Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 15D / d / Solution

Let $X|x⟩=|x+1⟩$ on the input register and let $N|y⟩=|-y⟩$ on the target register. Both are fixed unitary operators independent of $f$. Conjugating the query by the input shift givesbecause the intermediate query is made at $x+1$. Conjugating by target negation givessince $y\mapsto -y\mapsto -y+f(x)\mapsto y-f(x)$. Each construction uses $U_f$ exactly once.