 Solution

Letwhere $H$ is the Hadamard gate. For a computational-basis string $z$, define the phase reflectionThus $I_0$ changes the sign of $|0^n⟩$, while $I_{x_0}$ changes the sign of the marked state.

Conjugating $I_0$ by $H_n$ reflects in the hyperplane perpendicular to $|s⟩$, sois the Grover diffusion operator, the reflection about $|s⟩$. The Grover iteration operatoris the product of two reflections. It acts as a rotation through $2\theta$ in the plane spanned by the marked state $|x_0⟩$ and the normalized uniform superposition of unmarked states, whereIt is the identity up to sign on the orthogonal complement of that plane.