Qubit Golay code[1] 

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 \begin{align} \left(\begin{array}{ccccccccccccccccccccccc} 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 1 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) \tag*{(1)}\end{align}

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

Protection

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

Magic

Magic-state distillation scailing exponent \(\gamma=\log_2 23 \approx 4.52\)[3].

Encoding

Fault-tolerant depth-7 circuit consisting of 57 CNOT gates and preparing a logical-zero state [4].

Transversal Gates

All encoded Clifford gates by choosing \(\overline{U}=U^{\otimes 23}\) for every Clifford unitary \(U\) [4].

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 \(SH\) [5].

Fault Tolerance

Fault-tolerant depth-7 circuit consisting of 57 CNOT gates and preparing a logical-zero state [4].

Threshold

\(1.32\times 10^{-3}\)-per gate error rate for depolarizing noise upon recursive concatenation [4], improving previous lower bounds [68]. The first numerical study [6] found that the Golay code achieved the highest threshold among a dozen well-known codes at the time [7].

Notes

See Ref. [9] for more details.

Parent

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]
S. Prakash, “Magic state distillation with the ternary Golay code”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, (2020) arXiv:2003.02717 DOI
[4]
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
[5]
B. W. Reichardt, “Quantum Universality from Magic States Distillation Applied to CSS Codes”, Quantum Information Processing 4, 251 (2005) arXiv:quant-ph/0411036 DOI
[6]
A. M. Steane, “Overhead and noise threshold of fault-tolerant quantum error correction”, Physical Review A 68, (2003) arXiv:quant-ph/0207119 DOI
[7]
A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
[8]
B. Reichardt and Y. Ouyang. Unpublished (2006).
[9]
B. W. Reichardt, “Error-detection-based quantum fault tolerance against discrete Pauli noise”, (2006) arXiv:quant-ph/0612004
[10]
M. Sullivan, “Code conversion with the quantum Golay code for a universal transversal gate set”, (2023) arXiv:2307.14425
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Zoo Code ID: qubit_golay

Cite as:
“Qubit 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={Qubit 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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“Qubit 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.