Alternative names: Doubled semion model code.
Description
A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that realizes the 2D double semion topological phase. The code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [3].
This stabilizer code family is inequivalent to a CSS code via a Clifford circuit whose depth does not scale with \(n\) [4; Thm. 1.1]. This is because the double semion phase has a sign problem [4,5], and existence of such a Clifford circuit would allow one to construct a code Hamiltonian that is free of such a problem.
Cousins
- Modular-qudit surface code— The exchange statistics of the anyon for the double-semion code coincides with a subset of anyons in the \(\mathbb{Z}_4\) surface code, but the fusion rules are different. The double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [3] or by gauging [6–15] the one-form symmetry associated with said anyon [3; Footnote 20].
- Double-semion string-net code— The double-semion stabilizer code and the double-semion string-net code both realize the double semion topological phase, but the former is a modular-qudit stabilizer code, while the latter is an \(XS\) stabilizer code. A commuting-projector version of the double-semion string-net code can also be derived [16,17].
- X-cube model code— A non-stabilizer commuting-projector code constructed by stacking layers of the double-semion string-net model, called the semionic X-cube model [18], is equivalent to the X-cube model [19] (see also Refs. [20,21]).
- \(\mathbb{Z}_q^{(1)}\) subsystem code— The anyonic exchange statistics of \(\mathbb{Z}_4^{(1)}\) subsystem code resemble those of the double semion code, but its fusion rules realize the \(\mathbb{Z}_4\) group.
- Chiral semion subsystem code— The semion code can be obtained from the double-semion stabilizer code by gauging out the anyon \(\bar{s}\) [3; Fig. 15].
Primary Hierarchy
Abelian TQD stabilizer codeLattice stabilizer QLDPC Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Parents
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 [22]. 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 [3].
Double-semion stabilizer code
References
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- T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
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Page edit log
- Nathanan Tantivasadakarn (2024-03-26) — most recent
- Victor V. Albert (2023-11-28)
- Victor V. Albert (2021-12-29)
Cite as:
“Double-semion stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/double_semion