发表机构
University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了多项式深度的局部随机量子电路输出分布以逆多项式速率收敛到Porter-Thomas分布，为量子优势实验提供了理论依据。
AI 中文摘要
Porter-Thomas统计是随机量子态输出分布的一个特征，更广泛地说，是混沌量子多体系统的特征。收敛到Porter-Thomas分布在随机电路采样和量子优势的实验演示中起着核心作用，其中低深度随机量子电路的输出统计被期望近似为Porter-Thomas，尽管缺乏严格的收敛证明。我们证明了多项式深度的砖墙随机电路的输出分布在总变差距离上以逆多项式速度收敛到Porter-Thomas分布。具体来说，考虑由最近邻Haar随机门构造的局部随机量子电路在固定比特串上的输出概率分布。那么，对于任何$m \geq 0$，深度为$O(n^{2m+1}\log(n))$的电路所对应的分布与Porter-Thomas分布在总变差距离上的差距至多为$O(1/n^m)$。我们的证明使用了近似设计中的矩界、特征函数的解析估计以及逆矩的局部反集中性质。
英文摘要
Porter-Thomas statistics are a characteristic feature of the output distribution of random quantum states and, more broadly, chaotic quantum many-body systems. Convergence to Porter-Thomas plays a central role in random circuit sampling and experimental demonstrations of quantum advantage, where the output statistics of low-depth random quantum circuits are expected to be approximately Porter-Thomas, despite the absence of a rigorous proof of convergence. We show that the output distribution of polynomial-depth brickwork random circuits converges inverse-polynomially in total variation distance to the Porter-Thomas distribution. Specifically, consider the output probability distribution over a fixed bitstring of a local random quantum circuit, constructed from nearest-neighbor Haar random gates. Then, for any $m \geq 0$, the distribution corresponding to circuits of depth $O(n^{2m+1}\log(n))$ is at most $O(1/n^m)$ far in total variation distance from the Porter-Thomas distribution. Our proof uses moment bounds from approximate designs, analytic estimates for characteristic functions, and a local anticoncentration property for inverse moments.
Comments28 pages, 1 figure
