Subsystem Galois-qudit stabilizer code

Subsystem Galois-qudit stabilizer code[1]

Also known as Gauge Galois-qudit stabilizer code.

Description

Galois-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a Galois-qudit stabilizer code and assigning some of its logical qubits to be gauge qubits.

Protection

Bounds on code parameters are formulated in Ref. [2]. The Singleton bound has been extended to Galois-qudit subsystem codes [3].

Parent

Subsystem Galois-qudit code

Children

Subsystem qubit stabilizer code — Subsystem Galois-qudit stabilizer codes reduce to subsystem qubit stabilizer codes for qudit dimension \(q=2\).
Subsystem Galois-qudit CSS code

Cousins

Galois-qudit stabilizer code — Subsystem Galois-qudit stabilizer codes reduce to Galois-qudit stabilizer codes when there are no gauge qudits.
Quantum maximum-distance-separable (MDS) code — All pure MDS subsystem stabilizer codes are derived from MDS stabilizer codes [4].
Galois-qudit BCH code — Asymmetric quantum BCH codes have been constructed [6–8][5; Lemma 4.4], including subsystem BCH codes [9].
Hermitian Galois-qudit code — The Hermitian construction has been extended to subsystem Galois-qudit stabilizer codes [2].

References

[1]: A. Klappenecker and P. K. Sarvepalli, “Clifford Code Constructions of Operator Quantum Error Correcting Codes”, (2006) arXiv:quant-ph/0604161
[2]: S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
[3]: A. Klappenecker and P. K. Sarvepalli, “On subsystem codes beating the quantum Hamming or Singleton bound”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, 2887 (2007) arXiv:quant-ph/0703213 DOI
[4]: S. A. Aly and A. Klappenecker, “Subsystem Code Constructions”, (2008) arXiv:0712.4321
[5]: P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, “Asymmetric quantum codes: constructions, bounds and performance”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009) DOI
[6]: L. Ioffe and M. Mézard, “Asymmetric quantum error-correcting codes”, Physical Review A 75, (2007) arXiv:quant-ph/0606107 DOI
[7]: S. A. Aly, “Asymmetric quantum BCH codes”, 2008 International Conference on Computer Engineering & Systems (2008) DOI
[8]: G. G. La Guardia, “New families of asymmetric quantum BCH codes”, Quantum Information and Computation 11, 239 (2011) DOI
[9]: S. A. Aly, “Asymmetric and Symmetric Subsystem BCH Codes and Beyond”, (2008) arXiv:0803.0764

Page edit log

Victor V. Albert (2022-11-09) — most recent

Cite as:

“Subsystem Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_subsystem_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/subsystem/galois_subsystem_stabilizer.yml.