Distributed inner product estimation with limited quantum communication

We consider the task of distributed inner product estimation when allowed limited quantum communication. Here, Alice and Bob are given $k$ copies of an unknown $n$-qubit quantum states $\mathrm{\mid }\psi ⟩,\mathrm{\mid }\phi ⟩$ respectively. They are allowed to communicate $q$ qubits and unlimited classical communication, and their goal is to estimate $\mathrm{\mid }⟨\psi \mathrm{\mid }\phi ⟩{\mathrm{\mid }}^2$ up to constant accuracy. We show that $k=\mathrm{\Theta }(\sqrt{2^{n-q}})$ copies are essentially necessary and sufficient for this task (extending the work of Anshu, Landau and Liu (STOC’22) who considered the case when $q=0$). Additionally, we consider estimating $\mathrm{\mid }⟨\psi \mathrm{\mid }M\mathrm{\mid }\phi ⟩{\mathrm{\mid }}^2$, for arbitrary Hermitian $M$. For this task we show that certain norms on $M$ characterize the sample complexity of estimating $\mathrm{\mid }⟨\psi \mathrm{\mid }M\mathrm{\mid }\phi ⟩{\mathrm{\mid }}^2$ when using only classical~communication.