Explorations in Quantum Pseudorandomness

与随机数在经典信息理论中的基本作用类似，随机量子态和酉算子是量子信息科学的关键资源。不幸的是，产生一个随机的酉变换是非常低效的，激发了对量子伪随机性的一个较弱的概念的研究，这个概念可以有效地实现，同时仍然是实用的。在这个程序中，量子伪随机性的基本性质和含义将从数学，信息理论和物理的角度进行探索，重点是使用所谓的随机量子电路产生伪随机性。主要的重点将放在(i)在比目前考虑的更一般的条件下描述随机电路的行为；（ii）对实现误差和退相干的影响进行专门研究。在这样做的过程中，更大的目标是了解随机量子电路是否可以用来模拟一般多体量子系统的动力学，并将其与量子混沌领域的预期联系铭记在心。这个项目的科学影响将是双重的。首先，对使用随机量子电路产生伪随机状态和算子的复杂性的理解将得到实质性的提高。除了为假设随机性作为资源的量子信息协议提供坚实的理论基础之外，这最终可能会在增强的量子算法设计中得到应用。其次，还将获得关于复杂量子系统特性中表观随机性的出现和表征的额外基本见解。在量子信息理论、应用数学和多体物理的交叉学科边界上，将为受本基金资助的研究生提供密集的学科培训。

In analogy to the fundamental role that random numbers play in classical information theory, random quantum states and unitary operators are a key resource in quantum information science. Unfortunately, generating a random unitary transformation is intractably inefficient, motivating the investigation of a weaker notion of quantum pseudorandomness that may be achieved efficiently while remaining practically useful. In this program, fundamental properties and implications of quantum pseudorandomness will be explored from a mathematical, information-theoretic, and physical standpoint, by focusing on the generation of pseudorandomness using so-called random quantum circuits. Primary emphasis will be placed on (i) characterizing the behavior of random circuits under more general conditions than considered thus far; and (ii) performing a dedicated study of the effect of implementation errors and decoherence. In doing so, the bigger goal of understanding whether random quantum circuits may serve to model the dynamics of generic many-body quantum systems will be kept in mind, with anticipated connections to the field of quantum chaos.The scientific impact of this program will be twofold. First, a substantially improved understanding of the complexity involved in generating pseudorandom states and operators using random quantum circuits is expected to emerge. Beside providing a firm theoretical foundation for quantum information protocols that presume randomness as a resource, this could eventually find application in enhanced quantum algorithm design. Second, additional fundamental insights will also be gained on the emergence and characterization of apparent randomness in the properties of complex quantum systems. Intensive training on subjects at the cross-disciplinary boundary between quantum information theory, applied mathematics, and many-body physics will be provided to the graduate student supported by this grant.