Truncated trihexagonal (4.6.12) color code

Truncated trihexagonal (4.6.12) color code[1]

Description

Triangular color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling.

Stabilizer generators are shown in Figure I.

Figure I: Stabilizer generators of the 4.6.12 color code.

Protection

There is a \([[(3d^2+5)/2-3d, 1, d]]\) code family [1; Fig. 2].

Transversal Gates

CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.Transversal \(S\) gate [1,2].

Parent

2D color code

Cousin

\(A_2\) hexagonal lattice code — The 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling is obtained by applying a fattening procedure to the triangular lattice [3].

References

[1]: A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[2]: H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
[3]: H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI

Page edit log

Victor V. Albert (2024-03-29) — most recent
Eric Huang (2024-03-18)

Cite as:

“Truncated trihexagonal (4.6.12) color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/4612_color

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/2d_color/4612_color/4612_color.yml.