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.

翻译：尽管付出了巨大努力，量子机器学习领域在处理经典数据时，仅对受密码学启发的合成数据集展示了量子学习优势。本文致力于寻找量子学习算法能够实现相较经典学习算法具有可证明指数级加速的学习问题。我们反思了与该问题相关的计算学习理论概念，并讨论了定义中细微差异如何导致学习器需要满足和解决的截然不同的要求与任务。我们审视了现有具备可证明量子加速的学习问题，发现它们大多依赖于生成数据的函数在经典计算上的难解性，而非函数识别上的困难。为此，我们提出了两种新的学习分离，其中经典困难主要在于识别生成数据的函数。此外，我们探索了可在数据由量子生成场景下被利用来证明量子加速的计算困难假设，这暗示了在更广泛的自然场景（如凝聚态物理和高能物理）中可能存在量子优势。我们还讨论了经典阴影范式在学习分离背景下的局限性，以及如何将表征物质相和哈密顿量学习等物理驱动场景纳入计算学习框架。