Optimal multi-parameter quantum metrology for frequencies of magnetic field

Multi-parameter quantum estimation has attracted considerable attention due to its broad applications. Due to the complexity of quantum dynamics, existing research places significant emphasis on estimating parameters in time-independent Hamiltonians. Here, our work makes an effort to explore multi-parameter estimation with time-dependent Hamiltonians. In particular, we focus on the discrimination of two close frequencies of a magnetic field by using a single qubit. We optimize the quantum controls by employing both traditional optimization methods and reinforcement learning to improve the precision for estimating the frequencies of the two magnetic fields. In addition to the estimation precision, we also evaluate the robustness of the optimization schemes against the shift of the control parameters. The results demonstrate that the hybrid reinforcement learning approach achieves the highest estimation precision, and exhibits superior robustness. Moreover, a fundamental challenge in multi-parameter quantum estimation stems from the incompatibility of the optimal control strategies for different parameters. We demonstrate that the hybrid control strategies derived through numerical optimization remain effective in enhancing the precision of multi-parameter estimation in spite of the incompatibilities, thereby mitigating incompatibilities between control strategies on the estimation precision. Finally, we investigate the trade-offs in estimation precision among different parameters for different scenarios, revealing the inherent challenges in balancing the optimization of multiple parameters simultaneously and providing insights into the fundamental distinction between quantum single-parameter estimation and multi-parameter estimation.