量子模型泛化能力由有效维度决定，噪声可提升性能。
Effective Dimension Governs Generalization in Quantum Kernel Vision Models
- 用有效维度统一解释纠缠结构与噪声对泛化的影响。
- 噪声使测试准确率最高提升13%，在过拟合时效果显著。
- 适合研究量子机器学习泛化机制的学者参考。
近期量子视觉模型——量子视觉变换器与量子卷积网络——展现出两个令人困惑的实证现象：(i) 具有更多或更均匀分布纠缠的量子电路泛化能力更强；(ii) 注入量子噪声反而能提高测试准确率而非降低。这些现象目前被视为偶然发现，依赖网格搜索，且解释多为定性分析。本文表明，二者均为单一可度量量——噪声调制的量子特征核的有效维度 $d_{ m eff}$——的表现。以量子核视觉模型（量子特征映射由核分类器读出）为主要研究对象，我们提出一种谱分析框架，指出纠缠结构与量子噪声是调节 $d_{ m eff}$ 的两个调控旋钮；在过拟合区间，收缩 $d_{ m eff}$ 相当于岭回归正则化。通过精确分解去极化核 $K_p=(1-p)^2K+\tfrac{p(2-p)}{D}\mathbf{1}\mathbf{1}^\top$，其 $d_{ m eff}(K_p)\to1$，以及幅度阻尼的收缩结果、核机容量界、容量/对齐风险分解等机制分析，验证了纠缠实验中单调收缩的现象（非普遍证明）。在单参数去极化族中，收缩为构造上精确成立，仅用于在高达12量子比特下以机器精度验证核分解，不作为 $d_{ m eff}$ 的证据。幅度阻尼可收缩 $d_{ m eff}$，并使测试准确率提升达+13%，呈现倒U型最优区域；其效应符号在过拟合与欠拟合区间反转；噪声注入匹配显式谱滤波边界。本研究将两项零散观察整合为量子视觉模型设计的可度量原则。
原文摘要 · Abstract (English)
Recent quantum vision models-quantum vision transformers and quantum convolutional networks-report two striking but unexplained empirical phenomena: (i) ansatze with more, or more uniformly distributed, entanglement generalize better, and (ii) injecting quantum noise can improve test accuracy rather than degrade it. These observations are currently treated as curiosities, discovered by grid search and explained, if at all, by hand. We show that both are manifestations of a single, measurable quantity: the \emph{effective dimension} $d_{\rm eff}$ of the (noise-shaped) quantum feature kernel. Working primarily with quantum-kernel vision models-a quantum feature map read out by a kernel classifier-we give a spectral account in which entanglement structure and quantum noise are two knobs that move $d_{\rm eff}$; in an overfitting regime, contracting $d_{\rm eff}$ acts as ridge-like regularization. We analyze the mechanism: an \emph{exact} decomposition of the depolarized kernel $K_p=(1-p)^2K+\tfrac{p(2-p)}{D}\mathbf{1}\mathbf{1}^\top$ with $d_{\rm eff}(K_p)\to1$, a contraction result (and its boundary) for amplitude damping, a kernel-machine capacity bound, and a capacity/alignment risk decomposition; the monotone contraction operative in our entangled experiments is verified empirically, not proven in general. Along the one-parameter depolarizing family the collapse is instead exact by construction; we use it only to confirm the kernel decomposition to machine precision and at up to $12$ qubits, not as evidence for $d_{\rm eff}$. Amplitude damping contracts $d_{\rm eff}$ and lifts test accuracy by up to $+13\%$ along an inverted-U sweet spot; the effect's sign flips between the over- and under-fitting regimes; noise injection matches an explicit spectral-filtering frontier. Our results organize two reported anecdotes into a single measurable principle for designing quantum-vision models.
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