  Using the circulant symmetry of a Hamiltonian describing three qubits, we
realize the quantum Fourier transform. This symmetry allows us to construct a
set of eigenvectors independently on the magnitude of physical parameters
involved in the Hamiltonian and as a result, the entanglement will be
maintained. The realization will be leaned on trapped ions and the gate
implementation requires an adiabatic transition from each spin product state to
Fourier modes. The fidelity was numerically calculated and the results show
important values. Finally, we discuss the acceleration of the gate by using the
counter-driving field.
