Hybrid digital/analog precoding has been widely utilized in millimeter-wave (mmWave) multiple-input multiple-output (MIMO) systems. Compared to the digital precoder, it is more challenging to design the analog precoder because of the constant amplitude (CA) constraints on the entries. Although high-resolution analog phase shifters (APSs) can provide more choices in phase adjustment, it also adds the difficulty in designing effective codebooks for beam selection. In addition, the finer APSs are accompanied by the higher hardware cost and power consumption. Driven by the low hardware cost and power consumption, we employ one-bit APSs to perform analog precoding. A simple codebook for hierarchical searching is designed. Then a low-complexity hierarchical beam selection algorithm is devised to reduce the training overhead. An approximate expression of spectral efficiency after hybrid precoding in one-bit-APS system is also derived, which is close to the simulation results in medium signal-to-noise ratio (SNR). Despite a performance degradation, both the theoretical analysis and the simulation results show that the performance is still comparable to the existing alternatives with much lower training overhead and hardware cost. Therefore, the proposed one-bit-APS system can be an acceptable tradeoff in practical implementation.