Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 324 / 3 / c / Solution

Start with $(|a⟩+|b⟩)/\sqrt{2}$ and a zeroed second register. Apply the coherent construction from part (a)(iv) to obtainNow run the inverse of the phase-estimation map from part (b) on the two registers. It erases the first label in both branches:Discarding the zeroed register leaves the required quantum Fourier transform of the superposition. The coherent use of a computed label followed by its inverse is an uncomputation.