Group-based quantum code 

Root code for the Group Kingdom


Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(\ell^2\)-normalizable functions on a second-countable unimodular group \(G\), i.e., a \(G\)-valued qudit. In other words, a group-valued qudit is a vector space whose canonical basis states \(|g\rangle\) are labeled by elements \(g\) of a group \(G\). For \(K\)-dimensional logical subspace and for block codes defined on groups \(G^{n}\), can be denoted as \(((n,K))_G\). When the logical subspace is the Hilbert space of \(\ell^2\)-normalizable functions on \(G^{ k}\), can be denoted as \([[n,k]]_G\). Ideal codewords may not be normalizable, depending on whether \(G\) is continuous and/or noncompact, so approximate versions have to be constructed in practice.


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 \(\lambda\) [13]. 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.


Various gates for a single \(G\)-valued qudit are described in [2,5].


Group-based \(Z\)-type operators correspond to group-valued fields in the continuum limit [3].



  • 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.
  • Twisted quantum double (TQD) code
  • 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.



C. G. Brell, “Generalized cluster states based on finite groups”, New Journal of Physics 17, 023029 (2015) arXiv:1408.6237 DOI
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
V. V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096
C. Fechisin, N. Tantivasadakarn, and V. V. Albert, “Non-invertible symmetry-protected topological order in a group-based cluster state”, (2023) arXiv:2312.09272
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
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
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Zoo Code ID: group_quantum

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“Group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.
@incollection{eczoo_group_quantum, title={Group-based quantum code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.