Quantum Algorithms for Magic Square Diophantine Equations

TL;DR

This paper develops quantum algorithms to detect and reconstruct solutions to magic square Diophantine equations using period finding and Fourier transforms, with applications in quantum communication.

Contribution

It introduces a quantum framework for identifying and reconstructing magic square solutions, including new methods like shifted-oracle techniques and explicit bounds.

Findings

Explicit periodic characterizations for 3x3 magic squares.

Quantum algorithms reduce detection to period finding.

Finite bounds enable exhaustive solutions in some cases.

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.