量子核增强动态识别，解决系数偏移问题并实现精确方程恢复
Q-SINDy: Quantum-Kernel Sparse Identification of Nonlinear Dynamics with Provable Coefficient Debiasing
- 引入量子核增强的SINDy框架，通过正交化消除量子特征对系数的吞噬效应
- 在6个经典系统上验证，正交化后恢复精度媲美原始SINDy，未校正时真阳性率下降100%
- 提出可量化偏移严重性的诊断指标，适用于量子与经典核方法对比
量子特征映射为经典学习任务提供高维表达能力，将其融入稀疏非线性动力学识别（SINDy）是自然但尚未探索的方向。本文提出 extbf{Q-SINDy}，一种量子核增强的SINDy框架，并发现一种特定失效模式： extit{系数吞噬}，即量子特征吸收本应属于多项式基的系数质量，导致方程恢复失真。我们推导出精确的偏移公式 $Δξ_P = (P^ op P)^{-1}P^ op Q\ ildeξ_Q$，并证明在拟合时将量子特征正交化于多项式列空间可完全消除该偏差。数值验证在多个系统中达到机器精度（<10^{-12}）。实验覆盖六个典型动力系统（杜芬、范德波尔、洛伦兹、洛特卡-沃尔泰拉、三次振子、罗斯勒）和三种量子特征编码（ZZ角编码、IQP、数据重加载），正交化Q-SINDy始终保持与原始SINDy相当的结构恢复能力，而未校正增强使真阳性率降低高达100%。提出的动态感知诊断指标 $R^2_Q$ for $\ar X$ 在统计上显著预测吞噬程度（皮尔逊相关 $r=0.70$, $p=0.023$）。20种超参数配置下的RBF经典核对比显示其失效更严重，排除了特征数量的影响。正交化在每门2%去极化噪声下仍稳健，且框架无需修改即可扩展至伯格斯方程。
原文摘要 · Abstract (English)
Quantum feature maps offer expressive embeddings for classical learning tasks, and augmenting sparse identification of nonlinear dynamics (SINDy) with such features is a natural but unexplored direction. We introduce \textbf{Q-SINDy}, a quantum-kernel-augmented SINDy framework, and identify a specific failure mode that arises: \emph{coefficient cannibalization}, in which quantum features absorb coefficient mass that rightfully belongs to the polynomial basis, corrupting equation recovery. We derive the exact cannibalization-bias formula $Δξ_P = (P^\top P)^{-1}P^\top Q\,\hatξ_Q$ and prove that orthogonalizing quantum features against the polynomial column space at fit time eliminates this bias exactly. The claim is verified numerically to machine precision ($<10^{-12}$) on multiple systems. Empirically, across six canonical dynamical systems (Duffing, Van der Pol, Lorenz, Lotka-Volterra, cubic oscillator, Rössler) and three quantum feature map architectures (ZZ-angle encoding, IQP, data re-uploading), orthogonalized Q-SINDy consistently matches vanilla SINDy's structural recovery while uncorrected augmentation degrades true-positive rates by up to 100\%. A refined dynamics-aware diagnostic, $R^2_Q$ for $\dot X$, predicts cannibalization severity with statistical significance (Pearson $r=0.70$, $p=0.023$). An RBF classical-kernel control across 20 hyperparameter configurations fails more severely than any quantum variant, ruling out feature count as the cause. Orthogonalization remains robust under depolarizing hardware noise up to 2\% per gate, and the framework extends without modification to Burgers' equation.
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