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arXiv · 2308.13741

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

Kei Saito·Etsuo Segawa
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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.

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Kei Saito, Etsuo Segawa. 2023-08-26. A method of approximation of discrete Schr\"odinger equation with the normalized Laplacian by discrete-time quantum walk on graphs. https://arxiv.org/abs/2308.13741

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