Quantum Walks and Weak Measurements

量子信息处理(QIP)使用叠加、干涉和纠缠等量子现象来执行使用经典资源难以或不可能完成的信息处理任务。本研究项目研究了量子IP中的两个重要工具：量子游动和弱测量，以及它们之间的关系。量子游动类似于经典的随机游动。在图的一个顶点上的“步行者”重复地沿着边随机地移动到相邻的顶点。在量子版本中，步行者同时沿着每一条可能的边缘以叠加的方式移动，产生新的干涉效果。这些效应受到图形的整体对称性的强烈影响，为新的量子算法打开了可能性。然而，在量子力学测量中，步行器的位置会干扰系统的状态，并破坏干涉效应。为了避免这个问题，我们使用弱测量，它提供的信息较少，但对状态的干扰也较小。选择最优的测量强度在最小化预期命中时间的同时最大化发现系统处于所需状态的可能性。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漫步命中时间的影响，无限命中时间漫步的存在，以及这些如何允许一类新的量子算法。他们正在为图上的量子行走开发路径积分技术。他们还研究了如何将强广义测量分解为一系列弱(或连续)测量。在给定一族可能的弱测量的情况下，确定可以产生哪些广义测量是一个悬而未决的问题。

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.