An information theoretical analysis of quantum optimal control

We show that if an efficient classical representation of the dynamics exists, optimal control problems on many-body quantum systems can be solved efficiently with finite precision. We show that the size of the space of parameters necessary to solve quantum optimal control problems defined on pure, mixed states and unitaries is polynomially bounded from the size of the of the set of reachable states in polynomial time. We provide a bound for the minimal time necessary to perform the optimal process given the bandwidth of the control pulse, that is the continuous version of the Solovay-Kitaev theorem. We explore the connection between entanglement present in the system and complexity of the control problem, showing that one-dimensional slightly entangled dynamics can be efficiently controlled. Finally, we quantify how noise affects the presented results.