Abelian quantum-double stabilizer code[1]
Description
Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order with trivial cocycle. The corresponding anyon theory is defined by an Abelian group. The \(G=\mathbb{Z}_2\) instance on a torus is the toric code, and cyclic-group instances reduce to modular-qudit surface codes. All such codes can be realized by a stack of modular-qudit surface codes because all finite Abelian groups are direct products of cyclic groups.
There exists an invariant that can be computed to uniquely characterize the anyons of a state in an Abelian quantum-double topological phase [2].
Protection
Error-correcting properties established in Ref. [3] using operator algebra theory. Correcting the maximum number of correctable errors is \(NP\)-complete [4].Encoding
Any geometrically local unitary circuit connecting two quantum double models whose groups are not isomorphic must have depth at least linear in \(n\) [2].Decoding
Efficient decoder correcting below the code distance [4].Cousins
- 2D lattice stabilizer code— Translation-invariant 2D prime-qudit lattice stabilizer codes are equivalent to several copies of the prime-qudit surface code and a trivial code via a local constant-depth qudit Clifford circuit [5].
- 3D subsystem surface code— The 3D subsystem surface code Hamiltonian phase diagram exhibits \(\mathbb{Z}_2\) topological order [6].
- \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code— The \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code can be obtained from a stack of \(q=3\) and \(q=9\) square-lattice qudit surface codes by gauging out the anyons \(m_1^{-1}e_2^3\) and \(m_2^{-1}\) [7; Sec. 7.5].
- Galois-qudit color code— A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [8,10–13][9; Sec. 5.3]. Galois-qudit color codes yield Abelian quantum-double codes with Abelian-group topological order via this decomposition.
- Galois-qudit surface code— A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [8,10–13][9; Sec. 5.3]. Galois-qudit surface codes yield Abelian quantum-double codes with \(\mathbb{F}_{p^m}\cong \mathbb{Z}_p^m\) topological order via this decomposition.
Primary Hierarchy
Abelian TQD stabilizer codeLattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Parents
The anyon theory corresponding to Abelian quantum double codes is defined by an Abelian group and trivial cocycle. Stacks of Abelian quantum double models are the starting point for constructing all Abelian TQD stabilizer codes by condensing bosons; for \(G=\prod_i \mathbb{Z}_{N_i}\), it suffices to use \(\prod_i \mathbb{Z}_{N_i^2}\) quantum doubles [14]. Conversely, every Abelian anyon theory is a subtheory of some Abelian TQD [7; Sec. 6.2]. Upon gauging some symmetries [15–24], Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models [25,26].
The anyon theory corresponding to (Abelian) quantum double codes is defined by an (Abelian) group.
Abelian quantum-double stabilizer code
Children
When treated as ground states of the code Hamiltonian, states of the color code on a torus geometry realize \(\mathbb{Z}_2\times\mathbb{Z}_2\) topological order [27], equivalent to the phase realized by two copies of the toric code (i.e., the surface code on a torus) via a local constant-depth Clifford circuit [28]. This process can be viewed as an ungauging [15–24] of certain symmetries.
Matching codes were inspired by the \(\mathbb{Z}_2\) topological order phase of the Kitaev honeycomb model [29].
When treated as ground states of the code Hamiltonian, surface codewords realize \(\mathbb{Z}_2\) topological order, a topological phase of matter that also exists in \(\mathbb{Z}_2\) lattice gauge theory [30]. Local Clifford deformation preserves this topological order.
The XZZX surface code is an example of \(\mathbb{Z}_2\) topological order as manifest in the Wen plaquette model [31].
Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) topological order [32].
References
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Page edit log
- Victor V. Albert (2023-04-06) — most recent
Cite as:
“Abelian quantum-double stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_double_abelian