On a Quantum Algorithm for the Resolution of Systems of Linear Equations

The numerical resolution of systems of linear equations is an important problem which recurs continously in applied sciences. In particular, it represents an indispensable tool in applied mathematics which can be utilized as a foundation to more complicated problems (e.g. optimization problems, partial differential equations, eigenproblems, etc.). In this work, we introduce a solver for systems of linear equations based on quantum mechanics. More specifically, given a system of linear equations we introduce an equivalent optimization problem which objective function defines an electrostatic potential. Then, we evolve a many-body quantum system immersed in this potential and show that the corresponding Wigner quasi-distribution function converges to the global energy minimum. The simulations are performed by using the time-dependent, ab-initio, many-body Wigner Monte Carlo method. Finally, by numerically emulating the (random) process of measurement, we demonstrate that one can extract the solution of the original mathematical problem. As a proof of concept we solve 3 simple, but different, linear systems with increasing complexity. The outcomes clearly show that our suggested approach is a valid quantum algorithm for the resolution of systems of linear equations.