Ising Machines for Diophantine Problems in Physics

TL;DR

This paper introduces methods using Ising machines to solve complex Diophantine problems in physics, demonstrating their effectiveness on number theory and anomaly cancellation tasks, including discovering new Taxicab numbers.

Contribution

It presents a novel approach to solving Diophantine problems in physics using Ising models on annealers, applicable to both quantum and classical annealing methods.

Findings

Successfully solved Taxicab number problems, discovering new ones.

Applied the method to anomaly cancellation in $U(1)$ extensions of the Standard Model.

Demonstrated the effectiveness of Ising machines for high-order Diophantine problems.

Abstract

Diophantine problems arise frequently in physics, in for example anomaly cancellation conditions, string consistency conditions and so forth. We present methods to solve such problems to high order on annealers that are based on the quadratic Ising Model. This is the intrinsic framework for both quantum annealing and for common forms of classical simulated annealing. We demonstrate the method on so-called Taxicab numbers (discovering some apparently new ones), and on the realistic problem of anomaly cancellation in $U(1)$ extensions of the Standard Model.