Opportunities and challenges in learning how to control high dimensional quantum systems

The control of complex quantum system is of paramount importance for quantum technologies. One common approach for controlling quantum systems is through properly tailored electromagnetic pulses. In this talk, I will consider a more general picture where controls are simply given by tunable parameters that enter in the system evolution. Taking this view allows for analyzing the task of finding optimal control solutions for a wide range of applications, including variational quantum algorithms (VQAs). The ease of finding these control solutions is determined by the structure of the underlying optimization landscape, whose study has a long history in optimal control theory. Decades of numerical and experimental evidence, combined with recent analytical work, suggests that the control landscape structure typically favors finding optimal controls, i.e., iterative procedures tend to converge to global optima, provided that some assumptions on the properties of the control pulses are met. I will begin by revisiting these assumptions within the context of this more general control picture. I will then go on to focus on VQAs, where parameters in a quantum circuit serve as the controls for preparing the ground state of a desired Hamiltonian, and will argue that almost all VQAs converge to the ground state almost surely if the quantum circuit is described by sufficiently many control parameters. I will conclude by discussing the trade-off between obtaining convergence guarantees through overparameterization and the appearance of barren plateaus, where regions in the optimization landscape can be become exponentially flat.