <jats:p>We resolve an open problem in <jats:italic>quantum property testing</jats:italic> by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group.</jats:p>\n                </jats:list-item>\n                <jats:list-item>\n                  <jats:p>We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding <jats:italic>de Finetti theorems</jats:italic>, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well).</jats:p>\n                </jats:list-item>\n                <jats:list-item>\n                  <jats:p>We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related <jats:italic>mana</jats:italic>) – a measure relevant to fault-tolerant quantum computation. The result constitutes a <jats:italic>robust</jats:italic> generalization of the <jats:italic>discrete Hudson theorem</jats:italic>.</jats:p>\n                </jats:list-item>\n                <jats:list-item>\n                  <jats:p>We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.\n</jats:p>\n                </jats:list-item>\n              </jats:list></jats:p>","DOI":"10.1007/s00220-021-04118-7","page":"1325-1393","source":"Crossref","title":"Schur–Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations","volume":"385","author":[{"given":"David","family":"Gross","sequence":"first","affiliation":[]},{"given":"Sepehr","family":"Nezami","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Walter","sequence":"additional","affiliation":[]}],"container-title":"Communications in Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2021,6,29]]},"URL":"http://dx.doi.org/10.1007/s00220-021-04118-7","ISSN":["0010-3616","1432-0916"],"container-title-short":"Commun. Math. Phys.","note":"alternative-id:4118\narxivid:1712.08628"},{"id":"arxiv:2403.13632","type":"article-journal","author":[{"given":"Kaifeng","family":"Bu"}],"title":"Extremality of stabilizer states","issued":{"date-parts":[[2024,3,20]]},"note":"arxivid:2403.13632\narxiv_version_number:1"},{"id":"doi:10.1109/ACCESS.2018.2865918","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","DOI":"10.1109/access.2018.2865918","page":"45699-45706","source":"Crossref","title":"The Gilbert-Varshamov Bound for Stabilizer Codes Over &lt;inline-formula&gt; \n                     &lt;tex-math notation=\"LaTeX\"&gt;$\\mathbb{Z}_m$ &lt;/tex-math&gt;\n                  &lt;/inline-formula&gt;","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0001-8460-3527","authenticated-orcid":false,"given":"Nianqi","family":"Tang","sequence":"first","affiliation":[]},{"given":"Zhuo","family":"Li","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2967-8478","authenticated-orcid":false,"given":"Lijuan","family":"Xing","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-7714-3822","authenticated-orcid":false,"given":"Ming","family":"Zhang","sequence":"additional","affiliation":[]}],"container-title":"IEEE Access","original-title":[],"issued":{"date-parts":[[2018]]},"URL":"http://dx.doi.org/10.1109/ACCESS.2018.2865918","ISSN":["2169-3536"],"container-title-short":"IEEE Access"},{"id":"doi:10.1109/ISIT.2012.6283558","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/isit.2012.6283558","page":"165-169","source":"Crossref","title":"Stabilizer codes over Frobenius rings","author":[{"given":"Sushma","family":"Nadella","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[]}],"event":"2012 IEEE International Symposium on Information Theory - ISIT","container-title":"2012 IEEE International Symposium on Information Theory Proceedings","original-title":[],"issued":{"date-parts":[[2012,7]]},"URL":"http://dx.doi.org/10.1109/ISIT.2012.6283558"},{"id":"arxiv:2003.07164","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.102.042409","source":"Crossref","title":"Qutrit and ququint magic states","volume":"102","author":[{"given":"Akalank","family":"Jain","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2033-3091","authenticated-orcid":true,"given":"Shiroman","family":"Prakash","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2020,10,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.102.042409","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042409","note":"arxivid:2003.07164"},{"id":"arxiv:2106.08251","type":"article-journal","author":[{"given":"Eric","family":"Sabo"},{"given":"Arun B.","family":"Aloshious"},{"given":"Kenneth R.","family":"Brown"}],"title":"Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes","issued":{"date-parts":[[2022,1,17]]},"note":"arxivid:2106.08251\narxiv_version_number:2"},{"id":"arxiv:2209.01449","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>In this work, we compute the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> stabilizer codes made up of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>-dimensional qudits, for arbitrary positive integers <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>. In a seminal work by Gross \\cite{Gross2006} the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> stabilizer codes was computed for the case when <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is a prime (or the power of a prime, i.e., <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mi>m</mml:mi></mml:msup></mml:math>, but when the qudits are Galois-qudits). The proof in \\cite{Gross2006} is inapplicable to the non-prime case. For our proof, we introduce a group structure to <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> codes, and use this in conjunction with the Chinese remainder theorem to count the number of <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo stretchy=\"false\">[</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">]</mml:mo><mml:msub><mml:mo stretchy=\"false\">]</mml:mo><mml:mi>d</mml:mi></mml:msub></mml:math> codes. Our work overlaps with \\cite{Gross2006} when <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is a prime and in this case our results match exactly, but the results differ for the more generic case. Despite that, the overall order of magnitude of the number of stabilizer codes scales agnostic of whether the dimension is prime or non-prime. This is surprising since the method employed to count the number of stabilizer states (or more generally stabilizer codes) depends on whether <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math> is prime or not. The cardinality of stabilizer states, which was so far known only for the prime-dimensional case (and the Galois qudit prime-power dimensional case) plays an important role as a quantifier in many topics in quantum computing. Salient among these are the resource theory of magic, design theory, de Finetti theorem for stabilizer states, the study and optimisation of the classical simulability of Clifford circuits, the study of quantum contextuality of small-dimensional systems and the study of Wigner-functions. Our work makes available this quantifier for the generic case, and thus is an important step needed to place results for quantum computing with non-prime dimensional quantum systems on the same pedestal as prime-dimensional systems.</jats:p>","DOI":"10.22331/q-2023-07-06-1048","page":"1048","source":"Crossref","title":"Counting stabiliser codes for arbitrary dimension","volume":"7","author":[{"ORCID":"https://orcid.org/0000-0001-5986-248X","authenticated-orcid":false,"given":"Tanmay","family":"Singal","sequence":"first","affiliation":[{"name":"Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń, Poland"}]},{"given":"Che","family":"Chiang","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan"}]},{"given":"Eugene","family":"Hsu","sequence":"additional","affiliation":[{"name":"Quantum information center, Chung Yuan Christian University, No. 200, Zhongbei Rd., Zhongli Dist., Taoyuan City 320314, Taiwan"}]},{"given":"Eunsang","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of mathematical Data Science, Hanyang University, Ansan, Gyeonggi-do, 15588, Korea"}]},{"given":"Hsi-Sheng","family":"Goan","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan"},{"name":"Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan"},{"name":"Physics Division, National Center for Theoretical Sciences, Taipei, 10617, Taiwan"}]},{"ORCID":"https://orcid.org/0000-0002-3396-8427","authenticated-orcid":false,"given":"Min-Hsiu","family":"Hsieh","sequence":"additional","affiliation":[{"name":"Hon Hai Quantum Computing Research Center, Taipei, Taiwan"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2023,7,6]]},"URL":"http://dx.doi.org/10.22331/q-2023-07-06-1048","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2209.01449"},{"id":"manual:-M.-Grassl-private-communicat","type":"document","author":[{"family":"Grassl","given":"M."}],"genre":"private communication,","issued":"2024"},{"id":"arxiv:1506.08883","type":"journal-article","publisher":"AIP Publishing","issue":"1","abstract":"<jats:p>We consider wavefunctions which are non-negative in some tensor product basis. We study what possible teleportation can occur in such wavefunctions, giving a complete answer in some cases (when one system is a qubit) and partial answers elsewhere. We use this to show that a one-dimensional wavefunction which is non-negative and has zero correlation length can be written in a “coherent Gibbs state” form, as explained later. We conjecture that such holds in higher dimensions. Additionally, some results are provided on possible teleportation in general wavefunctions, explaining how Schmidt coefficients before measurement limit the possible Schmidt coefficients after measurement, and on the absence of a “generalized area law” [D. Aharonov et al., in Proceedings of Foundations of Computer Science (FOCS) (IEEE, 2014), p. 246; e-print arXiv.org:1410.0951] even for Hamiltonians with no sign problem. One of the motivations for this work is an attempt to prove a conjecture about ground state wavefunctions which have an “intrinsic” sign problem that cannot be removed by any quantum circuit. We show a weaker version of this, showing that the sign problem is intrinsic for commuting Hamiltonians in the same phase as the double semion model under the technical assumption that TQO-2 holds [S. Bravyi et al., J. Math. Phys. 51, 093512 (2010)].</jats:p>","DOI":"10.1063/1.4936216","source":"Crossref","title":"How quantum are non-negative wavefunctions?","volume":"57","author":[{"given":"M. B.","family":"Hastings","sequence":"first","affiliation":[{"name":"Microsoft Research Station Q, , Santa Barbara, California 93106-6105, USA and Quantum Architectures and Computation Group, , Redmond, Washington 98052, USA"},{"name":"Microsoft Research Station Q, , Santa Barbara, California 93106-6105, USA and Quantum Architectures and Computation Group, , Redmond, Washington 98052, USA"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2015,11,30]]},"URL":"http://dx.doi.org/10.1063/1.4936216","ISSN":["0022-2488","1089-7658"],"page":"015210","note":"arxivid:1506.08883"},{"id":"arxiv:2005.05343","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physrevresearch.2.033515","source":"Crossref","title":"Intrinsic sign problems in topological quantum field theories","volume":"2","author":[{"ORCID":"https://orcid.org/0000-0002-6744-4592","authenticated-orcid":true,"given":"Adam","family":"Smith","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2189-902X","authenticated-orcid":true,"given":"Omri","family":"Golan","sequence":"additional","affiliation":[]},{"given":"Zohar","family":"Ringel","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Research","original-title":[],"language":"en","issued":{"date-parts":[[2020,9,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevResearch.2.033515","ISSN":["2643-1564"],"container-title-short":"Phys. Rev. Research","page":"033515","note":"arxivid:2005.05343"},{"id":"arxiv:2509.03708","type":"article-journal","author":[{"given":"Leyna","family":"Shackleton"}],"title":"Twisted quantum doubles are sign problem-free","issued":{"date-parts":[[2026,3,16]]},"note":"arxivid:2509.03708\narxiv_version_number:2"},{"id":"arxiv:2404.08552","type":"article-journal","author":[{"given":"Shawn X.","family":"Cui"},{"given":"César","family":"Galindo"},{"given":"Diego","family":"Romero"}],"title":"Abelian Group Quantum Error Correction in Kitaev's Model","issued":{"date-parts":[[2024,4,12]]},"note":"arxivid:2404.08552\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0609070","type":"journal-article","publisher":"IOP Publishing","issue":"13","DOI":"10.1088/1751-8113/40/13/013","page":"3481-3505","source":"Crossref","title":"Qudit surface codes and gauge theory with finite cyclic