Exploring families of quantum controls for generating unitary transformations

A numerical method is presented for exploring the intertwined roles of the control-field structure and the final time T in determining the unitary evolution operator U(T, 0) for finite-level quantum control systems. The algorithm can (a) identify controls achieving a target unitary operator W at time T up to machine precision and (b) identify a continuous family of controls producing the same operator W over a continuous interval of final times. The high degree of precision is obtained, in part, by exploiting the geometry of the unitary group. In particular, geodesics of the unitary group are followed, both for tracking to a target transformation and for error management.