Explorations in Quantum Pseudorandomness

与随机数在经典信息论中所扮演的基本角色类似，随机量子态和酉算符是量子信息科学中的关键资源。不幸的是，生成随机么正变换的效率很低，这促使人们研究一种更弱的量子伪随机性概念，这种概念可以在保持实用的同时有效地实现。在这个项目中，我们将从数学、信息论和物理学的角度，通过使用所谓的随机量子电路产生伪随机性，来探索量子伪随机性的基本性质和含义。主要的重点将放在(I)表征随机电路在比目前所考虑的更一般的条件下的行为；以及(Ii)对实现误差和去相干的影响进行专门的研究。在这样做的时候，更大的目标是理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，并预期与量子混沌领域的联系。这个项目的科学影响将是双重的。首先，人们对使用随机量子电路产生伪随机态和算符所涉及的复杂性的理解有望有实质性的提高。除了为假设随机性为资源的量子信息协议提供坚实的理论基础外，这最终可能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属性中表观随机性的出现和表征的其他基本见解。在量子信息理论、应用数学和多体物理之间的交叉学科边界上的强化培训将提供给由该助学金资助的研究生。

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.