  Previous research on quantum annealing methods focused on effectively
modeling systems of linear equations by utilizing quadratic unconstrained
binary optimization (QUBO) formulations. These studies take part in enhancing
quantum computing algorithms, which extract properties of quantum computers
suitable for improving classical computational models. In this paper, we
further develop the QUBO formulations of systems of linear equations by
applying Sylvester's law of inertia, which explores matrix congruence of any
real symmetric matrix to a diagonal matrix. We expect that the proposed
algorithm can effectively implement higher dimensional systems of linear
equations on a quantum computer. Further experimental verification of the
proposed QUBO models as well as their comparisons to classical algorithms are
also made.
