发表机构
Muğla Sıtkı Koçman University（穆格拉·锡特基·科奇曼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对ZZ特征映射，证明小带宽下其诱导核可等效为闭式经典核，在光谱基准任务中，移除量子电路无明显预测损失，且性能与原量子核接近。
AI 中文摘要
量子核方法在调整编码带宽后会失去相对于经典核的优势，经带宽调整的量子核已被证明与径向基函数核极为相似。该观察的分析支撑基于可分编码电路，仅定性地捕捉纠缠电路。本文针对ZZ特征映射填补了这一空白：证明在小带宽 regime 下，诱导核的主导项是各向异性高斯核，其度量为M = I + π²Q，其中Q是纠缠图的无符号拉普拉斯矩阵，且二次结构在任意电路深度下均作为富比尼-施图迪度量的拉回而存在。各向异性完全取决于相位约定：在未移位约定下，度量与纠缠无关，为单位矩阵，这解释了此前报道的各向异性相似性。推导通过路径、环和完全纠缠图的直接模拟验证，相对误差低于10⁻³。对应的经典核无需拟合参数，也无需量子模拟。在两个近红外光谱基准、四个目标、五个预处理流程及每个单元100次重采样划分的场景中，与量子核相比，20个单元中有18个的配对95%自举区间包含零，测试误差的典型相对差异为2.7%，两种方法在96%的划分中选择相同的预处理流程。该简化失效的 regime 在两个数据集上始于相同带宽，且无鲁棒预测价值：将网格限制在经典 regime 可使平均测试误差提升3.8%，同等规模的随机限制无法复现该增益。因此，可移除量子电路而无可检测的预测损失，且能明确说明其计算的是哪个经典核。
英文摘要
Bandwidth-tuned quantum kernels have been shown to lose their advantage over classical kernels and to resemble radial basis function kernels, but the analytical support for that observation rests on separable encoding circuits and captures entangling circuits only qualitatively. We close this gap for the ZZ feature map. We prove that in the small-bandwidth regime the induced kernel has, to leading order, the anisotropic Mahalanobis geometry defined by M = I + pi^2 Q, where Q is the signless Laplacian of the entanglement graph, and that it admits a parameter-free Gaussian surrogate matching it through second order. The quadratic structure persists at any fixed depth as a pullback of the Fubini-Study metric. The anisotropy depends entirely on a phase convention: under the unshifted convention the metric is the identity irrespective of entanglement, which accounts analytically for the isotropic resemblance previously reported. The derivation is verified against direct simulation for path, cycle and complete graphs, with relative error below 10^-3. The corresponding classical kernel requires no fitted parameters and no quantum simulation. Compared with the quantum kernel on two near-infrared spectroscopic benchmarks, across four targets, five preprocessing pipelines and 100 resampled splits per cell, the paired 95% bootstrap interval contains zero in 18 of 20 cells, the typical relative difference in test error is 2.7%, and the two families select the same preprocessing pipeline in 96% of splits. The regime in which the reduction fails adds no robust predictive value: restricting the grid to the classical regime improves mean test error by 3.8%, a gain that an equally large random restriction does not reproduce.
Comments25 pages, 3 figures, 7 tables, including an appendix. Code and analysis outputs: https://doi.org/10.5281/zenodo.21978922
