Quantum locally recoverable code (QLRC)[1] 

Description

A QLRC of locality \(r\) is a block quantum code whose code states can be recovered after a single erasure by performing a recovery map on at most \(r\) subsystems.

Protection

A Singleton-like QLRC bound states that an \(((n,K,d))_q\) QLRC of locality \(r\) and rate \(R = \frac{\log_q K}{n}\) must have relative distance [1; Thm. 4] \begin{align} \delta = \frac{d}{n} \leq \frac{1-R}{2} - \Omega\left(\frac{1}{r}\right)~, \tag*{(1)}\end{align} implying that locality restricts the distance of the code. Random QLRCs with qudit dimension \(q = 2^{O(r)}\) achieve a relative distance that is order \(O(1/r)\) below the bound [1; Prop. 5]. Codes constructed with the help of AEL distance amplification [2,3] admit a gap of order \(O(1/r^{1/4})\) [1; Prop. 6].

Decoding

Codes constructed with the help of AEL distance amplification [2,3] admit efficient decoders [1].

Parent

Children

  • Quantum LDPC (QLDPC) code — QLDPC codes are stabilizer QLRCs whose locality \(r \leq w\), the maximum number of subsystems that a stabilizer generator participates in [1].
  • Quantum Tamo-Barg (QTB) code — Folded versions of QTB codes defined on qudits of dimension \(q = n^{O(r^2)}\) yield explicit examples of QLRCs of arbitrary locality \(r\) [1; Thm. 2].

Cousins

References

[1]
L. Golowich and V. Guruswami, “Quantum Locally Recoverable Codes”, (2023) arXiv:2311.08653
[2]
N. Alon, J. Edmonds, and M. Luby, “Linear time erasure codes with nearly optimal recovery”, Proceedings of IEEE 36th Annual Foundations of Computer Science DOI
[3]
N. Alon and M. Luby, “A linear time erasure-resilient code with nearly optimal recovery”, IEEE Transactions on Information Theory 42, 1732 (1996) DOI
[4]
J. Briët and R. de Wolf, “Locally Decodable Quantum Codes”, (2008) arXiv:0806.2101
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Zoo Code ID: quantum_locally_recoverable

Cite as:
“Quantum locally recoverable code (QLRC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_locally_recoverable
BibTeX:
@incollection{eczoo_quantum_locally_recoverable, title={Quantum locally recoverable code (QLRC)}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_locally_recoverable} }
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“Quantum locally recoverable code (QLRC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_locally_recoverable

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/stabilizer/qldpc/quantum_locally_recoverable.yml.