Quantum Algorithms for Magic Square Diophantine Equations

• URL: http://arxiv.org/abs/2605.04106v1
• Date: Mon, 04 May 2026 10:02:12 GMT
• Title: Quantum Algorithms for Magic Square Diophantine Equations
• Abstract summary: Magic-square constraints define Diophantine systems whose solutions exhibit rigid periodic structure.<n>We study this structure in an oracle setting, where a marked set of integers is given by black-box access and the goal is to decide whether it encodes a magic square.<n>For $3times 3$ magic squares and weighted variants, we prove explicit periodic characterizations that reduce detection to period finding.<n>For larger orders, we identify a class of solutions built from repeated arithmetic patterns, which can be detected via the quantum Fourier transform.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Magic-square constraints define Diophantine systems whose solutions, in several natural families, exhibit rigid periodic structure. We study this structure in an oracle setting, where a marked set of integers is given by black-box access and the goal is to decide whether it encodes a magic square. For $3\times 3$ magic squares and weighted variants, we prove explicit periodic characterizations that reduce detection to period finding. For larger orders, we identify a class of solutions built from repeated arithmetic patterns, which can be detected via the quantum Fourier transform. We then introduce a shifted-oracle method, based on interference between an oracle and its translates, that helps reconstruct solutions in structured cases. Together, these ingredients give a quantum framework for detecting and reconstructing certain magic-square solutions under suitable assumptions. We also derive finite bounds that make some instances exhaustively solvable and obtain Shor-based criteria for certifying non-existence in restricted number-theoretic settings. As an application, we sketch a quantum communication protocol based on an oracle encoding of a large magic-square solution.

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