A method of approximation of discrete Schr\"odinger equation with the normalized Laplacian by discrete-time quantum walk on graphs

TL;DR

This paper introduces a new class of continuous-time quantum walk models on graphs derived from discrete-time models, providing a framework to approximate the Schrödinger equation using normalized Laplacian and analyzing spectral properties.

Contribution

It presents a novel method to approximate continuous-time quantum walks from discrete-time models on graphs, including infinite cases, with spectral analysis and error bounds.

Findings

Discrete-time quantum walks can approximate continuous-time walks with small error.

The induced continuous-time quantum walk extends the Schrödinger equation with matrix-valued elements.

Spectral properties of the induced walk are thoroughly analyzed.

Abstract

We propose a class of continuous-time quantum walk models on graphs induced by a certain class of discrete-time quantum walk models with the parameter $\epsilon\in [0,1]$. Here the graph treated in this paper can be applied both finite and infinite cases. The induced continuous-time quantum walk is an extended version of the (free) discrete-Schr\"odinger equation driven by the normalized Laplacian: the element of the weighted Hermitian takes not only a scalar value but also a matrix value depending on the underlying discrete-time quantum walk. We show that each discrete-time quantum walk with an appropriate setting of the parameter $\epsilon$ in the long time limit identifies with its induced continuous-time quantum walk and give the running time for the discrete-time to approximate the induced continuous-time quantum walk with a small error $\delta$. We also investigate the detailed spectral information on the induced continuous-time quantum walk.