Inhomogeneous Quantum Walks

TL;DR

This paper investigates inhomogeneous quantum walks with position-dependent transition probabilities, analyzing their boundedness, asymptotic speed, and comparing to existing models, including a quantum Polya Urn example.

Contribution

It introduces a general framework for inhomogeneous quantum walks, analyzing periodic cases and their properties, which advances understanding of quantum walk dynamics.

Findings

Periodic inhomogeneous walks can be bounded or unbounded.

Unbounded walks exhibit specific asymptotic speeds.

A quantum Polya Urn model is constructed as an example.

Abstract

We study a natural construction of a general class of inhomogeneous quantum walks (namely walks whose transition probabilities depend on position). Within the class we analyze walks that are periodic in position and show that, depending on the period, such walks can be bounded or unbounded in time; in the latter case we analyze the asymptotic speed. We compare the construction to others in the existing literature. As an example we give a quantum version of a non-irreducible classical walk: the Polya Urn.