The Quantum Walk of F. Riesz

TL;DR

This paper introduces a novel method to associate quantum walks with probability measures on the unit circle, exemplified by the Riesz measure, expanding the traditional framework of quantum walks.

Contribution

It presents a new approach linking quantum walks to arbitrary measures on the unit circle, including unconventional transition steps, with a detailed example involving the Riesz measure.

Findings

Established a method to connect quantum walks with measures on the unit circle

Demonstrated the approach using the Riesz measure as a key example

Extended the framework of quantum walks to include non-traditional transitions

Abstract

We exhibit a way to associate a quantum walk (QW) on the non-negative integers to any probability measure on the unit circle. This forces us to consider one step transitions that are not traditionally allowed. We illustrate this in the case of a very interesting measure, originally proposed by F. Riesz for a different purpose.