Increasing the dimension of linear systems solved by classical or quantum binary optimization: A new method to solve large linear equation systems

Recently, binary optimization has become an attractive research topic due to the development of quantum computing and specialized classical systems inspired by quantum computing. These hardware systems promise to speed up the computation significantly. In this work, we propose a new method to solve linear systems written as a binary optimization problem. The procedure solves the problem efficiently and allows it to handle large linear systems. Our approach is founded on the geometry of the original linear problem and resembles the gradient conjugate method. The conjugated directions used can significantly improve the algorithm's convergence rate. We also show that a partial knowledge of the intrinsic geometry of the problem can divide the original problem into independent sub-problems of smaller dimensions. These sub-problems can then be solved using quantum or classical solvers. Although determining the geometry of the problem has an additional computational cost, it can substantially improve the performance of our method compared to previous implementations.