P\'olya number of continuous-time quantum walks

TL;DR

This paper introduces a new way to measure the recurrence of continuous-time quantum walks using a Pólya number, analyzing various graphs and measurement timings to understand their recurrence behavior.

Contribution

It defines the Pólya number for continuous-time quantum walks and explores how measurement timing affects recurrence properties across different graph structures.

Findings

Recurrence depends on the decay rate of the probability at the origin.

Poisson and regular measurement timings are both considered.

The Pólya number effectively characterizes recurrence in quantum walks.

Abstract

We propose a definition for the P\'olya number of continuous-time quantum walks to characterize their recurrence properties. The definition involves a series of measurements on the system, each carried out on a different member from an ensemble in order to minimize the disturbance caused by it. We examine various graphs, including the ring, the line, higher dimensional integer lattices and a number of other graphs and calculate their P\'olya number. For the timing of the measurements a Poisson process as well as regular timing are discussed. We find that the speed of decay for the probability at the origin is the key for recurrence.