Random quantum circuits have been utilized in the contexts of quantum supremacy demonstrations, variational quantum algorithms for chemistry and machine learning, and blackhole information. The ability of random circuits to approximate any random unitaries has consequences on their complexity, expressibility, and trainability. To study this property of random circuits, we develop numerical protocols for estimating the frame potential, the distance between a given ensemble and the exact randomness. Our tensor-network-based algorithm has polynomial complexity for shallow circuits and is high-performing using CPU and GPU parallelism. We study 1. local and parallel random circuits to verify the linear growth in complexity as stated by the Brown-Susskind conjecture, and; 2. hardware-efficient ans\"atze to shed light on its expressibility and the barren plateau problem in the context of variational algorithms. Our work shows that large-scale tensor network simulations could provide important hints toward open problems in quantum information science.

翻译：摘要：随机量子电路已在量子霸权演示、化学与机器学习领域的变分量子算法以及黑洞信息等场景中得到应用。随机电路近似任意随机酉算子的能力对其复杂度、可表达性和可训练性具有重要影响。为研究随机电路的这一特性，我们开发了数值协议来估计框架势——即给定系综与精确随机性之间的距离。我们的基于张量网络的算法对浅层电路具有多项式复杂度，并能通过CPU和GPU并行实现高性能计算。我们研究了：1. 局域并行随机电路，以验证布朗-萨斯坎德猜想中提出的复杂度线性增长规律；2. 硬件高效拟设，以阐明其在变分算法中的可表达性及贫瘠高原问题。我们的工作表明，大规模张量网络模拟可为量子信息科学中的开放问题提供重要启示。