Polynomial-time solvability of b-bit quantized MIMO beamforming optimization

Determine whether a polynomial-time algorithm exists that is guaranteed to obtain the global optimum of the received SNR objective for b-bit quantized MIMO beamforming, specifically maximizing |g† H f|^2 over pre-coding and post-coding vectors f ∈ S^{N_T} and g ∈ S^{N_R}, where S is the set of available b-bit phase shifts.

Background

The paper formulates the MIMO beamforming design with b-bit quantized phase shifters at both transmitter and receiver, where the objective is to maximize the received SNR ρ(g, f) = |g† H f|^2 under discrete phase constraints. The authors emphasize that the problem is combinatorial and NP-hard due to the discrete nature of the pre/post-coding vectors.

While the paper proposes QAOA-based heuristic solutions, it explicitly notes that no polynomial-time algorithm is known to guarantee global optimality for this discrete optimization problem. Establishing polynomial-time solvability (or proving impossibility under standard complexity assumptions) would fundamentally impact algorithmic design for quantized MIMO beamforming.