Title: Quantum machine learning advantages beyond hardness of evaluation
Primary area: learning theory
Keywords: Quantum machine learning, Quantum-classical learning separations, Learning theory

Abstract: Recent years have seen rigorous proofs of quantum advantages in machine learning, particularly when data is labeled by cryptographic or inherently quantum functions. These results typically rely on the infeasibility of classical polynomial-sized circuits to evaluate the true labeling function. While broad in scope, these results however reveal little about advantages stemming from the actual learning process itself. This motivates the study of the so-called identification task, where the goal is to ``just'' identify the labeling function behind a dataset, making the learning step the only possible source of advantage. The identification task also has natural applications, which we discuss. Yet, such identification advantages remain poorly understood. So far they have only been proven in cryptographic settings by leveraging random-generatability, the ability to efficiently generate labeled data. However, for quantum functions this property is conjectured not to hold, leaving identification advantages unexplored. In this work, we provide the first proofs of identification learning advantages for quantum functions under complexity-theoretic assumptions. Our main result relies on a new proof strategy, allowing us to show that for a broad class of quantum identification tasks there exists an exponential quantum advantage unless BQP is in a low level of the polynomial hierarchy. Along the way we prove a number of more technical results including the aforementioned conjecture that quantum functions are not random generatable (subject to plausible complexity-theoretic assumptions), which shows a new proof strategy was necessary. These findings suggest that for many quantum-related learning tasks, the entire learning process—not just final evaluation—gains significant advantages from quantum computation

TL;DR: We establish a quantum-classical separation in supervised learning for identifying the target labeling function, without the need for direct evaluation, when the target function is quantum.