Positive-sign cyclic quantum Fourier transform

For an integer $N\ge 1$, let ${\mathcal{F}}_N$ be the negative-sign, $N^{-1\mathrm{/}2}$-normalized transform of The unitary discrete Fourier transform on $\mathbb{Z}\mathrm{/}N\mathbb{Z}$. Define the cyclic quantum Fourier transform by ${\mathrm{QFT}}_N:={\mathcal{F}}_N^{-1}$. The inverse exists by Finite Fourier inversion for the unitary transform on $\mathbb{Z}\mathrm{/}N\mathbb{Z}$, and its matrix on the standard basis indexed by $0,\dots ,N-1$ is

${\mathrm{QFT}}_N\mathrm{\mid }x⟩=N^{-1\mathrm{/}2}{\sum }_{y=0}^{N-1}{e}^{2\pi ixy\mathrm{/}N}\mathrm{\mid }y⟩,0\le x<N\mathrm{.}$

Here the standard-basis ket convention is the one specified by Dirac Notation Conversion. Thus the transform has the positive sign and the same $N^{-1\mathrm{/}2}$ normalization as the inverse Fourier formula. At $N=1$ it is the identity on the one-dimensional register. In the binary specialization $N=2^n$, the circuit page writes each index in its unique base-two expansion.