Exponential quantum advantages in learning quantum observables from classical data

npj | quantum information Article Published in partnership with The University of New South Wales https://doi.org/10.1038/s41534-025-01162-2 Exponential quantum advantages in learning quantum observables from classical data Check for updates 1,2 1234567890():,; 1234567890():,; Riccardo Molteni 1,2,3 , Casper Gyurik 1,2 & Vedran Dunjko Quantum computers are believed to bring computational advantages in simulating quantum manybody 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. The very first proposed application of quantum computers can be traced back to Feynman’s idea of simulating quantum physics on a quantum device. Together with factoring1, simulation of quantum many-body systems stands as the clearest example of the dramatic advantages of quantum computers2. Machine learning is another much newer area where quantum computers are believed to possibly bring advantages in certain learning problems, and in fact, there are provable speed-ups achieved by a quantum algorithm3–7 for specific machine learning problems. Relating back to the original Feynman’s idea of simulating quantum physics, machine learning problems in quantum many-body physics seem a natural scenario where learning advantages could arise. However, perhaps surprisingly, it was recently shown that access to data exemplifying what the hard-to-compute function does can drastically change the hardness of the computational task, questioning the role of quantum computation in machine-learning scenarios8–10. If every quantum computation could be replicated classically, provided access to data, such results would confine the practical application of quantum computers solely to the data acquisition stage. This is, however, not the case, as was shown already in the examples considered in refs. 3,4. In such cases, the unknown function was cryptographic in nature 1 and not related to genuine quantum simulation problems. Nonetheless,11 introduced the first methods for establishing learning separations beyond cryptographic tasks. In particular, they demonstrated that learning problems with provable speed-ups can be constructed from any BQP-complete function, thereby enabling stronger links to physical scenarios. However, the work in ref. 11 had two notable shortcomings. First, the classical nonlearnability results were based on relatively strong assumptions about distributional problems, which, while plausible, are less well-understood than their decision problem counterparts. Second, the study did not introduce any significant quantum learning algorithms for general settings. Instead, quantum learnability was only shown in somewhat artificial scenarios where the concept class was small (polynomially bounded), allowing for the application of straightforward brute-force learning methods. The results presented in this work effectively address both issues, enabling the establishment of clear boundaries between classical and quantum learning algorithms when dealing with data generated by quantum processes. Specifically, we consider learning problems where one is interested in predicting expectation values of an unknown observable from measurements on input quantum states, which can either be ground states Applied Quantum Algorithms, Leiden University, Leiden, Netherlands. 2LIACS, Universiteit Leiden, Leiden, Netherlands. 3Pasqal SAS, Massy, France. e-mail: ; ; npj Quantum Information | (2026)12:19 1 Article https://doi.org/10.1038/s41534-025-01162-2 of local Hamiltonians or time-evolved states. As motivation for the learning task, we have in mind experimentally plausible settings where learning an unknown observable may arise, such as in phase classification with an unknown order parameter or when dealing with real devices where the implemented measurement may be influenced by noise or external factors. In this scenario, our result is a proof of learning advantages for different types of quantum observables. The main contributions of this work are as follows: • We prove an exponential quantum advantage in learning observables that are formed as a linear combination of local Pauli strings acting on time-evolved or ground states of local Hamiltonians. Importantly, by considering more general learning settings and leveraging results from classical learning theory, we base all of our learning advantage results on the more widely studied and generally accepted assumption that BQP-hard processes cannot be simulated by a polynomial-size classical circuit (i.e., BQP ⊈ P=poly), rather than on the less well-understood assumptions related to distributional problems. • Assuming BQP ⊈ P=poly, we show how to construct learning problems that provide a quantum learning advantage in the general case of learning unitarily-parametrized observables, whenever an efficient procedure for learning an unknown class of unitaries through query access exists. We then provide a concrete example by connecting to recent results on learning shallow unitaries. We also examine the Hamiltonian learning problem where the identification of the target function is demonstrated to be classically easy. We summarize the learning settings and the achieved separations in Table 1, and provide practical examples of where these learning settings may arise in Section Discussion. Results To make our work more self-contained and accessible, we include the necessary preliminaries from learning theory and complexity theory. PAC learning The definition of learnability in this work aligns directly with the widely adopted probably approximately correct (PAC) learning framework12,13. In the case of supervised learning, a learning problem in the PAC framework is defined by a concept class F which, for eac (...truncated)