Cyclic shift operator

On a basis indexed by ${\mathbb{Z}}_N$, a cyclic shift operator has the form $U(a)|b⟩=|a+b\mathrm{mod}N⟩$. The quantum Fourier transform simultaneously diagonalizes all such shifts.

For the cyclic shift $S|j⟩=|j+1\mathrm{mod}N⟩$ and the conventionthe Fourier states obeyConsequently $S={\mathrm{QFT}}_{N}D{\mathrm{QFT}}_N^{-1}$, where $D|k⟩=e^{-2\pi ik/N}|k⟩$.