Quantum List Recovery and Decoding: Achievability and Limitations

2026-09-30

AI Summary

This study addresses the list blowup caused by X/Z sector separation in quantum list recovery and decoding, as well as candidate collapse following stabilizer quotienting, by investigating the combinatorial bounds of balanced folded quantum Reed–Solomon (RS) codes and random CSS codes. Methodologically, this work proposes a joint candidate pairing lemma that preserves single-sector coefficients to avoid product losses, establishes a quantum generalized Singleton bound via classical projections alongside an adapted stabilizer distinguishing construction, and integrates combinatorial coding, probabilistic analysis, and folded RS techniques. The primary contributions include deriving precise asymptotic upper bounds on list sizes for quantum list recovery and quantum list decoding, thereby establishing asymptotically tight trade-offs for quantum list-decoding radii.

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.

Problem

• Quantum list decoding
• Quantum list recovery
• CSS codes
• Quantum Singleton bound
• Folded quantum Reed-Solomon codes

Innovation

• Quantum List Decoding
• Quantum List Recovery
• Pairing Lemma
• Quantum Singleton Bound
• CSS Codes