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 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 our results to a broader case of unitarily parametrized observables. For each case, we delineate sharp boundaries separating physically relevant tasks that admit efficient classical learning from those for which a quantum computer remains necessary for data analysis. Unlike previous works, our classical hardness results rely only on the weaker assumption that $$BQP$$ hard processes cannot be simulated by polynomial-size classical circuits, and we also provide a nontrivial quantum learning algorithm. Our results clarify when quantum resources are useful for learning problems in quantum many-body physics, and suggest practical directions in which quantum learning improvements may emerge.

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Nature

Researchers prove exponential quantum advantages for learning quantum observables from classical measurement data, establishing clear boundaries between tasks that can be efficiently solved classically versus those requiring quantum computers. The work demonstrates quantum learning advantages for linear combinations of Pauli strings and unitarily parametrized observables in physically relevant scenarios, unlike previous results limited to cryptographic functions. The classical hardness results rely on weaker assumptions than prior work (BQP-hard processes cannot be simulated by polynomial-size classical circuits), and the authors provide practical quantum learning algorithms. These findings clarify when quantum resources are necessary for analyzing quantum many-body physics problems and suggest directions for practical quantum learning improvements.

Exponential quantum advantages in learning quantum observables from classical data