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We present a general theoretical framework for finding the time-optimalunitary evolution of the quantum systems when the Hamiltonian is subject toarbitrary constraints. Quantum brachistochrone (QB) is such a framework basedon the variational principle, whose drawback is that it deals with equalityconstraints only. While inequality constraints can be reduced to equality onesin some situations, there are situations where they cannot, especially when adrift field is present in the Hamiltonian. The drift which we cannot controlappears in a wide range of systems. We first develop a framework based onPontryagin's maximum principle (MP) in order to deal with inequalityconstraints as well. The new framework contains QB as a special case, and theirdetailed correspondence is given. Second, using this framework, we discussgeneral relations among the drift, the singular controls, and the inequalityconstraints. The singular controls are those that satisfy MP trivially so as tocause a trouble in determining the optimal protocol. Third, to overcome thisissue, we derive an additional necessary condition for a singular protocol tobe optimal by applying the generalized Legendre-Clebsch condition. Thiscondition in particular reveals the physical meaning of singular controls.Finally, we demonstrate how our framework and results work in some examples.

A general formulation of time-optimal quantum control and optimality of singular protocols