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

Global phase and minimum time of quantum Fourier transform for qudits represented by quadrupole nuclei