Adaptivity can help exponentially for shadow tomography
Sitan Chen, Weiyuan Gong, Zhihan Zhang
TL;DR
This work challenges the prevailing view that adaptivity offers little advantage in learning quantum data by proving an exponential separation for Pauli shadow tomography with two-copy measurements: adaptive protocols require only $O(n)$ copies, while any nonadaptive protocol requires $\Omega(2^{n/2})$ copies. The key technique reduces shadow tomography to a hypothesis-testing problem between ensembles $\sigma_a^+$ and $\sigma_a^-$ and uses Le Cam's lemma to bound the average total variation distance for nonadaptive two-copy measurements, showing it cannot be made small unless the number of copies is exponential in $n$. The result hinges on standard Pauli group identities, the SWAP operator, and representations of two-copy measurements as rank-1 POVMs, with a detailed nonadaptive lower bound proved for the first half of the main theorem. This establishes the first known exponential advantage of adaptivity in shadow tomography, informing future work on the role of adaptivity rounds and the computational feasibility of adaptive strategies in quantum learning.
Abstract
In recent years there has been significant interest in understanding the statistical complexity of learning from quantum data under the constraint that one can only make unentangled measurements. While a key challenge in establishing tight lower bounds in this setting is to deal with the fact that the measurements can be chosen in an adaptive fashion, a recurring theme has been that adaptivity offers little advantage over more straightforward, nonadaptive protocols. In this note, we offer a counterpoint to this. We show that for the basic task of shadow tomography, protocols that use adaptively chosen two-copy measurements can be exponentially more sample-efficient than any protocol that uses nonadaptive two-copy measurements.