Quantum walks: a Markovian perspective

TL;DR

This paper explores the conditions under which classical Markov processes can emulate quantum walks, highlighting the differences in propagation rates and addressing numerical challenges in modeling quantum effects.

Contribution

It introduces a Markovian framework to approximate quantum walks and discusses numerical methods to incorporate quantum causality effects.

Findings

Quantum walks propagate quadratically faster than classical walks.

A Markovian approach can emulate certain quantum walk behaviors.

Numerical techniques are proposed to account for ensemble effects in quantum emulation.

Abstract

For a continuous-time quantum walk on a line the variance of the position observable grows quadratically in time, whereas, for its classical counterpart on the same graph, it exhibits a linear, diffusive, behaviour. A quantum walk, thus, propagates at a rate which is linear in time, as compared to the square root rate for a classical random walk. Indeed, it has been suggested that there are graphs that can be traversed by a quantum walker exponentially faster than by the classical random analogue. In this note we adopt the approach of exploring the conditions to impose on a Markov process in order to emulate its quantum counterpart: the central issue that emerges is the problem of taking into account, in the numerical generation of each sample path, the causative effect of the ensemble of trajectories to which it belongs. How to deal numerically with this problem is shown in a paradigmatic example.