Researchers have developed a new algorithm for online quantum shadow tomography that matches the efficiency of classical methods. This breakthrough addresses a fundamental problem in the characterization of quantum states, where the goal is to estimate the properties of an unknown state from adaptive measurements. Previously, quantum approaches to this problem were suboptimal compared to their classical counterparts, limiting the efficiency in reconstructing quantum information.

Online shadow tomography involves a quantum system (a $d$-dimensional state $\rho$) being subjected to a sequence of adaptively proposed observables $A^{(1)}, \ldots, A^{(m)}$. After each measurement, the expected value $\mathrm{Tr}(A^{(t)}\rho)$ must be estimated to within $\pm \epsilon$. The primary goal is to minimize the number of copies of the quantum state required for this estimation. Prior results for online shadow tomography showed suboptimal dependencies on the parameters $m$, $d$, and $\epsilon$.

The new work closes this gap by presenting a pair of algorithms that achieve classical efficiency rates. One of the bounds obtained is the first to achieve an $o(\log^2 m)$ dependence along with $\mathrm{poly}(\log(d)/\epsilon)$, improving exponents even in the offline shadow tomography setting. The other bound is optimal among those independent of $d$, surpassing the best prior result by a factor of $\sqrt{m} \log m$. The key to these advances lies in a new framework for quantifying post-measurement damage, based on the quantum Efron-Stein decomposition.