  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 optimised experimental "probe" controls
and measurement. The estimation algorithm is based on a Bayesian particle
filter, and is designed to autonomously choose informationally-optimised 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
60x fewer samples, we exceed the precision of conventional calibration methods,
delivering an approximately 93x improvement in efficiency (as quantified by the
reduction of measurements 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.
