On Universality of Quantum Fourier Transform

A methodology is presented to obtain the basis of qudits which are admissible to quantum Fourier transform (QFT) in the sense that the set of such kets are related by the QFT in the same way as the kets of the computational basis. We first study this method for qubits to characterize the ensemble that works for the Hadamard transformation (QFT for two dimension). In this regard we identify certain incompleteness in the result of Maitra and Parashar (Int. J. Quantum Inform. 4 (2006) 653). Next we characterize the ensemble of qutrits for which QFT is possible. Further, some theoretical results related to higher dimensions are also discussed. Considering the unitary matrix U_n related to QFT, the issue boils down to the problem of characterizing matrices that commute with U_n.