  We demonstrate the relation between a global phase of the quantum gate and
the layout of energy levels of its effective Hamiltonian required for
implementing the gate for minimum time. By an example of the quantum Fourier
transform gate for a qudit represented by a quadrupole nucleus with the spin I
= 1, the effective Hamiltonians and minimum implementation times for different
global phases are found. Using numerical optimal control methods, the problem
of the global phase in searching for the optimal pulse shape is considered in
detail for the quantum Fourier transform gate at I = 1, 3/2, 2, and 5/2. It is
shown that at the constrained control time the gradient algorithms can converge
to the solutions corresponding to different global phases or the same global
phase with different minimum times of the gate implementation.
