## Description

Code whose projection is onto an irreducible representation of a subgroup \(G\) of a group of canonical or distinguished unitary operations, e.g., transversal gates in the case of block quantum codes, Gaussian operations in the case of bosonic codes, or \(SU(2)\) operations in the case of single-spin codes.

## Protection

Error correction ability is not guaranteed, but can be searched in the multiplicity space of the irrep in case there is more than one copy present.

## Encoding

General encoding map [3; Lemma 1].

## Gates

By definition, a group \(G\) of gates can be realized on the code using the unitary operations used to define the code.

## Parent

- Quantum error-correcting code (QECC) — Group-representation code projections are onto an irrep of a subgroup of canonical or distinguished unitary operations on a Hilbert space. Removing the restriction to distinguished operations and allowing all operations, every code projection on an \(N\)-dim Hilbert space can be expressed as a projection onto the irrep formed by the code-preserving subgroup of \(U(N)\). The same idea holds when \(N\) is taken to infinity. In other words, while all codes are covariant w.r.t. some group, group-representation codes are covariant w.r.t. a canonical or distinguished subgroup.

## Children

- Cat code — Cat codes are group representation codes with \(G\) being a cyclic group [3].
- Clifford group-representation QSC — The Clifford group-representation QSC is a group-representation code with \(G\) being the binary octahedral subgroup of \(SU(2)\).
- One-hot quantum code — One-hot quantum codes are group-representation codes with the \(G = SU(q)\) subgroup of Gaussian rotations [3].
- Covariant block quantum code — Covariant codes are block group-representation codes [3; Lemma 2].
- Knill code — Knill codes project onto a single irrep of the normalizer of a normal subgroup of the group formed by a nice error basis [4; Lemma 3.1].
- Five-qubit perfect code — The five-qubit code is a group-representation code with \(G\) being the \(2T\) subgroup of \(SU(2)\) [3].
- \([[7,1,3]]\) Steane code — The Steane code is a group-representation code with \(G\) being the \(2O\) subgroup of \(SU(2)\) [3].
- Twisted \(1\)-group code — Twisted \(1\)-group codes are group-representation codes with \(G\) being a twisted \(1\)-group.
- Clifford spin code — Clifford spin codes are group-representation codes with \(G\) being a subgroup of \(SU(2)\) [3].
- \(SU(3)\) spin code — \(SU(3)\) spin codes are group-representation codes with \(G\) being a subgroup of \(SU(3)\) [3].

## Cousins

- Small-distance block quantum code — See Ref. [5] for tables of distance-two codes with various families of transversal gates.
- Quantum spherical code (QSC) — QSCs should be able to be formulated as group-representation codes whose group is that formed by the permutation representation of the code polytope symmetry group, but this representation may be reducible.

## References

- [1]
- J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021) arXiv:2005.10910 DOI
- [2]
- E. Kubischta and I. Teixeira, “Family of Quantum Codes with Exotic Transversal Gates”, Physical Review Letters 131, (2023) arXiv:2305.07023 DOI
- [3]
- A. Denys and A. Leverrier, “Quantum error-correcting codes with a covariant encoding”, (2024) arXiv:2306.11621
- [4]
- E. Knill, “Group Representations, Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608049
- [5]
- E. Kubischta and I. Teixeira, “Quantum Codes and Irreducible Products of Characters”, (2024) arXiv:2403.08999

## Page edit log

- Victor V. Albert (2024-02-21) — most recent

## Cite as:

“Group-representation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/group_representation