局部与多尺度策略以缓解量子核中的指数集中
Local and Multi-Scale Strategies to Mitigate Exponential Concentration in Quantum Kernels
AI总结：
本文提出局部和多尺度策略以缓解量子核中的指数集中问题，并通过实验验证了这些方法在不同数据集上的有效性。
摘要：
基于保真度的量子核提供了一个量子特征映射与经典核方法之间的直接接口，但它们可能会表现出指数集中：随着系统规模或电路表达性的增加，Gram矩阵趋近于单位矩阵并抑制了信息相似性结构。我们对Qiskit中实现的两种缓解策略进行了实证研究：(i) 局部（块状）核，用于聚合子系统相似性，以及(ii) 多尺度核，用于在不同块粒度上混合局部和全局相似性。我们对几个表格数据集进行了基准测试，使用匹配的预处理、分割和SVM协议，扫描特征维度d∈{4,6,…,20}。我们报告了基于非对角核统计的集中诊断、通过有效秩的谱丰富性以及与标签的中心对齐。在所有数据集中，局部和多尺度构造一致地缓解了集中问题，并相对于全局保真度基线产生了更丰富的核谱，而对分类准确率的影响则取决于数据集和维度。
英文摘要：
Fidelity-based quantum kernels provide a direct interface between quantum feature maps and classical kernel methods, but they can exhibit exponential concentration: with increasing system size or circuit expressivity, the Gram matrix approaches the identity and suppresses informative similarity structure. We present an empirical study of two mitigation strategies implemented in Qiskit: (i) local (patch-wise) kernels that aggregate subsystem similarities, and (ii) multi-scale kernels that mix local and global similarity across patch granularities. We benchmark baseline, local, and multi-scale kernels under matched preprocessing, splits, and SVM protocols on several tabular datasets, sweeping the feature dimension $d\in\{4,6,\dots,20\}$. We report concentration diagnostics based on off-diagonal kernel statistics, spectral richness via effective rank, and centered alignment with labels. Across datasets, local and multi-scale constructions consistently mitigate concentration and yield richer kernel spectra relative to the global fidelity baseline, while the impact on classification accuracy depends on the dataset and dimension.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
