Tetrahedral color code

Tetrahedral color code[1,2]

Description

3D color code defined on select tetrahedra of a 3D tiling. Qubits are placed on the vertices, edges, triangles, and in the center of each tetrahedron. The code has both string-like and sheet-like logical operators [3].

Transversal Gates

A \([[5d^3-12d^2+16,3,d]]\) close relative of this code admits a logical CCZ gate via single-qubit rotations [4].

Fault Tolerance

Fault-tolerant quantum computation designed for a 2D architecture [5].

Threshold

\(0.46\%\) with clustering decoder [3].\(1.9\%\) for 1D string-like logical operators and \(27.6\%\) for 2D sheet-like operators for 3D codes with noise models using optimal decoding and perfect measurements [3].

Cousin

Body-centered cubic (bcc) lattice— The tetrahedral color code is defined on a lattice of tetrahedra that are carved out of a BCC lattice.

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

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Quantum rainbow code

Quantum pin codeStabilizer Hamiltonian-based Qubit QECC Quantum

Color code

3D color codeQubit Generalized homological-product Lattice stabilizer QLDPC CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

3D color code

Tetrahedral color code

Children

\([[15,1,3]]\) quantum Reed-Muller code

The \([[15,1,3]]\) code is a tetrahedral color code defined on a single tetrahedron.

References

[1]: H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
[2]: A. Kubica and M. E. Beverland, “Universal transversal gates with color codes: A simplified approach”, Physical Review A 91, (2015) arXiv:1410.0069 DOI
[3]: A. Kubica, M. E. Beverland, F. Brandão, J. Preskill, and K. M. Svore, “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018) arXiv:1708.07131 DOI
[4]: D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
[5]: H. Bombin, “2D quantum computation with 3D topological codes”, (2018) arXiv:1810.09571

Page edit log

Victor V. Albert (2024-05-03) — most recent

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/3d_color/tetrahedral_color.yml.