Quantum Golay code[1] 

Also known as Qubit Golay code.

Description

A \([[23, 1, 7]]\) self-dual CSS code with eleven stabilizer generators of each type, and with each generator being weight eight.

The code's 11-by-23 stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix of the Golay code.

The automorphism group of the code is \(M_{23}\) [2].

Protection

Detects up to 6-qubit errors and corrects up to 3-qubit errors.

Encoding

Fault-tolerant depth-7 circuit consisting of 57 CNOT gates and preparing a logical-zero state [3].

Transversal Gates

All encoded Clifford gates by choosing \(\overline{U}=U^{\otimes 23}\) for every Clifford unitary \(U\) [3].

Gates

The Golay code can be used to perform magic-state distillation for the magic state defined as \(|T\rangle\langle T|=\frac{1}{2}(I+\frac{1}{\sqrt{3}}(X+Y+Z))\), where \(|T\rangle\) is an eigenstate of the Clifford operator \(SH\) [4].

Fault Tolerance

Fault-tolerant depth-7 circuit consisting of 57 CNOT gates and preparing a logical-zero state [3].

Threshold

\(1.32\times 10^{-3}\)-per gate error rate for depolarizing noise upon recursive concatenation [3], improving previous lower bounds [57]. The first numerical study [5] found that the Golay code achieved the highest threshold among a dozen well-known codes at the time [6].

Notes

See Ref. [8] for more details.

Parents

Cousins

References

[1]
A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[2]
H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
[3]
A. Paetznick and B. W. Reichardt, “Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code”, (2013) arXiv:1106.2190
[4]
B. W. Reichardt, “Quantum Universality from Magic States Distillation Applied to CSS Codes”, Quantum Information Processing 4, 251 (2005) arXiv:quant-ph/0411036 DOI
[5]
A. M. Steane, “Overhead and noise threshold of fault-tolerant quantum error correction”, Physical Review A 68, (2003) arXiv:quant-ph/0207119 DOI
[6]
A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
[7]
B. Reichardt and Y. Ouyang. Unpublished (2006).
[8]
B. W. Reichardt, “Error-detection-based quantum fault tolerance against discrete Pauli noise”, (2006) arXiv:quant-ph/0612004
[9]
C.-Y. Lai and C.-C. Lu, “A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices”, IEEE Transactions on Information Theory 57, 7163 (2011) arXiv:0712.0103 DOI
[10]
A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Quantum Data-Syndrome Codes”, IEEE Journal on Selected Areas in Communications 38, 449 (2020) arXiv:1907.01393 DOI
[11]
M. Sullivan, “Code conversion with the quantum Golay code for a universal transversal gate set”, Physical Review A 109, (2024) arXiv:2307.14425 DOI
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Zoo Code ID: qubit_golay

Cite as:
“Quantum Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qubit_golay
BibTeX:
@incollection{eczoo_qubit_golay, title={Quantum Golay code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qubit_golay} }
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“Quantum Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qubit_golay

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/magic/qubit_golay.yml.