A Quantum Diophantine Equation Solution Finder

TL;DR

This paper proposes a quantum algorithm leveraging Grover's search to efficiently find solutions to Diophantine equations, demonstrating potential for faster solution discovery compared to classical brute force methods.

Contribution

It introduces a novel quantum approach using Grover's algorithm to solve Diophantine equations more efficiently than traditional brute force techniques.

Findings

Demonstrates quantum search applied to Diophantine equations

Shows potential for exponential speedup over classical methods

Provides a proof-of-concept example for the simplest case

Abstract

Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer solutions and check for equality. Grover's algorithm is a quantum search algorithm which can find marked indices in a list very efficiently. By treating the indices as the integer variables in the diophantine equation, Grover's algorithm can be used to find solutions in brute force way more efficiently than classical methods. We present an example for the simplest possible diophantine equation.