Polynomial-time solvability of the b-bit quantized pre/post-coding MIMO beamforming problem

Determine whether there exists a polynomial-time algorithm guaranteed to find the global optimum of the b-bit quantized pre-coding and post-coding design problem in multiple-input multiple-output (MIMO) systems, namely maximizing |g^H H f|^2 subject to f ∈ S^{N_T×1} and g ∈ S^{N_R×1}, where S is the set of b-bit phase-shift values and H is the complex channel matrix.

Background

The paper formulates the beamforming design for MIMO systems with b-bit phase shifters at both transmitter and receiver as maximizing the received SNR |g^H H f|^2 over quantized pre- and post-coding vectors. Because f and g are drawn from a finite set of b-bit phase values, the problem is combinatorial.

The authors emphasize that this optimization is NP-hard, and they explicitly state that no polynomial-time algorithm is known to guarantee the global optimum for this discrete optimization problem. This motivates their exploration of QAOA and alternating optimization as scalable heuristic approaches.