Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 324 / 1 / b / i / Solution

Put $P_1={\sigma }_1\otimes {\sigma }_2$ and $|\psi ⟩=|A{⟩}^{\otimes 2}$. After the first Hadamard gate and the two controlled Pauli gates, the joint state isThe final Hadamard gate changes this toConditioned on ancilla outcome $s\in \{0,1\}$, the normalized data state is therefore

Write the input in the two eigenspaces of $P_1$ aswhere the displayed eigenstates are normalized. A direct PBC measurement giveswith probabilities $|\alpha {|}^2$ and $|\beta {|}^2$. Hence the ancilla circuit and the Pauli measurement have identical outcome distributions and conditional data states after identifying the Pauli outcome with $(-1{)}^s$.