发表机构
Università degli Studi di Palermo; Università degli Studi di Milano; Queen’s University Belfast（巴勒莫大学; 米兰大学; 贝尔法斯特女王大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析量子线性回归中有限采样导致的病态问题，针对量子极限学习机揭示采样噪声带来的误差规律，指出数据集大小与单态统计量的权衡关系，为量子学习训练提供基准。
AI 中文摘要
量子保真度核（quantum fidelity kernels）与量子极限学习机（quantum extreme learning machines）具有相同的基本架构：一个固定的量子设备生成特征，这些特征通过经典训练的线性模型进行组合。我们将这种广泛的非变分设置称为量子线性回归。这些模型的一个决定性特征是，量子特征无法直接获取，必须通过有限次数的采样（shots）来估计。我们证明，有限的统计量会使量子线性回归不可避免地出现病态问题。理想的特征向量被限制在由希尔伯特空间维度决定的子空间中，而有限采样的估计值通常会获得该子空间之外的分量。训练过程随后会试图拟合在无限统计量极限下消失的方向，导致学习到的模型对采样噪声越来越敏感。我们针对在$n_{\rm tr}$个态上训练、每个态使用$N$次采样的量子极限学习机分析了这一机制。尽管存在病态问题，但当训练目标是精确的时，预测仍然可控。有限采样会产生系统性的预测误差，该误差随$N$呈二次方减小，而由随机训练测量引起的波动则随总训练预算呈反比减小。因此，与特征完全已知的标准线性回归不同，仅增加数据集大小无法消除采样噪声导致的误差。如果训练目标也存在噪声（例如由于态制备误差），原本隐藏的方向会变得活跃，测试误差随后会随$N$增大，我们将这种行为归因于一种量子过拟合形式。我们的结果揭示了非变分量子学习中数据集大小与单态统计量之间的基本权衡，并为资源感知的训练和正则化策略提供了基准。
英文摘要
Quantum fidelity kernels and quantum extreme learning machines share the same basic architecture: a fixed quantum device generates features that are combined through a classically trained linear model. We refer to this broad non-variational setting as quantum linear regression. A defining aspect of these models is that the quantum features are not directly available, but must be estimated from a finite number of shots. We show that finite statistics makes quantum linear regression unavoidably ill-conditioned. Ideal feature vectors are confined to a subspace determined by the dimension of the Hilbert space, whereas finite-shot estimates generically acquire components outside this subspace. Training then attempts to fit directions that disappear in the infinite-statistics limit, making the learned model increasingly sensitive to sampling noise. We analyze this mechanism for quantum extreme learning machines trained on $n_{\rm tr}$ states with $N$ shots per state. Despite the ill-conditioning, predictions remain controlled when the training targets are exact. Finite sampling produces a systematic prediction error that decreases quadratically with $N$, while fluctuations induced by the random training measurements decrease inversely with the total training budget. Thus, unlike in standard linear regression with exactly known features, increasing the dataset size alone cannot remove the error caused by sampling noise. If the training targets are also noisy, due for instance to state-preparation errors, the otherwise hidden directions become active. The test error can then increase with $N$, a behavior that we ascribe to a form of quantum overfitting. Our results expose a fundamental tradeoff between dataset size and per-state statistics in non-variational quantum learning and provide a baseline for resource-aware training and regularization strategies.
CommentsMain: 15 pages, 6 figures. Supplementary: 37 pages, 6 figures
