For every quantum walk there is a (classical) lifted Markov chain with faster mixing time

TL;DR

This paper demonstrates that for any quantum walk, there exists a classical lifted Markov chain with a faster, polynomial-time computable mixing time that matches the quantum walk's average mixing distribution, linking quantum and classical mixing efficiencies.

Contribution

It constructs a classical lifted Markov chain that exactly replicates the quantum walk's average mixing distribution with a faster, polynomial-time computable mixing time, establishing a direct classical analogue.

Findings

Constructed a lifted Markov chain with $n^2 D(G)$ vertices

Chain mixes in $D(G)$ steps, matching quantum walk distribution

Transition probabilities computable in polynomial time

Abstract

Quantum walks on graphs have been shown in certain cases to mix quadratically faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. Here, we construct a lifted Markov chain on a graph with $n^2 D(G)$ vertices that mixes exactly to the average mixing distribution of a quantum walk on the graph $G$ with $n$ vertices, where $D(G)$ is the diameter of $G$. Moreover, the mixing time of this chain is $D(G)$ timesteps, and we prove that computing the transition probabilities for the lifted chain takes time polynomial in $n$. As an immediate consequence, for every quantum walk there is a lifted Markov chain with a faster mixing time that is polynomial-time computable, as the quantum mixing time is trivially lower bounded by the graph diameter. The result is based on a lifting presented by Apers, Ticozzi and Sarlette (arXiv:1705.08253).