Circuit equation of Grover walk

TL;DR

This paper develops a mathematical framework using a twisted gradient operator to analyze the stationary states and scattering properties of Grover walks on infinite graphs with internal subgraphs, linking potential functions to Poisson equations.

Contribution

It introduces a novel circuit equation for Grover walks that relates stationary states to potential functions satisfying Poisson equations, advancing understanding of quantum walk scattering.

Findings

Characterizes stationary states via a twisted gradient and potential functions.

Derives a Poisson equation relating potential functions to a generalized Laplacian.

Analyzes scattering and energy penetration in specific graph structures, including complete graphs.

Abstract

We consider the Grover walk on the infinite graph in which an internal finite subgraph receives the inflow from the outside with some frequency and also radiates the outflow to the outside. To characterize the stationary state of this system, which is represented by a function on the arcs of the graph, we introduce a kind of discrete gradient operator twisted by the frequency. Then we obtain a circuit equation which shows that (i) the stationary state is described by the twisted gradient of a potential function which is a function on the vertices; (ii) the potential function satisfies the Poisson equation with respect to a generalized Laplacian matrix. Consequently, we characterize the scattering on the surface of the internal graph and the energy penetrating inside it. Moreover, for the complete graph as the internal graph, we illustrate the relationship of the scattering and the internal energy to the frequency and the number of tails.