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

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 the “box algorithm.” The box algorithm is demonstrated by solving a truss mechanics problem on a D-Wave quantum computer.

Figure 1

Illustration of the procedure for solving a differential equation.

Figure 2

Connectivity of (a) nodal graph and (b) element graph.

Figure 3

Self-interaction and site-site interaction parameters for nodal subgraph.

Figure 4

Assembled graph for a domain discretized with (a) one element and (b) four elements.

Figure 5

Approximation of $\sqrt{x}$ function using boxed domain: (a) exact fit with a slack of 0.2 (loose fit), (b) exact fit with a slack of 0.02 (tight fit), (c) inexact fit but bounded in a box size of 0.2, and (d) inexact fit and unbounded by a slack of 0.2.

Figure 6

An illustration of a two-node approximation in $a_1-a_2$ vector space with contour plot of the functional, ${\mathrm{\Pi }}_2(a_1,a_2)$, and a representative box with center at ${\mathbf{u}}^{\mathbf{c}}$ and a box size of $r$.

Figure 7

Axial deformation of a bar with (a) a discontinuous cross section with a tip displacement and (b) a continuously varying cross section with a body force and a tip displacement.