# Box algorithm for the solution of differential equations on a quantum annealer

## TL;DR

This paper presents the 'box algorithm', a novel quantum annealer-based method for solving differential equations by reformulating finite element models as Ising Hamiltonians, demonstrated on a truss mechanics problem.

## Contribution

It introduces a graph coloring based iterative approach to handle differential quantities in quantum annealing solutions of differential equations.

## Key findings

- Successfully solved a truss mechanics problem on a D-Wave quantum computer.
- Demonstrated the feasibility of the box algorithm for differential equations.
- Addressed challenges of defining gradients in Ising models.

## Abstract

Differential equations are ubiquitous in models of physical phenomena. Applications like steady-state analysis of heat flow and deflection in elastic bars often admit to a second order differential equation. In this paper, we discuss the use of a quantum annealer to solve such differential equations by recasting a finite element model in the form of an Ising hamiltonian. The discrete variables involved in the Ising model introduce complications when defining differential quantities, for instance, gradients involved in scientific computations of solid and fluid mechanics. To address this issue, a graph coloring based methodology is proposed which searches iteratively for solutions in a subspace of weak solutions defined over a graph, hereafter called as the 'box algorithm.' The box algorithm is demonstrated by solving a truss mechanics problem on the D-Wave quantum computer.