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

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

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: quantum_cyclic

Cite as:
“Cyclic quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_cyclic
BibTeX:
@incollection{eczoo_quantum_cyclic,
  title={Cyclic quantum code},
  booktitle={The Error Correction Zoo},
  year={2021},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/quantum_cyclic}
}
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/quantum_cyclic

Cite as:

“Cyclic quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_cyclic

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