## Description

Modular-qudit stabilizer code with qudit dimension \(q=4\) that is characterized by the 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]. Originally formulated as a non-stabilizer qubit code [1].

## Parent

- Abelian TQD stabilizer code — When treated as ground states of the code Hamiltonian, the code states realize double-semion topological order, a topological phase of matter that also exists in twisted \(\mathbb{Z}_2\) gauge theory [4].

## Cousins

- Kitaev surface code — The double semion phase also has a realization in terms of qubits [1] that can be compared to the qubit surface code. There is a logical basis for both the toric and double-semion codes where each codeword is a superposition of states corresponding to all noncontractible loops of a particular homotopy type. The superposition is equal for the toric code, whereas some loops appear with a \(-1\) coefficient for the double semion.
- Modular-qudit surface code — The anyon fusion rules for the double-semion code and the \(\mathbb{Z}_4\) surface code are the same, but exchange statistics 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 out the one-form symmetry associated with said anyon [3; Footnote 18].
- \(\mathbb{Z}_q^{(1)}\) 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].
- Abelian TQD stabilizer code — 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].

## References

- [1]
- M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 DOI
- [2]
- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
- [3]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
- [4]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI

## Page edit log

- Victor V. Albert (2021-12-29) — most recent

## Cite as:

“Double-semion stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/double_semion