\([[12,2,4]]\) carbon code[1]
Alternative names: \(C_{12}\) code.
Description
Twelve-qubit CSS code resulting from the concatenation of the \([[4,2,2]]\) code and the \(C_6\) code that has simplified fault-tolerant state preparation circuits for the \(00\) and \(++\) states.Transversal Gates
Two-block CNOT gates are transversal because the code is CSS.Automorphism groups of the underlying classical codes can yield transversal Clifford gates when combined with qubit permutations [2]. In particular, logical Hadamard is realized by a transversal physical Hadamard followed by a qubit permutation, and a logical one-block CNOT is implemented by a qubit permutation [1,3].Decoding
Syndrome extraction circuit based on Knill error correction (a.k.a. telecorrection [4]), but using only one ancillary code block instead of two [1; Fig. 5].Realizations
Trapped-ion devices: Three rounds of error correction and post-selected fault-tolerant logical Bell-state preparation with logical error rates at least 5 times lower than physical rate on a quantum charge-coupled device (QCCD) [5] by Microsoft and Quantinuum [1].Cousins
- \([[4,2,2]]\) Four-qubit code— The carbon code is a concatenation of the \([[4,2,2]]\) code and the \(C_6\) code.
- \([[6,2,2]]\) \(C_6\) code— The carbon code is a concatenation of the \([[4,2,2]]\) code and the \(C_6\) code.
Primary Hierarchy
Parents
The carbon code is a concatenation of the \([[4,2,2]]\) code and the \(C_6\) code.
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[12,2,4]]\) carbon code
References
- [1]
- A. Paetznick et al., “Demonstration of logical qubits and repeated error correction with better-than-physical error rates”, (2024) arXiv:2404.02280
- [2]
- M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum codes for fault-tolerant quantum computation”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1302.1035 DOI
- [3]
- E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
- [4]
- C. M. Dawson, H. L. Haselgrove, and M. A. Nielsen, “Noise thresholds for optical cluster-state quantum computation”, Physical Review A 73, (2006) arXiv:quant-ph/0601066 DOI
- [5]
- S. A. Moses et al., “A Race-Track Trapped-Ion Quantum Processor”, Physical Review X 13, (2023) arXiv:2305.03828 DOI
Page edit log
- Victor V. Albert (2025-02-04) — most recent
- Marcus P da Silva (2025-02-04)
- Victor V. Albert (2024-04-04)
- Victory Omole (2024-04-10)
Cite as:
“\([[12,2,4]]\) carbon code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/carbon