Distributed inner product estimation with limited quantum communication

TL;DR

This paper investigates the quantum communication limits for estimating the inner product of distributed quantum states, establishing the necessary number of copies and communication resources for accurate estimation.

Contribution

It extends previous work by characterizing the sample complexity of inner product estimation with limited quantum communication, including generalizations to Hermitian operators.

Findings

Sample complexity is   for estimating inner products.

Quantum communication cost influences the number of state copies needed.

Norms on Hermitian operators determine the complexity of estimating  with classical communication.

Abstract

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 $\vert \psi \rangle,\vert \phi \rangle$ respectively. They are allowed to communicate $q$ qubits and unlimited classical communication, and their goal is to estimate $|\langle \psi|\phi\rangle|^2$ up to constant accuracy. We show that $k=\Theta(\sqrt{2^{n-q}})$ 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 $|\langle \psi|M|\phi\rangle|^2$, for arbitrary Hermitian $M$. For this task we show that certain norms on $M$ characterize the sample complexity of estimating $|\langle \psi|M|\phi\rangle|^2$ when using only classical~communication.