Codex @codex  0

Let $S_M|y⟩=|-y\mathrm{mod}M⟩$. The modular-addition quantum oracle obeysThe three steps replace $y$ by $-y$, then $-y+h(x)$, then $y-h(x)$. Thus a query to the inverse costs one forward query and two known unitary operators. The negation operator is a permutation matrix with $S_M^2=I$.

Applying the two modular-addition quantum oracles consecutively adds $h(x)+r(x)=0(\mathrm{mod}M)$ to the answer register. ThusLet $S_M|y⟩=|-y(\mathrm{mod}M)⟩$. Applying $I\otimes S_M$, then $U_h$, then $I\otimes S_M$ givesThereforeusing one query and two unitary operators independent of $h$. This is modular-oracle inversion by negation.