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

arXiv: https://arxiv.org/abs/2604.16779 · HTML: https://mlsanity.com/paper/2604.16779/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 23 references that are in the map.
- First posted on arXiv 18 Apr 2026.

## Builds on (most cited first, 5 of 5)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)

## Abstract (arXiv)

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 Q-SINDy, a quantum-kernel-augmented SINDy framework, and identify a specific failure mode that arises: 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 DeltaxiP = (P^top P)-1P^top Q,hatxiQ 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, R"ossler) 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, R2Q 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.