Measuring overlap of two quantum states

Question

I have a qubit state $|\psi\rangle = a|0\rangle + b|1\rangle$, and another qubit state $|\psi'\rangle = a'|0\rangle + b'| 1 \rangle$. I want, through quantum gates, to measure the overlap $|\langle \psi | \psi'\rangle|^2$ which is given by a projective measurement $\text{tr}\left[|\psi\rangle\langle \psi||\psi'\rangle\langle\psi'|\right]$. I want to know how to implement this measurement using quantum gates. Also, how would one generalise the measurement to a multi qubit state?

Answer 1

You are looking for the SWAP test.

The basic idea is the following: First, the overlap $\omega=\mathrm{tr}[\rho\sigma]$ of two quantum states (e.g. the pure states in your question) is equal to $$ \omega=\mathrm{tr}[(\rho\otimes\sigma)\mathbb F]\ , $$ with $\mathbb F$ the SWAP gate.

On the other hand, the expectation value of any unitary gate $U$ in a state $\rho$, $\mathrm{tr}[\rho U]$, can be estimated using phase estimation. The simplest version is to apply a controlled-U gate, where the control qubit is initialized in $|+\rangle$ and subsequently measured in the $|\pm\rangle$ basis. It is an easy exercise to see that the probabilities of the two measurement outcomes are directly related to $\omega$.