Hitting time for the continuous quantum walk

TL;DR

This paper introduces a definition for hitting time in continuous quantum walks, derives an explicit formula, and analyzes how measurement rate affects the walk's behavior, including conditions for infinite hitting times linked to graph symmetry.

Contribution

It provides a new formalism for hitting time in continuous quantum walks and explores the effects of measurement rate and graph symmetry on hitting times.

Findings

Hitting time diverges as measurement rate approaches 0 or infinity.

Infinite hitting times are linked to graph symmetry.

Explicit formula for continuous quantum walk hitting time derived.

Abstract

We define the hitting (or absorbing) time for the case of continuous quantum walks by measuring the walk at random times, according to a Poisson process with measurement rate $\lambda$. From this definition we derive an explicit formula for the hitting time, and explore its dependence on the measurement rate. As the measurement rate goes to either 0 or infinity the hitting time diverges; the first divergence reflects the weakness of the measurement, while the second limit results from the Quantum Zeno effect. Continuous-time quantum walks, like discrete-time quantum walks but unlike classical random walks, can have infinite hitting times. We present several conditions for existence of infinite hitting times, and discuss the connection between infinite hitting times and graph symmetry.