Unidirectional quantum walks: evolution and exit times

TL;DR

This paper studies a unidirectional quantum walk model, deriving explicit wave functions and analyzing exit times, offering computational advantages and insights into quantum walk dynamics.

Contribution

It introduces a unidirectional quantum walk framework using Discrete Fourier Transforms, enabling explicit wave function expressions and efficient computation of exit probabilities.

Findings

Explicit wave functions derived using Fourier analysis

Efficient algorithms for computing walk probabilities

Analysis of exit-time probabilities from fixed intervals

Abstract

In this paper we focus our attention on a particle that follows a unidirectional quantum walk, an alternative version of the nowadays widespread discrete-time quantum walk on a line. Here the walker at each time step can either remain in place or move in a fixed direction, e.g., rightward or upward. While both formulations are essentially equivalent, the present approach leads to consider Discrete Fourier Transforms, which eventually results in obtaining explicit expressions for the wave functions in terms of finite sums, and allows the use of efficient algorithms based on the Fast Fourier Transform. The wave functions here obtained govern the probability of finding the particle at any given location, but determine as well the exit-time probability of the walker from a fixed interval, which is also analyzed.