Cyclic shift diagonalization by the quantum Fourier transform
= Cyclic shift diagonalization by the quantum Fourier transform
For the cyclic shift $S|j\rangle=|j+1\bmod N\rangle$ and the convention
$$
\operatorname{QFT}_N|k\rangle=N^{-1/2}\sum_j e^{2\pi ijk/N}|j\rangle,
$$
the Fourier states obey
$$
S\operatorname{QFT}_N|k\rangle=e^{-2\pi ik/N}\operatorname{QFT}_N|k\rangle.
$$
Consequently $S=\operatorname{QFT}_N D\operatorname{QFT}_N^{-1}$, where $D|k\rangle=e^{-2\pi ik/N}|k\rangle$.