Online Shadow Tomography Matching the Classical Bounds

2026-07-31

AI Summary

This work addresses the problem of online quantum shadow tomography: efficiently and accurately estimating the expectation values of a sequence of observables adaptively chosen by an adversary, using as few copies of an unknown quantum state as possible. The authors introduce a novel analytical framework based on the quantum Efron–Stein decomposition, achieving—for the first time—a sample complexity bound that simultaneously depends on the number of observables $m$ and the Hilbert space dimension $d$ through a combination of $o(\log^2 m)$ and $\mathrm{poly}(\log d / \varepsilon)$. In the regime where the bound is independent of $d$, their algorithm matches the known optimal sample complexity. It improves upon prior methods in both online and offline settings, matching or nearly attaining the optimal rates of classical adaptive estimation across all key parameters.

Abstract

In \emph{Online Shadow Tomography}, we are given copies of an unknown $d$-dimensional quantum state $ρ$, an adversary (adaptively) proposes a sequence of bounded observables $A^{(1)},\ldots,A^{(m)}$, and after each $A^{(t)}$ is given we must estimate $\Tr(A^{(t)}ρ)$ to within $\pm ε$. This is the direct quantum generalization of the classical problem of \emph{Adaptive Data Analysis}. %The ``offline'' case, in which $A^{(1)}, \ldots, A^{(m)}$ are given upfront, is also a well-studied problem. The main goal is to minimize the number of copies, $n$, required. Prior results for online Shadow Tomography were suboptimal in all three parameters $m, d, ε$, lagging behind the best known and classical rates~\cite{bassily2021algorithmic}, for which there is some evidence of optimality. In this work, we finally close this gap, giving a pair of algorithms matching the classical rates. The bound on the left is the first to achieve $o(\log^2 m)$-dependence together with $\poly(\log(d)/\eps)$; moreover, it improves all three exponents even in the \emph{Offline} Shadow Tomography setting. The bound on the right is known to be optimal among bounds independent of~$d$, and improves the best prior result by a $\sqrt{m} \log m$ factor. The key to our proof is a new framework for quantifying post-measurement damage, based on the quantum Efron--Stein decomposition.

Problem

• Online Shadow Tomography
• Quantum State Estimation
• Adaptive Data Analysis
• Sample Complexity
• Bounded Observables

Innovation

• Online Shadow Tomography
• Quantum Efron–Stein decomposition
• Adaptive Data Analysis
• Sample complexity
• Post-measurement disturbance