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 \(\mathbb{Z}_n\).
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
- Braunstein five-mode code
- Permutation-invariant 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
- Frobenius code
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
- \([[5,1,3]]_q\) Galois-qudit code
Cousins
- Cyclic code
- Generalized bicycle (GB) code — Given a canonical generating polynomial \(g(x)\) of a cyclic quantum code \([[n,k,d]]\), its generator matrix is a cyclic matrix \(G=g(P)\). Here \(P\) is the permutation matrix of one-step length-\(n\) cyclic shift.
References
- [1]
- S. Dutta and P. P. Kurur, “Quantum Cyclic Code”, (2010) arXiv:1007.1697
Page edit log
- Victor V. Albert (2021-12-16) — most recent
- Nolan Coble (2021-12-15)
Cite as:
“Cyclic quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_cyclic