Estimation of time-optimal control for two-level quantum system with bounded amplitude

A systematic scheme is proposed to numerically estimate the quantum speed limit and temporal shape of time-optimal control. Two quantum state transitions in the two-level quantum system with constrained control field are studied as illustration. Comparisons between numerical results and analytical results are made, and deviations are significantly small. The shape of optimized control field is simple and does not switch frequently, thus are easy to implement in experiment. Our scheme is of importance in estimating the quantum speed limit and time-optimal controls in cases in which analytical solution is absent