Past exam of the introduction to quantum information course of the University of Oxford / 2015 / 3 / a / Solution

Use the conventional phase gate $P_{\phi }=\mathrm{diag}(1,e^{i\phi })$ in computational basis order $0,1$. The single-qubit Hadamard gate and the successive quantum states areThe final Hadamard gate recombines the two components, givingThe leading factor in the last expression is a global phase. The Born rule therefore yieldsAt $\phi =0$ the output is certainly zero, whereas at $\phi =\pi$ it is certainly one. The phase gate changes no intermediate outcome probabilities; the final Hadamard gate converts a relative quantum phase into an observable quantum interference fringe.