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Sample-optimal classical shadows for pure states

Daniel Grier, Hakop Pashayan, Luke Schaeffer

TL;DR

This work addresses how many copies of an unknown pure quantum state are needed to produce a classical shadow sufficient for estimating expectation values of observables. It shows that, using joint measurements, one can achieve a sample complexity of $\tilde{O}((\sqrt{B})/\epsilon + 1/\epsilon^2)$ (up to $\log(1/\delta)$ factors), with matching lower bounds, and that independent measurements require $\tilde{O}((\sqrt{Bd})/\epsilon + 1/\epsilon^2)$ samples; moreover, random Clifford measurements are not optimal for pure states, though the same measurements can be repurposed with a different estimator. The results illuminate compression limits for observable estimation and demonstrate that accessing the relevant information in pure states via shadows is near-optimal under joint measurements. These bounds improve the understanding of sample efficiency in shadow tomography for pure states and guide practical protocols for estimating many observables from few copies.

Abstract

We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state $ρ$ in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate $\mathrm{Tr}(O ρ)$ for any Hermitian observable $O$ to within additive error $ε$ provided $\mathrm{Tr}(O^2)\leq B$ and $\lVert O \rVert = 1$. Our main result applies to the joint measurement setting, where we show $\tildeΘ(\sqrt{B}ε^{-1} + ε^{-2})$ samples of $ρ$ are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of $ρ$ can be compressed for observable estimation. In the independent measurement setting, we show that $\mathcal O(\sqrt{Bd} ε^{-1} + ε^{-2})$ samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator.

Sample-optimal classical shadows for pure states

TL;DR

This work addresses how many copies of an unknown pure quantum state are needed to produce a classical shadow sufficient for estimating expectation values of observables. It shows that, using joint measurements, one can achieve a sample complexity of (up to factors), with matching lower bounds, and that independent measurements require samples; moreover, random Clifford measurements are not optimal for pure states, though the same measurements can be repurposed with a different estimator. The results illuminate compression limits for observable estimation and demonstrate that accessing the relevant information in pure states via shadows is near-optimal under joint measurements. These bounds improve the understanding of sample efficiency in shadow tomography for pure states and guide practical protocols for estimating many observables from few copies.

Abstract

We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate for any Hermitian observable to within additive error provided and . Our main result applies to the joint measurement setting, where we show samples of are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of can be compressed for observable estimation. In the independent measurement setting, we show that samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator.
Paper Structure (6 sections, 7 theorems, 32 equations, 1 figure, 2 algorithms)

This paper contains 6 sections, 7 theorems, 32 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. The classical shadows task
  3. Summary of results
  4. Preliminaries
  5. Joint Measurement Upper Bound
  6. Analysis

Key Result

Theorem 5

$\mathsf{Shadows}(B,\epsilon) = \tilde{\Theta}\mathopen{}\mathclose{\left(\frac{\sqrt{B}}{\epsilon} + \frac{1}{\epsilon^2}\right)$ provided $B \le \epsilon d$.

Figures (1)

  • Figure 1: Second moment calculation in the special case $s = 1$.

Theorems & Definitions (19)

  • Definition 1
  • Definition 2: Classical Shadows Task
  • Definition 3: Sample Complexity of the Classical Shadows Task
  • Definition 4: Classical Shadows with Independent Measurements
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Definition 8: permutation operator
  • Definition 9: symmetric subspace
  • Lemma 10
  • ...and 9 more