Description
Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(L^2\)-normalizable functions on a second-countable unimodular group \(G\), i.e., a \(G\)-valued qudit or \(G\)-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 \(L^2\)-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.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.
Quantum weight enumerators, linear programming bounds, and Rains shadow enumerators have been extended to quantum codes defined on multiplicity-free two-point homogeneous spaces [5].
Gates
Various gates for a single \(G\)-valued qudit are described in [2,6].The Clifford hierarchy can be extended to arbitrary Abelian \(G\) [7].Notes
Group-based \(Z\)-type operators correspond to group-valued fields in the continuum limit [3].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
Member of code lists
Primary Hierarchy
References
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- 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, D. Aasen, W. Xu, W. Ji, J. Alicea, and J. Preskill, “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096
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- C. Fechisin, N. Tantivasadakarn, and V. V. Albert, “Non-invertible symmetry-protected topological order in a group-based cluster state”, (2024) arXiv:2312.09272
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- R. Okada, “A Quantum Analog of Delsarte’s Linear Programming Bounds”, (2025) arXiv:2502.14165
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- E. J. Gustafson, H. Lamm, F. Lovelace, and D. Musk, “Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral”, Physical Review D 106, (2022) arXiv:2208.12309 DOI
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- B. Yoshida, “Gapped boundaries, group cohomology and fault-tolerant logical gates”, Annals of Physics 377, 387 (2017) arXiv:1509.03626 DOI
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- Yu. I. Manin, “Some remarks on Koszul algebras and quantum groups”, Annales de l’institut Fourier 37, 191 (1987) DOI
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- Y. I. Manin, “Quantized Theta-Functions”, Progress of Theoretical Physics Supplement 102, 219 (2013) DOI
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- Yu. I. Manin, “Functional equations for quantum theta functions”, (2003) arXiv:math/0307393
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- E. Chang-Young and H. Kim, “Theta vectors and quantum theta functions”, Journal of Physics A: Mathematical and General 38, 4255 (2005) arXiv:math/0402401 DOI
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- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
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- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
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- R. Zucchini, “Calibrated hypergraph states: II calibrated hypergraph state construction and applications”, (2025) arXiv:2501.18968
- [15]
- Y. Zhang, “Quantum Fourier Transform Over Galois Rings”, (2009) arXiv:0904.2560
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.