groups","volume":"40","author":[{"given":"Stephen S","family":"Bullock","sequence":"first","affiliation":[]},{"given":"Gavin K","family":"Brennen","sequence":"additional","affiliation":[]}],"container-title":"Journal of Physics A: Mathematical and Theoretical","original-title":[],"issued":{"date-parts":[[2007,3,14]]},"URL":"http://dx.doi.org/10.1088/1751-8113/40/13/013","ISSN":["1751-8113","1751-8121"],"container-title-short":"J. Phys. A: Math. Theor.","note":"alternative-id:S1751-8113(07)33084-9\narxivid:quant-ph/0609070"},{"id":"arxiv:2110.08680","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.105.022621","source":"Crossref","title":"Quantum computation with charge-and-color-permuting twists in qudit color codes","volume":"105","author":[{"ORCID":"https://orcid.org/0000-0002-6298-7037","authenticated-orcid":true,"given":"Manoj G.","family":"Gowda","sequence":"first","affiliation":[]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,2,24]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.022621","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022621","note":"arxivid:2110.08680"},{"id":"arxiv:1701.02335","type":"journal-article","publisher":"IOP Publishing","issue":"11","DOI":"10.1088/1367-2630/aa939a","page":"113022","source":"Crossref","title":"Comparison of memory thresholds for planar qudit geometries","volume":"19","author":[{"given":"Jacob","family":"Marks","sequence":"first","affiliation":[]},{"given":"Tomas","family":"Jochym-O’Connor","sequence":"additional","affiliation":[]},{"given":"Vlad","family":"Gheorghiu","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2017,11,15]]},"URL":"http://dx.doi.org/10.1088/1367-2630/aa939a","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1701.02335"},{"id":"arxiv:2211.00299","type":"journal-article","publisher":"AIP Publishing","issue":"5","abstract":"<jats:p>Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.</jats:p>","DOI":"10.1063/5.0134010","source":"Crossref","title":"Ground state degeneracy on torus in a family of ZN toric code","volume":"64","author":[{"ORCID":"https://orcid.org/0000-0002-8112-021X","authenticated-orcid":false,"given":"Haruki","family":"Watanabe","sequence":"first","affiliation":[{"name":"Department of Applied Physics, The University of Tokyo 1 , Tokyo, Japan"}]},{"given":"Meng","family":"Cheng","sequence":"additional","affiliation":[{"name":"Department of Physics, Yale University 2 , New Haven, Connecticut 06520, USA"}]},{"ORCID":"https://orcid.org/0000-0002-3381-6642","authenticated-orcid":false,"given":"Yohei","family":"Fuji","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, The University of Tokyo 1 , Tokyo, Japan"}]}],"container-title":"Journal of Mathematical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2023,5,1]]},"URL":"http://dx.doi.org/10.1063/5.0134010","ISSN":["0022-2488","1089-7658"],"page":"051901","note":"arxivid:2211.00299"},{"id":"arxiv:2507.02035","type":"journal-article","publisher":"American Physical Society (APS)","issue":"16","DOI":"10.1103/f63q-vqz3","source":"Crossref","title":"Hall-on-Toric State: Descendant Laughlin State in the Chiral\n                    <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\">\n                      <mml:mrow>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n                          </mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>p</mml:mi>\n                          </mml:mrow>\n                        </mml:msub>\n                      </mml:mrow>\n                    </mml:math>\n                    Toric Code","volume":"136","author":[{"ORCID":"https://orcid.org/0000-0001-9728-2371","authenticated-orcid":true,"given":"Robin","family":"Schäfer","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05qwgg493","id-type":"ROR","asserted-by":"publisher"}],"name":"Boston University"},{"name":"Helmholtz-Zentrum Berlin für Materialien und Energie"},{"name":"Freie Universität Berlin"}]},{"ORCID":"https://orcid.org/0000-0002-8275-2024","authenticated-orcid":true,"given":"Claudio","family":"Chamon","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02dqehb95","id-type":"ROR","asserted-by":"publisher"}],"name":"Purdue University"},{"id":[{"id":"https://ror.org/02dqehb95","id-type":"ROR","asserted-by":"publisher"}],"name":"Purdue University"}]},{"ORCID":"https://orcid.org/0000-0001-8979-756X","authenticated-orcid":true,"given":"Chris R.","family":"Laumann","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05qwgg493","id-type":"ROR","asserted-by":"publisher"}],"name":"Boston University"},{"id":[{"id":"https://ror.org/01bf9rw71","id-type":"ROR","asserted-by":"publisher"}],"name":"Max-Planck-Institut für Physik komplexer Systeme"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2026,4,22]]},"URL":"http://dx.doi.org/10.1103/f63q-vqz3","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"166603","note":"arxivid:2507.02035"},{"id":"arxiv:1311.4895","type":"journal-article","publisher":"IOP Publishing","issue":"6","DOI":"10.1088/1367-2630/16/6/063038","page":"063038","source":"Crossref","title":"Fast decoders for qudit topological codes","volume":"16","author":[{"given":"Hussain","family":"Anwar","sequence":"first","affiliation":[]},{"given":"Benjamin J","family":"Brown","sequence":"additional","affiliation":[]},{"given":"Earl T","family":"Campbell","sequence":"additional","affiliation":[]},{"given":"Dan E","family":"Browne","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2014,6,17]]},"URL":"http://dx.doi.org/10.1088/1367-2630/16/6/063038","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1311.4895"},{"id":"arxiv:2411.04185","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title>\n          <jats:p>The development of programmable quantum devices can be measured by the complexity of many-body states that they are able to prepare. Among the most significant are topologically ordered states of matter, which enable robust quantum information storage and processing. While topological orders are more readily accessible with qudits, experimental realizations have thus far been limited to lattice models of qubits. Here, we prepare and measure a ground state of the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${{\\mathbb{Z}}}_{3}$$</jats:tex-math>\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>Z</mml:mi>\n                    </mml:mrow>\n                    <mml:mrow>\n                      <mml:mn>3</mml:mn>\n                    </mml:mrow>\n                  </mml:msub>\n                </mml:math>\n              </jats:alternatives>\n            </jats:inline-formula> toric code state on 24 qutrits (obtained by encoding one qutrit into two qubits) in a trapped ion quantum processor with fidelity per qutrit exceeding 96.5(3)%. We manipulate two types of defects which go beyond the conventional qubit toric code: a parafermion, and its bound state which is related to charge conjugation symmetry. We further demonstrate defect fusion and the transfer of entanglement between anyons and defects, which we use to control topological qutrits. Our work opens up the space of long-range entangled states with qudit degrees of freedom for use in quantum simulation and universal error-correcting codes.</jats:p>","DOI":"10.1038/s41467-025-61391-z","source":"Crossref","title":"Qutrit toric code and parafermions in trapped ions","volume":"16","author":[{"given":"Mohsin","family":"Iqbal","sequence":"first","affiliation":[]},{"given":"Anasuya","family":"Lyons","sequence":"additional","affiliation":[]},{"given":"Chiu Fan Bowen","family":"Lo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5295-2124","authenticated-orcid":false,"given":"Nathanan","family":"Tantivasadakarn","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9226-203X","authenticated-orcid":false,"given":"Joan","family":"Dreiling","sequence":"additional","affiliation":[]},{"given":"Cameron","family":"Foltz","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0009-0004-6338-4221","authenticated-orcid":false,"given":"Thomas M.","family":"Gatterman","sequence":"additional","affiliation":[]},{"given":"Dan","family":"Gresh","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1021-0795","authenticated-orcid":false,"given":"Nathan","family":"Hewitt","sequence":"additional","affiliation":[]},{"given":"Craig A.","family":"Holliman","sequence":"additional","affiliation":[]},{"given":"Jacob","family":"Johansen","sequence":"additional","affiliation":[]},{"given":"Brian","family":"Neyenhuis","sequence":"additional","affiliation":[]},{"given":"Yohei","family":"Matsuoka","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-2560-4129","authenticated-orcid":false,"given":"Michael","family":"Mills","sequence":"additional","affiliation":[]},{"given":"Steven A.","family":"Moses","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0145-2899","authenticated-orcid":false,"given":"Peter","family":"Siegfried","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-6306-2263","authenticated-orcid":false,"given":"Ashvin","family":"Vishwanath","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-0385-7363","authenticated-orcid":false,"given":"Ruben","family":"Verresen","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-1480-6406","authenticated-orcid":false,"given":"Henrik","family":"Dreyer","sequence":"additional","affiliation":[]}],"container-title":"Nature Communications","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,8]]},"URL":"http://dx.doi.org/10.1038/s41467-025-61391-z","ISSN":["2041-1723"],"container-title-short":"Nat Commun","page":"6301","note":"alternative-id:61391\narxivid:2411.04185"},{"id":"arxiv:1810.10037","type":"journal-article","publisher":"IOP Publishing","issue":"3","DOI":"10.1088/1367-2630/ab0199","page":"033028","source":"Crossref","title":"Code deformation and lattice surgery are gauge fixing","volume":"21","author":[{"ORCID":"https://orcid.org/0000-0002-3445-0179","authenticated-orcid":false,"given":"Christophe","family":"Vuillot","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-6870-5670","authenticated-orcid":false,"given":"Lingling","family":"Lao","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-9959-6462","authenticated-orcid":false,"given":"Ben","family":"Criger","sequence":"additional","affiliation":[]},{"given":"Carmen","family":"García Almudéver","sequence":"additional","affiliation":[]},{"given":"Koen","family":"Bertels","sequence":"additional","affiliation":[]},{"given":"Barbara M","family":"Terhal","sequence":"additional","affiliation":[]}],"container-title":"New Journal of Physics","original-title":[],"issued":{"date-parts":[[2019,3,28]]},"URL":"http://dx.doi.org/10.1088/1367-2630/ab0199","ISSN":["1367-2630"],"container-title-short":"New J. Phys.","note":"arxivid:1810.10037"},{"id":"arxiv:1405.1780","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physrevlett.114.026401","source":"Crossref","title":"Generalized Kitaev Models and Extrinsic Non-Abelian Twist Defects","volume":"114","author":[{"given":"Maissam","family":"Barkeshli","sequence":"first","affiliation":[]},{"given":"Hong-Chen","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Ronny","family":"Thomale","sequence":"additional","affiliation":[]},{"given":"Xiao-Liang","family":"Qi","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2015,1,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.114.026401","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"026401","note":"arxivid:1405.1780"},{"id":"arxiv:1105.5872","type":"article-journal","author":[{"given":"Leng","family":"Riguang"},{"given":"Ma","family":"Zhi"}],"title":"Non-binary Entanglement-assisted Stabilizer Quantum Codes","issued":{"date-parts":[[2011,5,30]]},"note":"arxivid:1105.5872\narxiv_version_number:1"},{"id":"doi:10.1103/PhysRevA.103.042420","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.103.042420","source":"Crossref","title":"Quantum error correction architecture for qudit stabilizer codes","volume":"103","author":[{"ORCID":"https://orcid.org/0000-0002-1351-2959","authenticated-orcid":true,"given":"Priya J.","family":"Nadkarni","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2459-1445","authenticated-orcid":true,"given":"Shayan Srinivasa","family":"Garani","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2021,4,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.103.042420","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"042420"},{"id":"arxiv:2010.07902","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","DOI":"10.1109/tit.2022.3149291","page":"3942-3950","source":"Crossref","title":"Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes","volume":"68","author":[{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"first","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University