Localization of the Grover walks on spidernets and free Meixner laws

TL;DR

This paper studies quantum walks on spidernet graphs, deriving spectral properties and identifying conditions for localization using free Meixner laws and quantum probabilistic analysis.

Contribution

It introduces the Grover walk on spidernets, derives an integral representation of transition amplitudes, and characterizes spidernets exhibiting localization.

Findings

Derived integral representation of transition amplitudes.

Identified class of spidernets with localization.

Connected spectral distribution to free Meixner laws.

Abstract

A spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discrete-time quantum walks on graphs. We introduce the Grover walk on a spidernet and its one-dimensional reduction. We derive an integral representation of the $n$-step transition amplitude in terms of the free Meixner law which appears as the spectral distribution. As an application we determine the class of spidernets which exhibit localization. Our method is based on quantum probabilistic spectral analysis of graphs.