arXiv:2503.17020quant-phcs.LG2025-03被引 2

提出新型量子核函数,实现过拟合但依然泛化良好。

Benign Overfitting with Quantum Kernels

  • 将局部与全局量子测量结合构造新核函数
  • 理论与实证证明其具备良性过拟合特性
  • 适合追求量子优势的机器学习研究者

核方法通过特征映射比较输入数据。量子核函数遵循相同原理:将输入数据编码为量子态,生成希尔伯特空间中的量子特征表示,再通过合适的量子电路测量估计这些态之间的内积得到核值。因此,量子核函数在经典计算机上可能难以计算,但在量子硬件上可高效实现,有望带来量子优势。然而,设计有效的量子核函数仍是重大挑战。许多量子核函数(如保真度核)存在指数集中问题,导致核矩阵接近单位矩阵,无法捕捉有意义的数据相关性,引发过拟合和泛化性能差。本文提出一种新策略,构建具有良好泛化能力的量子核函数,借鉴经典机器学习中的良性过拟合现象。引入局部-全局量子核概念,由基于小子系统测量的局部量子核与基于全系统测量的全局量子核组成。为验证该构造的有效性,我们从理论上和实验上证明了局部-全局量子核表现出良性过拟合。

原文摘要 · Abstract (English)

Kernel methods compare inputs through feature maps. Quantum kernels follow the same principle: input data are encoded into quantum states, which define quantum feature representations in Hilbert spaces. Kernel values are then obtained by estimating inner products between these states using suitable quantum circuit measurements. As a result, quantum kernels may be intractable to compute classically while remaining efficiently computable on quantum hardware, potentially leading to a quantum advantage. However, designing effective quantum kernels remains a major challenge. Many quantum kernels, such as the fidelity kernel, suffer from exponential concentration. This results in near-identity kernel matrices that fail to capture meaningful data correlations and lead to overfitting and poor generalization. In this paper, we propose a novel strategy for constructing quantum kernels that achieve good generalization performance, drawing inspiration from benign overfitting in classical machine learning. We introduce the concept of Local-Global quantum kernels, which combine two components: a local quantum kernel based on measurements of small subsystems, and a global quantum kernel derived from full-system measurements. To support the effectiveness of the proposed construction, we show theoretically and empirically that Local-Global quantum kernels exhibit benign overfitting.

量子机器学习核方法过拟合

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