Online Shadow Tomography Matching the Classical Bounds

In \emph{Online Shadow Tomography}, we are given copies of an unknown $d$-dimensional quantum state $\rho$, an adversary (adaptively) proposes a sequence of bounded observables $A^{(1)},\ldots,A^{(m)}$, and after each $A^{(t)}$ is given we must estimate $\Tr(A^{(t)}\rho)$ to within $\pm \epsilon$. 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 $m, d, \epsilon$, 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.