发表机构
National Taiwan University（国立台湾大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对量子态同一性检验问题，提出两种算法：常数深度并行交换测试和基于Schur变换的样本最优算法，并证明匹配下界，实现最优样本复杂度。
AI 中文摘要
我们研究以下量子态同一性问题：给定对 $n$ 个未知纯量子态的副本访问，目标是确定是否每一对态的保真度至少为 $1-\varepsilon$，或者是否存在某一对态的保真度至多为 $\varepsilon$。此问题放宽了先前工作中考虑的相同或正交承诺。对于任意固定的重叠参数 $\varepsilon\in(0,1/4)$，我们提出两种量子算法，它们在电路深度和样本复杂度上具有互补的保证。第一种算法使用并行交换测试，以恒定深度运行，并需要 $O(\log(n/\delta)\log(2/\delta))$ 个每个输入态的副本，以至少 $1-\delta$ 的概率成功。第二种算法使用 Schur 变换，通过将 Jucys--Murphy 元素与 Schur 采样相关联的分析，将样本复杂度降低到 $O(\log(n/\delta))$。我们还证明了任何失败概率至多为 $\delta\le 1/2-c$（对于任意常数 $c>0$）的量子算法的匹配样本复杂度下界为 $\Omega(\log(n/\delta))$，从而确立了第二种算法在固定 $\varepsilon$ 下对 $n$ 和 $\delta$ 均达到最优依赖。
英文摘要
We study the following quantum state identity problem: given access to copies of $n$ unknown pure quantum states, the goal is to determine whether every pair has fidelity at least $1-\varepsilon$ or some pair has fidelity at most $\varepsilon$. This problem relaxes the identical-or-orthogonal promise considered in previous work. For any fixed overlap parameter $\varepsilon\in(0,1/4)$, we present two quantum algorithms with complementary guarantees on circuit depth and sample complexity. The first runs in constant depth using parallel swap tests and takes $O(\log(n/δ)\log(2/δ))$ copies of each input state to succeed with probability at least $1-δ$. The second uses the Schur transform to reduce the sample complexity to $O(\log(n/δ))$, through an analysis relating Jucys--Murphy elements to Schur sampling. We also prove a matching sample complexity lower bound of $Ω(\log(n/δ))$ for any quantum algorithm with failure probability at most $δ\le 1/2-c$, for any constant $c>0$, establishing that the second algorithm achieves the optimal dependence on both $n$ and $δ$ for fixed $\varepsilon$.
Comments24 pages