of Gdansk, Gda&#x0144;sk, Poland"}]},{"ORCID":"https://orcid.org/0000-0002-3856-4018","authenticated-orcid":false,"given":"Felix","family":"Huber","sequence":"additional","affiliation":[{"name":"ICFO&#x2014;Institut de Ci&#x00E8;ncies Fot&#x00F2;niques, Castelldefels, Barcelona, Spain"}]},{"ORCID":"https://orcid.org/0000-0001-6344-4870","authenticated-orcid":false,"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[{"name":"Departament de F&#x00ED;sica, Grup d&#x2019;Informaci&#x00F3; Qu&#x00E0;ntica, Universitat Aut&#x00F2;noma de Barcelona, Bellaterra, Barcelona, Spain"}]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2022,6]]},"URL":"http://dx.doi.org/10.1109/TIT.2022.3149291","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:2010.07902"},{"id":"doi:10.1007/s11128-021-03028-w","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","DOI":"10.1007/s11128-021-03028-w","source":"Crossref","title":"Entanglement-assisted Reed–Solomon codes over qudits: theory and architecture","volume":"20","author":[{"ORCID":"https://orcid.org/0000-0002-1351-2959","authenticated-orcid":false,"given":"Priya J.","family":"Nadkarni","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2459-1445","authenticated-orcid":false,"given":"Shayan Srinivasa","family":"Garani","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2021,4]]},"URL":"http://dx.doi.org/10.1007/s11128-021-03028-w","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","page":"129","note":"alternative-id:3028"},{"id":"arxiv:1008.2598","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.88.012320","source":"Crossref","title":"Entanglement increases the error-correcting ability of quantum error-correcting codes","volume":"88","author":[{"given":"Ching-Yi","family":"Lai","sequence":"first","affiliation":[]},{"given":"Todd A.","family":"Brun","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,7,19]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.88.012320","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012320","note":"arxivid:1008.2598"},{"id":"doi:10.1007/s11128-019-2234-5","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"4","abstract":"<jats:title>Abstract</jats:title><jats:p>We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert–Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.</jats:p>","DOI":"10.1007/s11128-019-2234-5","source":"Crossref","title":"Entanglement-assisted quantum error-correcting codes over arbitrary finite fields","volume":"18","author":[{"ORCID":"https://orcid.org/0000-0002-3908-4462","authenticated-orcid":false,"given":"Carlos","family":"Galindo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9758-2152","authenticated-orcid":false,"given":"Fernando","family":"Hernando","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5085-8879","authenticated-orcid":false,"given":"Ryutaroh","family":"Matsumoto","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7304-0087","authenticated-orcid":false,"given":"Diego","family":"Ruano","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2019,3,4]]},"URL":"http://dx.doi.org/10.1007/s11128-019-2234-5","ISSN":["1570-0755","1573-1332"],"container-title-short":"Quantum Inf Process","page":"116","note":"updated-by:{\"DOI\":\"10.1007/s11128-021-03066-4\",\"type\":\"correction\",\"label\":\"Correction\",\"source\":\"publisher\",\"updated\":{\"date-parts\":[[2021,6,24]],\"date-time\":\"2021-06-24T00:00:00Z\",\"timestamp\":1624492800000}}\nalternative-id:2234"},{"id":"arxiv:quant-ph/0308044","type":"journal-article","publisher":"American Physical Society (APS)","issue":"23","DOI":"10.1103/physrevlett.93.230504","source":"Crossref","title":"A Family of Quantum Protocols","volume":"93","author":[{"given":"Igor","family":"Devetak","sequence":"first","affiliation":[]},{"given":"Aram W.","family":"Harrow","sequence":"additional","affiliation":[]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2004,12,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.93.230504","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"230504","note":"arxivid:quant-ph/0308044"},{"id":"arxiv:quant-ph/0512015","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"10","DOI":"10.1109/tit.2008.928980","page":"4587-4618","source":"Crossref","title":"A Resource Framework for Quantum Shannon Theory","volume":"54","author":[{"given":"Igor","family":"Devetak","sequence":"first","affiliation":[]},{"given":"Aram W.","family":"Harrow","sequence":"additional","affiliation":[]},{"given":"Andreas J.","family":"Winter","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2008,10]]},"URL":"http://dx.doi.org/10.1109/TIT.2008.928980","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0512015"},{"id":"arxiv:quant-ph/0211014","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"05","abstract":"<jats:p> We present two methods for the construction of quantum circuits for quantum error- correcting codes (QECC). The underlying quantum systems are tensor products of subsystems (qudits) of equal dimension which is a prime power. For a QECC encoding k qudits into n qudits, the resulting quantum circuit has O(n(n - k)) gates. The running time of the classical algorithm to compute the quantum circuit is O(n(n - k)<jats:sup>2</jats:sup>). </jats:p>","DOI":"10.1142/s0129054103002011","page":"757-775","source":"Crossref","title":"EFFICIENT QUANTUM CIRCUITS FOR NON-QUBIT QUANTUM ERROR-CORRECTING CODES","volume":"14","author":[{"given":"MARKUS","family":"GRASSL","sequence":"first","affiliation":[{"name":"Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, 76 128 Karlsruhe, Germany"},{"name":"The Mathematical Sciences Research Institute, 1000 Centennial Drive, #5070, Berkeley, CA 94720-5070, USA"}]},{"given":"MARTIN","family":"RÖTTELER","sequence":"additional","affiliation":[{"name":"Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, 76 128 Karlsruhe, Germany"},{"name":"Department of Combinatorics and Optimization,  Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada,  N2L 3G1, Canada"}]},{"given":"THOMAS","family":"BETH","sequence":"additional","affiliation":[{"name":"Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, 76 128 Karlsruhe, Germany"}]}],"container-title":"International Journal of Foundations of Computer Science","original-title":[],"language":"en","issued":{"date-parts":[[2003,10]]},"URL":"http://dx.doi.org/10.1142/S0129054103002011","ISSN":["0129-0541","1793-6373"],"container-title-short":"Int. J. Found. Comput. Sci.","note":"alternative-id:10.1142/S0129054103002011\narxivid:quant-ph/0211014"},{"id":"arxiv:2506.03397","type":"article-journal","author":[{"given":"Vahid","family":"Nourozi"}],"title":"Reinforcement Learning Enhanced Greedy Decoding for Quantum Stabilizer Codes over $\\mathbb{F}_q$","issued":{"date-parts":[[2025,6,3]]},"note":"arxivid:2506.03397\narxiv_version_number:1"},{"id":"arxiv:1208.4907","type":"article-journal","author":[{"given":"Hari Dilip","family":"Kumar"}],"title":"On the Fourier Transform Approach to Quantum Error Control","issued":{"date-parts":[[2012,8,24]]},"note":"arxivid:1208.4907\narxiv_version_number:1"},{"id":"arxiv:1205.4253","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.88.022328","source":"Crossref","title":"Quantum error-correcting codes over mixed alphabets","volume":"88","author":[{"given":"Zhuo","family":"Wang","sequence":"first","affiliation":[]},{"given":"Sixia","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Heng","family":"Fan","sequence":"additional","affiliation":[]},{"given":"C. H.","family":"Oh","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2013,8,22]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.88.022328","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"022328","note":"arxivid:1205.4253"},{"id":"doi:10.1007/978-3-642-20901-7_9","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/978-3-642-20901-7_9","page":"142-158","source":"Crossref","title":"Variations on Encoding Circuits for Stabilizer Quantum Codes","author":[{"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2011]]},"ISBN":["9783642209000","9783642209017"],"URL":"http://dx.doi.org/10.1007/978-3-642-20901-7_9","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1016/j.disc.2010.06.043","type":"journal-article","publisher":"Elsevier BV","issue":"21","DOI":"10.1016/j.disc.2010.06.043","page":"2935-2945","source":"Crossref","title":"Constructions of new families of nonbinary CSS codes","volume":"310","author":[{"given":"Giuliano G.","family":"La Guardia","sequence":"first","affiliation":[]},{"suffix":"Jr.","given":"Reginaldo","family":"Palazzo","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2010,11]]},"URL":"http://dx.doi.org/10.1016/j.disc.2010.06.043","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X10002578"},{"id":"doi:10.1103/PhysRevA.80.042331","type":"journal-article","publisher":"American Physical Society (APS)","issue":"4","DOI":"10.1103/physreva.80.042331","source":"Crossref","title":"Constructions of new families of nonbinary quantum codes","volume":"80","author":[{"given":"Giuliano G.","family":"La Guardia","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2009,10,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.80.042331","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"042331"},{"id":"arxiv:quant-ph/0501126","type":"article-journal","author":[{"given":"Salah","family":"Aly"},{"given":"Andreas","family":"Klappenecker"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Primitive Quantum BCH Codes over Finite Fields","issued":{"date-parts":[[2006,1,18]]},"note":"arxivid:quant-ph/0501126\narxiv_version_number:2"},{"id":"doi:10.1007/11750321_63","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/11750321_63","page":"675-683","source":"Crossref","title":"On Non-binary Quantum BCH Codes","author":[{"given":"Zhi","family":"Ma","sequence":"first","affiliation":[]},{"given":"Xin","family":"Lu","sequence":"additional","affiliation":[]},{"given":"Keqin","family":"Feng","sequence":"additional","affiliation":[]},{"given":"Dengguo","family":"Feng","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2006]]},"ISBN":["9783540340218","9783540340225"],"URL":"http://dx.doi.org/10.1007/11750321_63","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1109/TIT.2006.890730","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"3","DOI":"10.1109/tit.2006.890730","page":"1183-1188","source":"Crossref","title":"On Quantum and Classical BCH Codes","volume":"53","author":[{"given":"Salah A.","family":"Aly","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2007,3]]},"URL":"http://dx.doi.org/10.1109/TIT.2006.890730","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"doi:10.26421/QIC13.1-2-3","type":"journal-article","publisher":"Rinton Press","issue":"1&2","abstract":"<jats:p>Let $q\\geq 3$ be a prime power. Maximal designed distances of primitive Hermitian dual containing $q^{2}$-ary BCH codes (narrow-sense or non-narrow-sense) are determined by a careful analysis of properties of cyclotomic cosets. Non-narrow-sense BCH codes which achieve these maximal designed distances are presented, and a sequence of nested non-narrow-sense BCH codes that contain these BCH codes with maximal designed distances are constructed and their parameters are computed. Consequently, new nonbinary quantum BCH codes are derived from these non-narrow-sense BCH codes. The nonbinary quantum BCH codes presented here have better parameters than those quantum BCH codes available in the literature.