Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 51 / 3 / b

Write the normalized input quantum state as $|\psi ⟩=a|0⟩+b|1⟩$. The fresh quantum ancilla is $|+⟩$, so the Controlled-Z gate createsApplying the measurement bra $⟨v_k(\theta )|$ to the first qubit leaves the unnormalized second-qubit vectorSince $|+⟩,|-⟩$ form an orthonormal basis,Also $U(\theta )|\psi ⟩=a|+⟩+e^{-i\theta }b|-⟩$, while the Pauli X gate fixes $|+⟩$ and changes the sign of $|-⟩$. Dividing by the branch norm therefore givesThis is one-bit teleportation with the present positive-exponent measurement vectors; the transferred logical gate is $U(\theta )=J(-\theta )$.