Quantum computing with qudits, quantum systems with
d>2 levels, offers a powerful extension beyond qubits, expanding the computational possibilities of quantum systems, allowing the simplification of the implementation of several algorithms and, possibly, providing a foundation for optimised error correction. In this work, we propose a novel formulation of qudit gates that is universally applicable for any number of levels
d, without restrictions on the dimensionality. By extending the mathematical framework of quantum gates to arbitrary dimensions, we derive explicit gate operations that form a universal set for quantum computation on qudits of any size. We demonstrate the validity of our approach through the implementation of the Quantum Fourier Transform (QFT) for arbitrary
d, verifying both the correctness and utility of our generalized gates. This novel methodology broadens the design space for quantum algorithms and fault-tolerant architectures, paving the way for advancements in qudit-based quantum computing.
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.