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].Cousins
- Dihedral \(G=D_m\) quantum-double code— Upon gauging some symmetries [7–9,9], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [10–12].
- XS stabilizer code— TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [13].
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 [4]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.
Abelian TQDs realize all modular gapped Abelian topological orders [4]. Conversely, every Abelian anyon theory is a subtheory of some TQD [14; 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 [15,16][14; Appx. H].
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].
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 [17]. 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 [14].
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 [14]. Upon gauging some symmetries [7–9,9], Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models [10–12].
References
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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