Singleton-bound approaching AQECC[1]
Description
Approximate quantum code of rate \(R\) that can tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\). The formulation of such codes relies on a notion of quantum list decoding [1,2]. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability.
Protection
For any \(\gamma>0\) and rate \(0<R<1\), these approximate quantum \([[n,R \cdot n]]_q\) codes have constant Galois-qudit dimension \(q=q(\gamma)\) and correct errors acting on \((1-R-\gamma) \cdot n/2\) registers, up to a recovery error of \(2^{-\Omega(n)}\).
Rate
Given rate \(R\), tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\).
Encoding
Efficient encoding.
Decoding
Quantum list decodable [1].
Parents
Cousins
- Quantum maximum-distance-separable (MDS) code — Singleton-bound approaching AQECCs saturate the quantum Singleton bound.
- Folded quantum Reed-Solomon (FQRS) code — Singleton-bound approaching AQECCs utilize FQRS codes.
- Approximate secret-sharing code — Quantum secret-sharing codes have asymptotically decaying rate and require qudit dimension to increase exponentially with \(n\), while Singleton-bound approaching AQECCs have constant rate and qudit dimension.
References
- [1]
- T. Bergamaschi, L. Golowich, and S. Gunn, “Approaching the Quantum Singleton Bound with Approximate Error Correction”, (2022) arXiv:2212.09935
- [2]
- D. Leung and G. Smith, “Communicating over adversarial quantum channels using quantum list codes”, (2007) arXiv:quant-ph/0605086
Page edit log
- Victor V. Albert (2023-01-08) — most recent
- Sam Gunn (2022-01-08)
Cite as:
“Singleton-bound approaching AQECC”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_singleton