</jats:p>","DOI":"10.26421/qic13.1-2-3","page":"21-35","source":"Crossref","title":"Hermitian dual containing BCH codes and Construction of new quantum codes","volume":"13","author":[{"given":"Ruihu","family":"Li","sequence":"first","affiliation":[]},{"given":"Fei","family":"Zou","sequence":"additional","affiliation":[]},{"given":"Yang","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Zongben","family":"Xu","sequence":"additional","affiliation":[]}],"container-title":"Quantum Information and Computation","original-title":[],"issued":{"date-parts":[[2013,1]]},"URL":"http://dx.doi.org/10.26421/QIC13.1-2-3","ISSN":["1533-7146","1533-7146"],"container-title-short":"QIC"},{"id":"arxiv:1705.00239","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"8","DOI":"10.1007/s10773-017-3399-2","page":"2479-2484","source":"Crossref","title":"Quantum Codes Derived from Cyclic Codes","volume":"56","author":[{"given":"Giuliano G.","family":"La Guardia","sequence":"first","affiliation":[]}],"container-title":"International Journal of Theoretical Physics","original-title":[],"language":"en","issued":{"date-parts":[[2017,4,29]]},"URL":"http://dx.doi.org/10.1007/s10773-017-3399-2","ISSN":["0020-7748","1572-9575"],"container-title-short":"Int J Theor Phys","note":"alternative-id:3399\narxivid:1705.00239"},{"id":"arxiv:2007.13309","type":"article-journal","author":[{"given":"X.","family":"Zhao"},{"given":"X.","family":"Li"},{"given":"Q.","family":"Wang"},{"given":"T.","family":"Yan"}],"title":"Hermitian dual-containing constacyclic BCH codes and related quantum codes of length $\\frac{q^{2m}-1}{q+1}$","issued":{"date-parts":[[2020,7,27]]},"note":"arxivid:2007.13309\narxiv_version_number:1"},{"id":"arxiv:2110.08414","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.105.032450","source":"Crossref","title":"Graph-theoretic approach to quantum error correction","volume":"105","author":[{"ORCID":"https://orcid.org/0000-0002-5560-0498","authenticated-orcid":true,"given":"Robert R.","family":"Vandermolen","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-3670-2413","authenticated-orcid":true,"given":"Duncan","family":"Wright","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2022,3,29]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.105.032450","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032450","note":"arxivid:2110.08414"},{"id":"arxiv:quant-ph/9911011","type":"article-journal","author":[{"given":"Ryutaroh","family":"Matsumoto"},{"given":"Tomohiko","family":"Uyematsu"}],"title":"Constructing quantum error-correcting codes for p^m-state systems from classical error-correcting codes","issued":{"date-parts":[[2000,6,26]]},"note":"arxivid:quant-ph/9911011\narxiv_version_number:3"},{"id":"doi:10.1016/j.disc.2007.08.038","type":"journal-article","publisher":"Elsevier BV","issue":"14","DOI":"10.1016/j.disc.2007.08.038","page":"3115-3124","source":"Crossref","title":"Nonbinary quantum error-correcting codes from algebraic curves","volume":"308","author":[{"given":"Jon-Lark","family":"Kim","sequence":"first","affiliation":[]},{"given":"Judy","family":"Walker","sequence":"additional","affiliation":[]}],"container-title":"Discrete Mathematics","original-title":[],"language":"en","issued":{"date-parts":[[2008,7]]},"URL":"http://dx.doi.org/10.1016/j.disc.2007.08.038","ISSN":["0012-365X"],"container-title-short":"Discrete Mathematics","note":"alternative-id:S0012365X07006619"},{"id":"arxiv:2208.05353","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"1","abstract":"<jats:title>Abstract</jats:title><jats:p>We present a new propagation rule for CSS codes. Starting with a CSS code <jats:inline-formula><jats:alternatives><jats:tex-math>$$[\\![n,k,d]\\!]_q$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mo>[</mml:mo>\n                      <mml:mspace/>\n                      <mml:mrow>\n                        <mml:mo>[</mml:mo>\n                        <mml:mi>n</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>d</mml:mi>\n                        <mml:mo>]</mml:mo>\n                      </mml:mrow>\n                      <mml:mspace/>\n                      <mml:mo>]</mml:mo>\n                    </mml:mrow>\n                    <mml:mi>q</mml:mi>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula>, we construct a CSS code with parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$$[\\![n-2,k,d-1]\\!]_q$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mo>[</mml:mo>\n                      <mml:mspace/>\n                      <mml:mrow>\n                        <mml:mo>[</mml:mo>\n                        <mml:mi>n</mml:mi>\n                        <mml:mo>-</mml:mo>\n                        <mml:mn>2</mml:mn>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>d</mml:mi>\n                        <mml:mo>-</mml:mo>\n                        <mml:mn>1</mml:mn>\n                        <mml:mo>]</mml:mo>\n                      </mml:mrow>\n                      <mml:mspace/>\n                      <mml:mo>]</mml:mo>\n                    </mml:mrow>\n                    <mml:mi>q</mml:mi>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula>. In general, one would only obtain a code with parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$$[\\![n-2,k,d-2]\\!]_q$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mo>[</mml:mo>\n                      <mml:mspace/>\n                      <mml:mrow>\n                        <mml:mo>[</mml:mo>\n                        <mml:mi>n</mml:mi>\n                        <mml:mo>-</mml:mo>\n                        <mml:mn>2</mml:mn>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>k</mml:mi>\n                        <mml:mo>,</mml:mo>\n                        <mml:mi>d</mml:mi>\n                        <mml:mo>-</mml:mo>\n                        <mml:mn>2</mml:mn>\n                        <mml:mo>]</mml:mo>\n                      </mml:mrow>\n                      <mml:mspace/>\n                      <mml:mo>]</mml:mo>\n                    </mml:mrow>\n                    <mml:mi>q</mml:mi>\n                  </mml:msub>\n                </mml:math></jats:alternatives></jats:inline-formula>. The construction applies to asymmetric quantum codes from the CSS construction as well.</jats:p>","DOI":"10.1007/s11128-023-03835-3","source":"Crossref","title":"New quantum codes from CSS codes","volume":"22","author":[{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"first","affiliation":[]}],"container-title":"Quantum Information Processing","original-title":[],"language":"en","issued":{"date-parts":[[2023,1,23]]},"URL":"http://dx.doi.org/10.1007/s11128-023-03835-3","ISSN":["1573-1332"],"container-title-short":"Quantum Inf Process","page":"86","note":"alternative-id:3835\narxivid:2208.05353"},{"id":"arxiv:0712.3223","type":"article-journal","author":[{"given":"Pedro J.","family":"Salas"}],"title":"Simple fault-tolerant encoding over q-ary CSS quantum codes","issued":{"date-parts":[[2007,12,19]]},"note":"arxivid:0712.3223\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0605086","type":"article-journal","author":[{"given":"Debbie","family":"Leung"},{"given":"Graeme","family":"Smith"}],"title":"Communicating over adversarial quantum channels using quantum list codes","issued":{"date-parts":[[2007,4,17]]},"note":"arxivid:quant-ph/0605086\narxiv_version_number:2"},{"id":"doi:10.1007/s12095-018-0299-0","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s12095-018-0299-0","page":"325-335","source":"Crossref","title":"Quantum codes over Fp from cyclic codes over Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉","volume":"11","author":[{"given":"Mohammad","family":"Ashraf","sequence":"first","affiliation":[]},{"given":"Ghulam","family":"Mohammad","sequence":"additional","affiliation":[]}],"container-title":"Cryptography and Communications","original-title":[],"language":"en","issued":{"date-parts":[[2018,4,4]]},"URL":"http://dx.doi.org/10.1007/s12095-018-0299-0","ISSN":["1936-2447","1936-2455"],"container-title-short":"Cryptogr. Commun.","note":"alternative-id:299"},{"id":"doi:10.1142/S0219498819500774","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"04","abstract":"<jats:p> In this work, skew quasi cyclic codes over [Formula: see text], where [Formula: see text] are considered. The generating set for one generator skew quasi cyclic codes over [Formula: see text] is also determined. We discuss a sufficient condition for one generator skew quasi cyclic codes to be free. Furthermore, a BCH type bound is given for free one generator skew quasi cyclic codes. We investigate the dual of skew quasi cyclic codes over [Formula: see text]. We give a necessary and sufficient condition for skew cyclic codes over [Formula: see text] to contain its dual. Moreover, we construct quantum codes from skew cyclic codes over [Formula: see text]. By using computer search we give some examples about skew quasi cyclic codes and list some quantum parameters in the table. </jats:p>","DOI":"10.1142/s0219498819500774","page":"1950077","source":"Crossref","title":"Skew quasi cyclic codes over 𝔽q + v𝔽q","volume":"18","author":[{"given":"Mehmet","family":"Özen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sakarya University, Sakarya, Turkey"}]},{"given":"N.","family":"Tuğba Özzaim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sakarya University, Sakarya, Turkey"}]},{"given":"Halit","family":"İnce","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sakarya University, Sakarya, Turkey"}]}],"container-title":"Journal of Algebra and Its Applications","original-title":[],"language":"en","issued":{"date-parts":[[2019,3,25]]},"URL":"http://dx.doi.org/10.1142/S0219498819500774","ISSN":["0219-4988","1793-6829"],"container-title-short":"J. Algebra Appl.","note":"alternative-id:10.1142/S0219498819500774"},{"id":"arxiv:2310.15092","type":"article-journal","author":[{"given":"Nadja","family":"Willenborg"},{"given":"Martino","family":"Borello"},{"given":"Anna-Lena","family":"Horlemann"},{"given":"Habibul","family":"Islam"}],"title":"Dihedral Quantum Codes","issued":{"date-parts":[[2025,5,4]]},"note":"arxivid:2310.15092\narxiv_version_number:3"},{"id":"arxiv:quant-ph/0601117","type":"article-journal","author":[{"given":"Salah A.","family":"Aly"},{"given":"Andreas","family":"Klappenecker"},{"given":"Pradeep Kiran","family":"Sarvepalli"}],"title":"Remarkable Degenerate Quantum Stabilizer Codes Derived from Duadic Codes","issued":{"date-parts":[[2006,1,18]]},"note":"arxivid:quant-ph/0601117\narxiv_version_number:1"},{"id":"arxiv:0711.2050","type":"article-journal","author":[{"given":"Kenza","family":"Guenda"}],"title":"Two Families of Quantum Codes Derived from Cyclic Codes","issued":{"date-parts":[[2007,11,13]]},"note":"arxivid:0711.2050\narxiv_version_number:1"},{"id":"doi:10.1142/S0219749909004979","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"01","abstract":"<jats:p> We construct new families of duadic codes and derive from them degenerate quantum codes. This provides a solution to a question raised by Aly, Klappenecker and Sarvepalli on the existence of quantum codes from duadic codes when the multiplicative order of q modulo n is even. We also characterize the affine-invariant maximal extended cyclic codes. Then by the CSS construction, we give a family of quantum code. </jats:p>","DOI":"10.1142/s0219749909004979","page":"373-384","source":"Crossref","title":"QUANTUM DUADIC AND AFFINE-INVARIANT CODES","volume":"07","author":[{"given":"KENZA","family":"GUENDA","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, University of Science and Technology, Houari Boumediene, B. P. 32 El Alia Bab Ezzouar, 16111 Algiers, Algeria"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2009,2]]},"URL":"http://dx.doi.org/10.1142/S0219749909004979","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749909004979"},{"id":"arxiv:quant-ph/0006061","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.63.032311","source":"Crossref","title":"Asymptotically good quantum codes","volume":"63","author":[{"given":"Alexei","family":"Ashikhmin","sequence":"first","affiliation":[]},{"given":"Simon","family":"Litsyn","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Tsfasman","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2001,2]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.63.032311","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"032311","note":"arxivid:quant-ph/0006061"},{"id":"arxiv:quant-ph/0107129","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","DOI":"10.1109/tit.2002.1013156","page":"2122-2124","source":"Crossref","title":"Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes","volume":"48","author":[{"given":"R.","family":"Matsumoto","sequence":"first","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"language":"en","issued":{"date-parts":[[2002,7]]},"URL":"http://dx.doi.org/10.1109/TIT.2002.1013156","ISSN":["0018-9448"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:quant-ph/0107129"},{"id":"arxiv:math/9910151","type":"article-journal","author":[{"given":"J. I.","family":"Farran"}],"title":"Decoding Algebraic Geometry codes by a key equation","issued":{"date-parts":[[1999,10,27]]},"note":"arxivid:math/9910151\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0107102","type":"article-journal","author":[{"given":"Hao","family":"Chen"}],"title":"Some Good Error-Correcting Codes from Algebraic-Geometric