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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 [24,4], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [57].
  • 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 [24,4], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [57].
  • 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
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Zoo Code ID: xs_stabilizer

Cite as:
“XS stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xs_stabilizer
BibTeX:
@incollection{eczoo_xs_stabilizer, title={XS stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/xs_stabilizer} }
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Permanent link:
https://errorcorrectionzoo.org/c/xs_stabilizer

Cite as:

“XS stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/xs_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/nonstabilizer/xp_stabilizer/xs_stabilizer.yml.