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Recently, binary optimization has become an attractive research topic due tothe development of quantum computing and specialized classical systems inspiredby quantum computing. These hardware systems promise to speed up thecomputation significantly. In this work, we propose a new method to solvelinear systems written as a binary optimization problem. The procedure solvesthe problem efficiently and allows it to handle large linear systems. Ourapproach is founded on the geometry of the original linear problem andresembles the gradient conjugate method. The conjugated directions used cansignificantly improve the algorithm's convergence rate. We also show that apartial knowledge of the intrinsic geometry of the problem can divide theoriginal problem into independent sub-problems of smaller dimensions. Thesesub-problems can then be solved using quantum or classical solvers. Althoughdetermining the geometry of the problem has an additional computational cost,it can substantially improve the performance of our method compared to previousimplementations.

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