Twist-defect color code

Twist-defect color code[1–3]

Alternative names: Color code with a twist.

Description

A non-CSS extension of the 2D color code whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. These twists terminate domain walls that permute color labels, Pauli labels, or interchange the two.

For lattices with dislocations and rotational disclinations, twist-defect stabilizer generators are placed at the location of the dislocations. Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and any twist defects present.

Protection

Code properties depend on the number and size of the twist defects. There are 72 types of twist defects in the 2D color code, organized by the \(S_{3}\wr\mathbb{Z}_{2}\) symmetry of the anyons [4].

Gates

Clifford gates can be implemented via twist-based lattice surgery [5] or braiding twist defects [6]. Domino twists [7].

Cousins

Abelian topological code— Twist-defect color codes realize \(\mathbb{Z}_2 \times \mathbb{Z}_2\) topological order with twist defects.
Twist-defect surface code— Twist-defect color codes and twist-defect surface codes both encode using twist defects in topological stabilizer codes.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Parents

Qubit QLDPC code

2D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

Twist-defect color code

Children

\([[4,1,2]]\) twist-defect code

A small 6.6.6 twist-defect color code admits three weight-three stabilizer generators [4; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [8] via a single-qubit Clifford circuit.

2D color code

Twist-defect color codes reduce to 2D color codes when there are no defects. See Ref. [9] for an alternative non-CSS extension of 2D color codes.

Stellated color code

XYZ color code

References

[1]: H. Bombin, “Topological Order with a Twist: Ising Anyons from an Abelian Model”, Physical Review Letters 105, (2010) arXiv:1004.1838 DOI
[2]: H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011) arXiv:1006.5260 DOI
[3]: J. C. Y. Teo, A. Roy, and X. Chen, “Unconventional fusion and braiding of topological defects in a lattice model”, Physical Review B 90, (2014) arXiv:1306.1538 DOI
[4]: M. S. Kesselring, F. Pastawski, J. Eisert, and B. J. Brown, “The boundaries and twist defects of the color code and their applications to topological quantum computation”, Quantum 2, 101 (2018) arXiv:1806.02820 DOI
[5]: D. Litinski and F. von Oppen, “Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes”, Quantum 2, 62 (2018) arXiv:1709.02318 DOI
[6]: M. G. Gowda and P. K. Sarvepalli, “Color codes with twists: Construction and universal-gate-set implementation”, Physical Review A 104, (2021) arXiv:2104.03669 DOI
[7]: M. G. Gowda, “Color codes with domino twists: construction, logical measurements, and computation”, Quantum Information Processing 24, (2025) arXiv:2411.05402 DOI
[8]: S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[9]: P. Padmanabhan, A. Chowdhury, F. Sugino, M. G. Majumdar, and K. K. Sabapathy, “Non-CSS color codes on 2D lattices : Models and Topological Properties”, (2022) arXiv:2112.13617

Page edit log

Victor V. Albert (2024-03-29) — most recent

Cite as:

“Twist-defect color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/twist_defect_color

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/non-css/twist_defect_color.yml.