 Solution

The two stated formulas show that $Q$ rotates the $(|\beta ⟩,|\alpha ⟩)$ plane by $2\theta$. Starting from $|\xi ⟩$, whose angle from $|\beta ⟩$ is $\theta$, two iterations give angleHence $Q^2|\xi ⟩=|\alpha ⟩$. The reflection about $|\xi ⟩$ is implementable asThus apply $U$ to $|0\cdots 0⟩$, followed by two applications of $-I_{\xi }{I}_A$. This prepares $|\alpha ⟩$ exactly and deterministically.