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 \(T\leq \frac{\pi d^{2}(d-1)}{2g_{\text{min}}}\) where \(g_{\text{min}}\) 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.