A mathematical framework for maze solving using quantum walks

TL;DR

This paper introduces a quantum walk-based mathematical framework for maze solving, demonstrating that it effectively identifies shortest paths in various maze structures by analyzing stationary states.

Contribution

It develops a novel quantum walk approach with absorbing holes for maze solving, highlighting its effectiveness in different graph structures.

Findings

Probability amplitude concentrates on shortest paths in trees.

Maximized amplitude near shortest paths in ladder-like structures.

Framework provides a new quantum method for maze pathfinding.

Abstract

We provide a mathematical framework for identifying the shortest path in a maze using a Grover walk, which becomes non-unitary by introducing absorbing holes. In this study, we define the maze as a network with vertices connected by unweighted edges. Our analysis of the stationary state of the Grover walk on finite graphs, where we strategically place absorbing holes and self-loops on specific vertices, demonstrates that this approach can effectively solve mazes. By setting arbitrary start and goal vertices in the underlying graph, we obtain the following long-time results: (i) in tree structures, the probability amplitude is concentrated exclusively along the shortest path between start and goal; (ii) in ladder-like structures with additional paths, the probability amplitude is maximized near the shortest path.