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\(((5,3,2))_3\) qutrit code[1]

Description

Smallest qutrit block code realizing the \(\Sigma(360\phi)=3.A_6\) subgroup of \(SU(3)\) transversally. The next smallest code is \(((7,3,2))_3\).

Transversal Gates

\(\Sigma(360\phi)=3.A_6\) group gates can be realized transversally.

Cousin

Member of code lists

Primary Hierarchy

Quantum Domain
Spin Kingdom
Spin codeQECC Quantum
Twisted \(1\)-group codePI Cyclic quantum Small-distance block quantum QECC Quantum
Parents
Twisted \(1\)-group code
The \(((5,3,2))_3\) qutrit code admits a transversal representation of the twisted \(1\)-group \(\Sigma(360\phi)=3.A_6\) [1].
\(((5,3,2))_3\) qutrit code

References

[1]
E. Kubischta and I. Teixeira, “Quantum Codes from Twisted Unitary t -Groups”, Physical Review Letters 133, (2024) arXiv:2402.01638 DOI
[2]
A. Aydin, V. V. Albert, and A. Barg, “Quantum Error Correction beyond <mml:math xmlns:mml=”http://www.w3.org/1998/Math/MathML” display=”inline”> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy=”false”>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=”false”>)</mml:mo> </mml:math>  : Spin, Bosonic, and Permutation-Invariant Codes from Convex Geometry”, PRX Quantum 7, (2026) arXiv:2509.20545 DOI
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Zoo Code ID: su3_sigma360

Cite as:
“\(((5,3,2))_3\) qutrit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/su3_sigma360
BibTeX:
@incollection{eczoo_su3_sigma360, title={\(((5,3,2))_3\) qutrit code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/su3_sigma360} }
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Permanent link:
https://errorcorrectionzoo.org/c/su3_sigma360

Cite as:

“\(((5,3,2))_3\) qutrit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/su3_sigma360

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/spins/many_spin/su3_sigma360.yml.