Modular-addition quantum oracle ($U_h$)

For $h:{\mathbb{Z}}_N\to {\mathbb{Z}}_M$, its modular-addition oracle is the unitary operatorFor each fixed $x$, addition by $h(x)$ permutes the answer register's computational basis. This preserves inner products, even when $h$ is not injective.

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 $S_M^2=I$.