Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 49 / 2 / c / Solution

First expand the result of the first gate in the computational basis:Using $1+e^{-i\alpha }=2e^{-i\alpha /2}\cos (\alpha /2)$ and $1-e^{-i\alpha }=2i{e}^{-i\alpha /2}\sin (\alpha /2)$, this becomesApply $U(\beta )|0⟩=|+⟩$ and $U(\beta )|1⟩=e^{-i\beta }|-⟩$ to obtainFor example, its quantum measurement in the computational basis has probabilities $\mathbb{P}(k=0)=(1+\sin \alpha \sin \beta )/2$ and $\mathbb{P}(k=1)=(1-\sin \alpha \sin \beta )/2$, obtained by squaring the two basis amplitudes.