Truncated trihexagonal (4.6.12) color code

Truncated trihexagonal (4.6.12) color code[1]

Description

2D color code defined on a (typically triangular) patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling.

Stabilizer generators are shown in Fig. I.

Figure I: Stabilizer generators of the 4.6.12 color code.

Protection

For triangular patches, 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.

Cousins

Honeycomb tiling— The 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling is obtained by applying a fattening procedure to the honeycomb tiling [2].
\([[2m,2m-2,2]]\) error-detecting code— The \([[2m,2m-2,2]]\) error-detecting code for \(m=4\) is a color code defined on a single octagon of the 6.6.6 or 4.6.12 tilings.

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

Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

2D color codeTwist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

2D color code

Truncated trihexagonal (4.6.12) color code

Children

\([[4,2,2]]\) Four-qubit code

The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.6.12 tiling [3,4]. Concatenating the \([[4,2,2]]\) code with two copies of the surface code yields the self-dual 4.6.12 color code [5].

\([[6,4,2]]\) error-detecting code

The \([[6,4,2]]\) error-detecting code is a color code defined on a single hexagon of the 6.6.6 or 4.6.12 tilings.

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, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
[3]: M. S. Kesselring, J. C. Magdalena de la Fuente, F. Thomsen, J. Eisert, S. D. Bartlett, and B. J. Brown, “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
[4]: R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
[5]: B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 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.