An upper bound on the time required to implement unitary operations

We derive an upper bound for the time needed to implement a generic unitary transformation in a d dimensional quantum system using d control fields. We show that given the ability to control the diagonal elements of the Hamiltonian, which allows for implementing any unitary transformation under the premise of controllability, the time T needed is upper bounded by where g_min is the smallest coupling constant present in the system. We study the tightness of the bound by numerically investigating randomly generated systems, with specific focus on a system consisting of d energy levels that interact in a tight-binding like manner.