Computing a Sparse Approximate Inverse on Quantum Annealing Machines

TL;DR

This paper investigates using quantum annealing machines to compute sparse approximate inverses for large linear systems, potentially improving preconditioning in iterative solvers like conjugate gradient.

Contribution

It introduces a novel approach to compute SPAI using quantum annealing by solving QUBO problems, demonstrating feasibility on different linear system conditions.

Findings

Quantum annealing can be used to compute SPAI for linear systems.

The method works on well-conditioned and poorly-conditioned systems.

Preconditioning with SPAI improves convergence in iterative methods.

Abstract

Many engineering problems involve solving large linear systems of equations. Conjugate gradient (CG) is one of the most popular iterative methods for solving such systems. However, CG typically requires a good preconditioner to speed up convergence. One such preconditioner is the sparse approximate inverse (SPAI). In this paper, we explore the computation of an SPAI on quantum annealing machines by solving a series of quadratic unconstrained binary optimization (QUBO) problems. Numerical experiments are conducted using both well-conditioned and poorly-conditioned linear systems arising from a 2D finite difference formulation of the Poisson problem.