Multiple-input multiple-output (MIMO) is critical for 6G communication, offering improved spectral efficiency and reliability. However, conventional fully digital designs face significant challenges due to high hardware complexity and power consumption. Low-bit MIMO architectures, such as those employing b-bit quantized phase shifters, provide a cost-effective alternative but introduce NP-hard combinatorial problems in the pre- and post-coding design. This paper explores the use of the Quantum Approximate Optimization Algorithm (QAOA) and alternating optimization to address the problem of b-bit quantized phase shifters both at the transmitter and the receiver. We demonstrate that the structure of this quantized beamforming problem aligns naturally with hybrid-classical methods like QAOA, as the phase shifts used in beamforming can be directly mapped to rotation gates in a quantum circuit. Notably, this paper is the first to show that theoretical connection. Then, the Hamiltonian derivation analysis for the b-bit case is presented, which could have applications in different fields, such as integrated sensing and communication, and emerging quantum algorithms such as quantum machine learning. In addition, a warm-start QAOA approach is studied which improves computational efficiency. Numerical results highlight the effectiveness of the proposed methods in achieving an improved quantized beamforming gain over their classical optimization benchmarks from the literature.