Improving the convergence of an iterative algorithm for solving arbitrary linear equation systems using classical or quantum binary optimization

TL;DR

This paper introduces a novel binary optimization-based method for solving linear systems, enhancing convergence especially for large condition numbers by leveraging problem geometry and quantum or classical solvers.

Contribution

It proposes a new approach that transforms linear systems into binary optimization problems, improving convergence rates and enabling decomposition into smaller sub-problems for efficient solving.

Findings

Accelerates convergence for large condition number systems.

Enables problem decomposition into smaller sub-problems.

Demonstrates performance gains over existing methods.

Abstract

Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this work, we propose a novel method for solving linear systems. Our approach leverages binary optimization, making it particularly well-suited for problems with large condition numbers. We transform the linear system into a binary optimization problem, drawing inspiration from the geometry of the original problem and resembling the conjugate gradient method. This approach employs conjugate directions that significantly accelerate the algorithm's convergence rate. Furthermore, we demonstrate that by leveraging partial knowledge of the problem's intrinsic geometry, we can decompose the original problem into smaller, independent sub-problems. These sub-problems can be efficiently tackled using either quantum or classical solvers. While determining the problem's geometry introduces some additional computational cost, this investment is outweighed by the substantial performance gains compared to existing methods.