EA Galois-qudit stabilizer code

EA Galois-qudit stabilizer code[1]

Description

A Galois-qudit stabilizer code constructed using a variation of the stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver. A code is typically denoted as \([[n,k;e]]_q\) or \([[n,k,d;e]]_q\), where \(d\) is the distance of the underlying non-EA \([[n,k,d]]_q\) code, and \(e\) is the number of required pre-shared maximally entangled Galois-qudit maximally entangled states.

Decoding

Syndrome extraction and computation based on classical additive codes [2].

Parent

EA Galois-qudit code

Children

EA qubit stabilizer code — EA Galois-qudit stabilizer codes reduce to EA qubit stabilizer codes for \(q=2\).
EA quantum LCD code
Maximal-entanglement EA Galois-qudit stabilizer code

Cousins

Galois-qudit stabilizer code — EA Galois-qudit stabilizer codes utilize additional ancillary Galois-qudits in a pre-shared entangled state, but reduce to Galois-qudit stabilizer codes when said qudits are interpreted as noiseless physical qudits. Pure Galois-qudit codes can be used to make EA Galois-qudit stabilizer codes [1][3; Thm. 10].
Galois-qudit GRS code — Galois-qudit GRS codes can be used to construct EA Galois-qudit stabilizer codes [4,5].
Concatenated quantum code — Concatenated EA Galois-qudit stabilizer codes have been studied [6,7].
Algebraic-geometry (AG) code — Certain AG codes can be used to construct EA Galois-qudit stabilizer codes [8].

References

[1]: L. Riguang and M. Zhi, “Non-binary Entanglement-assisted Stabilizer Quantum Codes”, (2011) arXiv:1105.5872
[2]: P. J. Nadkarni and S. S. Garani, “Quantum error correction architecture for qudit stabilizer codes”, Physical Review A 103, (2021) DOI
[3]: M. Grassl, F. Huber, and A. Winter, “Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 68, 3942 (2022) arXiv:2010.07902 DOI
[4]: K. Guenda, S. Jitman, and T. A. Gulliver, “Constructions of Good Entanglement-Assisted Quantum Error Correcting Codes”, (2016) arXiv:1606.00134
[5]: P. J. Nadkarni and S. S. Garani, “Entanglement-assisted Reed–Solomon codes over qudits: theory and architecture”, Quantum Information Processing 20, (2021) DOI
[6]: J. Fan et al., “Entanglement-assisted concatenated quantum codes”, Proceedings of the National Academy of Sciences 119, (2022) arXiv:2202.08084 DOI
[7]: T. Sidana and N. Kashyap, “Entanglement-Assisted Quantum Error-Correcting Codes over Local Frobenius Rings”, (2023) arXiv:2202.00248
[8]: L. Sok, “On linear codes with one-dimensional Euclidean hull and their applications to EAQECCs”, (2021) arXiv:2101.06461

Page edit log

Victor V. Albert (2023-07-18) — most recent

Cite as:

“EA Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ea_galois_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/ea_stabilizer/ea_galois_stabilizer.yml.