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Double-semion stabilizer code[1,2]

Alternative names: Doubled semion model code.

Description

A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that realizes the 2D double semion topological phase. The code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [3].

This stabilizer code family is inequivalent to a CSS code via a Clifford circuit whose depth does not scale with \(n\) [4; Thm. 1.1]. This is because the double semion phase has a sign problem [4,5], and existence of such a Clifford circuit would allow one to construct a code Hamiltonian that is free of such a problem.

Cousins

  • Modular-qudit surface code— The exchange statistics of the anyon for the double-semion code coincides with a subset of anyons in the \(\mathbb{Z}_4\) surface code, but the fusion rules are different. The double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [3] or by gauging [615] the one-form symmetry associated with said anyon [3; Footnote 20].
  • Double-semion string-net code— The double-semion stabilizer code and the double-semion string-net code both realize the double semion topological phase, but the former is a modular-qudit stabilizer code, while the latter is an \(XS\) stabilizer code. A commuting-projector version of the double-semion string-net code can also be derived [16,17].
  • X-cube model code— A non-stabilizer commuting-projector code constructed by stacking layers of the double-semion string-net model, called the semionic X-cube model [18], is equivalent to the X-cube model [19] (see also Refs. [20,21]).
  • \(\mathbb{Z}_q^{(1)}\) subsystem code— The anyonic exchange statistics of \(\mathbb{Z}_4^{(1)}\) subsystem code resemble those of the double semion code, but its fusion rules realize the \(\mathbb{Z}_4\) group.
  • Chiral semion subsystem code— The semion code can be obtained from the double-semion stabilizer code by gauging out the anyon \(\bar{s}\) [3; Fig. 15].

Primary Hierarchy

Parents
When treated as ground states of the code Hamiltonian, the double-semion stabilizer code states realize 2D double-semion topological order, a topological phase of matter that exists as the deconfined phase of the 2D twisted \(\mathbb{Z}_2\) gauge theory [22]. All Abelian TQD codes can be realized as modular-qudit stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and condensing certain bosonic anyons [3].
Double-semion stabilizer code

References

[1]
M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 DOI
[2]
T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
[3]
T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
[4]
M. B. Hastings, “How quantum are non-negative wavefunctions?”, Journal of Mathematical Physics 57, (2015) arXiv:1506.08883 DOI
[5]
A. Smith, O. Golan, and Z. Ringel, “Intrinsic sign problems in topological quantum field theories”, Physical Review Research 2, (2020) arXiv:2005.05343 DOI
[6]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[7]
J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
[8]
S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[9]
D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[10]
L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
[11]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[12]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[13]
K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[14]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[15]
D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[16]
G. Dauphinais, L. Ortiz, S. Varona, and M. A. Martin-Delgado, “Quantum error correction with the semion code”, New Journal of Physics 21, 053035 (2019) arXiv:1810.08204 DOI
[17]
J. C. Magdalena de la Fuente, N. Tarantino, and J. Eisert, “Non-Pauli topological stabilizer codes from twisted quantum doubles”, Quantum 5, 398 (2021) arXiv:2001.11516 DOI
[18]
H. Ma, E. Lake, X. Chen, and M. Hermele, “Fracton topological order via coupled layers”, Physical Review B 95, (2017) arXiv:1701.00747 DOI
[19]
W. Shirley, K. Slagle, and X. Chen, “Fractional excitations in foliated fracton phases”, Annals of Physics 410, 167922 (2019) arXiv:1806.08625 DOI
[20]
T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
[21]
S. Pai and M. Hermele, “Fracton fusion and statistics”, Physical Review B 100, (2019) arXiv:1903.11625 DOI
[22]
R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
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Zoo Code ID: double_semion

Cite as:
“Double-semion stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/double_semion
BibTeX:
@incollection{eczoo_double_semion, title={Double-semion stabilizer code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/double_semion} }
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“Double-semion stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/double_semion

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/topological/double_semion.yml.