Description
A 3D color code defined on a colored tetrahedron cut from a suitably colored BCC lattice [1]. Qubits are placed on tetrahedra, on the triangles covering the tetrahedron faces, on the edges along the tetrahedron edges, and on the tetrahedron vertices. The code has both string-like and sheet-like logical operators [3].Rate
The tetrahedral family with linear size parameter \(L\) has \(n=1+4L+6L^2+4L^3\) physical qubits and encodes one logical qubit [1].Transversal Gates
A \([[5d^3-12d^2+16,3,d]]\) close relative of this code admits a logical \(CCZ\) gate via single-qubit rotations; for this family, stabilizers remain asymptotically constant-weight and can be gauge-reduced to weight at most six [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].Cousins
- Body-centered cubic (bcc) lattice— The tetrahedral color code is defined on a lattice of tetrahedra carved out of a suitably colored BCC lattice [1].
- 3D surface code— A tetrahedral 3D color code with four differently colored boundaries is equivalent, via a local Clifford circuit, to three 3D surface codes attached along one boundary, with condensation of a composite electric charge on that attached boundary [6].
Member of code lists
- 3D stabilizer codes
- Color code and friends
- Hamiltonian-based codes and friends
- Quantum codes
- Quantum codes based on homological products
- Quantum codes with a rate
- Quantum codes with fault-tolerant gadgets
- Quantum codes with other thresholds
- Quantum codes with transversal gates
- Quantum CSS codes
- Quantum LDPC codes
- Qubit QLDPC codes
- Stabilizer codes
- Topological codes
Primary Hierarchy
3D color codeQubit CSS Generalized homological-product Lattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Parents
Tetrahedral color code
Children
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 Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
- [5]
- H. Bombin, “2D quantum computation with 3D topological codes”, (2018) arXiv:1810.09571
- [6]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
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