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Group-based quantum code

Root code for the Group Kingdom

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.

A notion of Gaussian states and Hudson's theorem have been developed for arbitrary locally compact Abelian \(G\) [1]. A Wigner function formalism has also been developed [2].

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 \(\lambda\) [3–5]. 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) [6] 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 [4; 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 [7].

Gates

Various gates for a single \(G\)-valued qudit are described in [4,8].The Clifford hierarchy can be extended to arbitrary Abelian \(G\) [9–11].

Notes

See Refs. [12,13] for introductions to Hilbert spaces for Abelian groups.Group-based \(Z\)-type operators correspond to group-valued fields in the continuum limit [5] and Wilson link operators in lattice gauge theory [14].

Cousins

Member of code lists

Primary Hierarchy

Quantum Domain
Group Kingdom
Parents
Quantum error-correcting code (QECC)Quantum
Group-based quantum code
Children
Dijkgraaf-Witten gauge theory codeCDSC Kitaev surface
Group GKP codeCDSC Quantum RM code Color Homological Quantum QR code Kitaev surface BB
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.
Stabilizer codeCSS QLDPC Generalized homological-product Lattice stabilizer Bosonic stabilizer Analog stabilizer Quantum lattice Fracton stabilizer BB Homological Fermion-into-qubit Color Twist-defect color CDSC Twist-defect surface Kitaev surface Quantum QR code Quantum RM code
Stabilizer codes are constructed out of Pauli strings, modular-qudit Pauli strings, Galois-qudit Pauli strings, oscillator displacement operators, or rotor generalized Pauli strings. All of these are examples of the Weyl-Heisenberg group on Manin's quantum plane, which is defined on a configuration space that is generally a free Abelian group [16–19].
Bosonic codeFock-state bosonic Bosonic stabilizer Quantum lattice Analog stabilizer
Group quantum codes whose physical spaces are constructed using the group of the reals \(\mathbb{R}\) under addition are bosonic codes.
Modular-qudit codeQubit Fermion Self-complementary qubit Hastings-Haah Floquet Fermion-into-qubit Twist-defect color Twist-defect surface Fracton stabilizer Color BB Homological Quantum RM code CDSC Kitaev surface
Group quantum codes whose physical spaces are constructed using modular-integer groups \(\mathbb{Z}_q\) are modular-qudit codes.
Galois-qudit codeQubit Hastings-Haah Floquet Fermion Self-complementary qubit Fermion-into-qubit Twist-defect color CDSC Twist-defect surface Color Homological BB Kitaev surface Quantum QR code Quantum RM code
A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [20]; see Sec. 5.3 of Ref. [21]. Interpreted this way, Galois-qudit codes are group quantum codes whose physical spaces are constructed using Galois fields \(GF(q)\) as groups. More general versions of such qudits can be valued in a Galois ring [22], over which there also exists a Fourier transform [23].

References

[1]
C. Beny, J. Crann, H. H. Lee, S.-J. Park, and S.-G. Youn, “Gaussian quantum information over general quantum kinematical systems I: Gaussian states”, (2022) arXiv:2204.08162
[2]
S. J. Park, C. Beny, and H. H. Lee, “Twisted Fourier analysis and pseudo-probability distributions”, (2020) arXiv:2004.13860
[3]
C. G. Brell, “Generalized cluster states based on finite groups”, New Journal of Physics 17, 023029 (2015) arXiv:1408.6237 DOI
[4]
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
[5]
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
[6]
C. Fechisin, N. Tantivasadakarn, and V. V. Albert, “Noninvertible Symmetry-Protected Topological Order in a Group-Based Cluster State”, Physical Review X 15, (2025) arXiv:2312.09272 DOI
[7]
R. Okada, “A Quantum Analog of Delsarte’s Linear Programming Bounds”, (2025) arXiv:2502.14165
[8]
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
[9]
D. Gottesman and I. L. Chuang, “Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations”, Nature 402, 390 (1999) arXiv:quant-ph/9908010 DOI
[10]
F. H. E. Watson, E. T. Campbell, H. Anwar, and D. E. Browne, “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
[11]
B. Yoshida, “Gapped boundaries, group cohomology and fault-tolerant logical gates”, Annals of Physics 377, 387 (2017) arXiv:1509.03626 DOI
[12]
R. F. Werner, “Uncertainty relations for general phase spaces”, (2016) arXiv:1601.03843
[13]
V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
[14]
K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[15]
Y. Hu, Y. Wan, and Y.-S. Wu, “Twisted quantum double model of topological phases in two dimensions”, Physical Review B 87, (2013) arXiv:1211.3695 DOI
[16]
Yu. I. Manin, “Some remarks on Koszul algebras and quantum groups”, Annales de l’institut Fourier 37, 191 (1987) DOI
[17]
Y. I. Manin, “Quantized Theta-Functions”, Progress of Theoretical Physics Supplement 102, 219 (2013) DOI
[18]
Yu. I. Manin, “Functional equations for quantum theta functions”, (2003) arXiv:math/0307393
[19]
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
[20]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[21]
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
[22]
R. Zucchini, “Calibrated hypergraph states: II calibrated hypergraph state construction and applications”, (2025) arXiv:2501.18968
[23]
Y. Zhang, “Quantum Fourier Transform Over Galois Rings”, (2009) arXiv:0904.2560
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Zoo Code ID: group_quantum

Cite as:
“Group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/group_quantum
BibTeX:
@incollection{eczoo_group_quantum, title={Group-based quantum code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_quantum} }
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“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.