Brickwork \(XS\) stabilizer code

Brickwork \(XS\) stabilizer code[1]

Description

An \(XS\) stabilizer code that realizes the topological order of the Type-III \(G=\mathbb{Z}^3_2\) TQD model [2,3], which is the same topological order as the \(G=D_4\) quantum double [4]. 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 non-Abelian Type-III \(G=\mathbb{Z}^3_2\) TQD model [2,3], which is the same topological order as the ordinary \(G=D_4\) quantum double [4].
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 [1,5].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Clifford-hierarchy stabilizer code

XP stabilizer code

XS stabilizer code

Parents

XS stabilizer code

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

Abelian TQD codeTopological Hamiltonian-based QECC Quantum

The ground-state subspace of the brickwork \(XS\) stabilizer code realizes the topological order of the Type-III \(G=\mathbb{Z}^3_2\) TQD model [2,3], which is the same topological order as the \(G=D_4\) quantum double [4].

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 and Transversal Non-Clifford Gate Beyond the n \({}^{\text{1/3}}\) Distance Barrier”, PRX Quantum 6, (2025) arXiv:2405.11719 DOI
[4]: M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
[5]: B. 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/clifford_hierarchy/xp/brickwork.yml.