Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 324 / 4 / b / Solution

Write $|\psi ⟩=\alpha |0⟩+\beta |1⟩$ and $t=e^{i\theta }$. Before measurement, the three controlled-NOT gates map a computational-basis component $|x,y,0⟩$ toThe measured bit is therefore $b=y\oplus x$. For $b=0$, the unnormalized state of the first qubit isFor $b=1$, it isEach branch has probability $1/2$; after normalization and removal of the irrelevant global phase $t$, the two outputs arewith equal probability. This is probabilistic phase-gate injection.