  A common way to manipulate a quantum system, for example spins or artificial
atoms, is to use properly tailored control pulses. In order to accomplish
quantum information tasks before coherence is lost, it is crucial to implement
the control in the shortest possible time. Here we report the near time-optimal
preparation of a Bell state with fidelity higher than $99\%$ in an NMR
experiment, which is feasible by combining the synergistic capabilities of
modelling and experiments operating in tandem. The pulses preparing the Bell
state are found by experiments that are recursively assisted with a
gradient-based optimization algorithm working with a model. Thus, we explore
the interplay between model-based numerical optimal design and
experimental-based learning control. Utilizing the balanced synergism between
the dual approaches should have broad applications for accelerating the search
for optimal quantum controls.