Codes","issued":{"date-parts":[[2001,7,20]]},"note":"arxivid:quant-ph/0107102\narxiv_version_number:1"},{"id":"arxiv:2502.01864","type":"article-journal","author":[{"given":"Zhiyang","family":"He"},{"given":"Vinod","family":"Vaikuntanathan"},{"given":"Adam","family":"Wills"},{"given":"Rachel Yun","family":"Zhang"}],"title":"Quantum Codes with Addressable and Transversal Non-Clifford Gates","issued":{"date-parts":[[2025,7,7]]},"note":"arxivid:2502.01864\narxiv_version_number:3"},{"id":"doi:10.1007/11617983_13","type":"book-chapter","publisher":"Springer Berlin Heidelberg","DOI":"10.1007/11617983_13","page":"136-143","source":"Crossref","title":"Nonbinary Quantum Codes from Hermitian Curves","author":[{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"first","affiliation":[]},{"given":"Andreas","family":"Klappenecker","sequence":"additional","affiliation":[]}],"container-title":"Lecture Notes in Computer Science","original-title":[],"issued":{"date-parts":[[2006]]},"ISBN":["9783540314233","9783540314240"],"URL":"http://dx.doi.org/10.1007/11617983_13","ISSN":["0302-9743","1611-3349"],"note":"publisher-location:Berlin, Heidelberg"},{"id":"doi:10.1142/S0219749919500060","type":"journal-article","publisher":"World Scientific Pub Co Pte Lt","issue":"01","abstract":"<jats:p> Hermitian dual-containing codes play an important role in the constructing quantum codes. In this paper, we present a new criterion of Hermitian dual-containing code based on the rank of generator matrix for a linear code. Then, using the criterion, we construct a class of new quantum maximum-distance-separable (MDS) codes and some new quantum codes. </jats:p>","DOI":"10.1142/s0219749919500060","page":"1950006","source":"Crossref","title":"New quantum codes from Hermitian dual-containing codes","volume":"17","author":[{"given":"Xiusheng","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China"}]},{"given":"Long","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China"}]},{"given":"Hualu","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China"}]}],"container-title":"International Journal of Quantum Information","original-title":[],"language":"en","issued":{"date-parts":[[2019,2]]},"URL":"http://dx.doi.org/10.1142/S0219749919500060","ISSN":["0219-7499","1793-6918"],"container-title-short":"Int. J. Quantum Inform.","note":"alternative-id:10.1142/S0219749919500060"},{"id":"doi:10.1109/TIT.2010.2054174","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","DOI":"10.1109/tit.2010.2054174","page":"4735-4740","source":"Crossref","title":"Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes","volume":"56","author":[{"given":"Lingfei","family":"Jin","sequence":"first","affiliation":[]},{"given":"San","family":"Ling","sequence":"additional","affiliation":[]},{"given":"Jinquan","family":"Luo","sequence":"additional","affiliation":[]},{"given":"Chaoping","family":"Xing","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2010,9]]},"URL":"http://dx.doi.org/10.1109/TIT.2010.2054174","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory"},{"id":"arxiv:0812.4514","type":"journal-article","publisher":"American Physical Society (APS)","issue":"1","DOI":"10.1103/physreva.77.012308","source":"Crossref","title":"Quantum generalized Reed-Solomon codes: Unified framework for quantum maximum-distance-separable codes","volume":"77","author":[{"given":"Zhuo","family":"Li","sequence":"first","affiliation":[]},{"given":"Li-Juan","family":"Xing","sequence":"additional","affiliation":[]},{"given":"Xin-Mei","family":"Wang","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2008,1,9]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.77.012308","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"012308","note":"arxivid:0812.4514"},{"id":"arxiv:2306.13399","type":"article-journal","author":[{"given":"Manabu","family":"Hagiwara"}],"title":"Quantum Deletion Codes Derived From Quantum Reed-Solomon Codes","issued":{"date-parts":[[2023,6,23]]},"note":"arxivid:2306.13399\narxiv_version_number:1"},{"id":"arxiv:2411.03632","type":"article-journal","author":[{"given":"Quynh T.","family":"Nguyen"},{"given":"Christopher A.","family":"Pattison"}],"title":"Quantum fault tolerance with constant-space and logarithmic-time overheads","issued":{"date-parts":[[2025,8,14]]},"note":"arxivid:2411.03632\narxiv_version_number:2"},{"id":"arxiv:2510.07269","type":"article-journal","author":[{"given":"Christine","family":"Li"},{"given":"John","family":"Preskill"},{"given":"Qian","family":"Xu"}],"title":"Transversal dimension jump for product qLDPC codes","issued":{"date-parts":[[2026,3,2]]},"note":"arxivid:2510.07269\narxiv_version_number:2"},{"id":"arxiv:2411.04464","type":"article-journal","author":[{"given":"Louis","family":"Golowich"},{"given":"Venkatesan","family":"Guruswami"}],"title":"Decoding Quasi-Cyclic Quantum LDPC Codes","issued":{"date-parts":[[2024,11,7]]},"note":"arxivid:2411.04464\narxiv_version_number:1"},{"id":"manual:-G.-V.-Kalachev-and-P.-A.-Pan","type":"article-journal","author":[{"family":"Kalachev","given":"G.V."},{"family":"Panteleev","given":"P.A."}],"title":"On the minimum distance in one class of quantum LDPC codes","volume":"24","container-title":"Intelligent systems. Theory and applications","issue":"4","issued":"2020","page":"87–117"},{"id":"manual:-M.-B.-Hastings-LR-codes-priv","type":"document","author":[{"family":"Hastings","given":"M.B."}],"title":"LR codes, private communication","issued":"2014"},{"id":"arxiv:2305.06890","type":"article-journal","author":[{"given":"Renyu","family":"Wang"},{"given":"Hsiang-Ku","family":"Lin"},{"given":"Leonid P.","family":"Pryadko"}],"title":"Abelian and non-abelian quantum two-block codes","issued":{"date-parts":[[2023,7,31]]},"note":"arxivid:2305.06890\narxiv_version_number:2"},{"id":"manual:-H.-K.-Lin-and-L.-Pryadko-pri","type":"document","author":[{"family":"Lin","given":"H.-K."},{"family":"Pryadko","given":"L."}],"genre":"private communication,","issued":"2024"},{"id":"arxiv:2311.16980","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/qce65121.2025.00080","page":"688-699","source":"Crossref","title":"Matching Generalized-Bicycle Codes to Neutral Atoms for Low-Overhead Fault-Tolerance","author":[{"given":"Joshua","family":"Viszlai","sequence":"first","affiliation":[{"name":"University of Chicago,USA"}]},{"given":"Willers","family":"Yang","sequence":"additional","affiliation":[{"name":"University of Chicago,USA"}]},{"given":"Sophia Fuhui","family":"Lin","sequence":"additional","affiliation":[{"name":"University of Chicago,USA"}]},{"given":"Junyu","family":"Liu","sequence":"additional","affiliation":[{"name":"University of Chicago,USA"}]},{"given":"Natalia","family":"Nottingham","sequence":"additional","affiliation":[{"name":"University of Chicago,USA"}]},{"given":"Jonathan M","family":"Baker","sequence":"additional","affiliation":[{"name":"University of Texas, Austin,USA"}]},{"given":"Frederic T","family":"Chong","sequence":"additional","affiliation":[{"name":"University of Chicago,USA"}]}],"event":"2025 IEEE International Conference on Quantum Computing and Engineering (QCE)","container-title":"2025 IEEE International Conference on Quantum Computing and Engineering (QCE)","original-title":[],"issued":{"date-parts":[[2025,8,30]]},"URL":"http://dx.doi.org/10.1109/QCE65121.2025.00080","note":"arxivid:2311.16980"},{"id":"arxiv:2412.21171","type":"article-journal","author":[{"given":"Daiki","family":"Komoto"},{"given":"Kenta","family":"Kasai"}],"title":"Quantum Error Correction near the Coding Theoretical Bound","issued":{"date-parts":[[2025,9,8]]},"note":"arxivid:2412.21171\narxiv_version_number:4"},{"id":"arxiv:2102.10030","type":"article-journal","author":[{"given":"M. B.","family":"Hastings"}],"title":"On Quantum Weight Reduction","issued":{"date-parts":[[2023,7,27]]},"note":"arxivid:2102.10030\narxiv_version_number:3"},{"id":"arxiv:2012.11998","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"5","abstract":"<jats:title>Abstract</jats:title><jats:p>Many <jats:italic>q</jats:italic>-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal <jats:inline-formula><jats:alternatives><jats:tex-math>$$q^2$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msup>\n                    <mml:mi>q</mml:mi>\n                    <mml:mn>2</mml:mn>\n                  </mml:msup>\n                </mml:math></jats:alternatives></jats:inline-formula>-ary linear codes. This result can be generalized to <jats:inline-formula><jats:alternatives><jats:tex-math>$$q^{2 m}$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:msup>\n                    <mml:mi>q</mml:mi>\n                    <mml:mrow>\n                      <mml:mn>2</mml:mn>\n                      <mml:mi>m</mml:mi>\n                    </mml:mrow>\n                  </mml:msup>\n                </mml:math></jats:alternatives></jats:inline-formula>-ary linear codes, <jats:inline-formula><jats:alternatives><jats:tex-math>$$m &gt; 1$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>m</mml:mi>\n                    <mml:mo>&gt;</mml:mo>\n                    <mml:mn>1</mml:mn>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula>. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"uri\" xlink:href=\"http://www.codetables.de\">http://www.codetables.de</jats:ext-link>, 2020) and new <jats:italic>q</jats:italic>-ary ones, with <jats:inline-formula><jats:alternatives><jats:tex-math>$$q \\ne 2$$</jats:tex-math><mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\n                  <mml:mrow>\n                    <mml:mi>q</mml:mi>\n                    <mml:mo>≠</mml:mo>\n                    <mml:mn>2</mml:mn>\n                  </mml:mrow>\n                </mml:math></jats:alternatives></jats:inline-formula>, improving others in the literature.</jats:p>","DOI":"10.1007/s10623-022-01018-2","page":"1103-1112","source":"Crossref","title":"On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes","volume":"90","author":[{"ORCID":"https://orcid.org/0000-0002-3908-4462","authenticated-orcid":false,"given":"Carlos","family":"Galindo","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9758-2152","authenticated-orcid":false,"given":"Fernando","family":"Hernando","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2022,3,21]]},"URL":"http://dx.doi.org/10.1007/s10623-022-01018-2","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:1018\narxivid:2012.11998"},{"id":"arxiv:1002.4088","type":"journal-article","publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"8","DOI":"10.1109/tit.2011.2159040","page":"5536-5550","source":"Crossref","title":"Additive Asymmetric Quantum Codes","volume":"57","author":[{"given":"Martianus Frederic","family":"Ezerman","sequence":"first","affiliation":[]},{"given":"San","family":"Ling","sequence":"additional","affiliation":[]},{"given":"Patrick","family":"Sole","sequence":"additional","affiliation":[]}],"container-title":"IEEE Transactions on Information Theory","original-title":[],"issued":{"date-parts":[[2011,8]]},"URL":"http://dx.doi.org/10.1109/TIT.2011.2159040","ISSN":["0018-9448","1557-9654"],"container-title-short":"IEEE Trans. Inform. Theory","note":"arxivid:1002.4088"},{"id":"doi:10.1007/s10623-014-9934-8","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"2","DOI":"10.1007/s10623-014-9934-8","page":"417-424","source":"Crossref","title":"Quantum codes from nearly self-orthogonal quaternary linear codes","volume":"73","author":[{"given":"Petr","family":"Lisoněk","sequence":"first","affiliation":[]},{"given":"Vijaykumar","family":"Singh","sequence":"additional","affiliation":[]}],"container-title":"Designs, Codes and Cryptography","original-title":[],"language":"en","issued":{"date-parts":[[2014,2,21]]},"URL":"http://dx.doi.org/10.1007/s10623-014-9934-8","ISSN":["0925-1022","1573-7586"],"container-title-short":"Des. Codes Cryptogr.","note":"alternative-id:9934"},{"id":"manual:-P.-Lisonek-and-R.-Dastbasteh","type":"chapter","author":[{"family":"Lisonek","given":"P."},{"family":"Dastbasteh","given":"R."}],"title":"Constructions of quantum codes","container-title":"3rd International Workshop on Boolean Functions and their Applications, loen, norway","issued":"2018","URL":"https://org.uib.no/selmer/workshops/BFA2018/Slides/Lisonek.pdf","note":"Available at \\url{https://org.uib.no/selmer/workshops/BFA2018/Slides/Lisonek.pdf}."