Grover diffusion operator

For $|s⟩=H^{\otimes n}|0^n⟩$, the diffusion operator is $2|s⟩⟨s|-I=-H^{\otimes n}{I}_0{H}^{\otimes n}$, where $I_0=I-2|0^n⟩⟨0^n|$.

The diffusion operator sends every computational-basis amplitude $a_x$ to $2\overline{a}-a_x$, where $\overline{a}$ is their arithmetic mean.