XS stabilizer code[1]
Description
A type of stabilizer code where stabilizer generators are elements of the group \( \{\alpha I, X, \sqrt{Z}\}^{\otimes n} \), with \( \sqrt{Z} = \text{diag} (1, i)\). The codespace is a joint \(+1\) eigenspace of a set of stabilizer generators, which need not commute to define a valid codespace.
The phases in the qubit-basis expansion of an XS stabilizer state are polynomials of degree three or below [1].
Cousins
- Abelian TQD code— Abelian TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [1]. Upon gauging some symmetries [2–11], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [12–14].
- Twisted quantum double (TQD) code— Abelian TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [1]. Upon gauging some symmetries [2–11], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [12–14].
- 3D color code— The 3D color code on a particular lattice admits XS stabilizers; see talk by M. Kesselring at the 2020 FTQC conference.
Member of code lists
Primary Hierarchy
Parents
The XP stabilizer formalism reduces to the XS formalism at \(N=4\).
XS stabilizer code
Children
The brickwork \(XS\) stabilizer code is an \(XS\) stabilizer code [15].
The double-semion string-net code is an \(XS\) stabilizer code [1; Fig. 1].
The \([[8,3,2]]\) code can be viewed as an XS stabilizer code [16; Exam. 6.10].
The \([[15,1,3]]\) code can be viewed as an XS stabilizer code [16; Exam. 6.4].
References
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- D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
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- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
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- D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
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- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
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- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
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- L. Lootens, B. Vancraeynest-De Cuiper, N. Schuch, and F. Verstraete, “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
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- M. Davydova, A. Bauer, J. C. M. de la Fuente, M. Webster, D. J. Williamson, and B. J. Brown, “Universal fault tolerant quantum computation in 2D without getting tied in knots”, (2025) arXiv:2503.15751
- [16]
- M. A. Webster, B. J. Brown, and S. D. Bartlett, “The XP Stabiliser Formalism: a Generalisation of the Pauli Stabiliser Formalism with Arbitrary Phases”, Quantum 6, 815 (2022) arXiv:2203.00103 DOI
Page edit log
- Victor V. Albert (2022-04-19) — most recent
Cite as:
“XS stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xs_stabilizer