  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.
