Brickwork \(XS\) stabilizer code

Brickwork \(XS\) stabilizer code[1]

Description

An \(XS\) stabilizer code that realizes the topological order of the \(G=\mathbb{Z}_3^2\) TQD model [2,3], which is the same topological order as the \(G=D_4\) quantum double [4,5]. Its qubits are placed on a 2D square lattice, and the stabilizers are defined using two overlapping rectangular tilings.

Decoding

Just-in-time decoder [1].

Cousins

Dihedral \(G=D_m\) quantum-double code— The ground-state subspace of the brickwork \(XS\) stabilizer code realizes the topological order of the \(G=\mathbb{Z}_3^2\) TQD model [2,3], which is the same topological order as the \(G=D_4\) quantum double [4,5].
3D color code— The brickwork \(XS\) stabilizer code can be obtained from a 3D color code [1].
Hexagonal \(CZ\) code— The brickwork \(XS\) stabilizer code and the hexagonal \(CZ\) code realize the same topological phases and are equivalent via a local unitary [1,6].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

XP stabilizer code

XS stabilizer code

Parents

XS stabilizer code

The brickwork \(XS\) stabilizer code is an \(XS\) stabilizer code [1].

Abelian TQD codeAbelian topological Topological Hamiltonian-based QECC Quantum

The ground-state subspace of the brickwork \(XS\) stabilizer code realizes the topological order of the \(G=\mathbb{Z}_3^2\) TQD model [2,3], which is the same topological order as the \(G=D_4\) quantum double [4,5].

Brickwork \(XS\) stabilizer code

Children

\([[4,2,2]]\) Four-qubit code

The \([[4,2,2]]\) code can be interpreted as a brickwork code on a square of the overlapping rectangular tilings [1].

References

[1]: 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
[2]: B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
[3]: P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
[4]: M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
[5]: 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
[6]: Benjamin J. Brown, private communication, 2025

Page edit log

Victor V. Albert (2025-07-03) — most recent
Benjamin J. Brown (2025-07-03)

Cite as:

“Brickwork \(XS\) stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/brickwork

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