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

Prepare $|\xi ⟩=Q^{-1/2}{\sum }_b|b⟩$ in the second register by part (ii), while preserving the first register $|a⟩$. Applying the diagonal unitary $V$ from part (iii) givesThe preparation can be made coherent using amplitude amplification in place of postselection when this map is needed as a subroutine.