\([[11,1,5]]_3\) qutrit Golay code[1] 

Description

An \([[11,1,5]]_3\) CSS code constructed from the ternary Golay code.

The code's 11-by-23 stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both \begin{align} \begin{pmatrix} 2&1&1&2&2&0&1&0&0&0&0\\ 2&1&2&1&0&2&0&1&0&0&0\\ 2&2&1&0&1&2&0&0&1&0&0\\ 2&2&0&1&2&1&0&0&0&1&0\\ 2&0&2&2&1&1&0&0&0&0&1 \end{pmatrix}. \tag*{(1)}\end{align}

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

Cousins

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

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: qutrit_golay

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
BibTeX:
@incollection{eczoo_qutrit_golay, title={\([[11,1,5]]_3\) qutrit Golay code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qutrit_golay} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/qutrit_golay

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/small/qutrit_golay.yml.