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arXiv · 2411.12191

A mathematical framework for maze solving using quantum walks

Leo Matsuoka·Hiromichi Ohno·Etsuo Segawa
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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.

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Leo Matsuoka, Hiromichi Ohno, Etsuo Segawa. 2024-11-19. A mathematical framework for maze solving using quantum walks. https://arxiv.org/abs/2411.12191

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