# Quantum annealing for systems of polynomial equations

## TL;DR

This paper introduces a quantum annealing-based direct method for solving systems of polynomial equations, demonstrating its application to linear regression and analyzing its scaling behavior and iterative refinement capabilities.

## Contribution

It presents a novel quantum annealing approach for directly solving polynomial systems, validated on real hardware, with analysis of scaling and iterative improvements.

## Key findings

- Successfully solved second-order polynomial equations on a quantum annealer.
- Demonstrated application to linear regression problems.
- Showed iterative annealing can achieve high-precision solutions.

## Abstract

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of $10^{-8}$.