  Control of quantum systems is a central element of high-precision experiments
and the development of quantum technological applications. Control pulses that
are typically temporally or spatially modulated are often designed based on
theoretical simulations. As we gain control over larger and more complex
quantum systems, however, we reach the limitations of our capabilities of
theoretical modeling and simulations, and learning how to control a quantum
system based exclusively on experimental data can help us to exceed those
limitations. Due to the intrinsic probabilistic nature of quantum mechanics, it
is fundamentally necessary to repeat measurements on individual quantum systems
many times in order to estimate the expectation value of an observable with
good accuracy. Control algorithms requiring accurate data can thus imply an
experimental effort that negates the benefits of avoiding theoretical modeling.
We present a control algorithm based on Bayesian optimization that finds
optimal control solutions in the presence of large measurement shot noise and
even in the limit of single-shot measurements. With several numerical and
experimental examples we demonstrate that this method is capable of finding
excellent control solutions with minimal experimental effort
