Description
TQD code whose codewords realize a 2D Abelian twisted-quantum-double topological order. For Abelian TQDs, the corresponding anyon theory is defined by an Abelian group and a group cocycle built from Type-I, Type-II, or Type-III 3-cocycles [2–4]. Abelian TQDs with Type-I and -II cocycles account for all 2D Abelian topological orders that admit gapped boundaries [1]. Abelian TQDs with Type-III cocycles may admit non-Abelian topological orders.
Type-I and -II Abelian TQD codes can be realized as modular-qudit lattice stabilizer codes on composite-dimensional qudits by starting from a stack of Abelian quantum double models (it suffices to take \(\prod_i \mathbb{Z}_{N_i^2}\) toric codes for \(G=\prod_i \mathbb{Z}_{N_i}\)) and condensing certain bosonic anyons [3]. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [5,6].
Encoding
Fault-tolerant state-preparation circuits for all non-chiral abelian topological phases [4].Fault Tolerance
Fault-tolerant state-preparation circuits for all non-chiral abelian topological phases [4].Cousins
- Commuting-projector Hamiltonian code— Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [6].
- Abelian topological code— Abelian TQDs with Type-I and -II cocycles account for all 2D Abelian topological orders that admit gapped boundaries [1]. 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].
- Quantum-double code— A Type-III \(\mathbb{Z}_2^3\) Abelian TQD realizes the same topological order as the \(G=D_4\) quantum double model [10,11]. There is a sufficient condition for when a Type-III TQD can be realized as a quantum double model [12].
- Dihedral \(G=D_m\) quantum-double code— A Type-III \(\mathbb{Z}_2^3\) Abelian TQD realizes the same topological order as the \(G=D_4\) quantum double model [10,11].
- XS stabilizer code— Abelian TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [13]. Upon gauging some symmetries [14–23], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [10,11].
Primary Hierarchy
Parents
The anyon theory corresponding to Abelian TQD codes is defined by an Abelian group and a Type-I, Type-II, or Type-III 3-cocycle. Abelian TQDs with Type-I and -II cocycles account for all 2D Abelian topological orders that admit gapped boundaries [1].
Abelian TQD code
Children
When treated as ground states of the code Hamiltonian, the double-semion string-net 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 [25].
Every Abelian TQD code with Type-I and -II cocycles can be realized as a modular-qudit Pauli stabilizer code by starting from a stack of Abelian quantum double models (it suffices to take \(\prod_i \mathbb{Z}_{N_i^2}\) toric codes) and condensing certain bosonic anyons [3].
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