A Factorisation Algorithm in Adiabatic Quantum Computation

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.