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
- 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
- Five-qubit perfect code — The five-qubit code is the smallest known example of quantum cyclic code. The full automorphism group of the code is the dihedral group of order 10 [2].
- Frobenius code
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
- \([[5,1,3]]_q\) Galois-qudit code
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
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