Predicting adaptively chosen observables in quantum systems

TL;DR

This paper explores the challenges and solutions for adaptively predicting properties of quantum systems, revealing that adaptivity can significantly increase the sample complexity for certain observables.

Contribution

It establishes lower bounds on sample complexity for adaptive predictions and introduces efficient algorithms for different classes of quantum observables.

Findings

Adaptive prediction of local and Pauli observables requires (\u221a M) samples.

Efficient algorithms match the information-theoretic lower bounds.

For bounded-Frobenius-norm observables, only (\u221a  M) samples are needed, independent of system size.

Abstract

Recent advances have demonstrated that $\mathcal{O}(\log M)$ measurements suffice to predict $M$ properties of arbitrarily large quantum many-body systems. However, these remarkable findings assume that the properties to be predicted are chosen independently of the data. This assumption can be violated in practice, where scientists adaptively select properties after looking at previous predictions. This work investigates the adaptive setting for three classes of observables: local, Pauli, and bounded-Frobenius-norm observables. We prove that $\Omega(\sqrt{M})$ samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of $M$ adaptively chosen local and Pauli observables. We also present computationally-efficient algorithms that achieve this information-theoretic lower bound. In contrast, for bounded-Frobenius-norm observables, we devise an algorithm requiring only $\mathcal{O}(\log M)$ samples, independent of system size. Our results highlight the potential pitfalls of adaptivity in analyzing data from quantum experiments and provide new algorithmic tools to safeguard against erroneous predictions in quantum experiments.