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\([[12,2,4]]\) carbon code[1]

Description

Twelve-qubit CSS code for which \(H_X\) and \(H_Z\) are equal up to qubit permutations.

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].

Decoding

Syndrome extraction circuit based on Knill error correction (a.k.a. telecorrection [3]), but using only two code blocks instead of three [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) [4] by Microsoft and Quantinuum [1].

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
Parents
Qubit CSS 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]
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
[4]
S. A. Moses et al., “A Race-Track Trapped-Ion Quantum Processor”, Physical Review X 13, (2023) arXiv:2305.03828 DOI
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Zoo Code ID: carbon

Cite as:
“\([[12,2,4]]\) carbon code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/carbon
BibTeX:
@incollection{eczoo_carbon, title={\([[12,2,4]]\) carbon code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/carbon} }
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Permanent link:
https://errorcorrectionzoo.org/c/carbon

Cite as:

“\([[12,2,4]]\) carbon code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/carbon

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/carbon.yml.