Exponential speedup of quantum algorithms for the pathfinding problem

TL;DR

This paper presents a quantum algorithm that exponentially speeds up the pathfinding problem on certain graphs, outperforming classical algorithms which require subexponential time, highlighting quantum advantage in graph traversal tasks.

Contribution

The paper introduces a quantum algorithm for pathfinding on graphs based on welded trees, demonstrating exponential speedup over classical methods.

Findings

Quantum algorithm finds paths efficiently in specific graphs.

Classical algorithms require subexponential time.

Quantum advantage extends to broader graph classes.

Abstract

Given $x, y$ on an unweighted undirected graph $G$, the goal of the pathfinding problem is to find an $x$-$y$ path. In this work, we first construct a graph $G$ based on welded trees and define a pathfinding problem in the adjacency list oracle $O$. Then we provide an efficient quantum algorithm to find an $x$-$y$ path in the graph $G$. Finally, we prove that no classical algorithm can find an $x$-$y$ path in subexponential time with high probability. The pathfinding problem is one of the fundamental graph-related problems. Our findings suggest that quantum algorithms could potentially offer advantages in more types of graphs to solve the pathfinding problem.