A method of approximation of discrete Schrödinger equation with the normalized Laplacian by discrete-time quantum walk on graphs
Kei Saito, Etsuo Segawa
TL;DR
The paper builds a rigorous bridge between discrete-time and continuous-time quantum walks on graphs by introducing a mobility parameter $\varepsilon$ in a Szegedy walk framework. It proves that the discrete-time evolution with time step $t/N$ converges to the continuous-time evolution generated by $H = (S_o + C S_o C)/2$, connecting the discrete dynamics to a discrete Schrödinger equation driven by the discriminant operator $T$ on $\ell^2(V)$. A key corollary shows that the continuous-time Szegedy walk reproduces the standard continuous-time quantum walk driven by the normalized Laplacian, with a precise running-time bound for approximation accuracy. The paper further provides a thorough spectral analysis, decomposing the spectrum into inherited and birth eigenspaces and clarifying the roles of $T$, $S_o$, and $C$ in shaping both the point and continuous spectra, including localization phenomena on certain graphs.
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 $ε\in [0,1]$. 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-Schrödinger 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 $ε$ 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 $δ$. We also investigate the detailed spectral information on the induced continuous-time quantum walk.