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arXiv:2405.02027 · npj Quantum Information

Exponential quantum advantages in learning quantum observables from classical data

Riccardo Molteni, Casper Gyurik, Vedran Dunjko

Published Jan 10, 2026• 6 citations
arXivHTMLPDF

Abstract

Abstract 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 $$BQP$$ B Q P 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.

DOI
10.1038/s41534-025-01162-2

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Abstract 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 BQPBQPBQP B Q P 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.
An overarching milestone of quantum machine learning (QML) is to demonstrate the advantage of QML over all possible classical learning methods in accelerating a common type of learning task, as represented by supervised learning from classical data. However, the provable advantages of QML in supervised learning have been known so far only for the learning tasks designed for using the advantage of specific quantum algorithms, i.e., Shor’s algorithms. Here, we explicitly construct an unprecedentedly broader family of supervised learning tasks with classical data to offer the provable advantage of QML based on general quantum computational advantages, progressing beyond Shor’s algorithms. Our learning task is feasibly achievable by executing a general class of functions that can be computed efficiently in polynomial time for a large fraction of inputs by arbitrary quantum algorithms, but not by any classical algorithm. We prove the hardness of achieving this learning task for any possible polynomial-time classical learning method. We also clarify protocols for preparing the classical data to demonstrate this learning task in experiments. These results provide a way to exploit a variety of quantum advantages in computing functions for the realization of provably advantageous QML.
Despite significant effort, the quantum machine learning community has only demonstrated quantum learning advantages for artificial cryptography-inspired datasets when dealing with classical data. In this paper we address the challenge of finding learning problems where quantum learning algorithms can achieve a provable exponential speedup over classical learning algorithms. We reflect on computational learning theory concepts related to this question and discuss how subtle differences in definitions can result in significantly different requirements and tasks for the learner to meet and solve. We examine existing learning problems with provable quantum speedups and find that they largely rely on the classical hardness of evaluating the function that generates the data, rather than identifying it. To address this, we present two new learning separations where the classical difficulty primarily lies in identifying the function generating the data. Furthermore, we explore computational hardness assumptions that can be leveraged to prove quantum speedups in scenarios where data is quantum-generated, which implies likely quantum advantages in a plethora of more natural settings (e.g., in condensed matter and high energy physics). We also discuss the limitations of the classical shadow paradigm in the context of learning separations, and how physically-motivated settings such as characterizing phases of matter and Hamiltonian learning fit in the computational learning framework.
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One of the key challenges in quantum machine learning is finding relevant machine learning tasks with a provable quantum advantage. A natural candidate for this is learning unknown Hamiltonian dynamics. Here, we tackle the supervised learning version of this problem, where we are given random examples of the inputs to the dynamics as classical data, paired with the expectation values of some observable after the time evolution, as corresponding output labels. The task is to replicate the corresponding input-output function. We prove that this task can yield provable exponential classical-quantum learning advantages under common complexity assumptions in natural settings. To design our quantum learning algorithms, we introduce a new method, which we term \textit{\subroutine}~algorithm for parametrized circuit functions, and which may be of independent interest. Furthermore, we discuss the limitations of generalizing our method to arbitrary quantum dynamics while maintaining provable guarantees. We explain that significant generalizations are impossible under certain complexity-theoretic assumptions, but nonetheless, we provide a heuristic kernel method, where we trade-off provable correctness for broader applicability.
All the data stored and processed by our computers is encoded as sequences of zeros and ones, called bits. Quantum computers offer an alternative to this traditional way of encoding and manipulating information. The information in a collection of qubits can be a combination, also called a superposition, over many possible sequences of zeros and ones. Where classical computation consists of flipping zeros and ones, quantum computation consists of manipulating high-dimensional vectors called quantum states. Researchers have found ways to use this to engineer quantum algorithms that solve some tasks much faster than any classical computer would. These are called quantum advantages, and are proven for specific tasks, such as breaking some encryption schemes and solving certain large linear systems. The goal of this thesis was to find new tasks for which quantum computers provably have an advantage over classical computers. In this thesis, I present two such advantages and associated results. The first one revolves around a physically relevant learning task. In other words, I explore how quantum computers can be used to help solve specialized machine learning tasks. The second one addresses the simulation of classical systems. One of the first high-potential and natural uses of quantum computers was the simulation of quantum systems. A lot of real-world problems, however, are ruled by classical physics. It turns out that quantum computers are well suited for the simulation of some exponentially large classical systems. In this thesis, I add a new problem to this list.
Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning (ML) techniques to impressive results in regression, classification, data generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets is motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed up classical ML algorithms. Here we review the literature in quantum ML and discuss perspectives for a mixed readership of classical ML and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in ML are identified as promising directions for the field. Practical questions, such as how to upload classical data into quantum form, will also be addressed.