Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 51 / 1 / c / Solution

All four promised functions have $a=1$. Use quadratic Boolean phase cancellation: apply a Controlled-Z gate to remove the phase $(-1{)}^{xy}$, then apply the Walsh-Hadamard transform. With the rightmost gate acting first, chooseBecause $C_Z^2=I$ and the Controlled-Z gate commutes with the Pauli Z gates,The quantum measurement in the computational basis consequently producesEach answer occurs with probability one for its promised Boolean quantum oracle.