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Distributed inner product estimation with limited quantum communication

Srinivasan Arunachalam, Louis Schatzki

TL;DR

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

Distributed inner product estimation with limited quantum communication

TL;DR

It is shown that certain norms on characterize the sample complexity of estimating the sample complexity of estimating|\langle \psi|M|\phi\rangle|^2$ when using only classical~communication.

Abstract

We consider the task of distributed inner product estimation when allowed limited quantum communication. Here, Alice and Bob are given copies of an unknown -qubit quantum states respectively. They are allowed to communicate qubits and unlimited classical communication, and their goal is to estimate up to constant accuracy. We show that copies are essentially necessary and sufficient for this task (extending the work of Anshu, Landau and Liu (STOC'22) who considered the case when ). Additionally, we consider estimating , for arbitrary Hermitian . For this task we show that certain norms on characterize the sample complexity of estimating when using only classical~communication.

Paper Structure

This paper contains 19 sections, 21 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Proof sketch.
  4. Related works and open questions
  5. A natural open question.
  6. Acknowledgements.
  7. Preliminaries
  8. Quantum States and Measurements
  9. Subroutines
  10. Estimating Inner Product
  11. Lower bound in main theorem
  12. Upper bound in main theorem
  13. Bounds Depending on $\varepsilon$
  14. Generalized Distributed Inner Product Estimation
  15. Upper Bound
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose Alice and Bob are given $k$ copies of $n$-qubit states $\ket{\psi}$ and $\ket{\phi}$ respectively and can communication $\Theta(q)$ qubits along with performing arbitrary $\mathsf{LOCC}$. Then it is necessary and sufficient to obtain $k=\Theta\left( \sqrt{2^{n-q}} \right)$ copies of their st

Figures (2)

  • Figure 1: Protocol to estimate inner product using shared entanglement.
  • Figure 2: Protocol to estimate $| \langle \phi \vert M_\varepsilon \vert \psi \rangle |^2$ using only $\mathsf{LOCC}$.

Theorems & Definitions (43)

  • Theorem 1.1: Informal
  • Theorem 1.2: Informal
  • Definition 2.1: $\mathop{\mathrm{SWAP}}\nolimits$ test
  • Definition 2.2: Symmetric Subspace
  • Definition 2.3: Standard POVM on $\vee^k \mathbb{C}^d$
  • Definition 2.4: Quantum Teleportation bennett1993teleportingwerner2001all
  • Theorem 3.1
  • Theorem 3.2
  • Definition 3.1: Distributed Inner Product Estimation, Decision Version
  • Definition 3.2: Robustness of entanglement vidal1999robustness
  • ...and 33 more