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On the sample complexity of purity and inner product estimation

Weiyuan Gong, Jonas Haferkamp, Qi Ye, Zhihan Zhang

TL;DR

It is proved that arbitrary protocols with $k-qubit quantum memory that estimate purity to error $\epsilon$ require $\Omega(median\{1/\epsilon^2,2^{n/2}/\sqrt{\epsilon,2^{n-k}/\epsilon^2\})$ copies of $\rho$.

Abstract

We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate $tr(ρ^2)$ of an unknown quantum state $ρ$ to additive error $ε$. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate $tr(ρσ)$ to additive error $ε$ given copies of unknown quantum state $ρ$ and $σ$ using classical communication and restricted quantum communication. In this paper, we show a strong connection between the sample complexity of purity estimation with bounded quantum memory and inner product estimation with bounded quantum communication and unentangled measurements. We propose a protocol that solves quantum inner product estimation with $k$-qubit one-way quantum communication and unentangled local measurements using $O(median\{1/ε^2,2^{n/2}/ε,2^{n-k}/ε^2\})$ copies of $ρ$ and $σ$. Our protocol can be modified to estimate the purity of an unknown quantum state $ρ$ using $k$-qubit quantum memory with the same complexity. We prove that arbitrary protocols with $k$-qubit quantum memory that estimate purity to error $ε$ require $Ω(median\{1/ε^2,2^{n/2}/\sqrtε,2^{n-k}/ε^2\})$ copies of $ρ$. This indicates the same lower bound for quantum inner product estimation with one-way $k$-qubit quantum communication and classical communication, and unentangled local measurements. For purity estimation, we further improve the lower bound to $Ω(\max\{1/ε^2,2^{n/2}/ε\})$ for any protocols using an identical single-copy projection-valued measurement. Additionally, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed state and provide a lower bound on the sample complexity.

On the sample complexity of purity and inner product estimation

TL;DR

It is proved that arbitrary protocols with \epsilon\Omega(median\{1/\epsilon^2,2^{n/2}/\sqrt{\epsilon,2^{n-k}/\epsilon^2\})\rho$.

Abstract

We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. In purity estimation, we are to estimate of an unknown quantum state to additive error . Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate to additive error given copies of unknown quantum state and using classical communication and restricted quantum communication. In this paper, we show a strong connection between the sample complexity of purity estimation with bounded quantum memory and inner product estimation with bounded quantum communication and unentangled measurements. We propose a protocol that solves quantum inner product estimation with -qubit one-way quantum communication and unentangled local measurements using copies of and . Our protocol can be modified to estimate the purity of an unknown quantum state using -qubit quantum memory with the same complexity. We prove that arbitrary protocols with -qubit quantum memory that estimate purity to error require copies of . This indicates the same lower bound for quantum inner product estimation with one-way -qubit quantum communication and classical communication, and unentangled local measurements. For purity estimation, we further improve the lower bound to for any protocols using an identical single-copy projection-valued measurement. Additionally, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed state and provide a lower bound on the sample complexity.

Paper Structure

This paper contains 28 sections, 26 theorems, 81 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Cross-platform verification.
  4. Testing and learning properties of quantum states.
  5. Quantum memory and communication tradeoffs.
  6. Concurrent work.
  7. Outlook
  8. Arbitrary $k$-qubit communication.
  9. Tight bounds.
  10. Alternative models for restricted quantum communication.
  11. Other distributed quantum property estimation problems.
  12. Roadmap
  13. Preliminaries
  14. Basic results in quantum information
  15. Quantum measurements.
  16. ...and 13 more sections

Key Result

Theorem 1

Given unknown quantum states $\rho$ and $\sigma$, and error parameter $\varepsilon$, there exists a non-adaptive algorithm that solves inner product estimation of $\mathrm{tr}(\rho\sigma)$ with $k$-qubit one-way quantum communication (prob:inner_product_k) using $O(\mathop{\mathrm{median}}\nolimits\

Figures (1)

  • Figure 1: Comparison of the models for quantum inner product estimation with $k$-qubit one-way quantum communication and any classical one-way communication (\ref{['prob:inner_product_k']}) and purity estimation with $k$-qubit quantum memory (\ref{['prob:purity_k']}).

Theorems & Definitions (46)

  • Theorem 1: Informal, see \ref{['thm:partial_swap']}
  • Theorem 2: Informal, see \ref{['thm:purity_ip_lower']}
  • Theorem 3: Informal, see \ref{['thm:improved_lower']}
  • Theorem 4
  • Definition 1: Separable measurements
  • Theorem 5: Levy's lemma for Haar-random pure states
  • Lemma 1: See e.g. Ref. harrow2013church
  • Definition 2: Tree representation for $(c, \pazocal{M})$ learning protocols chen2024optimal
  • Definition 3: Likelihood ratio
  • Lemma 2: Toolbox for proving lower bounds
  • ...and 36 more