## 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]. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [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].

## Parents

- Modular-qudit stabilizer code
- 2D lattice stabilizer code — 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 [4]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.
- Abelian topological code — Abelian TQDs realize all modular gapped Abelian topological orders [4]. Conversely, every Abelian anyon theory is a subtheory of some TQD [7; 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 [8,9][7; 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 exists as the deconfined phase of the twisted \(\mathbb{Z}_2\) gauge theory in two dimensions [10].
- 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 [7]. Upon gauging, Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models [11–13].

## 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 [11–13].
- 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 [7].
- XS stabilizer code — TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [14].

## 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]
- 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 topological fault-tolerant circuits for all non-chiral abelian topological phases”, (2024) arXiv:2403.12119
- [7]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [8]
- 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
- [9]
- 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
- [10]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
- [11]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [12]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [13]
- 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
- [14]
- X. Ni, O. Buerschaper, and M. Van den Nest, “A non-commuting stabilizer formalism”, Journal of Mathematical Physics 56, (2015) arXiv:1404.5327 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