Quantum Fourier Transform II

In this chapter, we will first discuss another definition of quantum Fourier transform (QFT). We will discuss why there are two definitions (each is the inverse of the other) and thus gain a deeper understanding of the difference in the transformation of basis and the transformation of vectors. Then we will introduce a many-qubit SWAP gate and construct quantum circuits for a 3-qubit QFT and inverse-QFT (IQFT). We will realize these circuits on IBM-Q. Finally, we will introduce the circuit for an n-qubit QFT and IQFT.