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Given $s, t$ in an unweighted undirected graph $G$, the goal of thepathfinding problem is to find an $s$-$t$ path. In this work, we firstconstruct a graph $G$ based on welded trees and define a pathfinding problem inthe adjacency list oracle $O$. Then we provide an efficient quantum algorithmto find an $s$-$t$ path in the graph $G$. Finally, we prove that no classicalalgorithm can find an $s$-$t$ path in subexponential time with highprobability. The pathfinding problem is one of the fundamental graph-relatedproblems. Our findings suggest that quantum algorithms may potentially offeradvantages in more types of graphs to solve the pathfinding problem and open upnew possibilities for practical applications of quantum computations in variousfields.

Exponential speedup of quantum algorithms for the pathfinding problem