A comfortable graph structure for Grover walk

TL;DR

This paper analyzes a Grover walk on a finite graph connected to semi-infinite paths, characterizing its stationary state, scattering behavior, and introducing a measure of how well the graph supports quantum walkers.

Contribution

It provides a detailed scattering matrix depending on the graph's bipartiteness and introduces a comfortability function linked to the graph's combinatorial properties.

Findings

Scattering matrix explicitly derived for the Grover walk.

Comfortability function quantifies quantum walker retention.

Stationary state behavior is characterized by graph structure.

Abstract

We consider a Grover walk model on a finite internal graph, which is connected with a finite number of semi-infinite length paths and receives the alternative inflows along these paths at each time step. After the long time scale, we know that the behavior of such a Grover walk should be stable, that is, this model has a stationary state. In this paper our objectives are to give some characterization upon the scattering of the stationary state on the surface of the internal graph and upon the energy of this state in the interior. For the scattering, we concretely give a scattering matrix, whose form is changed depending on whether the internal graph is bipartite or not. On the other hand, we introduce a comfortability function of a graph for the quantum walk, which shows how many quantum walkers can stay in the interior, and we succeed in showing the comfortability of the walker in terms of combinatorial properties of the internal graph.