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 [5–7]. 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
- Qubit CSS code
- Quantum quadratic-residue (QR) code — The Golay code is a qubit quantum QR code [9,10].
- Quantum data-syndrome (QDS) code — There exists a set of stabilizer generators for the qubit Golay code that make it a QDS code [10].
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 [11].
- 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 \(\alpha > 1.2\) [12].
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
- [12]
- 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