\([[11,1,5]]_3\) qutrit Golay code[1]
Description
An \([[11,1,5]]_3\) constructed from the ternary Golay code via the CSS construction. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix of the ternary Golay code.
Magic
Magic-state distillation scailing exponent \(\gamma=\log_3(1728\times 11) \approx 8.97\), where the \(1728\) factor comes from the fact that one round of distillation succeeds with probability \(\approx 1/1728\) [1].
Transversal Gates
All single-qutrit encoded Clifford gates [1].
Gates
Magic-state distillation of the strange state \(|S\rangle=\frac{1}{\sqrt{2}}(|1\rangle-|2\rangle)\) and the Norell state \(|N\rangle=\frac{1}{\sqrt{2}}(|1\rangle+|2\rangle)\), with the former achieving a cubic error suppression [1].
Parents
- Modular-qudit CSS code
- Quantum quadratic-residue (QR) code — The qutrit Golay code is a qutrit quantum QR code since the ternary Golay code is a QR code.
- Small-distance block quantum code
Cousins
- Ternary Golay code — The qutrit Golay code is a CSS code constructed from the ternary Golay code.
- Graph quantum code — The qutrit Golay code can be realized as a graph quantum code [1; Fig. 2].
- \([[23, 1, 7]]\) Quantum Golay code
References
- [1]
- 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
Page edit log
- Yinchen Liu (2024-03-15) — most recent
- Victor V. Albert (2024-03-15)
Cite as:
“\([[11,1,5]]_3\) qutrit Golay code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/qutrit_golay