We demonstrate an efficient algorithm for inverse problems in time-dependent quantum dynamics based on feedback loops between Hamiltonian parameters and the solutions of the Schr{o}dinger equation. Our approach formulates the inverse problem as a target vector estimation problem and uses Bayesian surrogate models of the Schr{o}dinger equation solutions to direct the optimization of feedback loops. For the surrogate models, we use Gaussian processes with vector outputs and composite kernels built by an iterative algorithm with Bayesian information criterion (BIC) as a kernel selection metric. The outputs of the Gaussian processes are designed to model an observable simultaneously at different time instances. We show that the use of Gaussian processes with vector outputs and the BIC-directed kernel construction reduce the number of iterations in the feedback loops by, at least, a factor of 3. We also demonstrate an application of Bayesian optimization for inverse problems with noisy data. To demonstrate the algorithm, we consider the orientation and alignment of polyatomic molecules SO$_2$ and chiral propylene oxide (PPO) induced by strong laser pulses. We use simulated time evolutions of the orientation or alignment signals to determine the relevant components of the molecular polarizability tensors to within 1% accuracy. We show that, for the five independent components of the polarizability tensor of PPO, this can be achieved with as few as 30 quantum dynamics calculations.
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