  The performance of quantum gate operations is experimentally determined by
how correct operational parameters can be determined and set, and how stable
these parameters can be maintained. In addition, gates acting on different sets
of qubits require unique sets of control parameters. Thus, an efficient
multi-dimensional parameter estimation procedure is crucial to calibrate even
medium sized quantum processors. Here, we develop and characterize an efficient
calibration protocol to automatically estimate and adjust experimental
parameters of the widely used Molmer-Sorensen entangling gate operation in a
trapped ion quantum information processor. The protocol exploits Bayesian
parameter estimation methods which includes a stopping criterion based on a
desired gate infidelity. We experimentally demonstrate a median gate infidelity
of $1.3(1)\cdot10^{-3}$, requiring only $1200\pm500$ experimental cycles, while
completing the entire gate calibration procedure in less than one minute. This
approach is applicable to other quantum information processor architectures
with known or sufficiently characterized theoretical models.
