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.
