Quantum Walks and Weak Measurements

量子信息处理（QIP）利用叠加、干涉和纠缠等量子现象来完成传统资源难以或不可能完成的信息处理任务。本课题研究了QIP中的两个重要工具：量子行走和弱测量，以及它们之间的相互关系。量子行走类似于经典的随机行走。在图的一个顶点上的“步行者”会沿着边缘反复随机移动到邻近的顶点。在量子版本中，步行者同时沿着每条可能的边缘叠加移动，产生新的干涉效应。这些效应受到图的全局对称性的强烈影响，为新的量子算法开辟了可能性。然而，在量子力学中，测量步行者的位置会扰乱系统的状态，破坏干涉效应。为了避免这个问题，我们使用弱测量，提供更少的信息，但也更少地干扰状态。选择最佳的测量强度可以最大限度地使系统处于理想状态，同时使预期的命中时间最小化。研究人员解决了涉及量子行走和弱测量的几个重要问题。他们研究了对称性对连续时间量子行走的撞击时间的影响，无限撞击时间行走的存在，以及这些如何允许一类新的量子算法。他们正在开发图上量子行走的路径积分技术。他们还研究了如何将强广义测量分解成一系列弱（或连续）测量。给定一系列可能的弱测量值，确定可以产生哪些广义测量值是一个开放问题。

Quantum information processing (QIP) uses quantum phenomena such as superposition, interference, and entanglement to perform information-processing tasks that are difficult or impossible with classical resources. This research project studies two important tools in QIP, and their relationship to each other: quantum walks and weak measurements.Quantum walks are analogous to classical random walks. A "walker" at one of the vertices of a graph repeatedly moves randomly along the edges to neighboring vertices. In the quantum version the walker simultaneously moves in a superposition along every possible edge, producing novel interference effects. These effects are strongly influenced by the global symmetry of the graph, opening up the possibility of new quantum algorithms. However, in quantum mechanics measuring the location of the walker disturbs the state of the system and destroys the interference effects. To avoid this problem we use weak measurements that give less information but also disturb the state less. Choosing the optimal measurement strength maximizes the likelihood of finding the system in the desired state while minimizing the expected hitting time.The researchers address several important problems involving quantum walks and weak measurements. They study the effect of symmetry on hitting time for continuous-time quantum walks, the existence of infinite hitting-time walks, and how these allow a new class of quantum algorithms. They are developing path-integral techniques for quantum walks on graphs. They also study how to decompose strong generalized measurements into a sequence of weak (or continuous) measurements. It is an open problem, given a family of possible weak measurements, to determine which generalized measurements can be produced.