Description
Block group-representation code whose codespace is a copy of an irrep \(\lambda\) of a finite group \(\mathcal{G}\subset U(q)\) in \(n\) qudits, on which \(\mathcal{G}\) acts transversally as \(g^{\otimes n}\) [2; Sec. II.A]. If \(\lambda\otimes f\) is irreducible, with \(f\) the defining representation of \(\mathcal{G}\), then \(\mathcal{G}\) is a \(\lambda\)-twisted (unitary) \(1\)-group and every such code has distance at least two [1].
Twisted \(1\)-groups include the binary icosahedral group \(2I\), the \(\Sigma(72\phi)\) and \(\Sigma(360\phi)\) subgroups of \(SU(3)\), and the family \(\{Sp(2b,3), b\geq 2\}\) [2; Secs. III and VI]. When \(\lambda\) is not faithful, the transversal logical group is the image \(\lambda(\mathcal{G})\), e.g., \(PSp(2b,3)\) for \(Sp(2b,3)\) [2; Secs. III and V]. A \(((6,3,2))\) code of \(2I\) implements \(A_5\), a \(((6,2,2))_3\) code of \(\Sigma(216\phi)\) implements \(2T\), and a \(((6,5,2))_3\) code of \(\Sigma(360\phi)\) implements \(A_6\) transversally [2; Sec. VI].
Transversal and Permutation-Based Gates
All gates in the underlying twisted \(1\)-group [1; Thm. 1].Notes
See Ref. [2; Tables I and II] for codes from sporadic and other finite groups.Cousins
- Permutation-invariant (PI) code— Twisted \(1\)-group codes lying in the symmetric subspace, such as the \(((7,2,3))\) Pollatsek-Ruskai code, are PI codes [1].
- Unitary \(t\)-design— Twisted unitary \(t\)-groups [1] generalize the idea of unitary \(t\)-groups [3–5], which are subgroups of the unitary group that form unitary \(t\)-designs.
- Fock-state bosonic code— PI twisted \(1\)-group codes can be converted to constant-excitation Fock-state codes via the simplex mapping [6; Prop. V.2]. Any transversal gates are mapped to Gaussian gates on the Fock-state codes [6].
Primary Hierarchy
References
- [1]
E. Kubischta and I. Teixeira, “Quantum Codes from Twisted Unitary <mml:math xmlns:mml=”http://www.w3.org/1998/Math/MathML” display=”inline”> <mml:mi>t</mml:mi> </mml:math> -Groups”, Physical Review Letters 133, (2024) arXiv:2402.01638 DOI
- [2]
E. Kubischta and I. Teixeira, “Quantum codes and irreducible products of characters”, Designs, Codes and Cryptography 93, 2919 (2025) arXiv:2403.08999 DOI
- [3]
R. M. Guralnick and P. H. Tiep, “Decompositions of Small Tensor Powers and Larsen’s Conjecture”, (2005) arXiv:math/0502080
- [4]
A. Roy and A. J. Scott, “Unitary designs and codes”, Designs, Codes and Cryptography 53, 13 (2009) arXiv:0809.3813 DOI
- [5]
E. Bannai, G. Navarro, N. Rizo, and P. H. Tiep, “Unitary t-groups”, (2018) arXiv:1810.02507
- [6]
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
Page edit log
- Victor V. Albert (2026-10-29) — most recent
- Victor V. Albert (2026-06-08)
- Victor V. Albert (2024-03-20)
Cite as:
“Twisted \(1\)-group code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/t_group, arXiv:2606.11484
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/spins/many_spin/t_group.yml.