Quantum state driving along arbitrary trajectories

Starting with the quantum brachistochrone problem of the infinitesimal form, we solve the minimal time and corresponding time-dependent Hamiltonian to drive a pure quantum state with limited resources along arbitrary preassigned trajectories. It is also shown that out of all possible trajectories with limited resources, which are physically accessible and which are not. The solution is then generalized to the mixed quantum state cases and applied to trajectories parametrized by single or multiple parameters with a discrete or continuous spectrum. We then compare the solution to that of the counterdiabatic driving and show how the Berry phase is directly involved in both driving processes.