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Optimal high-precision shadow estimation

Sitan Chen, Jerry Li, Allen Liu

TL;DR

The paper addresses the problem of estimating many linear properties of an unknown $d$-dimensional quantum state from as few copies as possible, focusing on the high-accuracy regime where $\varepsilon = O(d^{-c})$. It introduces a representation-theoretic approach based on Keyl's POVM and Schur-Weyl duality, together with a splitting-based dimensionality-reduction that reduces to balanced states and linearizes around the maximally mixed state $I_d/d$. The main contributions are the first tight, nontrivial sample complexity bounds for shadow tomography and classical shadows in this regime, achieving a dimension-free rate $n = O(\log m / \varepsilon^2)$ for general observables and proving an equivalence between the quantum and classical shadow tasks under these conditions, along with reductions to the medium-accuracy regime. This advances the theoretical understanding of shadow estimation and yields practically efficient protocols for high-precision quantum state profiling with broad implications for quantum data analysis and benchmarking.

Abstract

We give the first tight sample complexity bounds for shadow tomography and classical shadows in the regime where the target error is below some sufficiently small inverse polynomial in the dimension of the Hilbert space. Formally we give a protocol that, given any $m\in\mathbb{N}$ and $ε\le O(d^{-12})$, measures $O(\log(m)/ε^2)$ copies of an unknown mixed state $ρ\in\mathbb{C}^{d\times d}$ and outputs a classical description of $ρ$ which can then be used to estimate any collection of $m$ observables to within additive accuracy $ε$. Previously, even for the simpler task of shadow tomography -- where the $m$ observables are known in advance -- the best known rates either scaled benignly but suboptimally in all of $m, d, ε$, or scaled optimally in $ε, m$ but had additional polynomial factors in $d$ for general observables. Intriguingly, we also show via dimensionality reduction, that we can rescale $ε$ and $d$ to reduce to the regime where $ε\le O(d^{-1/2})$. Our algorithm draws upon representation-theoretic tools recently developed in the context of full state tomography.

Optimal high-precision shadow estimation

TL;DR

The paper addresses the problem of estimating many linear properties of an unknown -dimensional quantum state from as few copies as possible, focusing on the high-accuracy regime where . It introduces a representation-theoretic approach based on Keyl's POVM and Schur-Weyl duality, together with a splitting-based dimensionality-reduction that reduces to balanced states and linearizes around the maximally mixed state . The main contributions are the first tight, nontrivial sample complexity bounds for shadow tomography and classical shadows in this regime, achieving a dimension-free rate for general observables and proving an equivalence between the quantum and classical shadow tasks under these conditions, along with reductions to the medium-accuracy regime. This advances the theoretical understanding of shadow estimation and yields practically efficient protocols for high-precision quantum state profiling with broad implications for quantum data analysis and benchmarking.

Abstract

We give the first tight sample complexity bounds for shadow tomography and classical shadows in the regime where the target error is below some sufficiently small inverse polynomial in the dimension of the Hilbert space. Formally we give a protocol that, given any and , measures copies of an unknown mixed state and outputs a classical description of which can then be used to estimate any collection of observables to within additive accuracy . Previously, even for the simpler task of shadow tomography -- where the observables are known in advance -- the best known rates either scaled benignly but suboptimally in all of , or scaled optimally in but had additional polynomial factors in for general observables. Intriguingly, we also show via dimensionality reduction, that we can rescale and to reduce to the regime where . Our algorithm draws upon representation-theoretic tools recently developed in the context of full state tomography.
Paper Structure (9 sections, 12 theorems, 56 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 12 theorems, 56 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Technical overview
  4. Outlook
  5. Representation Theory and Keyl's POVM
  6. Basic Facts
  7. Balanced Case
  8. Splitting Reduction
  9. Reducing to Small $\varepsilon$ via Random Projection

Key Result

Theorem 2.1

Let $m, d \in \mathbb{Z}$ be fixed, and let $\varepsilon = O(d^{-12})$. Then, there is an estimator which takes $n$ copies of an unknown mixed state $\rho \in {\mathbb{C}}^{d \times d}$, where and outputs a classical function $F: {\mathbb{C}}^{d \times d} \to [0, 1]$ so that for any fixed collection of $m$ observables $\{O_i\}_{i = 1}^m$, the function satisfies $| F(O_i) - \mathrm{tr} (O_i \rho)

Figures (1)

  • Figure 1: Overview of our shadow estimation algorithm

Theorems & Definitions (42)

  • Theorem 2.1
  • Theorem 2.2: informal, see Theorem \ref{['lem:reduce-dimension']}
  • Definition A.1
  • Definition A.2
  • Theorem A.5: Schur-Weyl Duality goodman2009symmetry
  • Definition A.6: Schur Subspace
  • Theorem A.7: Gelfand-Tsetlin Basis goodman2009symmetry
  • Definition A.8: Maximal-weight Vector
  • Remark A.9
  • Definition A.10: Weak Schur Sampling
  • ...and 32 more