发现量子核函数性能瓶颈的统一指标，可预测其泛化能力。
Spectral Anatomy of Quantum Gaussian Process Kernels
- 用归一化谱熵统一解释量子核的性能问题。
- 高熵适合平滑目标，低熵适配量子数据目标。
- 该指标在硬件与模拟器间高度一致，适合实际部署评估。
近期研究揭示了量子高斯过程（QGP）的两个关键现象：基于HHL的回归方法在典型条件下无法实现指数加速；同时高度表达的量子核会导致后验病态，破坏贝叶斯优化。本文证明，这两种看似无关的现象均由核矩阵的归一化谱熵 $S(K)//log n$ 统一调控。建立了Nyström近似误差的Cauchy--Schwarz尾界，导出了基于Bach自由度 $d_σ(K)$ 的有限样本方差收缩恒等式，并通过目标在核特征基中的内在维度刻画了目标相关最优熵。实证显示，该诊断具有核无关性：在去量化、ECE和方差收缩图中，硬件高效、匹配门、IQP以及RBF/Matérn/RFF/深度核等各类核均收敛至相同 $S//log n$ 曲线。对于平滑目标，最优负对数似然对应高熵；对带限量子数据则为低熵。该诊断从模拟器成功迁移至IBM Heron硬件，在24个配置下 $n_q = 4$ 时，中位绝对误差3.2%，均值5.2%；匹配门与IQP平均误差均低于5%，仅一个高保真配置出现30%异常，重跑后降至0.5%（归因校准漂移）；在另一赫龙后端均值误差2.7%，$n_q = 6$ 扩展实验均值误差1.7%。全程未使用误差缓解。
原文摘要 · Abstract (English)
Two recent results have reshaped quantum Gaussian processes (QGPs). On the one hand, \citet{lowe2025assessing} rule out the exponential speedups claimed by HHL-based QGP regression in the typical, well-conditioned regime; on the other, an independent line of work shows that highly expressive quantum kernels suffer posterior pathologies that break Bayesian optimization. We show that these seemingly unrelated phenomena are governed by the same quantity: the normalized spectral entropy $S(K)/\log n$ of the kernel Gram matrix. We prove a Cauchy--Schwarz tail bound on Nyström approximation error, a finite-sample variance-contraction identity in terms of Bach's degrees of freedom $d_σ(K)$, and a characterization of the \emph{target-dependent} optimal entropy via the intrinsic dimension of the target in the kernel eigenbasis. Empirically, the diagnostic is kernel-agnostic: hardware-efficient, matchgate, IQP \emph{and} RBF/Matérn/RFF/deep-kernel families all collapse onto identical $S/\log n$ curves on dequantization, ECE, and variance-contraction panels. The NLL sweet spot lives at high entropy for smooth targets and at low entropy for band-limited quantum-data targets. The diagnostic transfers from simulator to IBM Heron hardware with median absolute error $3.2\%$ and mean $5.2\%$ in $S/\log n$ across $24$ configurations at $n_q = 4$, with matchgate and IQP within $5\%$ mean and a single HE configuration returning a $30\%$ outlier that drops to $0.5\%$ on rerun (attributed to calibration drift); the same diagnostic transfers to a second Heron backend (mean error $2.7\%$) and to a $n_q = 6$ scale-up on the original backend (mean error $1.7\%$). No error mitigation is applied throughout.
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