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 stabilizer code— Abelian TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [1]. Upon gauging some symmetries [2–4,4], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [5–7].
- 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 [2–4,4], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [5–7].
- 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 \([[8,3,2]]\) code can be viewed as an XS stabilizer code [8; Exam. 6.10].
The \([[15,1,3]]\) code can be viewed as an XS stabilizer code [8; Exam. 6.4].
References
- [1]
- X. Ni, O. Buerschaper, and M. Van den Nest, “A non-commuting stabilizer formalism”, Journal of Mathematical Physics 56, (2015) arXiv:1404.5327 DOI
- [2]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [3]
- 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
- [4]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [5]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [6]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [7]
- 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
- [8]
- 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