Quantum Fast-Forwarding: Markov Chains and Graph Property Testing

TL;DR

This paper introduces quantum fast-forwarding (QFF), a novel quantum algorithmic tool that accelerates the transient dynamics of Markov chains, leading to faster and more space-efficient graph property testing algorithms.

Contribution

The paper presents QFF, enabling quadratic speedups in simulating Markov chain dynamics and improving classical graph property testing algorithms in both runtime and space complexity.

Findings

Quadratic speedup in random walk algorithms for graph expansion testing

Exponential reduction in space complexity for property testing

Efficient quantum solution for robust s-t connectivity problem

Abstract

We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with $P$ the Markov chain transition matrix and $D = \sqrt{P\circ P^T}$ its discriminant matrix ($D=P$ if $P$ is symmetric), we construct a quantum walk algorithm that for any quantum state $|v\rangle$ and integer $t$ returns a quantum state $\epsilon$-close to the state $D^t|v\rangle/\|D^t|v\rangle\|$. The algorithm uses $O\Big(\|D^t|v\rangle\|^{-1}\sqrt{t\log(\epsilon\|D^t|v\rangle\|)^{-1}}\Big)$ expected quantum walk steps and $O(\|D^t|v\rangle\|^{-1})$ expected reflections around $|v\rangle$. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of $s$-$t$ connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.