Swap test

The Swap test is a procedure in quantum computation that is used to check how much two quantum states differ.

Consider two states: $|\phi \rangle$ and $|\psi \rangle$. The state of the system at the beginning of the protocol is $|0,\phi ,\psi \rangle$. After the Hadamard gate, the state of the system is ${\frac {1}{\sqrt {2}}}(|0,\phi ,\psi \rangle +|1,\phi ,\psi \rangle )$. The controlled SWAP gate transforms the state into ${\frac {1}{\sqrt {2}}}(|0,\phi ,\psi \rangle +|1,\psi ,\phi \rangle )$. The second Hadamard gate results in ${\frac {1}{2}}(|0,\phi ,\psi \rangle +|1,\phi ,\psi \rangle +|0,\psi ,\phi \rangle -|1,\psi ,\phi \rangle )={\frac {1}{2}}|0\rangle (|\phi ,\psi \rangle +|\psi ,\phi \rangle )+{\frac {1}{2}}|1\rangle (|\phi ,\psi \rangle -|\psi ,\phi \rangle )$

The Measurement gate on the first qubit ensures that it's 0 with a probability of

$P({\text{First qubit}}=0)={\frac {1}{2}}{\Big (}\langle \phi |\langle \psi |+\langle \psi |\langle \phi |{\Big )}{\frac {1}{2}}{\Big (}|\phi \rangle |\psi \rangle +|\psi \rangle |\phi \rangle {\Big )}={\frac {1}{2}}+{\frac {1}{2}}{|\langle \psi |\phi \rangle |}^{2}$

when measured. If $\psi$ and $\phi$ are orthogonal $({|\langle \psi |\phi \rangle |}^{2}=0)$, then the probability that 0 is measured is ${\frac {1}{2}}$. If the states are equal $({|\langle \psi |\phi \rangle |}^{2}=1)$, then the probability that 0 is measured is 1.