Quantum list recovery and decoding limits for css and reed solomon codes
Quantum List Recovery and Decoding: Achievability and Limitations
Information Theory
Summary
Quantum error correction tries to find a small set of possible fixes for errors in quantum data. The authors study how big these sets must be and how close the error detection can get to theoretical limits for certain quantum codes. They resolve challenges when combining parts of quantum errors separately and find precise relationships between error bounds and list sizes. Their results explain how well quantum Reed--Solomon and random CSS codes perform in this decoding task.
What this means in practice
- •For quantum hardware engineers: Design error correction schemes with predictable limits on decoding ambiguity for fault-tolerant quantum processors.
- •For quantum software developers: Implement list decoding algorithms that optimize syndrome interpretation efficiency near capacity limits for quantum communication protocols.
A theory result. No direct application yet.
Authors
Fernando Granha Jeronimo, Xiaojuan Ma
Abstract
Quantum list recovery (QLR) and quantum list decoding (QLD) seek short lists of logically distinct Pauli corrections consistent with a syndrome and prescribed error constraints. For CSS codes, two issues arise: treating the \(X\)- and \(Z\)-sectors separately can multiply their output list sizes, while distinct classical candidates can collapse after stabilizer quotienting. We study combinatorial upper and lower bounds for balanced folded quantum Reed--Solomon (FQRS) codes and balanced random CSS codes, both studied by Bergamaschi, Golowich, and Gunn (STOC 24). Let \(R\in(0,1)\) be the quantum rate and \(R_1=(1+R)/2\) the common component rate. As the radius \(ρ=(1-R)/2-γ\) approaches the quantum Singleton bound, we have, deterministically for FQRS codes and w.h.p. for balanced random CSS codes, \[ L^\star_{\rm QLR} = \ell^{Θ(R_1/γ)}, \qquad L^\star_{\rm QLD} = Θ\!\left(\frac{1-R}γ\right) \qquad (γ\downarrow0). \] The asymptotically exact QLD radius tradeoff is \[ ρ_L^\star = \frac{L}{L+1}\frac{1-R}{2}. \] These conclusions extend to the average-radius setting. For achievability, building on the work of Brakensiek, Chen, Dhar, and Zhang (STOC 2026), we establish a pairing lemma for joint \(X/Z\) candidate lists that preserves the one-sector coefficient and avoids a product loss in list size. For the QLD converse, we prove a quantum generalized Singleton bound based on the classical projection-and-patching argument with stabilizer distinctness. For the QLR lower bounds, we adapt the folded Reed--Solomon construction of Chen and Zhang (STOC 2025) so that the candidates remain stabilizer distinct. For random CSS codes, we show classical bad lists survive the stabilizer quotient with high probability.