XS stabilizer code

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 \( S = \sqrt{Z} = diag (1, i)\). 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 [12,13].
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

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

XP stabilizer codeHamiltonian-based QECC Quantum

Parents

XP stabilizer code

The XP stabilizer formalism reduces to the XS formalism at \(N=4\).

Clifford-hierarchy stabilizer codeQubit QECC Quantum

XS stabilizer generators are products of Pauli \(X\) strings and diagonal gates built from \(S\), which lie at the second level of the Clifford hierarchy.

XS stabilizer code

Children

Brickwork \(XS\) stabilizer code

The brickwork \(XS\) stabilizer code is an \(XS\) stabilizer code [14].

Double-semion string-net code

The double-semion string-net code is an \(XS\) stabilizer code [1; Fig. 1].

\([[15,1,3]]\) quantum RM code

The \([[15,1,3]]\) code can be viewed as an XS stabilizer code [15; 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]
J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
[4]
S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[5]
D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[6]
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
[7]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[8]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[9]
K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[10]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[11]
D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, Nature Physics 22, 598 (2026) arXiv:2410.02213 DOI
[12]
M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
[13]
B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
[14]
M. Davydova, A. Bauer, J. C. Magdalena de la Fuente, M. Webster, D. J. Williamson, and B. J. Brown, “Universal Fault-Tolerant Quantum Computation in 2D without Getting Tied in Knots”, Physical Review X 16, (2026) arXiv:2503.15751 DOI
[15]
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 (2026-06-08) — most recent
Victor V. Albert (2022-04-19)

Cite as:

“XS stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xs_stabilizer, arXiv:2606.11484

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