## 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]. Abelian TQDs realize all modular gapped Abelian topological orders [4]. 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. Abelian TQDs realize all modular gapped Abelian topological orders [4].

## Children

- Double-semion stabilizer code — When treated as ground states of the code Hamiltonian, the 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 [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 some symmetries [11–13,13], Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models [14–16].

## Cousins

- 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].
- Dihedral \(G=D_m\) quantum-double code — Upon gauging some symmetries [11–13,13], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [14–16].
- XS stabilizer code — TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [17].

## References

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- A. Bauer, “Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases”, (2024) arXiv:2403.12119
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- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
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- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
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- 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
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- 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