XYZ color code[1]
Description
A variant of the 6.6.6 color code whose generators are \(XZXZXZ\) and \(ZYZYZY\) Pauli strings associated to each hexagonal in the hexagonal (6.6.6) tiling. A further variation called the domain wall color code admits generators of the form \(XXXZZZ\) and \(ZZZXXX\) [2]. While such codes are equivalent to CSS color codes with the same distance, other properties like noise-bias performance can differ significantly.Code Capacity Threshold
\(50\%\) threshold for noise infinitely biased towards \(X\) or \(Y\) or \(Z\) errors using cellular-automaton decoder [1].Independent \(X,Y\) noise: threshold value of the sum of both noise probabilities is between \(9\%\) and \(14\%\), depending on the noise bias [1].Cousins
- XZZX surface code— The XZZX surface (XYZ color) code is a non-CSS analogue of the rotated surface (6.6.6 color) code. On lattices with a checkerboard sublattice of qubits, each is related to its CSS counterpart by single-qubit Clifford rotations.
- Fracton stabilizer code— The XYZ color code resembles a Type-II fracton code in the limit of infinite noise bias [1].
- Asymmetric quantum code (AQC)— XYZ color codes perform well against biased noise [1].
- 3D XYZ color code— Both codes are non-CSS variants of color codes. The XYZ color code is equivalent to the 6.6.6 color code via single-qubit Clifford gates [1]. The 3D XYZ color code realizes a phase different from that of the 3D color code [3].
Member of code lists
- 2D stabilizer codes
- Asymmetric quantum codes and friends
- Color codes
- Fracton codes and friends
- Lattice qubit stabilizer codes
- Quantum codes
- Quantum codes based on homological products
- Quantum codes with code capacity thresholds
- Quantum codes with notable decoders
- Qubit CSS codes
- Surface code and friends
- Topological codes
Primary Hierarchy
2D color codeTwist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Honeycomb (6.6.6) color codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
The XYZ color code is a 6.6.6 color code presented in non-CSS form. It is obtained by applying a single-qubit Clifford rotation to every qubit on one sublattice of the honeycomb [1; Sec. 2.1]. This rotation maps the \(X\)- and \(Z\)-type generators to \(XZXZXZ\) and \(ZYZYZY\), respectively. The domain wall color code is likewise obtained by applying Hadamard gates to half of the qubits [2]. Since both codes are defined by such rotations, every instance is equivalent to a CSS 6.6.6 color code.
XYZ color code
References
- [1]
- J. F. S. Miguel, D. J. Williamson, and B. J. Brown, “A cellular automaton decoder for a noise-bias tailored color code”, Quantum 7, 940 (2023) arXiv:2203.16534 DOI
- [2]
- K. Tiurev, A. Pesah, P.-J. H. S. Derks, J. Roffe, J. Eisert, M. S. Kesselring, and J.-M. Reiner, “Domain Wall Color Code”, Physical Review Letters 133, (2024) arXiv:2307.00054 DOI
- [3]
- D. Lee and B. Yoshida, “Chiral Color Code : Single-shot error correction for exotic topological order”, (2025) arXiv:2509.18324
Page edit log
- Victor V. Albert (2026-10-13) — most recent
- Victor V. Albert (2026-06-08)
- Victor V. Albert (2024-03-29)
Cite as:
“XYZ color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/xyz_color, arXiv:2606.11484