Erick R. Castro ^1* Eldues O. Martins ^1,2 Roberto S. Sarthour ¹ Alexandre M. Souza ¹ Ivan S. Oliveira ¹

• ¹ Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, Brazil
• ² Leopoldo Miguez de Mello Research Center, Petrobras (Brazil), Rio de Janeiro, Brazil

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.