# 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. It can be punctured twice to obtain a \([[21,3,5]]\) code [2].

The automorphism group of the code is \(M_{23}\) [3].

## Protection

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

## Magic

Magic-state distillation scailing exponent \(\gamma=\log_2 23 \approx 4.52\)[4]. Various magic-state distillation protocols have been developed for this code [2,5] and its \([[21,3,5]]\) doubly punctured version [2].

## Encoding

Fault-tolerant depth-7 circuit consisting of 57 CNOT gates and preparing a logical-zero state [6].

## Transversal Gates

Single-qubit Clifford group by choosing \(\overline{U}=U^{\otimes 23}\) for every Clifford unitary \(U\) [6].

## 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 [6].

## Threshold

\(1.32\times 10^{-3}\)-per gate error rate for depolarizing noise upon recursive concatenation [6], improving previous lower bounds [7–9]. The first numerical study [7] found that the Golay code achieved the highest threshold among a dozen well-known codes at the time [8].

## Notes

See Ref. [10] for more details.

## Parents

- Qubit CSS code
- Quantum quadratic-residue (QR) code — The Golay code is a qubit quantum QR code [11,12].
- Quantum data-syndrome (QDS) code — There exists a set of stabilizer generators for the qubit Golay code that make it a QDS code [12].

## Cousins

- Golay code — The qubit Golay code is a CSS code constructed with the Golay code.
- \([[11,1,5]]_3\) qutrit Golay code
- Triorthogonal code — A \([[95,1,7]]\) triorthogonal code with a transversal \(T\) gate can be obtained from the qubit Golay code via the doubling transformation [13].
- Small-distance block quantum code — The quantum Golay code can be punctured twice to obtain a \([[21,3,5]]\) code.
- Concatenated cat code — Two-component cat codes concatenated with Steane and Golay codes are estimated to be fault tolerant against photon loss noise with rate \(\eta < 5\times 10^{-4}\) provided that \(\alpha > 1.2\) [14].

## References

- [1]
- A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
- [2]
- J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
- [3]
- H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
- [4]
- 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
- [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. Paetznick and B. W. Reichardt, “Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code”, (2013) arXiv:1106.2190
- [7]
- A. M. Steane, “Overhead and noise threshold of fault-tolerant quantum error correction”, Physical Review A 68, (2003) arXiv:quant-ph/0207119 DOI
- [8]
- A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
- [9]
- B. Reichardt and Y. Ouyang. Unpublished (2006).
- [10]
- B. W. Reichardt, “Error-detection-based quantum fault tolerance against discrete Pauli noise”, (2006) arXiv:quant-ph/0612004
- [11]
- 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
- [12]
- 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
- [13]
- M. Sullivan, “Code conversion with the quantum Golay code for a universal transversal gate set”, Physical Review A 109, (2024) arXiv:2307.14425 DOI
- [14]
- A. P. Lund, T. C. Ralph, and H. L. Haselgrove, “Fault-Tolerant Linear Optical Quantum Computing with Small-Amplitude Coherent States”, Physical Review Letters 100, (2008) arXiv:0707.0327 DOI

## Page edit log

- Yinchen Liu (2024-03-15) — most recent
- Victor V. Albert (2024-03-15)

## Cite as:

“Quantum Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qubit_golay