QUBO formulations for a system of linear equations

With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in modern science and engineering. The Harrow Hassidim-Lloyd algorithm, a monumental quantum algorithm for solving linear systems on gate model quantum computers, was invented and several advanced variations have been developed. The algorithm was difficult to apply to general linear equations because it required various conditions for the matrix. In this paper, we introduce a new algorithm that can be applied to all linear systems. For a given general square matrix $A\in R^{n\times n}$ and a vector $\overrightarrow{b}\in R^n$, we will find quadratic unconstrained binary optimization (QUBO) models for a vector $\overrightarrow{x}\in R^n$ that satisfies $A\overrightarrow{x}=\overrightarrow{b}$. To formulate QUBO models for solving linear systems, we made use of a linear least-square problem with binary representation of the solution. We validated those QUBO models on the d-Wave system and discussed the results. For a simple system, we provide a Python code to calculate the matrix characterizing the relationship between the variables, and to print the test code that can be used directly in the d-Wave system.