Cyclic quantum code

Cyclic quantum code[1]

Description

A block quantum code such that cyclic permutations of the subsystems leave the codespace invariant. In other words, the automorphism group of the code contains the cyclic group \(\mathbb{Z}_n\).

An example \([[17,9,3]]\) cyclic Hermitian qubit code has stabilizer generated by cyclic shifts of the Pauli operators \(XXYIXYZZYXIYXXIII\) and \(ZZXIZXYYXZIXZZIII\). An example \([[17,1,7]]\) cyclic Hermitian qubit code has stabilizer generated by cyclic shifts of the Pauli operators \(XYYIZZIYYXIIIIIII\) and \(ZXXIYYIXXZIIIIIII\).

Protection

Cyclic symmetry guarantees that if a single subsystem is protected against some noise, then all other subsystems are also.

Encoding

Linear feedback shift registers [2].

Decoding

Linear feedback shift registers [2].Adapted from the Berlekamp decoding algorithm for classical BCH codes [1].

Notes

Many examples have been found by computer algebra programs. Ref. [1] gives examples of \([[17,1,7]]\) and \([[17,9,3]]\) quantum cyclic codes.

Cousins

Cyclic code— Cyclic quantum codes are quantum analogues of cyclic codes.
1D lattice stabilizer code— A 1D lattice stabilizer code with periodic boundary conditions is a quantum cyclic code.
Asymmetric quantum code (AQC)— Cyclic quantum codes can be adapted for asymmetric noise [3].

Member of code lists

Primary Hierarchy

Quantum Domain

Quantum code

Operator-algebra QECC (OAQECC)

Quantum error-correcting code (QECC)

Block quantum code

Quasi-cyclic quantum code

Parents

Quasi-cyclic quantum code

Cyclic quantum code

Children

\([[5,1,3]]_{\mathbb{Z}}\) Five-rotor code

Group-based quantum repetition code

\([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code

Permutation-invariant (PI) code

The cyclic group of these codes is a subgroup of the \(S_n\) symmetric group used in permutation invariant codes.

\(((5,6,2))\) qubit code

\([[7,1,3]]\) Steane code

The Steane code is equivalent to a cyclic code via qubit permutations [4; Exam. 1].

\(((9,12,3))\) qubit code

Bipartite cyclic cluster (BCC) code

BCC codes are invariant under cyclic shifts by construction [5].

La-cross code

Twisted XZZX toric code

\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code

Frobenius code

\([[5,1,3]]_q\) Galois-qudit code

References

[1]: S. Dutta and P. P. Kurur, “Quantum Cyclic Code”, (2010) arXiv:1007.1697
[2]: M. Grassl and T. Beth, “Cyclic quantum error–correcting codes and quantum shift registers”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 456, 2689 (2000) arXiv:quant-ph/9910061 DOI
[3]: Z. Liang, F. Yang, Z. Yi, and X. Wang, “Quantum XYZ cyclic codes for biased noise”, (2025) arXiv:2501.16827
[4]: A. A. Kovalev, I. Dumer, and L. P. Pryadko, “Design of additive quantum codes via the code-word-stabilized framework”, Physical Review A 84, (2011) arXiv:1108.5490 DOI
[5]: M. B. Hastings, “A Class of Cyclic Quantum Codes”, (2025) arXiv:2509.06865

Page edit log

Victor V. Albert (2026-06-08) — most recent
Simon Burton (2024-08-12)
Victor V. Albert (2024-08-12)
Victor V. Albert (2021-12-16)
Nolan Coble (2021-12-15)

Cite as:

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

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