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We develop a methodology for performing approximate optimal controlsimulations for quantum systems with multiple interacting degrees of freedom.The quantum dynamics are modeled using the first-order Magnus approximation inthe interaction picture, where the interactions between different degrees offreedom are treated as the perturbation. We present a numerical procedure forimplementing this approximation, which leverages the separability of thezeroth-order time evolution operator and the pairwise nature of commoninteractions for a reduced computational cost. This formulation of thefirst-order Magnus approximation is suitable to be combined with gradient-freemethods for control field optimization; to this end, we adopt a Stochastic HillClimbing algorithm. The associated computational costs are analyzed andcompared with those of the exact simulation in the large N limit. For numericalillustrations, we perform approximate optimal control simulations for systemsof two and three dipole-dipole coupled molecular rotors under the influence ofa global control field. For the two rotor system, we optimize fields for bothorientation and entanglement control objectives. For the three rotor system, weoptimize fields for orienting rotors in the same direction as well as inopposite directions.

physical review. a/physical review, a

Optimal control of interacting quantum systems based on the first-order  Magnus approximation: Application to multiple dipole-dipole coupled molecular  rotors