  Minimum-time quantum control protocols can be obtained from the quantum
brachistochrone formalism [Carlini, Hosoya, Koike, and Okudaira, Phys. Rev.
Lett. 96, 06053, (2006)]. We point out that the original treatment implicitly
applied the variational calculus with fixed boundary conditions. We argue that
the genuine quantum brachistochrone problem involves a variational problem with
a movable endpoint, contrary to the classical brachistochrone problem. This
formulation not only simplifies the derivation of the quantum brachistochrone
equation but introduces an additional constraint at the endpoint due to the
boundary effect. We present the general solution to the full quantum
brachistochrone equation and discuss its main features. Using it, we prove that
the speed of evolution under constraints is reduced with respect to the
unrestricted case. In addition, we find that solving the quantum
brachistochrone equation is closely connected to solving the dynamics of the
Lagrange multipliers, which is in general governed by nonlinear differential
equations. Their numerical integration allows generating time-extremal
trajectories. Furthermore, when the restricted operators form a closed
subalgebra, the Lagrange multipliers become constant and the optimal
Hamiltonian takes a concise form. The new class of analytically solvable models
for the quantum brachistochrone problem opens up the possibility of applying it
to many-body quantum systems, exploring notions related to geometry such as
quantum speed limits, and advancing significantly the quantum state and gate
preparation for quantum information processing.
