Return probability and self-similarity of the Riesz walk

TL;DR

This paper investigates the return probability and self-similarity properties of the Riesz walk, a quantum walk driven by a singular continuous measure, expanding understanding of quantum walks with complex measures.

Contribution

It analyzes the return probability of the Riesz walk and proposes conjectures on its self-similar structure, a novel focus in quantum walk research with singular measures.

Findings

Analysis of return probability behavior

Proposed conjectures on self-similarity

Insights into quantum walks with singular measures

Abstract

The quantum walk is a counterpart of the random walk. The 2-state quantum walk in one dimension can be determined by a measure on the unit circle in the complex plane. As for the singular continuous measure, results on the corresponding quantum walk are limited. In this situation, we focus on a quantum walk, called the Riesz walk, given by the Riesz measure which is one of the famous singular continuous measures. The present paper is devoted to the return probability of the Riesz walk. Furthermore, we present some conjectures on the self-similarity of the walk.