给出混合量子-经典模型泛化能力的理论分析框架
Generalization Bounds in Hybrid Quantum-Classical Machine Learning Models
- 构建统一数学框架分析混合模型泛化性
- 提出含量子门数、数据量等参数的泛化界
- 揭示量子与经典组件的相互作用机制
混合经典-量子模型旨在结合量子计算与经典机器学习的优势，但其实际潜力仍不明确。本文建立了一个统一的数学框架，用于分析混合模型的泛化性能，揭示其从数据中学习的机制。我们推导出一个新颖的泛化界：$ ilde{/mathcal O}ig( frac{α^{k}}{\sqrt{N}}ig( k^{ frac{3}{2}} ext{√}m n ext{+} ext{√}T ext{log}Tig) ig)$，其中 $N$ 为训练数据点数，$T$ 为可训练量子门数，$n$ 为量子电路输出维度，$k$ 为具有有界弗罗比尼乌斯范数 $\|F_i\|_F \leq α$ 的 $k$ 个线性层（$F_i \in \mathbb{R}^{m \times n}$）及其间激活函数。该界可分解为量子与经典贡献，提供理论工具以分离并理解二者影响。同时指出经典统计学习理论在混合场景下的概念局限，并提出未来研究方向。
原文摘要 · Abstract (English)
Hybrid classical-quantum models aim to harness the strengths of both quantum computing and classical machine learning, but their practical potential remains poorly understood. In this work, we develop a unified mathematical framework for analyzing generalization in hybrid models, offering insight into how these systems learn from data. We establish a novel generalization bound of the form $\tilde{\mathcal O}\left( \tfrac{α^{k}}{\sqrt{N}}\, \big( k^{\tfrac{3}{2}}\sqrt{m n}\;+\;\sqrt{T\log T}\big) \right)$ for $N$ training data points, $T$ trainable quantum gates, $n$ dimensional quantum circuit output, and $k$ bounded linear layers $ \|F_i\|_F \leq α$ where $ i = 1, \dots, k $ and $F_i \in \mathbb{R}^{m \times n} $ interspersed with activation functions. This generalization bound decomposes into quantum and classical contributions, providing a theoretical framework to separate their influence and clarifying their interaction. Alongside the bound, we highlight conceptual limitations of applying classical statistical learning theory in the hybrid setting and suggest promising directions for future theoretical work.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。