Finding paths with quantum walks or quantum walking through a maze

TL;DR

This paper demonstrates how quantum walks can efficiently find paths in maze-like structures, improving search times from classical methods through analytical and iterative quantum strategies.

Contribution

It introduces a quantum walk-based method for pathfinding in maze structures, achieving improved efficiency with analytical and iterative search techniques.

Findings

Pathfinding in maze structures using quantum walks is feasible.

The method achieves an $O(M\sqrt{N})$ search complexity.

Successive quantum searches can optimize path recovery.

Abstract

We show that it is possible to use a quantum walk to find a path from one marked vertex to another. In the specific case of $M$ stars connected in a chain, one can find the path from the first star to the last one in $O(M\sqrt{N})$ steps, where $N$ is the number of spokes of each star. First we provide an analytical result showing that by starting in a phase-modulated highly superposed initial state we can find the path in $O(M\sqrt{N}\log M)$ steps. Next, we improve this efficiency by showing that the recovery of the path can also be performed by a series of successive searches when we start at the last known position and search for the next connection in $O(\sqrt{N})$ steps leading to the overall efficiency of $O(M\sqrt{N})$. For this result we use the analytical solution that can be obtained for a ring of stars of double the length of the chain.