Simulation of Quantum Walks and Fast Mixing with Classical Processes

TL;DR

This paper compares quantum walks to classical lifted Markov chains, showing classical processes can simulate quantum speedups in mixing, but quantum effects are not always the cause of faster transport.

Contribution

It demonstrates that classical lifted Markov chains can replicate quantum walk behavior, clarifying the role of quantum effects in mixing speedups.

Findings

Classical lifted Markov chains can simulate quantum walk mixing.

Graph topology bounds the mixing performance of both quantum and classical processes.

Quantum walks exhibit superdiffusive spreading more prominently.

Abstract

We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us to answer an open question on how the graph topology ultimately bounds their mixing performance, and that of any stochastic local evolution. The results highlight that speedups in mixing and transport phenomena are not necessarily diagnostic of quantum effects, although superdiffusive spreading is more prominent with quantum walks.