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Finding the solutions of the equations that describe the dynamics of a givenphysical system is crucial in order to obtain important information about itsevolution. However, by using estimation theory, it is possible to obtain, undercertain limitations, some information on its dynamics. The quantum-speed-limit(QSL) theory was originally used to estimate the shortest time in which aHamiltonian drives an initial state to a final one for a given fidelity. Usingthe QSL theory in a slightly different way, we are able to estimate the runningtime of a given quantum process. For that purpose, we impose the saturation ofthe Anandan-Aharonov bound in a rotating frame of reference where the state ofthe system travels slower than in the original frame (laboratory frame).Through this procedure it is possible to estimate the actual evolution time inthe laboratory frame of reference with good accuracy when compared to previousmethods. Our method is tested successfully to predict the time spent in theevolution of nuclear spins 1/2 and 3/2 in NMR systems. We find that theestimated time according to our method is better than previous approaches by upto four orders of magnitude. One disadvantage of our method is that we need tosolve a number of transcendental equations, which increases with the systemdimension and parameter discretization used to solve such equationsnumerically.

Estimating the time evolution of NMR systems via quantum speed limit-like expression