Quantum Walks and Electric Networks

TL;DR

This paper establishes a general quantum walk framework that detects marked elements in graphs efficiently, extending previous results to arbitrary initial distributions, and applies it to develop a quantum algorithm for 3-distinctness.

Contribution

It generalizes quantum walk detection bounds to any initial distribution and introduces a time-efficient quantum algorithm for 3-distinctness.

Findings

Quantum walks detect marked elements in $O(\sqrt{WR})$ steps.

Generalization of Szegedy's result to arbitrary initial distributions.

Development of a quantum algorithm for 3-distinctness.

Abstract

We prove that a quantum walk can detect the presence of a marked element in a graph in $O(\sqrt{WR})$ steps for any initial probability distribution on vertices. Here, $W$ is the total weight of the graph, and $R$ is the effective resistance. This generalizes the result by Szegedy that is only applicable if the initial distribution is stationary. We describe a time-efficient quantum algorithm for 3-distinctness based on these ideas.