Scaling limit of the time averaged distribution for continuous time quantum walk and Szegedy's walk on the path

TL;DR

This paper investigates the scaling limits of time-averaged distributions in continuous time quantum walks and Szegedy's walks on paths, revealing conditions under which their limits coincide based on spectral properties.

Contribution

It establishes a connection between the scaling limits of continuous and discrete quantum walks via spectral gap conditions on Jacobi matrices.

Findings

Scaling limit of continuous time quantum walk matches Szegedy's walk under spectral gap conditions.

Spectral gap on the Jacobi matrix is crucial for the convergence of distributions.

Provides a theoretical framework linking continuous and discrete quantum walks.

Abstract

In this paper, we consider Szegedy's walk, a type of discrete time quantum walk, and corresponding continuous time quantum walk related to the birth and death chain. We show that the scaling limit of time averaged distribution for the continuous time quantum walk induces that of Szegedy's walk if there exists the spectral gap on so-called the corresponding Jacobi matrix .