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arXiv:2504.15964

Quantum machine learning advantages beyond hardness of evaluation

Riccardo Molteni, Simon C. Marshall, Vedran Dunjko

Published Apr 22, 2025• 0 citations
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Abstract

The most general examples of quantum learning advantages involve data labeled by cryptographic or intrinsically quantum functions, where classical learners are limited by the infeasibility of evaluating the labeling functions using polynomial-sized classical circuits. While broad in scope, such results reveal little about advantages arising from the learning process itself. In cryptographic settings, further insight is possible via random-generatability - the ability to classically generate labeled data - enabling hardness proofs for identification tasks, where the goal is to identify the labeling function from a dataset, even when evaluation is classically intractable. These tasks are particularly relevant in quantum contexts, including Hamiltonian learning and identifying physically meaningful order parameters. However, for quantum functions, random-generatability is conjectured not to hold, leaving no known identification advantages in genuinely quantum regimes. In this work, we give the first proofs of quantum identification learning advantages under standard complexity assumptions. We confirm that quantum-hard functions are not random-generatable unless BQP is contained in the second level of the polynomial hierarchy, ruling out cryptographic-style data generation strategies. We then introduce a new approach: we show that verifiable identification - solving the identification task for valid datasets while rejecting invalid ones - is classically hard for quantum labeling functions unless BQP is in the polynomial hierarchy. Finally, we show that, for a broad class of tasks, solving the identification problem implies verifiable identification in the polynomial hierarchy. This yields our main result: a natural class of quantum identification tasks solvable by quantum learners but hard for classical learners unless BQP is in the polynomial hierarchy.

DOI
10.48550/arxiv.2504.15964

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The most general examples of quantum learning advantages involve data labeled by cryptographic or intrinsically quantum functions, where classical learners are limited by the infeasibility of evaluating the labeling functions using polynomial-sized classical circuits. While broad in scope, such results reveal little about advantages arising from the learning process itself. In cryptographic settings, further insight is possible via random-generatability - the ability to classically generate labeled data - enabling hardness proofs for identification tasks, where the goal is to identify the labeling function from a dataset, even when evaluation is classically intractable. These tasks are particularly relevant in quantum contexts, including Hamiltonian learning and identifying physically meaningful order parameters. However, for quantum functions, random-generatability is conjectured not to hold, leaving no known identification advantages in genuinely quantum regimes. In this work, we give the first proofs of quantum identification learning advantages under standard complexity assumptions. We confirm that quantum-hard functions are not random-generatable unless BQP is contained in the second level of the polynomial hierarchy, ruling out cryptographic-style data generation strategies. We then introduce a new approach: we show that verifiable identification - solving the identification task for valid datasets while rejecting invalid ones - is classically hard for quantum labeling functions unless BQP is in the polynomial hierarchy. Finally, we show that, for a broad class of tasks, solving the identification problem implies verifiable identification in the polynomial hierarchy. This yields our main result: a natural class of quantum identification tasks solvable by quantum learners but hard for classical learners unless BQP is in the polynomial hierarchy.
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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Morohoshi, Nakayama, Manabe, and Mitarai proposed a physically motivated quantum machine learning problem in which the goal is to predict quantities of the form Tr⁡[f(H)ρ]\operatorname{Tr}[f(H)ρ]Tr[f(H)ρ] from classical descriptions of a Hamiltonian HHH and a quantum state ρρρ, where fff is an unknown function. We call this problem Hamiltonian function learning in this paper. They constructed an efficient quantum learning algorithm under suitable conditions, while leaving a rigorous proof of average-case classical hardness open. In this paper, we rigorously prove the average-case classical hardness for two distribution-specific Hamiltonian function learning problems for fcos⁡,π(λ)=cos⁡(πλ)f_{\cos,π}(λ)=\cos(πλ)fcos,π​(λ)=cos(πλ) and fexp⁡,β(λ)=e−βλf_{\exp,β}(λ)=e^{-βλ}fexp,β​(λ)=e−βλ discussed in the paper of Morohoshi et al. under the assumption of the average-case hardness of factoring random RSA moduli. More specifically, we show that an efficient classical randomized learner under squared loss whose output hypotheses are evaluable in classical polynomial time for either problem would yield a classical randomized polynomial-time algorithm for factoring random RSA moduli.
This dissertation investigates the fundamental computational capabilities of quantum algorithms with respect to combinatorial optimization and distribution learning. We establish rigorous quantum advantages over classical algorithms for such problems under both widely accepted cryptographic assumptions and unconditionally. First, we demonstrate a super-polynomial quantum advantage for approximating solutions to combinatorial optimization problems, explicitly constructing advantage-bearing instances of Integer Linear Programming. Our proofs build upon approximation hardness results that instantiate a connection between the representation-independent hardness of learning and approximation hardness, as captured by Occam’s razor within computational learning theory. We show that the instances we construct can be efficiently approximated by quantum algorithms utilizing Shor’s factoring algorithm, yet remain classically intractable unless standard cryptographic assumptions fail. Second, we fill a previously open gap by proving a super-polynomial quantum advantage for learning the density function of probability distributions, complementing prior results established for generative distribution learning. We show that the hardness of density modeling can be established under weaker cryptographic assumptions involving pseudorandom functions. The quantum advantage for density modeling fundamentally arises from Shor’s algorithm for computing discrete logarithms. Moreover, we explicitly articulate known connections between Boolean function learning hardness and distribution learning hardness using established constructions. Third, addressing practical constraints of near-term quantum devices, we establish an unconditional quantum advantage for generative distribution learning using shallow, constant-depth quantum circuits. We find that shallow quantum circuits pose a superior class of hypotheses compared to classical counterparts. This result underscores the potential of near-term quantum hardware by highlighting the unique computational capabilities enabled by quantum entanglement and non-local correlations, offered even by constrained devices. Collectively, these contributions deepen our understanding of quantum computational advantages for combinatorial optimization and distribution learning, clarifying their foundations and pointing toward promising avenues for future exploration.
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.
Quantum machine learning (QML) is often listed as a promising candidate for useful applications of quantum computers, in part due to numerous proofs of possible quantum advantages. A central question is how small a role quantum computers can play while still enabling provable learning advantages over classical methods. We study an especially restricted setting in which a quantum computer is used only as a feature extractor: it acts independently on individual data points, without access to labels or global dataset information, is available only to augment the training set, and is not available at deployment. Training and deployment are therefore carried out by fully classical learners on a dataset augmented with quantum-generated features. We formalize this model by adapting the classical framework of Learning Under Privileged Information (LUPI) to the quantum case, which we call Learning Under Quantum Privileged Information (LUQPI). Within this framework, we show that even such minimally involved quantum feature extraction, available only during training, can yield exponential quantum-classical separations for suitable concept classes and data distributions under reasonable computational assumptions. We further situate LUQPI within a taxonomy of related quantum and classical learning settings and show how standard classical machinery, most notably the SVM+ algorithm, can exploit quantum-augmented data. Finally, we present numerical experiments in a physically motivated many-body setting, where privileged quantum features are expectation values of observables on ground states, and observe consistent performance gains for LUQPI-style models over strong classical baselines.