从函数可表示性出发，揭示量子机器学习优势的可行路径
Prospects for quantum advantage in machine learning from the representability of functions
- 将量子电路结构与可学习函数性质关联，分析深度和非克莱夫门数的影响
- 发现多数现有模拟方法共享可退化为经典计算的共性路径
- 区分三类模型：全可模拟、经典易处理、仍具量子优势，指明突破方向
在机器学习任务中展示量子优势需面对复杂的模型与算法格局。本文提出一个框架，将参数化量子电路的结构与它们实际能学习的函数数学性质相联系。该框架表明，电路深度和非克莱夫门数量等基本属性，直接决定了模型输出是否可被高效经典模拟或替代。我们论证，这种分析揭示了众多现有模拟方法背后的共同退化路径。更重要的是，它明确了三类模型的区别：完全可模拟、其函数空间经典可处理、仍保持稳健量子特性。这一视角为该领域提供了概念地图，澄清了不同模型与经典可模拟性的关系，并指出了量子优势可能存在的位置。
原文摘要 · Abstract (English)
Demonstrating quantum advantage in machine learning tasks requires navigating a complex landscape of proposed models and algorithms. To bring clarity to this search, we introduce a framework that connects the structure of parametrized quantum circuits to the mathematical nature of the functions they can actually learn. Within this framework, we show how fundamental properties, like circuit depth and non-Clifford gate count, directly determine whether a model's output leads to efficient classical simulation or surrogation. We argue that this analysis uncovers common pathways to dequantization that underlie many existing simulation methods. More importantly, it reveals critical distinctions between models that are fully simulatable, those whose function space is classically tractable, and those that remain robustly quantum. This perspective provides a conceptual map of this landscape, clarifying how different models relate to classical simulability and pointing to where opportunities for quantum advantage may lie.
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