发表机构
Siebel School of Computing and Data Science, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校赛贝尔计算与数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对量子列表恢复与解码，给出平衡折叠量子RS码和随机CSS码的列表大小上下界，并证明QLD半径权衡，通过配对引理和广义Singleton界实现可达性与逆命题。
AI 中文摘要
量子列表恢复（QLR）和量子列表解码（QLD）旨在寻找与综合征及规定错误约束相一致的逻辑上不同的泡利修正的短列表。对于CSS码，会出现两个问题：分别处理X扇区和Z扇区可能会使输出列表大小相乘，而不同的经典候选在稳定子商化后可能坍缩。我们研究了平衡折叠量子里德-所罗门（FQRS）码和平衡随机CSS码的组合上界和下界，这两种码均由Bergamaschi、Golowich和Gunn（STOC 24）研究。设R∈(0,1)为量子速率，R1=(1+R)/2为共同分量速率。当半径ρ=(1-R)/2-γ接近量子Singleton界时，对于FQRS码确定性地，对于平衡随机CSS码以高概率地，我们有L*_QLR = ℓ^{Θ(R1/γ)}，L*_QLD = Θ((1-R)/γ)（γ↓0）。渐近精确的QLD半径权衡为ρ*_L = (L/(L+1))·((1-R)/2)。这些结论扩展到平均半径设置。对于可达性，基于Brakensiek、Chen、Dhar和Zhang（STOC 2026）的工作，我们建立了一个联合X/Z候选列表的配对引理，该引理保留单扇区系数并避免列表大小的乘积损失。对于QLD的逆命题，我们基于经典投影和修补论证以及稳定子区分性证明了量子广义Singleton界。对于QLR下界，我们调整了Chen和Zhang（STOC 2025）的折叠里德-所罗门构造，使得候选保持稳定子区分性。对于随机CSS码，我们证明经典坏列表以高概率在稳定子商化后幸存。
英文摘要
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.