},{"id":"arxiv:2011.06996","type":"journal-article","publisher":"Informa UK Limited","issue":"4","DOI":"10.1080/23799927.2020.1850530","page":"243-259","source":"Crossref","title":"Algebraic quantum codes: linking quantum mechanics and discrete mathematics","volume":"6","author":[{"ORCID":"https://orcid.org/0000-0002-3720-5195","authenticated-orcid":false,"given":"Markus","family":"Grassl","sequence":"first","affiliation":[{"name":"International Centre for Theory of Quantum Technologies, University Gdansk, Gdansk, Poland"}]}],"container-title":"International Journal of Computer Mathematics: Computer Systems Theory","original-title":[],"language":"en","issued":{"date-parts":[[2020,12,10]]},"URL":"http://dx.doi.org/10.1080/23799927.2020.1850530","ISSN":["2379-9927","2379-9935"],"container-title-short":"International Journal of Computer Mathematics: Computer Systems Theory","note":"alternative-id:10.1080/23799927.2020.1850530\narxivid:2011.06996"},{"id":"doi:10.1109/CIG.2010.5592860","type":"proceedings-article","publisher":"IEEE","DOI":"10.1109/cig.2010.5592860","page":"1-5","source":"Crossref","title":"Topological color codes over higher alphabet","author":[{"given":"Pradeep","family":"Sarvepalli","sequence":"first","affiliation":[]}],"event":"2010 IEEE Information Theory Workshop (ITW 2010)","container-title":"2010 IEEE Information Theory Workshop","original-title":[],"issued":{"date-parts":[[2010,8]]},"URL":"http://dx.doi.org/10.1109/CIG.2010.5592860"},{"id":"arxiv:1202.3338","type":"article-journal","author":[{"given":"Iryna","family":"Andriyanova"},{"given":"Denise","family":"Maurice"},{"given":"Jean-Pierre","family":"Tillich"}],"title":"New constructions of CSS codes obtained by moving to higher alphabets","issued":{"date-parts":[[2012,2,15]]},"note":"arxivid:1202.3338\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0703213","type":"journal-article","publisher":"The Royal Society","issue":"2087","abstract":"<jats:p>\n            Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to\n            <jats:inline-formula>\n              <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"2887equ1.gif\" />\n            </jats:inline-formula>\n            <jats:sub>\n              <jats:italic>q</jats:italic>\n            </jats:sub>\n            -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal\n            <jats:inline-formula>\n              <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"2887equ2.gif\" />\n            </jats:inline-formula>\n            <jats:sub>\n              <jats:italic>q</jats:italic>\n            </jats:sub>\n            -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.\n          </jats:p>","DOI":"10.1098/rspa.2007.0028","page":"2887-2905","source":"Crossref","title":"On subsystem codes beating the quantum Hamming or Singleton bound","volume":"463","author":[{"given":"Andreas","family":"Klappenecker","sequence":"first","affiliation":[{"name":"Texas A&amp;M University, College StationTexas, TX 77843, USA"}]},{"given":"Pradeep Kiran","family":"Sarvepalli","sequence":"additional","affiliation":[{"name":"Texas A&amp;M University, College StationTexas, TX 77843, USA"}]}],"container-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","original-title":[],"language":"en","issued":{"date-parts":[[2007,8,21]]},"URL":"http://dx.doi.org/10.1098/rspa.2007.0028","ISSN":["1364-5021","1471-2946"],"container-title-short":"Proc. R. Soc. A.","note":"alternative-id:10.1098/rspa.2007.0028\narxivid:quant-ph/0703213"},{"id":"arxiv:2311.12324","type":"journal-article","publisher":"American Physical Society (APS)","issue":"26","DOI":"10.1103/physrevlett.133.260601","source":"Crossref","title":"Absorption-Emission Codes for Atomic and Molecular Quantum Information Platforms","volume":"133","author":[{"ORCID":"https://orcid.org/0000-0003-3409-1126","authenticated-orcid":true,"given":"Shubham P.","family":"Jain","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"}]},{"ORCID":"https://orcid.org/0000-0002-9978-9202","authenticated-orcid":true,"given":"Eric R.","family":"Hudson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046rm7j60","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California Los Angeles"},{"id":[{"id":"https://ror.org/046rm7j60","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California Los Angeles"}]},{"ORCID":"https://orcid.org/0000-0003-1578-906X","authenticated-orcid":true,"given":"Wesley C.","family":"Campbell","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/046rm7j60","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California Los Angeles"},{"id":[{"id":"https://ror.org/046rm7j60","id-type":"ROR","asserted-by":"publisher"}],"name":"University of California Los Angeles"}]},{"ORCID":"https://orcid.org/0000-0002-0335-9508","authenticated-orcid":true,"given":"Victor V.","family":"Albert","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/02048n894","id-type":"ROR","asserted-by":"publisher"}],"name":"Joint Center for Quantum Information and Computer Science"},{"id":[{"id":"https://ror.org/05xpvk416","id-type":"ROR","asserted-by":"publisher"}],"name":"NIST"},{"id":[{"id":"https://ror.org/047s2c258","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maryland"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2024,12,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.133.260601","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"260601","note":"arxivid:2311.12324"},{"id":"arxiv:2504.16746","type":"article-journal","author":[{"given":"Yi","family":"Li"},{"given":"Qingyuan","family":"Mei"},{"given":"Qing-Xuan","family":"Jie"},{"given":"Weizhou","family":"Cai"},{"given":"Yue","family":"Li"},{"given":"Zhiyuan","family":"Liu"},{"given":"Zi-Jie","family":"Chen"},{"given":"Zihan","family":"Xie"},{"given":"Xu","family":"Cheng"},{"given":"Xingyu","family":"Zhao"},{"given":"Zhenghao","family":"Luo"},{"given":"Mengxiang","family":"Zhang"},{"given":"Xu-Bo","family":"Zou"},{"given":"Chang-Ling","family":"Zou"},{"given":"Yiheng","family":"Lin"},{"given":"Jiangfeng","family":"Du"}],"title":"Beating the break-even point with autonomous quantum error correction","issued":{"date-parts":[[2025,4,23]]},"note":"arxivid:2504.16746\narxiv_version_number:1"},{"id":"arxiv:quant-ph/0006092","type":"journal-article","publisher":"American Physical Society (APS)","issue":"6","DOI":"10.1103/physreva.62.062310","source":"Crossref","title":"Chiral spin liquids and quantum error-correcting codes","volume":"62","author":[{"given":"N. E.","family":"Bonesteel","sequence":"first","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2000,11,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.62.062310","ISSN":["1050-2947","1094-1622"],"container-title-short":"Phys. Rev. A","page":"062310","note":"arxivid:quant-ph/0006092"},{"id":"arxiv:2503.12142","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>The implementation of practical error correction protocols is essential for deployment of quantum information technologies. Ways of exploiting high-spin nuclei, which have multilevel quantum resources, have attracted interest in this context because they offer additional Hilbert space dimensions in a spatially compact and theoretically efficient structure. We present a quantitative analysis of the performance of a spin-qudit-based error-correctable quantum memory, with reference to the actual Hamiltonians of several potential candidate systems. First, the ideal code-word implemented on a spin-7/2 nucleus, which provides first-order Pauli-<a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\"><a:mi>X</a:mi><a:mo>,</a:mo><a:mo> </a:mo><a:mi>Y</a:mi></a:math> and <b:math xmlns:b=\"http://www.w3.org/1998/Math/MathML\"><b:mi>Z</b:mi></b:math> error correction, has intrinsic infidelity due to mixed eigenstates under realistic conditions. We confirm that expansion to a spin-9/2 system with tailored code words can compensate for this infidelity. Second, we claim that electric field fluctuations, which are inevitable in real systems, should also be considered as a noise source, and we illustrate an encoding and decoding scheme for a multi-spin-qudit-based error correction code that can simultaneously compensate for both electric- and magnetic-field perturbations. Such strategies are important as we move beyond the current noisy-intermediate quantum era, and fidelities above two or three nines become crucial for the implementation of quantum technologies.</jats:p>","DOI":"10.1103/7q6l-d4qh","source":"Crossref","title":"Designing quantum error correction codes for practical spin qudit systems","volume":"112","author":[{"ORCID":"https://orcid.org/0000-0001-8954-8862","authenticated-orcid":true,"given":"Sumin","family":"Lim","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/052gg0110","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Oxford"}]},{"ORCID":"https://orcid.org/0000-0001-5521-2658","authenticated-orcid":true,"given":"Arzhang","family":"Ardavan","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/052gg0110","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Oxford"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2025,8,12]]},"URL":"http://dx.doi.org/10.1103/7q6l-d4qh","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022418","note":"arxivid:2503.12142"},{"id":"arxiv:2210.16957","type":"journal-article","publisher":"American Physical Society (APS)","issue":"3","DOI":"10.1103/physreva.107.032411","source":"Crossref","title":"Quantum error correction in the lowest Landau level","volume":"107","author":[{"ORCID":"https://orcid.org/0000-0002-0824-3758","authenticated-orcid":true,"given":"Yale","family":"Fan","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-9075-015X","authenticated-orcid":true,"given":"Willy","family":"Fischler","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-5451-0290","authenticated-orcid":true,"given":"Eric","family":"Kubischta","sequence":"additional","affiliation":[]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,3,13]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.107.032411","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"032411","note":"arxivid:2210.16957"},{"id":"arxiv:1205.4517","type":"article-journal","author":[{"given":"Christopher","family":"Bumgardner"}],"title":"Codes in W\\ast-metric Spaces: Theory and Examples","issued":{"date-parts":[[2012,5,21]]},"note":"arxivid:1205.4517\narxiv_version_number:1"},{"id":"doi:10.1016/0024-3795(93)90274-R","type":"journal-article","publisher":"Elsevier BV","DOI":"10.1016/0024-3795(93)90274-r","page":"107-127","source":"Crossref","title":"On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras","volume":"193","author":[{"given":"L.J.","family":"Landau","sequence":"first","affiliation":[]},{"given":"R.F.","family":"Streater","sequence":"additional","affiliation":[]}],"container-title":"Linear Algebra and its Applications","original-title":[],"language":"en","issued":{"date-parts":[[1993,11]]},"URL":"http://dx.doi.org/10.1016/0024-3795(93)90274-R","ISSN":["0024-3795"],"container-title-short":"Linear Algebra and its Applications","note":"special_numbering:C\nalternative-id:002437959390274R"},{"id":"arxiv:2401.04271","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>We construct a fault-tolerant quantum error-correcting protocol based on a qubit encoded in a large spin qudit using a spin-cat code, analogous to the continuous-variable cat encoding. With this, we can correct the dominant error sources, namely processes that can be expressed as error operators that are linear or quadratic in the components of angular momentum. Such codes tailored to dominant error sources can exhibit superior thresholds and lower resource overheads when compared to those designed for unstructured noise models. A key component is the  gate that preserves the rank of spherical tensor operators. Categorizing the dominant errors as phase and amplitude errors, we demonstrate how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Furthermore, we propose a measurement-free error-correction scheme to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical  gate errors, we establish that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. We consider a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\" overflow=\"scroll\"><a:msup><a:mi/><a:mn>87</a:mn></a:msup><a:mi>Sr</a:mi></a:math>, and show how to generate the universal gate set, including the rank-preserving  gate, using quantum control and the Rydberg blockade. These findings pave the way for encoding a qubit in a large spin with the potential to achieve fault tolerance, high threshold, and reduced resource overhead in quantum information processing.