Optimizing antenna beamforming with quantum computing

We consider beamforming with largescale antenna arrays in which the elements can transmit only in one of a small number of phase-shifts. This creates an NP-hard optimization problem, namely the maximization of the ratio of two Hermitian quadratic forms, with the state vector constrained to the set of allowed phase-shifts. We show how the maximization problem can be rigorously solved by reformulating it as a sequence of quadratic (but non-convex) minimization problems. These minimization problems can be solved exactly with integer linear programming when sufficiently small. When they are large there is no good classical solution method, but we show that they can be solved using quantum computers of the annealing type. Not only is there a large improvement in solution time with quantum computing, there is also the potential for energy saving.