Realization of quantum Fourier transform over ZN

Since the difficulty in preparing the equal superposition state of amplitude is 1/root N, we construct a quantile transform of quantum Fourier transform (QFT) over Z(N) based on the elementary transforms, such as Hadamard transform and Pauli transform. The QFT over Z(N) can then be realized by the quantile transform, and used to further design its quantum circuit and analyze the requirements for the quantum register and quantum gates. However, the transform needs considerable quantum computational resources and it is difficult to construct a high-dimensional quantum register. Hence, we investigate the design of t-bit quantile transform, and introduce the definition of t-bit semiclassical QFT over Z(N). According to probability amplitude, we prove that the transform can be used to realize QFT over Z(N) and further design its quantum circuit. For this transform, the requirements for the quantum register, the one-qubit gate, and two-qubit gate reduce obviously when compared with those for the QFT over Z(N).