Q-SINDy: Quantum-Kernel Sparse Identification of Nonlinear Dynamics with Provable Coefficient Debiasing

• URL: http://arxiv.org/abs/2604.16779v3
• Date: Wed, 22 Apr 2026 04:58:24 GMT
• Title: Q-SINDy: Quantum-Kernel Sparse Identification of Nonlinear Dynamics with Provable Coefficient Debiasing
• Abstract summary: Quantum feature maps offer expressive embeddings for classical learning tasks.<n>We introduce textbfQ-SINDy, a quantum kernelaugmented SINDy framework.<n>We identify a specific failure mode that arises: emphcoefficient cannibalization, in which quantum features absorb mass coefficient that rightfully belongs to the basis, corrupting equation recovery.
• Score: 1.4323566945483497
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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