Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 15D / b

Let $a=\Vert {\Pi }_A\xi \Vert$ and $b=\Vert {\Pi }_{A^{\perp }}\xi \Vert$. Since $a^2+b^2=1$, choose $\theta \in [0,\pi /2]$ with $a=\sin \theta$, $b=\cos \theta$. Normalizing the two nonzero projections givesAt an endpoint, the unused normalized vector may be chosen arbitrarily in its subspace.

Since $I_A|\alpha ⟩=-|\alpha ⟩$,Substituting the decomposition of $|\xi ⟩$ and using the double-angle identities gives

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.