Generalizable control for multiparameter quantum metrology

Quantum control can be employed in quantum metrology to improve the precision limit for the estimation of unknown parameters. The optimal control, however, typically depends on the actual values of the parameters and thus needs to be designed adaptively with the updated estimations of those parameters. Traditional methods, such as gradient ascent pulse engineering (GRAPE), need to be rerun for each new set of parameters encountered, making the optimization costly, especially when many parameters are involved. Here we study the generalizability of optimal control, namely, optimal controls that can be systematically updated across a range of parameters with minimal cost. In cases where control channels can completely reverse the shift in the Hamiltonian due to a change in parameters, we provide an analytical method which efficiently generates optimal controls for any parameter starting from an initial optimal control found by either GRAPE or reinforcement learning. When the control channels are restricted, the analytical scheme is invalid, but reinforcement learning still retains a level of generalizability, albeit in a narrower range. In cases where the shift in the Hamiltonian is impossible to decompose to available control channels, no generalizability is found for either the reinforcement learning or the analytical scheme. We argue that the generalization of reinforcement learning is through a mechanism similar to the analytical scheme. Our results provide insights into when and how the optimal control in multiparameter quantum metrology can be generalized, thereby facilitating efficient implementation of optimal quantum estimation of multiple parameters, particularly for an ensemble of systems with ranges of parameters.