Adaptivity can help exponentially for shadow tomography

TL;DR

This paper demonstrates that adaptivity in measurement choices can exponentially improve the efficiency of shadow tomography, challenging the notion that adaptivity offers limited benefits in quantum data learning.

Contribution

It shows that adaptive two-copy measurement protocols can exponentially outperform nonadaptive ones in shadow tomography, revealing the significant advantage of adaptivity.

Findings

Adaptive protocols are exponentially more sample-efficient.

Nonadaptive protocols are less effective for shadow tomography.

Adaptivity offers substantial benefits in quantum measurement strategies.

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.