发表机构
Boston University; Ruhr-University Bochum（波士顿大学; 波鸿鲁尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究探讨简单量子过程能否产生伪随机性，证明酉设计通常不导致伪随机性，并给出反例，提出新猜想以理解伪随机性的涌现。
AI 中文摘要
简单过程能否表现出高度复杂性？Gowers（Comb. Prob. Comp. '96）猜想，重复组合局部随机可逆操作可以产生与随机不可区分的全局置换。在本工作中，我们研究这一问题的酉量子类比，试图在这一长期猜想上取得新进展。我们的第一个结果表明，以酉设计形式进行的统计矩匹配通常不会导致伪随机性——即使对于最简单的量子过程也是如此：对于每个固定的$t$，我们给出一个高效可采样的单量子比特和双量子比特门上的分布族$\{\nu_n\}_n$，使得在$T=O_t(n^2\log^2 n)$步独立步骤之后，所得的$n$量子比特系综是一个近似酉$t$设计，误差可忽略不计$\exp(-\Omega(\log^2 n))$，然而一个高效的量子算法仅使用$O_t(\log^2 n)$次查询就能将其与随机区分开来。这反驳了Hoory--Magen--Myers--Rackoff猜想（ICALP '04）的酉类比。我们的第二个结果是在多项式有界矩下酉设计与伪随机酉之间的更强分离；然而，我们的反例需要高度结构化的系综，与之前的简单局部游走形成对比。这表明在使用酉设计模拟黑洞物理学中的信息扰乱时应保持谨慎，因为即使是最大程度扰乱的系统也可能表现出高效实验可访问的结构。受这些发现的启发，我们随后提出了关于伪随机性如何在简单量子过程（如随机量子电路）中合理出现的新猜想。
英文摘要
Can simple processes appear highly complex? Gowers (Comb. Prob. Comp. '96) conjectured that repeatedly composing local random reversible operations can yield global permutations that are indistinguishable from random. In this work, we study the unitary quantum analog of this question, in an attempt to make new progress on this longstanding conjecture. Our first result shows that statistical moment matching in the form of unitary designs does not generically lead to pseudorandomness---even for the simplest quantum processes: for every fixed $t$, we give an efficiently samplable family $\{ν_n\}_n$ of distributions on one- and two-qubit gates such that, after $T=O_t(n^2\log^2 n)$ independent steps, the resulting $n$-qubit ensemble is an approximate unitary $t$-design with negligible error $\exp(-Ω(\log^2 n))$, yet an efficient quantum algorithm distinguishes it from random using only $O_t(\log^2 n)$ queries. This refutes the unitary analog of the Hoory--Magen--Myers--Rackoff conjecture (ICALP '04) for permutations. Our second result is a stronger separation between unitary designs and pseudorandom unitaries at polynomially bounded moments; our counterexample, however, requires highly structured ensembles, in contrast with the simple local walks from before. This suggests caution when using unitary designs to model information scrambling in black-hole physics, as even maximally scrambled systems can exhibit structure which is accessible to efficient experiments. Motivated by these findings, we then propose new conjectures for how pseudorandomness can plausibly emerge within simple quantum processes, such as random quantum circuits.
