\([[49,1,5]]\) 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 [1–3]. 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 [4].
- 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 [4].
- 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 PI code — The \(((27,2,5))\) binary dihedral PI 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]
- K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) DOI
- [4]
- S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
Page edit log
- Victor V. Albert (2021-12-16) — most recent
- Benjamin Quiring (2021-12-16)
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