# Double-semion code[1]

## Description

Stub.

## Parents

- Modular-qudit stabilizer code — Double-semion code can be realized as a modular-qudit stabilizer code with \(q=4\) [2].
- Abelian topological 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 [3].

## Cousin

- Kitaev surface code — There is a logical basis for 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.

## Zoo code information

## References

- [1]
- M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005). DOI; cond-mat/0404617
- [2]
- Tyler D. Ellison et al., “Pauli stabilizer models of twisted quantum doubles”. 2112.11394
- [3]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990). DOI

## Cite as:

“Double-semion code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/double_semion

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/double_semion.yml.