发表机构
RIKEN Hakubi Research Team, RIKEN Center for Quantum Computing (RQC)（理化学研究所博慧研究团队，理化学中心量子计算部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究解决了$U(1)$对称随机电路中酉设计形成速率的最优标度问题，发现双粒子相遇速率而非电荷输运构成瓶颈，并提供了通过对称性破缺门规避瓶颈的协议。
AI 中文摘要
理解守恒律下的随机性形成，对于表征孤立量子系统中的混沌现象以及受对称性约束的量子信息处理都至关重要。在$N$量子比特系统上的$U(1)$对称随机电路为此问题提供了一个最小可解的设置，其中酉设计的形成速率曾被推测由底层电路几何中的电荷输运所主导。然而，电荷输运是否确实设定了最慢的弛豫模式，一直是一个未解问题，原因在于难以推导出对称设计形成速率的紧致下界。在本工作中，我们通过确定在包括晶格、扩展图以及全对全相互作用在内的广泛电路几何中，对于$k\leq O(\sqrt{\log N})$的酉$k$-设计形成速率的最优标度，解决了这一问题。特别地，对于$k\geq 2$，我们发现形成速率由双粒子相遇速率——即在给定电路几何中两个随机游走者相遇的速率——所主导。在许多几何中，该相遇速率在参数上小于单粒子输运速率，从而否定了先前的猜想。我们进一步通过一种基于辅助子群系综的新颖证明技术推导其匹配的下界，确立了这一相遇机制确实设定了设计形成速率。尽管这一相遇模式在$U(1)$对称随机电路中给设计形成施加了稳健的瓶颈，我们也展示了通过使用破坏对称性的局部门可以规避此瓶颈，并提供了一种实现更快设计形成的显式协议。这些结果确立了守恒律下随机性形成的新图景，并为表征和高效生成对称性约束的随机性提供了指导原则。
英文摘要
Understanding randomness formation under conservation laws is important both for characterizing chaotic phenomena in isolated quantum systems and for symmetry-constrained quantum information processing. $U(1)$-symmetric random circuits on an $N$-qubit system provide a minimal tractable setting for this problem, in which the formation rate of unitary designs was conjectured to be governed by charge transport in the underlying circuit geometry. However, whether charge transport indeed sets the slowest relaxation mode has remained an open problem, due to the difficulty in deriving a tight lower bound on the symmetric design formation rate. In this work, we resolve this problem by determining the optimal scaling of the unitary $k$-design formation rate for $k\leq O(\sqrt{\log N})$ across a wide range of circuit geometries, including lattices, expander graphs, and all-to-all interactions. In particular, for $k\geq 2$, we find that the formation rate is governed by a two-particle encounter rate---the rate at which two random walkers encounter each other in a given circuit geometry. This encounter rate is parametrically smaller than the single-particle transport rate in many geometries, thereby disproving the previous conjecture. We further establish that this encounter mechanism indeed sets the design formation rate by deriving its matching lower bound with a novel proof technique based on an auxiliary subgroup ensemble. Although this encounter mode imposes a robust bottleneck on design formation in $U(1)$-symmetric random circuits, we also show that this bottleneck can be circumvented by using symmetry-breaking local gates and provide an explicit protocol that achieves faster design formation. These results establish a new picture of randomness formation under conservation laws and provide guiding principles for characterizing and efficiently generating symmetry-constrained randomness.
Comments93 pages, 4 figures
