Optimal Quantum Control with Poor Statistics

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. Because of 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 that finds optimal control solutions in the presence of large measurement shot noise and even in the limit of single-shot measurements. The algorithm builds up on Bayesian optimization that is well suited to handle noisy data; but since the commonly used assumption of Gaussian noise is not appropriate for projective measurements with a low number of repetitions, we develop Bayesian inference that respects the binomial nature of shot noise. With several numerical and experimental examples we demonstrate that this method is capable of finding excellent control solutions with minimal experimental effort.