Identifying and calibrating quantitative dynamical models for physical quantum systems is important for a variety of applications. Here we present a closed-loop Bayesian learning algorithm for estimating multiple unknown parameters in a dynamical model, using optimized experimental "probe" controls and measurement. The estimation algorithm is based on a Bayesian particle filter, and is designed to autonomously choose informationally optimized probe experiments with which to compare to model predictions. We demonstrate the performance of the algorithm in both simulated calibration tasks and in an experimental single-qubit ion-trap system. Experimentally, we find that, with 60 times fewer samples, we exceed the precision of conventional calibration methods, delivering an approximately 93 times improvement in efficiency (as quantified by the reduction of measurements and resets required to achieve a target residual uncertainty and multiplied by the increase in accuracy). In simulated and experimental demonstrations, we see that successively longer pulses are selected as the posterior uncertainty iteratively decreases, leading to an exponential improvement in the accuracy of model parameters with the number of experimental queries, and a commensurate increase in the per-query time.
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