Properties of long quantum walks in one and two dimensions

TL;DR

This paper provides a detailed numerical analysis of discrete-time quantum walks in one and two dimensions, revealing their probability distribution structures, scaling behaviors, and effects of entanglement, with implications for quantum computation.

Contribution

It offers new insights into the structure, scaling, and entanglement effects of quantum walks in 1D and 2D, including experimental considerations.

Findings

Probability distributions exhibit oscillatory features at all scales.

Maximal entanglement leads to speed-up and semi-classical behavior.

Scaling characteristics are analyzed up to one million steps.

Abstract

The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena by performing a detailed numerical study of discrete-time quantum walks. In one dimension (1D), we compute the structure of the probability distribution, which is not a smooth curve but shows oscillatory features on all length scales. By analyzing walks up to $N$ = 1000000 steps, we discuss the scaling characteristics and limiting forms of the QW in both real and Fourier space. In 2D, with a view to ready experimental realization, we consider two types of QW, one based on a four-faced coin and the other on sequential flipping of a single two-faced coin. Both QWs may be generated using two two-faced coins, which in the first case are completely unentangled and in the second are maximally entangled. We draw on our 1D results to characterize the properties of both walks, demonstrating maximal speed-up and emerging semi-classical behavior in the maximally entangled QW.