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

The roles of drift and control field constraints upon quantum control speed limits