A Factorisation Algorithm in Adiabatic Quantum Computation

TL;DR

This paper reformulates integer factorization as an optimization problem using Diophantine polynomials and proposes an adiabatic quantum algorithm to solve it, aiming to improve quantum factorization methods.

Contribution

It introduces a novel optimization-based formulation of factorization and an adiabatic quantum algorithm tailored for this problem.

Findings

Unique solutions for the Diophantine polynomial formulations.

The algorithm distinguishes prime from composite numbers.

Potential for quantum speedup in factorization tasks.

Abstract

The problem of factorising positive integer $N$ into two integer factors $x$ and $y$ is first reformulated as an optimisation problem over the positive integer domain of either of the Diophantine polynomials $Q_N(x,y)=N^2(N-xy)^2 + x(x-y)^2$ or $R_N(x,y) = N^2(N-xy)^2 + (x-y)^2 + x$, of each of which the optimal solution is unique with $x\le \sqrt{N} \le y$, and $x=1$ if and only if $N$ is prime. An algorithm in the context of Adiabatic Quantum Computation is then proposed for the general factorisation problem.