## Description

## Protection

### Group-based error basis

A convenient error set is the group-based analogue of the Pauli string set for qubit codes. For a single group-valued qudit, this set consists of products of \(X\)-type operators labeled by group elements \(g\), and \(Z\)-type operators labeled by matrix elements of \(G\)-irreps \(\lambda\) [1–3]. The outline below is for finite groups, but can be extended to compact unimodular groups or to oscillators and rotors by substituting the sum over the group with a group integral.

Group-based error basis: There are two types of \(X\)-type operators, corresponding to left and right group multiplication. These act on computational basis states \(|h\rangle\) as \begin{align} \overrightarrow{X}_{g}|h\rangle&=|gh\rangle\tag*{(1)}\\ \overleftarrow{X}_{g}|h\rangle&=|hg^{-1}\rangle \tag*{(2)}\end{align} for any group elements \(h,g\). The \(Z\)-type operators can be thought of as matrix-product operators (MPOs) [4] whose virtual dimension is the dimension \(d_{\lambda}\) of their corresponding irrep. The are diagonal in the group-valued basis, yielding the \(d_{\lambda}\)-dimensional irrep matrix \(Z_{\lambda}(g)\) evaluated at the given group element, \begin{align} \hat{Z}_{\lambda}\otimes|g\rangle=Z_{\lambda}(g)\otimes|g\rangle~. \tag*{(3)}\end{align} Each matrix element of this irrep matrix is a generally non-unitary operator on the group-valued qudit. For one-dimensional irreps, the matrix reduces to a single unitary \(Z\)-type operator, and the direct-product symbol is no longer needed.

Products of either left- or right-multiplication \(X\)-type operators with all \(Z\)-type operators form a basis for linear operators on the group-valued qudit space that is complete and orthonormal under the Hilbert-Schmidt inner product [2; Eq. (123)]. In particular, \begin{align} \text{tr}(\overrightarrow{X}_{g}^{\dagger}\overrightarrow{X}_{h})=\delta_{g,h}^{G}~, \tag*{(4)}\end{align} where the group Kronecker delta function \(\delta^{G}_{g,h}=1\) if \(g=h\) and zero otherwise.

## Gates

## Notes

## Parent

## Children

- Dijkgraaf-Witten gauge theory code
- Graph quantum code
- Group GKP code
- Rotor code — Group quantum codes whose physical spaces are constructed using either the group of the integers \(\mathbb{Z}\) or the circle group \(U(1)\) are rotor codes.
- Bosonic code — Group quantum codes whose physical spaces are constructed using the group of the reals \(\mathbb{R}\) under addition are bosonic codes.
- Modular-qudit code — Group quantum codes whose physical spaces are constructed using modular-integer groups \(\mathbb{Z}_q\) are modular-qudit codes.
- Galois-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [6]; see Sec. 5.3 of Ref. [7]. Interpreted this way, Galois-qudit codes are group quantum codes whose physical spaces are constructed using Galois fields \(GF(q)\) as groups.

## Cousins

- Category-based quantum code — Category quantum codes whose physical spaces are constructed using a finite group as the category are group codes.
- Group-alphabet code

## References

- [1]
- C. G. Brell, “Generalized cluster states based on finite groups”, New Journal of Physics 17, 023029 (2015) arXiv:1408.6237 DOI
- [2]
- V. V. Albert, J. P. Covey, and J. Preskill, “Robust Encoding of a Qubit in a Molecule”, Physical Review X 10, (2020) arXiv:1911.00099 DOI
- [3]
- V. V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096
- [4]
- C. Fechisin, N. Tantivasadakarn, and V. V. Albert, “Non-invertible symmetry-protected topological order in a group-based cluster state”, (2024) arXiv:2312.09272
- [5]
- E. J. Gustafson et al., “Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral”, Physical Review D 106, (2022) arXiv:2208.12309 DOI
- [6]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [7]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074

## Page edit log

- Victor V. Albert (2021-12-03) — most recent

## Cite as:

“Group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/group_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/group_quantum.yml.