Description
TQD 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 [3; Sec. IV.A]. Abelian TQDs realize all modular gapped Abelian topological orders [3].
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 [3]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [4,5].
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
- Commuting-projector Hamiltonian code— Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [5].
- Quantum-double code— A Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [7–9]. There is a sufficient condition for when a Type-III TQD can be realized as a quantum double model [10].
- Dihedral \(G=D_m\) quantum-double code— A Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [7–9].
- XS stabilizer code— Abelian TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [11]. Upon gauging some symmetries [12–21], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [7–9].
Primary Hierarchy
References
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Page edit log
- Victor V. Albert (2023-04-06) — most recent
Cite as:
“Abelian TQD code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/tqd_abelian