Employing typicality in optimal control theory: Addressing large Hilbert spaces

Controlling the dynamics of quantum systems is a crucial task in quantum science and technology. Obtaining the driving field that transforms the quantum systems into its objective is a typical control task. This task is hard, scaling unfavorably with the size of the quantum map. To tackle this issue we employ typicality to assist in finding the control field for such transformation. To demonstrate the method we choose the control task of cooling the fine-structure states of the aluminum monofluoride molecule, at relatively high temperature. As a result high-rotational states are occupied, meaning a high effective Hilbert space. Using quantum typicality, we demonstrate that we can simulate an ensemble of states, enabling a control task addressing simultaneously many states. We employ this method to find a control field for cooling molecules with a large number of internal sates, corresponding to high initial temperatures.