An analytic solution for one-dimensional quantum walks

TL;DR

This paper derives the first general analytic solutions for one-dimensional quantum walks, revealing new symmetry features and providing insights into their behavior, which enhances modeling of quantum phenomena similar to classical random walks.

Contribution

It presents the first comprehensive analytic solutions for 1D quantum walks in position and momentum space, uncovering new symmetry properties and deepening understanding of their dynamics.

Findings

New symmetry features of quantum walk probability densities

Analytic expressions for quantum walk probability distributions

Enhanced modeling of quantum phenomena using these solutions

Abstract

The first general analytic solutions for the one-dimensional walk in position and momentum space are derived. These solutions reveal, among other things, new symmetry features of quantum walk probability densities and further insight into the behaviour of their moments. The analytic expressions for the quantum walk probability distributions provide a means of modelling quantum phenomena that is analogous to that provided by random walks in the classical domain.