Abelian TQD stabilizer code[13] 

Description

Modular-qudit stabilizer 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 [4; Sec. IV.A].

Parents

  • Modular-qudit stabilizer code — 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 [5].
  • Abelian topological code — Abelian TQDs realize all modular gapped Abelian topological orders [4]. Conversely, every abelian anyon theory is a subtheory of some TQD [5; 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 [6,7][5; Appx. H].
  • Twisted quantum double (TQD) code — The anyon theory corresponding to (abelian) TQD codes is defined by an (abelian) group and a Type III cocycle.

Children

  • Double-semion stabilizer code — When treated as ground states of the code Hamiltonian, the code states realize double-semion topological order, a topological phase of matter that also exists in twisted \(\mathbb{Z}_2\) gauge theory [8].
  • Abelian quantum double stabilizer code — The anyon theory corresponding to abelian quantum double codes is defined by an abelian group and trivial cocycle. 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 [5]. Upon gauging, Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain abelian quantum double models [911].

Cousins

  • Dihedral \(G=D_m\) quantum-double code — Upon gauging, a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [911].
  • Double-semion stabilizer code — 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 [5].
  • Translationally invariant stabilizer code — Translationally-invariant stabilizer codes can realize 2D modular gapped abelian topological orders [5]. Conversely, abelian TQD codes need not be translationally invariant, and can realize multiple topological phases on one lattice.
  • XS stabilizer code — TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [12].

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]
J. Kaidi et al., “Higher central charges and topological boundaries in 2+1-dimensional TQFTs”, SciPost Physics 13, (2022) arXiv:2107.13091 DOI
[4]
T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
[5]
T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
[6]
J. Haah, “Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D”, Journal of Mathematical Physics 62, 092202 (2021) arXiv:1907.02075 DOI
[7]
W. Shirley et al., “Three-Dimensional Quantum Cellular Automata from Chiral Semion Surface Topological Order and beyond”, PRX Quantum 3, (2022) arXiv:2202.05442 DOI
[8]
R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
[9]
M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
[10]
B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
[11]
L. Lootens et al., “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
[12]
X. Ni, O. Buerschaper, and M. Van den Nest, “A non-commuting stabilizer formalism”, Journal of Mathematical Physics 56, 052201 (2015) arXiv:1404.5327 DOI
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Zoo Code ID: tqd_abelian

Cite as:
“Abelian TQD stabilizer 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 stabilizer code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/tqd_abelian} }
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“Abelian TQD stabilizer 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/tree/main/codes/quantum/qudits/topological/tqd_abelian.yml.