## Description

A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that is characterized by 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]. Originally formulated as the ground-state space of a Hamiltonian with non-commuting terms [1], which can be extended to other spatial dimensions [4], and later as a commuting-projector code [5,6].

This stabilizer code family is inequivalent to a CSS code via a Clifford circuit whose depth does not scale with \(n\) [7; Thm. 1.1]. This is because the double semion phase has a sign problem [7,8], and existence of such a Clifford circuit would allow one to construct a code Hamiltonian that is free of such a problem.

## Parent

- Abelian TQD stabilizer code — When treated as ground states of the code Hamiltonian, the 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 [9].

## Cousins

- Toric code — The double semion phase also has a realization in terms of qubits [1] that can be compared to the toric 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 an odd number of loops appear with a \(-1\) coefficient for the double semion.
- Modular-qudit surface code — The exchange statistics of the anyon for the double-semion code coincides with a subset of anyons in the \(\mathbb{Z}_4\), 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 [10–12,12] the one-form symmetry associated with said anyon [3; Footnote 20].
- 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].
- \(\mathbb{Z}_q^{(1)}\) 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].

## 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”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [4]
- M. H. Freedman and M. B. Hastings, “Double Semions in Arbitrary Dimension”, Communications in Mathematical Physics 347, 389 (2016) arXiv:1507.05676 DOI
- [5]
- G. Dauphinais et al., “Quantum error correction with the semion code”, New Journal of Physics 21, 053035 (2019) arXiv:1810.08204 DOI
- [6]
- J. C. Magdalena de la Fuente, N. Tarantino, and J. Eisert, “Non-Pauli topological stabilizer codes from twisted quantum doubles”, Quantum 5, 398 (2021) arXiv:2001.11516 DOI
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- M. B. Hastings, “How quantum are non-negative wavefunctions?”, Journal of Mathematical Physics 57, (2015) arXiv:1506.08883 DOI
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- A. Smith, O. Golan, and Z. Ringel, “Intrinsic sign problems in topological quantum field theories”, Physical Review Research 2, (2020) arXiv:2005.05343 DOI
- [9]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
- [10]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [11]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [12]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI

## 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