An excellent introduction and description of Quantum Fourier transforms and applications of this algorithm in quantum computing.
Quantum Computers Will Speed Up the Internet’s Most Important Algorithm. Fast Fourier transforms provide a sandbox for practical quantum computing:
https://spectrum.ieee.org/computing/software/quantum-computers-will-speed-up-the-internets-most-important-algorithm#.X-YCsDWopfw
Quantum circuit for the fast Fourier transform:
https://link.springer.com/article/10.1007/s11128-020-02776-5