Cyclic quantum code[1]

Description

A code \(C\) constructed in a physical space consisting of a tensor product of \(n\) identical subsystems (e.g., qubits, modular qudits, or Galois qudits) 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

Cousin

References

[1]
S. Dutta and P. P. Kurur, “Quantum Cyclic Code”, (2010) arXiv:1007.1697
[2]
H. Hao, “Investigations on Automorphism Groups of Quantum Stabilizer Codes”, (2021) arXiv:2109.12735
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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} }
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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.