Modular multiplication eigenstates ($|u_s⟩=r^{-1/2}{\sum }_{j=0}^{r-1}{e}^{-2\pi isj/r}|x^j\mathrm{mod}n⟩$)

For $\gcd (x,n)=1$ and order $r$, multiplication by $x$ modulo $n$ is a unitary operator. On the orbit of one it cyclically permutes $r$ basis states. The displayed orthonormal eigenstates have eigenvalues $e^{2\pi is/r}$, and $|1⟩=r^{-1/2}{\sum }_s|u_s⟩$. Thus quantum phase estimation on this efficiently prepared state samples a uniform $s$ without knowing $r$. Controlled powers are implemented by quantum modular exponentiation. Sufficient phase resolution allows continued-fraction recovery in quantum order finding; the reduced denominator equals $r$ only when $s$ is coprime to $r$.