Abelian TQD stabilizer code[1]
Description
Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order.Primary Hierarchy
Parents
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 [2]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.
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 [2]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.
Abelian TQD stabilizer code
Children
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 [3]. 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 [4].
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 [4]. Upon gauging some symmetries [5–7,7], Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models [8–10].
References
- [1]
- J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, “Higher central charges and topological boundaries in 2+1-dimensional TQFTs”, SciPost Physics 13, (2022) arXiv:2107.13091 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]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
- [4]
- 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
- [5]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [6]
- 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
- [7]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [8]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [9]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [10]
- 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
Page edit log
- Victor V. Albert (2023-04-06) — most recent
Cite as:
“Abelian TQD stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/tqd_abelian_stabilizer