\([[3k + 8, k, 2]]\) triorthogonal code[1; Appx. B]
Description
Member of the \([[3k + 8, k, 2]]\) family (for even \(k\)) of triorthogonal and quantum divisible codes that admit a transversal \(T\) gate and are relevant for magic-state distillation.
Magic
The family yields the asymptotic exponent \(\gamma = \log_2 \frac{3k+8}{k} \to \log_2 3 \approx 1.6\) for sufficiently large \(k\) [2; Box 2]; see [3; Table V].
Transversal Gates
The code admits a transversal \(T\) gate [1].
Parents
Cousin
- \([[k+4,k,2]]\) H code — The H code \([[k+4,k,2]]\) family yields the \([[3k + 8, k, 2]]\) family of triorthogonal codes when level-lifted [4; Sec. VI.C].
References
- [1]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
- [2]
- E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
- [3]
- Quantum Information and Computation 18, (2018) arXiv:1709.02789 DOI
- [4]
- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
Page edit log
- Victor V. Albert (2024-02-28) — most recent
- Victor V. Albert (2021-12-16)
- Benjamin Quiring (2021-12-16)
Cite as:
“\([[3k + 8, k, 2]]\) triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/small_triorthogonal