</jats:p>\n          <jats:sec>\n            <jats:title/>\n            <jats:supplementary-material>\n              <jats:permissions>\n                <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement>\n                <jats:copyright-year>2024</jats:copyright-year>\n              </jats:permissions>\n            </jats:supplementary-material>\n          </jats:sec>","DOI":"10.1103/prxquantum.5.020355","source":"Crossref","title":"Fault-Tolerant Quantum Computation Using Large Spin-Cat Codes","volume":"5","author":[{"ORCID":"https://orcid.org/0000-0002-6229-7087","authenticated-orcid":true,"given":"Sivaprasad","family":"Omanakuttan","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"},{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"}]},{"ORCID":"https://orcid.org/0000-0002-1592-5626","authenticated-orcid":true,"given":"Vikas","family":"Buchemmavari","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"},{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"}]},{"ORCID":"https://orcid.org/0000-0001-7049-6206","authenticated-orcid":true,"given":"Jonathan A.","family":"Gross","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00njsd438","id-type":"ROR","asserted-by":"publisher"}],"name":"Google Quantum AI"}]},{"given":"Ivan H.","family":"Deutsch","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"},{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"}]},{"given":"Milad","family":"Marvian","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"},{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"},{"id":[{"id":"https://ror.org/05fs6jp91","id-type":"ROR","asserted-by":"publisher"}],"name":"University of New Mexico"}]}],"container-title":"PRX Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2024,6,7]]},"URL":"http://dx.doi.org/10.1103/PRXQuantum.5.020355","ISSN":["2691-3399"],"container-title-short":"PRX Quantum","page":"020355","note":"arxivid:2401.04271"},{"id":"arxiv:2408.04421","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>We present a biased atomic qubit, universally implementable across all atomic platforms, encoded as a “spin cat” within ground state Zeeman levels. The key characteristic of our configuration is the coupling of the ground state spin manifold of size <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mrow><a:msub><a:mrow><a:mi>F</a:mi></a:mrow><a:mrow><a:mi>g</a:mi></a:mrow></a:msub><a:mo>≫</a:mo><a:mn>1</a:mn></a:mrow></a:math> to an excited Zeeman spin manifold of size <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mrow><c:msub><c:mrow><c:mi>F</c:mi></c:mrow><c:mrow><c:mi>e</c:mi></c:mrow></c:msub><c:mo>=</c:mo><c:msub><c:mrow><c:mi>F</c:mi></c:mrow><c:mrow><c:mi>g</c:mi></c:mrow></c:msub><c:mo>−</c:mo><c:mn>1</c:mn></c:mrow></c:math> using light. This coupling results in eigenstates of the driven atom that include exactly two dark states in the ground state manifold, which are decoupled from light and immune to spontaneous emission from the excited states. These dark states constitute the spin cat, leading to the designation “dark spin cat.” We demonstrate that under strong Rabi drive and for large <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:msub><e:mi>F</e:mi><e:mi>g</e:mi></e:msub></e:math>, the dark spin cat is autonomously stabilized against common noise sources and encodes a qubit with significantly biased noise. Specifically, the bit-flip error rate decreases exponentially with <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:msub><g:mi>F</g:mi><g:mi>g</g:mi></g:msub></g:math> relative to the dephasing rate. We provide an analysis of dark spin cats and their robustness to noise, and we discuss bias-preserving single qubit and entangling gates, exemplified on a Rydberg tweezer platform.</jats:p>","DOI":"10.1103/w9zh-jwsx","source":"Crossref","title":"Dark Spin-Cat States as Biased Qubits","volume":"135","author":[{"ORCID":"https://orcid.org/0000-0002-2420-648X","authenticated-orcid":true,"given":"Andreas","family":"Kruckenhauser","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"},{"id":[{"id":"https://ror.org/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences"},{"name":"PlanQC GmbH"}]},{"ORCID":"https://orcid.org/0000-0002-5625-6481","authenticated-orcid":true,"given":"Ming","family":"Yuan","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering"},{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0002-8375-5833","authenticated-orcid":true,"given":"Han","family":"Zheng","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering"},{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0002-9775-6407","authenticated-orcid":true,"given":"Mikhail","family":"Mamaev","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering"},{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0003-4016-0706","authenticated-orcid":true,"given":"Pei","family":"Zeng","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering"},{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0009-0005-7945-3970","authenticated-orcid":true,"given":"Xuanhui","family":"Mao","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering"},{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0002-8738-9420","authenticated-orcid":true,"given":"Qian","family":"Xu","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering"},{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0003-3549-7160","authenticated-orcid":true,"given":"Torsten V.","family":"Zache","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"},{"id":[{"id":"https://ror.org/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences"}]},{"ORCID":"https://orcid.org/0000-0002-0000-9342","authenticated-orcid":true,"given":"Liang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Pritzker School of Molecular Engineering"},{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0002-0979-2521","authenticated-orcid":true,"given":"Rick","family":"van Bijnen","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"},{"id":[{"id":"https://ror.org/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences"},{"name":"PlanQC GmbH"}]},{"ORCID":"https://orcid.org/0000-0003-4014-1505","authenticated-orcid":true,"given":"Peter","family":"Zoller","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/054pv6659","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Innsbruck"},{"id":[{"id":"https://ror.org/056hzt889","id-type":"ROR","asserted-by":"publisher"}],"name":"Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2025,7,8]]},"URL":"http://dx.doi.org/10.1103/w9zh-jwsx","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"020601","note":"arxivid:2408.04421"},{"id":"arxiv:2101.03662","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.127.093602","source":"Crossref","title":"Generating Long-Lived Macroscopically Distinct Superposition States in Atomic Ensembles","volume":"127","author":[{"ORCID":"https://orcid.org/0000-0003-1766-8245","authenticated-orcid":true,"given":"Wei","family":"Qin","sequence":"first","affiliation":[{"name":"Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan"}]},{"ORCID":"https://orcid.org/0000-0002-8222-9268","authenticated-orcid":true,"given":"Adam","family":"Miranowicz","sequence":"additional","affiliation":[{"name":"Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan"},{"name":"Institute of Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland"}]},{"ORCID":"https://orcid.org/0000-0001-5091-2057","authenticated-orcid":true,"given":"Hui","family":"Jing","sequence":"additional","affiliation":[{"name":"Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan"},{"name":"Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China"}]},{"ORCID":"https://orcid.org/0000-0003-3682-7432","authenticated-orcid":true,"given":"Franco","family":"Nori","sequence":"additional","affiliation":[{"name":"Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan"},{"name":"Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109, USA"}]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2021,8,23]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.127.093602","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"093602","note":"arxivid:2101.03662"},{"id":"arxiv:2503.13908","type":"article-journal","author":[{"given":"Kyle","family":"DeBry"},{"given":"Nadine","family":"Meister"},{"given":"Agustin Valdes","family":"Martinez"},{"given":"Colin D.","family":"Bruzewicz"},{"given":"Xiaoyang","family":"Shi"},{"given":"David","family":"Reens"},{"given":"Robert","family":"McConnell"},{"given":"Isaac L.","family":"Chuang"},{"given":"John","family":"Chiaverini"}],"title":"Error correction of a logical qubit encoded in a single atomic ion","issued":{"date-parts":[[2025,3,18]]},"note":"arxivid:2503.13908\narxiv_version_number:1"},{"id":"arxiv:2405.15494","type":"journal-article","publisher":"Springer Science and Business Media LLC","issue":"3","DOI":"10.1038/s41567-024-02745-0","page":"362-367","source":"Crossref","title":"Schrödinger cat states of a nuclear spin qudit in silicon","volume":"21","author":[{"ORCID":"https://orcid.org/0009-0006-1137-3031","authenticated-orcid":false,"given":"Xi","family":"Yu","sequence":"first","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5013-7852","authenticated-orcid":false,"given":"Benjamin","family":"Wilhelm","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7040-5538","authenticated-orcid":false,"given":"Danielle","family":"Holmes","sequence":"additional","affiliation":[]},{"given":"Arjen","family":"Vaartjes","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-5715-6140","authenticated-orcid":false,"given":"Daniel","family":"Schwienbacher","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8165-5591","authenticated-orcid":false,"given":"Martin","family":"Nurizzo","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7470-4965","authenticated-orcid":false,"given":"Anders","family":"Kringhøj","sequence":"additional","affiliation":[]},{"given":"Mark R. van","family":"Blankenstein","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-2794-404X","authenticated-orcid":false,"given":"Alexander M.","family":"Jakob","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1791-6296","authenticated-orcid":false,"given":"Pragati","family":"Gupta","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-0134-3657","authenticated-orcid":false,"given":"Fay E.","family":"Hudson","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8153-4893","authenticated-orcid":false,"given":"Kohei M.","family":"Itoh","sequence":"additional","affiliation":[]},{"given":"Riley J.","family":"Murray","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-8134-948X","authenticated-orcid":false,"given":"Robin","family":"Blume-Kohout","sequence":"additional","affiliation":[]},{"given":"Thaddeus D.","family":"Ladd","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-4116-4581","authenticated-orcid":false,"given":"Namit","family":"Anand","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0003-1389-5096","authenticated-orcid":false,"given":"Andrew S.","family":"Dzurak","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0002-8326-8912","authenticated-orcid":false,"given":"Barry C.","family":"Sanders","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7733-6715","authenticated-orcid":false,"given":"David N.","family":"Jamieson","sequence":"additional","affiliation":[]},{"ORCID":"https://orcid.org/0000-0001-7445-699X","authenticated-orcid":false,"given":"Andrea","family":"Morello","sequence":"additional","affiliation":[]}],"container-title":"Nature Physics","original-title":[],"language":"en","issued":{"date-parts":[[2025,1,14]]},"URL":"http://dx.doi.org/10.1038/s41567-024-02745-0","ISSN":["1745-2473","1745-2481"],"container-title-short":"Nat. Phys.","note":"alternative-id:2745\narxivid:2405.15494"},{"id":"arxiv:2405.15695","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","abstract":"<jats:p>Spins and oscillators are foundational to much of physics and applied sciences. For quantum information, a spin <a:math xmlns:a=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><a:mrow><a:mn>1</a:mn><a:mo>/</a:mo><a:mn>2</a:mn></a:mrow></a:math> exemplifies the most basic unit, a qubit. High angular momentum spins (HAMSs) and harmonic oscillators provide multilevel manifolds which have the potential for hardware-efficient protected encodings of quantum information and simulation of many-body quantum systems. In this work, we demonstrate a new quantum control protocol that conceptually merges these disparate hardware platforms. Namely, we show how to modify a harmonic oscillator on demand to implement a continuous range of generators to accomplish linear and nonlinear HAMS dynamics. The spinlike dynamics are verified by demonstration of linear spin coherent [SU(2)] rotations, nonlinear spin rotations, and comparison to other manifolds like simply truncated oscillators. Our scheme allows universal control of a spin cat logical qubit encoding with interpretable drive pulses: We use linear operations to accomplish four logical gates and further show that nonlinear spin rotations can complete the logical gate set. Our results show how motion on a closed Hilbert space can be useful for quantum information processing and opens the door to superconducting circuit simulations of higher angular momentum quantum magnetism.