\([[49,1,5]]\) triorthogonal code[1; Appx. B] 

Description

Triorthogonal and quantum divisible code which is the smallest distance-five stabilizer code to admit a transversal \(T\) gate [1,2]. Its \(X\)-type stabilizers form a triply-even linear binary code in the symplectic representation.

Magic

The code yields an exponent \(\gamma = \log 49 / \log 5 \approx 2.42\).

Transversal Gates

The code admits a transversal \(T\) gate [1; Appx. B].

Parents

Cousins

  • Doubled color code — The \([[49,1,5]]\) triorthogonal code can be viewed as a (gauge-fixed) doubled color code obtained from the \([[17,1,5]]\) 2D color code via the doubling transformation [3].
  • Square-octagon (4.8.8) color code — The \([[49,1,5]]\) triorthogonal code can be viewed as a (gauge-fixed) doubled color code obtained from the \([[17,1,5]]\) 4.8.8 color code via the doubling transformation [3].
  • Divisible code — The \([[49,1,5]]\) triorthogonal code stabilizer generator matrix can be obtained from a triply-even linear binary code [1; Appx. B].
  • Binary dihedral permutation-invariant code — The \(((27,2,5))\) binary dihedral permutation-invariant code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[49,1,5]]\) triorthogonal code.

References

[1]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
[2]
K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
[3]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
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Zoo Code ID: stab_49_1_5

Cite as:
\([[49,1,5]]\) triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/stab_49_1_5
BibTeX:
@incollection{eczoo_stab_49_1_5, title={\([[49,1,5]]\) triorthogonal code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_49_1_5} }
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Cite as:

\([[49,1,5]]\) triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/stab_49_1_5

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/magic/triorthogonal/stab_49_1_5.yml.