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 \([[17,9,4]]\) cyclic Hermitian qubit code is spanned by cyclic shifts of the Pauli operators \(XXYIXYZZYXIYXXIII\) and \(ZZXIZXYYXZIXZZIII\). An example \([[17,1,7]]\) Hermitian qubit code is spanned 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.
Decoding
Adapted from the Berlekamp decoding algorithm for classical BCH codes [1].
Notes
Many examples have been found by computer algebra programs. Ref. [1] give examples of \([[17,1,7]]\) and \([[17,9,3]]\) quantum cyclic codes.
Parent
Children
- Five-rotor 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
- \(((9,12,3))\) qubit code
- 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
Cousins
- Cyclic code
- \([[7,1,3]]\) Steane code — The Steane code is equivalent to a cyclic code via qubit permutations [2; Exam. 1].
References
- [1]
- S. Dutta and P. P. Kurur, “Quantum Cyclic Code”, (2010) arXiv:1007.1697
- [2]
- 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
Page edit log
- Simon Burton (2024-08-12) — most recent
- 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.), 2024. https://errorcorrectionzoo.org/c/quantum_cyclic