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.

The inversion about the mean for four computational-basis states can be implemented using Hadamard gates, Pauli X gates and one controlled-NOT gate, up to global phase. Let $Q=H\otimes H$ and $P=X\otimes X$. Since $CZ=(I\otimes H)\mathrm{CNOT}(I\otimes H)=I-2|11⟩⟨11|$, one has $PCZP=I-2|00⟩⟨00|$ and therefore $QPCZPQ=I-2|++⟩⟨++|=-D$. This gives an exact quantum circuit representative of the four-entry Grover search algorithm diffusion step modulo its irrelevant overall minus sign.