 Solution

Choose $\alpha =0$ and prepare $|{\chi }_1⟩$. The circuit from part (iii) returnsso a quantum measurement in the computational basis of the first register reveals the discrete logarithm $\eta$ exactly. More generally, any $\beta$ that is invertible modulo $N$ returns $\beta \eta \mathrm{mod}N$, from which $\eta$ follows by multiplication by the modular inverse of $\beta$.