Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 15C / d / iii

This circuit is the swap test. If the two pure states are identical up to a physically irrelevant global phase, then $|⟨\psi |\varphi ⟩|=1$ and outcome one never occurs. If they are distinct, then $|⟨\psi |\varphi ⟩|<1$ and each run produces outcome one with positive probability $p_1$.

Run the circuit independently on fresh copies. Observing even one outcome one proves that the states differ. If no one occurs in $N$ runs, that event has probabilitywhen the states differ, which tends to zero as $N\to \infty$. Repetition therefore distinguishes equality from inequality with arbitrarily high probability.