Online Shadow Tomography Matching the Classical Bounds

In 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),…,A^(m), and after each A^(t) is given we must estimate (A^(t)ρ) to within ± ε. This is the direct quantum generalization of the classical problem of Adaptive Data Analysis. Prior results for online Shadow Tomography were suboptimal in all three parameters m, d, ε, lagging behind the best known and classical rates , 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 (log(d)/); moreover, it improves all three exponents even in the 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 √(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.