Elfs, trees and quantum walks

TL;DR

This paper introduces the elfs process, a Markov process based on electric flow sampling, and demonstrates its connection to random walks and quantum walks, with a focus on efficient sampling and hitting times on trees.

Contribution

It presents a novel electric flow sampling process, analyzes its properties, and develops a quantum walk algorithm that efficiently samples from the random walk distribution, especially on trees.

Findings

Electric hitting time on trees is logarithmic in size and weights.

The quantum walk algorithm on trees requires quadratically fewer steps than classical methods.

The elfs process connects electric flows with random walk distributions.

Abstract

We study an elementary Markov process on graphs based on electric flow sampling (elfs). The elfs process repeatedly samples from an electric flow on a graph. While the sinks of the flow are fixed, the source is updated using the electric flow sample, and the process ends when it hits a sink vertex. We argue that this process naturally connects to many key quantities of interest. E.g., we describe a random walk coupling which implies that the elfs process has the same arrival distribution as a random walk. We also analyze the electric hitting time, which is the expected time before the process hits a sink vertex. As our main technical contribution, we show that the electric hitting time on trees is logarithmic in the graph size and weights. The initial motivation behind the elfs process is that quantum walks can sample from electric flows, and they can hence implement this process very naturally. This yields a quantum walk algorithm for sampling from the random walk arrival distribution, which has widespread applications. It complements the existing line of quantum walk search algorithms which only return an element from the sink, but yield no insight in the distribution of the returned element. By our bound on the electric hitting time on trees, the quantum walk algorithm on trees requires quadratically fewer steps than the random walk hitting time, up to polylog factors.