参数化交互式量子分类器的傅里叶分析
Fourier Analysis of Parametrized Interactive Quantum Classifiers
- Federal Rural University of Pernambuco（伯南布哥联邦农村大学）
- Federal University of Pernambuco（伯南布哥联邦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文推导了参数化交互式量子分类器的闭式信道表达式，揭示哈密顿量参数对输出的傅里叶控制，提出广义矩阵编码，并在基准上提升非线性分类性能。
AI中文摘要：
交互式量子分类器（IQCs）构成了一类受开放量子系统启发的量子机器学习模型家族，其中目标量子比特与环境之间的相互作用由哈密顿量描述。先前的工作引入了替代的哈密顿量参数化，并经验性地表明它们可以提高分类性能，但这些参数在所得分类器中的作用仍鲜为人知。在本工作中，我们推导了具有单个目标量子比特的参数化IQC所生成的约化量子信道的闭式表达式。该解析解明确揭示了哈密顿量参数如何控制分类器输出的常数、正弦和余弦分量，从而建立了所诱导特征映射的傅里叶解释。这一分析激发了一类广义的哈密顿量编码家族，包括矩阵参数化的环境哈密顿量，其傅里叶分量依赖于输入特征的线性组合，从而能够实现不可分离的傅里叶结构。在合成和真实世界数据集上的数值实验表明，所提出的模型可以在多个非线性基准上提高分类性能。广义矩阵编码在评估的基准中实现了最强的总体性能，而一个更简单的四参数扩展通常以显著更少的可训练参数达到相当的性能。我们还使用基于保真度的标准可表达性度量来表征生成的状态系综，发现全局可表达性并不能直接预测分类性能。我们的结果为交互式量子分类器中的参数化哈密顿量提供了解析表征，并确立了傅里叶分析作为理解和设计开放系统启发的量子学习模型的有用框架。
英文摘要：
Interactive Quantum Classifiers (IQCs) constitute a family of quantum machine learning models inspired by open quantum systems, in which the interaction between a target qubit and an environment is described by a Hamiltonian. Previous works introduced alternative Hamiltonian parameterizations and showed empirically that they can improve classification performance, but the role of these parameters in the resulting classifier remains poorly understood. In this work, we derive a closed-form expression for the reduced quantum channel generated by a parametrized IQC with a single target qubit. The analytical solution explicitly reveals how the Hamiltonian parameters control the constant, sine, and cosine components of the classifier output, establishing a Fourier interpretation of the induced feature map. This analysis motivates a generalized family of Hamiltonian encodings, including matrix-parameterized environmental Hamiltonians whose Fourier components depend on linear combinations of input features, thereby enabling non-separable Fourier structures. Numerical experiments on synthetic and real-world datasets show that the proposed models can improve classification performance on several nonlinear benchmarks. The generalized matrix encoding achieves the strongest aggregate performance in the evaluated benchmark, while a simpler four-parameter extension often attains comparable performance with substantially fewer trainable parameters. We additionally characterize the generated state ensembles using the standard fidelity-based expressibility measure, finding that global expressibility does not directly predict classification performance. Our results provide an analytical characterization of parametrized Hamiltonians in Interactive Quantum Classifiers and establish Fourier analysis as a useful framework for understanding and designing open-system-inspired quantum learning models.
