Description
Stabilizer code on a four-colorable 3D lattice that realizes a single copy of the 3D fermionic surface code phase [2]. Qubits sit on vertices, and each face carries one generator whose Pauli type is set by the colors of the two volumes it borders.
Faces colored \(AB\) or \(CD\) carry \(X\)-type generators, \(AC\) or \(BD\) faces \(Y\)-type, and \(AD\) or \(BC\) faces \(Z\)-type [2]. Faces that share a single vertex have complementary colors and hence the same Pauli type, so all generators commute [1,2]. The model of Ref. [1] is a version of the 3D XYZ color code on the bitruncated cubic honeycomb [1,2]. The \(\mathbb{Z}_d^{(\alpha)}\) chiral color code generalizes the 3D XYZ color code to qudits [2; Sec. 5].
Protection
On a closed 3-manifold, the code encodes \(k=b_2\) logical qubits, where \(b_2\) is the second Betti number over \(\mathbb{Z}_2\) [1][2; Sec. 3.4]. On the 15-qubit tetrahedral lattice with four colored boundaries, the code has parameters \([[15,1,3]]\) [2; Sec. 4.3].Fault Tolerance
Single-shot error correction via gauge fixing from the 3D subsystem color code, which the code lacks as a standalone stabilizer code [2; Sec. 3.3].Cousins
- 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 [3]. The 3D XYZ color code realizes a phase different from that of the 3D color code [2].
- 3D fermionic surface code— The 3D XYZ color code realizes a single copy of the 3D fermionic surface code phase [2].
- 3D color code— The 3D XYZ color code is defined on the same lattices as the 3D color code, with \(X\)-, \(Y\)-, or \(Z\)-type face generators fixed by color labels. On a closed 3-manifold, it encodes \(b_2\) logical qubits, while the 3D color code encodes \(3b_2\) [2; Sec. 3.4].
- 3D subsystem color code— The 3D XYZ color code is obtained by gauge fixing the 3D subsystem color code, since its stabilizer group lies in the latter’s gauge group [2; Sec. 3.1]. On a closed manifold, single-shot error correction in the 3D subsystem color code prepares a particular codeword of the 3D XYZ color code, stabilized by its membrane logical operators [2; Sec. 3.5].
- Single-shot code— The 3D XYZ color code admits single-shot error correction when realized by gauge fixing the 3D subsystem color code, but not as a standalone stabilizer code [2; Sec. 3.3].
Primary Hierarchy
References
- [1]
- I. H. Kim, “Local non-CSS quantum error correcting code on a three-dimensional lattice”, (2013) arXiv:1012.0859
- [2]
- D. Lee and B. Yoshida, “Chiral Color Code : Single-shot error correction for exotic topological order”, (2025) arXiv:2509.18324
- [3]
- 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
Page edit log
- Victor V. Albert (2026-10-13) — most recent
Cite as:
“3D XYZ color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/3d_xyz_color, arXiv:2606.11484