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

TL;DR

This paper explores how the global phase affects the energy level layout and minimum implementation time of the quantum Fourier transform gate for qudits represented by quadrupole nuclei, using numerical optimal control methods.

Contribution

It establishes the relationship between global phase and energy level configuration, and analyzes the impact on minimum gate implementation time for various nuclear spins.

Findings

Global phase influences energy level layout of the effective Hamiltonian.

Different global phases can lead to different minimum implementation times.

Gradient algorithms may converge to solutions with varying global phases and times.

Abstract

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.