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

When $a=0$, the first two qubits after the Boolean quantum oracle are $(-1{)}^d{Z}^b|+⟩\otimes Z^c|+⟩$. The Hadamard gate obeys $H{Z}^t|+⟩=|t⟩$ for $t\in \{0,1\}$. Applying the Walsh-Hadamard transform therefore givesConsequently the quantum measurement in the computational basis has the deterministic outcomesThese four distinct answers identify the four promised Boolean quantum oracles with one query and certainty. The constant term does not affect the measurement in quantum mechanics, because a global phase does not change its probabilities.