Inner product estimation with limited quantum communication

In this work, we consider the following task: suppose Alice and Bob are given $k$ copies of an unknown $n$-qubit quantum state $\ket{\psi},\ket{\phi}$ respectively, are allowed to send $q$ qubits to one another, and the task is to estimate $|\langle \psi|\phi\rangle|^2$ upto constant accuracy. We show that $k=\sqrt{2^{n-q}}$ copies are essentially necessary and sufficient. Additionally, we also consider the task when the goal is to estimate $|\langle \psi|M|\rangle|^2$ and give tight sample complexity bounds for this task.