Twisted \(1\)-group code[1,2] 


Block group-representation code realizing particular irreps of particular groups such that a distance of two is automatically guaranteed. Groups which admit irreps with this property are called twisted (unitary) \(1\)-groups and include the binary icosahedral group \(2I\), the \(\Sigma(360\phi)\) subgroup of \(SU(3)\), the family \(\{PSp(2b, 3), b \geq 1\}\), and the alternating groups \(A_{5,6}\). Groups whose irreps are images of the appropriate irreps of twisted \(1\)-groups also yield such properties, e.g., the binary tetrahedral group \(2T\) or qutrit Pauli group \(\Sigma(72\phi)\).

A \(((3,2,2))_3\) code can implement the qutrit Pauli group \(\Sigma(72\phi)\) transversally, a \(((6,3,2))\) code can implement \(A_5\) transversally, a \(((6,2,2))_3\) implements \(2T\) transversally, and a \(((6,5,2))_3\) code implements \(A_6\) transverally.

Transversal Gates

All gates in the underlying twisted \(1\)-group. See [2; Table II] for other notable groups including the sporadic groups.




  • Design — Twisted unitary \(t\)-groups [1] generalize the idea of unitary \(t\)-groups [3,4], which are subgroups of the unitary group that form unitary \(t\)-designs.


E. Kubischta and I. Teixeira, “Free Quantum Codes from Twisted Unitary \(t\)-groups”, (2024) arXiv:2402.01638
E. Kubischta and I. Teixeira, “Quantum Codes and Irreducible Products of Characters”, (2024) arXiv:2403.08999
R. M. Guralnick and P. H. Tiep, “Decompositions of Small Tensor Powers and Larsen’s Conjecture”, (2005) arXiv:math/0502080
E. Bannai et al., “Unitary t-groups”, (2018) arXiv:1810.02507
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“Twisted \(1\)-group code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_t_group, title={Twisted \(1\)-group code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Twisted \(1\)-group code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.