[Jump to code hierarchy]

Abelian TQD code[1,2]

Description

TQD code whose codewords realize 2D modular gapped Abelian topological order. The corresponding anyon theory is defined by an Abelian group and a Type-III group cocycle that can be decomposed as a product of Type-I and Type-II group cocycles; see [3; Sec. IV.A]. Abelian TQDs realize all modular gapped Abelian topological orders [3].

All Abelian TQD codes can be realized as modular-qudit lattice 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]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [4,5].

Encoding

Fault-tolerant circuits for all non-chiral abelian topological phases and the \(\mathbb{Z}_2^3\) code with a type-III cocycle [6].

Fault Tolerance

Fault-tolerant circuits for all non-chiral abelian topological phases and the \(\mathbb{Z}_2^3\) code with a type-III cocycle [6].

Cousins

  • Commuting-projector Hamiltonian code— Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [5].
  • Quantum-double code— A Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [7–9]. There is a sufficient condition for when a Type-III TQD can be realized as a quantum double model [10].
  • Dihedral \(G=D_m\) quantum-double code— A Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [7–9].
  • XS stabilizer code— Abelian TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [11]. Upon gauging some symmetries [12–21], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [7–9].

Member of code lists

Primary Hierarchy

Quantum Domain
Group Kingdom
Group-based quantum codeQECC Quantum
Dijkgraaf-Witten gauge theory codeTopological Hamiltonian-based QECC Quantum
Twisted quantum double (TQD) codeTopological Hamiltonian-based QECC Quantum
Parents
Twisted quantum double (TQD) code
The anyon theory corresponding to (Abelian) TQD codes is defined by an (Abelian) group and a Type III cocycle. Abelian TQDs realize all modular gapped Abelian topological orders [3].
Abelian topological codeTopological Hamiltonian-based QECC Quantum
Abelian TQDs realize all modular gapped Abelian topological orders [3]. Conversely, every Abelian anyon theory is a subtheory of some TQD [22; Sec. 6.2]. Any Abelian anyon theory \(A\) can be realized at one of the surfaces of a 3D Walker-Wang model whose underlying theory is an Abelian TQD containing \(A\) as a subtheory [23,24][22; Appx. H].
Abelian TQD code
Children
Hexagonal \(CZ\) code
The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [8,25], which is the same topological order as the \(G=D_4\) quantum double [7,9].
Brickwork \(XS\) stabilizer code
The ground-state subspace of the brickwork \(XS\) stabilizer code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [8,25], which is the same topological order as the \(G=D_4\) quantum double [7,9].
Double-semion string-net code
When treated as ground states of the code Hamiltonian, the double-semion string-net 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 [26].
Abelian TQD stabilizer codeCDSC Kitaev surface
All Abelian TQD codes can be realized as modular-qudit lattice 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]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.

References

[1]
A. Kapustin and N. Saulina, “Topological boundary conditions in abelian Chern–Simons theory”, Nuclear Physics B 845, 393 (2011) arXiv:1008.0654 DOI
[2]
Y. Hu, Y. Wan, and Y.-S. Wu, “Twisted quantum double model of topological phases in two dimensions”, Physical Review B 87, (2013) arXiv:1211.3695 DOI
[3]
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
[4]
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
[5]
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
[6]
A. Bauer, “Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases”, Quantum 9, 1673 (2025) arXiv:2403.12119 DOI
[7]
M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
[8]
B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
[9]
L. Lootens, B. Vancraeynest-De Cuiper, N. Schuch, and F. Verstraete, “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
[10]
B. Li and G. Liu, “On Gauge Equivalence of Twisted Quantum Doubles”, (2024) arXiv:2408.09353
[11]
X. Ni, O. Buerschaper, and M. Van den Nest, “A non-commuting stabilizer formalism”, Journal of Mathematical Physics 56, (2015) arXiv:1404.5327 DOI
[12]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[13]
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
[14]
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
[15]
D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[16]
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
[17]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[18]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[19]
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
[20]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[21]
D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[22]
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
[23]
J. Haah, “Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D”, Journal of Mathematical Physics 62, (2021) arXiv:1907.02075 DOI
[24]
W. Shirley, Y.-A. Chen, A. Dua, T. D. Ellison, N. Tantivasadakarn, and D. J. Williamson, “Three-Dimensional Quantum Cellular Automata from Chiral Semion Surface Topological Order and beyond”, PRX Quantum 3, (2022) arXiv:2202.05442 DOI
[25]
P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
[26]
R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: tqd_abelian

Cite as:
“Abelian TQD code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/tqd_abelian
BibTeX:
@incollection{eczoo_tqd_abelian, title={Abelian TQD code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/tqd_abelian} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/tqd_abelian

Cite as:

“Abelian TQD code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/tqd_abelian

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/categories/gauge/tqd_abelian.yml.