A mathematical framework for maze solving using quantum walks
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910705205116928 |
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| author | Matsuoka, Leo Ohno, Hiromichi Segawa, Etsuo |
| author_facet | Matsuoka, Leo Ohno, Hiromichi Segawa, Etsuo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12191 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A mathematical framework for maze solving using quantum walks Matsuoka, Leo Ohno, Hiromichi Segawa, Etsuo Quantum Physics Mathematical Physics 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. |
| title | A mathematical framework for maze solving using quantum walks |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2411.12191 |