  In this work we derive a lower bound for the minimum time required to
implement a target unitary transformation through a classical time-dependent
field in a closed quantum system. The bound depends on the target gate, the
strength of the internal Hamiltonian and the highest permitted control field
amplitude. These findings reveal some properties of the reachable set of
operations, explicitly analyzed for a single qubit. Moreover, for fully
controllable systems, we identify a lower bound for the time at which all
unitary gates become reachable. We use numerical gate optimization in order to
study the tightness of the obtained bounds. It is shown that in the single
qubit case our analytical findings describe the relationship between the
highest control field amplitude and the minimum evolution time remarkably well.
Finally, we discuss both challenges and ways forward for obtaining tighter
bounds for higher dimensional systems, offering a discussion about the
mathematical form and the physical meaning of the bound.
