Exponential quantum advantages in learning quantum observables from classical data

Quantum computers are believed to bring computational advantages in simulating quantum many body systems. However, recent works have shown that classical machine learning algorithms are able to predict numerous properties of quantum systems with classical data. Despite various examples of learning tasks with provable quantum advantages being proposed, they all involve cryptographic functions and do not represent any physical scenarios encountered in laboratory settings. In this paper we prove quantum advantages for the physically relevant task of learning quantum observables from classical (measured out) data. We consider two types of observables: first we prove a learning advantage for linear combinations of Pauli strings, then we extend the result for a broader case of unitarily parametrized observables. For each type of observable we delineate the boundaries that separate physically relevant tasks which classical computers can solve using data from quantum measurements, from those where a quantum computer is still necessary for data analysis. Differently from previous works, we base our classical hardness results on the weaker assumption that $\mathsf{BQP}$ hard processes cannot be simulated by polynomial-size classical circuits and provide a non-trivial quantum learning algorithm. Our results shed light on the utility of quantum computers for machine learning problems in the domain of quantum many body physics, thereby suggesting new directions where quantum learning improvements may emerge.