Group-representation code[13] 


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.


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.


General encoding map [3; Lemma 1].


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


  • 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.



  • 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.


J. A. Gross, “Designing Codes around Interactions: The Case of a Spin”, Physical Review Letters 127, (2021) arXiv:2005.10910 DOI
E. Kubischta and I. Teixeira, “Family of Quantum Codes with Exotic Transversal Gates”, Physical Review Letters 131, (2023) arXiv:2305.07023 DOI
A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621
E. Knill, “Group Representations, Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608049
E. Kubischta and I. Teixeira, “Quantum Codes and Irreducible Products of Characters”, (2024) arXiv:2403.08999
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Zoo Code ID: group_representation

Cite as:
“Group-representation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_group_representation, title={Group-representation code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Group-representation code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.