\([[3k + 8, k, 2]]\) 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

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
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Zoo Code ID: small_triorthogonal

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
BibTeX:
@incollection{eczoo_small_triorthogonal, title={\([[3k + 8, k, 2]]\) triorthogonal code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/small_triorthogonal} }
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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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/magic/triorthogonal/small_triorthogonal.yml.