Double-semion stabilizer code[1][2]

Description

Modular-qudit stabilizer code with qudit dimension \(q=4\) that realizes 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].

Parents

Cousins

  • 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 1-form symmetry associated with said anyon [3; Footnote 18].
  • 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.

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
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Zoo Code ID: double_semion

Cite as:
“Double-semion stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/double_semion
BibTeX:
@incollection{eczoo_double_semion, title={Double-semion stabilizer code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/double_semion} }
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“Double-semion stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/double_semion

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