Fourier transform has taken place in different areas and applications, in this paper has been revised an important new form of application in the paradigm of quantum computing. Quantum Fourier transforms have gained increased attention with the rise of quantum walks, boson sampling, and quantum metrology [2]. In the Shor’s Algorithm it is used for find discrete logarithms on a quantum computer with two modular exponentiations and two quantum Fourier transforms [1]. Our propose consist in show a new quantum gate that can perform the Fractional Fourier Transform defined by Namias as a tool to solve the differential equation of the quantum mechanical oscillator [3], which it can satisfy the condition of to be unitary. We apply the quantum gate in different states through of a real quantum computer offered by IBM for check that it carry out the transform successfully, it results were compared with results of Quantum Fourier Transform for can understand its application. In the geometric point of view showed in the figure (2a) and (2c) through of Bloch Sphere, we can see the difference between the application of Quantum Fourier Transform (QFT) and Quantum Fractional Fourier Transform (with our quantum gate) to the basis state |0〉, while in the figure (2b) and (2d) is possible see with a theoretical result that probability density is equal for both.
The quantum Fourier transform for discrete variable (dvQFT) is an efficient algorithm for several applications. It is usually considered for the processing of quantum bits (qubits) and its efficient implementation is obtained with two elementary components: the Hadamard gate and the controlled--phase gate. In this paper, the quantum Fourier transform operating with continuous variables (cvQFT) is considered. Thus, the environment becomes the Hilbert space, where the natural definition of the cvQFT will be related to rotation operators, which in the
N--mode are completely specified by unitary matrices of order
N. Then the cvQFT is defined as the rotation operator whose rotation matrix is given by the discrete Fourier transform (DFT) matrix. For the implementation of rotation operators with primitive components (single--mode rotations and beam splitters), we follow the well known Murnaghan procedure, with appropriate modifications. Moreover, algorithms related to the fast Fourier transform (FFT) are applied to reduce drastically the implementation complexity. The final part is concerned with the application of the cvQFT to general Gaussian states. In particular, we show that cvQFT has the simple effect of transforming the displacement vector by a one-dimensional DFT, the squeeze matrix by a two-dimensional DFT, and the rotation matrix by a Fourier-like similarity transform.