</jats:p>","DOI":"10.1103/physrevx.15.021009","source":"Crossref","title":"Synthetic High Angular Momentum Spin Dynamics in a Microwave Oscillator","volume":"15","author":[{"ORCID":"https://orcid.org/0000-0003-3466-8589","authenticated-orcid":true,"given":"Saswata","family":"Roy","sequence":"first","affiliation":[{"id":[{"id":"https://ror.org/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University"}]},{"ORCID":"https://orcid.org/0000-0002-7757-3822","authenticated-orcid":true,"given":"Alen","family":"Senanian","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University"}]},{"ORCID":"https://orcid.org/0000-0001-7295-5648","authenticated-orcid":true,"given":"Christopher S.","family":"Wang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/024mw5h28","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Chicago"}]},{"ORCID":"https://orcid.org/0000-0003-4527-6863","authenticated-orcid":true,"given":"Owen C.","family":"Wetherbee","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University"}]},{"given":"Luojia","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University"}]},{"given":"B.","family":"Cole","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/025r5qe02","id-type":"ROR","asserted-by":"publisher"}],"name":"Syracuse University"}]},{"given":"C. P.","family":"Larson","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/025r5qe02","id-type":"ROR","asserted-by":"publisher"}],"name":"Syracuse University"}]},{"given":"E.","family":"Yelton","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/025r5qe02","id-type":"ROR","asserted-by":"publisher"}],"name":"Syracuse University"}]},{"ORCID":"https://orcid.org/0000-0003-0495-6360","authenticated-orcid":true,"given":"Kartikeya","family":"Arora","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/01kh5gc44","id-type":"ROR","asserted-by":"publisher"}],"name":"Indian Institute of Technology (Banaras Hindu University)"},{"id":[{"id":"https://ror.org/00kybxq39","id-type":"ROR","asserted-by":"publisher"}],"name":"Université de Sherbrooke"}]},{"ORCID":"https://orcid.org/0000-0002-1177-9887","authenticated-orcid":true,"given":"Peter L.","family":"McMahon","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University"}]},{"ORCID":"https://orcid.org/0000-0001-9890-8532","authenticated-orcid":true,"given":"B. L. T.","family":"Plourde","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/025r5qe02","id-type":"ROR","asserted-by":"publisher"}],"name":"Syracuse University"}]},{"given":"Baptiste","family":"Royer","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/00kybxq39","id-type":"ROR","asserted-by":"publisher"}],"name":"Université de Sherbrooke"}]},{"ORCID":"https://orcid.org/0000-0003-3648-7706","authenticated-orcid":true,"given":"Valla","family":"Fatemi","sequence":"additional","affiliation":[{"id":[{"id":"https://ror.org/05bnh6r87","id-type":"ROR","asserted-by":"publisher"}],"name":"Cornell University"}]}],"container-title":"Physical Review X","original-title":[],"language":"en","issued":{"date-parts":[[2025,4,8]]},"URL":"http://dx.doi.org/10.1103/PhysRevX.15.021009","ISSN":["2160-3308"],"container-title-short":"Phys. Rev. X","page":"021009","note":"arxivid:2405.15695"},{"id":"arxiv:2211.05181","type":"journal-article","publisher":"American Physical Society (APS)","issue":"2","DOI":"10.1103/physreva.108.022428","source":"Crossref","title":"Spin-squeezed Gottesman-Kitaev-Preskill codes for quantum error correction in atomic ensembles","volume":"108","author":[{"ORCID":"https://orcid.org/0000-0002-6229-7087","authenticated-orcid":true,"given":"Sivaprasad","family":"Omanakuttan","sequence":"first","affiliation":[{"name":"Center for Quantum Information and Control and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico, USA"}]},{"given":"T. J.","family":"Volkoff","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA"}]}],"container-title":"Physical Review A","original-title":[],"language":"en","issued":{"date-parts":[[2023,8,28]]},"URL":"http://dx.doi.org/10.1103/PhysRevA.108.022428","ISSN":["2469-9926","2469-9934"],"container-title-short":"Phys. Rev. A","page":"022428","note":"arxivid:2211.05181"},{"id":"arxiv:1412.4687","type":"journal-article","publisher":"American Physical Society (APS)","issue":"9","DOI":"10.1103/physrevlett.114.090502","source":"Crossref","title":"Simulating Hamiltonian Dynamics with a Truncated Taylor Series","volume":"114","author":[{"given":"Dominic W.","family":"Berry","sequence":"first","affiliation":[]},{"given":"Andrew M.","family":"Childs","sequence":"additional","affiliation":[]},{"given":"Richard","family":"Cleve","sequence":"additional","affiliation":[]},{"given":"Robin","family":"Kothari","sequence":"additional","affiliation":[]},{"given":"Rolando D.","family":"Somma","sequence":"additional","affiliation":[]}],"container-title":"Physical Review Letters","original-title":[],"language":"en","issued":{"date-parts":[[2015,3,3]]},"URL":"http://dx.doi.org/10.1103/PhysRevLett.114.090502","ISSN":["0031-9007","1079-7114"],"container-title-short":"Phys. Rev. Lett.","page":"090502","note":"arxivid:1412.4687"},{"id":"arxiv:1610.06546","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>We present the problem of approximating the time-evolution operator<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math>to error<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>ϵ</mml:mi></mml:math>, where the Hamiltonian<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">⟨</mml:mo><mml:mi>G</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mo>⊗</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>G</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo><mml:mo>⊗</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">I</mml:mi></mml:mrow><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:math>is the projection of a unitary oracle<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>onto the state<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mi>G</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">⟩</mml:mo></mml:math>created by another unitary oracle. Our algorithm solves this with a query complexity<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi></mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mml:mo></mml:mrow><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>log</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mml:mo></mml:mrow></mml:math>to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>d</mml:mi></mml:math>-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>in an invariant<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mtext>SU</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:math>subspace. A large class of operator functions of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow></mml:math>can then be computed with optimal query complexity, of which<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:math>is a special case.</jats:p>","DOI":"10.22331/q-2019-07-12-163","page":"163","source":"Crossref","title":"Hamiltonian Simulation by Qubitization","volume":"3","author":[{"ORCID":"https://orcid.org/0000-0002-6934-1052","authenticated-orcid":false,"given":"Guang Hao","family":"Low","sequence":"first","affiliation":[{"name":"Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA"}]},{"given":"Isaac L.","family":"Chuang","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering and Computer Science, Department of Physics, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2019,7,12]]},"URL":"http://dx.doi.org/10.22331/q-2019-07-12-163","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:1610.06546"},{"id":"arxiv:2203.08882","type":"journal-article","publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","abstract":"<jats:p>Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with Hamiltonians<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>V</mml:mi></mml:math>. Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>at inverse temperature<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo>&amp;#x2265;</mml:mo><mml:mn>0</mml:mn></mml:math>starting from a purification of the thermal state of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>. The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mover><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi><mml:mo stretchy=\"false\">&amp;#x007E;</mml:mo></mml:mover></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">&amp;#x0394;</mml:mi><mml:mspace width=\"negativethinmathspace\" /><mml:mi>A</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:math>, where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x0394;</mml:mi><mml:mspace width=\"negativethinmathspace\" /><mml:mi>A</mml:mi></mml:math>is the free-energy difference between<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:math>and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:math>is a work cutoff that depends on the properties of the work distribution and the approximation error<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi><mml:mo>&amp;#x003E;</mml:mo><mml:mn>0</mml:mn></mml:math>. If the non-equilibrium process is trivial, this complexity is exponential in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math>, where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math>is the spectral norm of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>V</mml:mi></mml:math>. This represents a significant improvement of prior quantum algorithms that have complexity exponential in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03B2;</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math>in the regime where<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mi>V</mml:mi><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:mo>&amp;#x226A;</mml:mo><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo fence=\"false\" stretchy=\"false\">&amp;#x2016;</mml:mo></mml:math>. The dependence of the complexity in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mi>&amp;#x03F5;</mml:mi></mml:math>varies according to the structure of the quantum systems. It can be exponential in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math>in general, but we show it to be sublinear in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math>if<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>commute, or polynomial in<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mn>1</mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mi>&amp;#x03F5;</mml:mi></mml:math>if<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>and<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>are local spin systems. The possibility of applying a unitary that drives the system out of equilibrium allows one to increase the value of<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:msub><mml:mi>w</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:math>and improve the complexity even further. To this end, we analyze the complexity for preparing the thermal state of the transverse field Ising model using different non-equilibrium unitary processes and see significant complexity improvements.</jats:p>","DOI":"10.22331/q-2022-10-06-825","page":"825","source":"Crossref","title":"Quantum algorithms from fluctuation theorems: Thermal-state preparation","volume":"6","author":[{"given":"Zoe","family":"Holmes","sequence":"first","affiliation":[{"name":"Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Gopikrishnan","family":"Muraleedharan","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Rolando D.","family":"Somma","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Yigit","family":"Subasi","sequence":"additional","affiliation":[{"name":"Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]},{"given":"Burak","family":"Şahinoğlu","sequence":"additional","affiliation":[{"name":"Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA"}]}],"container-title":"Quantum","original-title":[],"language":"en","issued":{"date-parts":[[2022,10,6]]},"URL":"http://dx.doi.org/10.22331/q-2022-10-06-825","ISSN":["2521-327X"],"container-title-short":"Quantum","note":"arxivid:2203.08882"}]