Swap Test

The swap test allows us to measure the angle between 2 Qubits $|\psi ⟩$ and $|\varphi ⟩$.

Algorithm

Having $|q_0,q_1,q_2⟩$ initialized to $|0,\psi ,\varphi ⟩$, the above algorithm results in the state $\frac{1}{2}|0⟩\left(|\psi ,\varphi ⟩+|\varphi ,\psi ⟩\right)+\frac{1}{2}|1⟩\left(|\psi ,\varphi ⟩-|\psi ,\varphi ⟩\right)$. As such the probability of measuring $|0⟩$ in the first Qubit is given by

$$\begin{array}{rl}{\left(\frac{1}{2}||\psi ,\varphi ⟩+|\varphi ,\psi ⟩|\right)}^2& =\frac{1}{4}(⟨\psi ,\varphi |+⟨\varphi ,\psi |)(|\psi ,\varphi ⟩+|\varphi ,\psi ⟩)\\ & =\frac{1}{4}(⟨\psi ,\varphi |\psi ,\varphi ⟩+⟨\varphi ,\psi |\psi ,\varphi ⟩+⟨\psi ,\varphi |\varphi ,\psi ⟩+⟨\varphi ,\psi |\varphi ,\psi ⟩)\\ & =\frac{1}{4}(1+2{⟨\varphi |\psi ⟩}^2+1)\\ & =\frac{1}{2}(1+{⟨\varphi |\psi ⟩}^2)\end{array}$$

Hence the swap gate allows us to measure the between two Quantum states.