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Quasi-cyclic quantum code[1]

Description

A block code on \(n\) subsystems such that cyclic shifts of the subsystems by \(\ell\) positions leave the codespace invariant.

Cousin

  • Quasi-cyclic code— Quasi-cyclic quantum codes are quantum analogues of quasi-cyclic codes.

Member of code lists

Primary Hierarchy

Quantum Domain
Quantum code
Operator-algebra QECC (OAQECC)
Quantum error-correcting code (QECC)
Block quantum code
Parents
Block quantum code
Quasi-cyclic quantum code
Children
Quasi-cyclic QLDPC (QC-QLDPC) code
Cyclic quantum codePI
Bipartite cyclic cluster (BCC) code
The translation \((m,j)\mapsto(m+1,j)\) is a shift by \(k\) physical qubits and consists of \(k\) cycles of length \(n_0\) [2].
Multivariate multicycle (MM) codeBB
After ordering coordinates by translation orbits, translation by any cyclic factor of the Abelian group simultaneously shifts every group-algebra block, making every MM code quasi-cyclic [3,4].

References

[1]
M. Hagiwara and H. Imai, “Quantum Quasi-Cyclic LDPC Codes”, 2007 IEEE International Symposium on Information Theory 806 (2007) arXiv:quant-ph/0701020 DOI
[2]
M. B. Hastings, “A Class of Cyclic Quantum Codes”, (2025) arXiv:2509.06865
[3]
H.-K. Lin, P. K. Lim, A. A. Kovalev, and L. P. Pryadko, “Abelian multi-cycle codes for single-shot error correction”, (2026) arXiv:2506.16910
[4]
F. A. Mian, O. Gwilliam, and S. Krastanov, “Multivariate Multicycle Codes for Complete Single-Shot Decoding”, (2026) arXiv:2601.18879
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Zoo Code ID: quantum_quasi_cyclic

Cite as:
“Quasi-cyclic quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/quantum_quasi_cyclic, arXiv:2606.11484
BibTeX:
@incollection{eczoo_quantum_quasi_cyclic,
title={Quasi-cyclic quantum code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/quantum_quasi_cyclic}
}
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Permanent link:
https://errorcorrectionzoo.org/c/quantum_quasi_cyclic

Cite as:

“Quasi-cyclic quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/quantum_quasi_cyclic, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/symmetric/quantum_quasi_cyclic.yml.