[2604.02697] LieTrunc-QNN: Lie Algebra Truncation and Quantum Expressivity Phase Transition from LiePrune to Provably Stable Quantum Neural Networks

Computer Science > Machine Learning arXiv:2604.02697 (cs) [Submitted on 3 Apr 2026] Title:LieTrunc-QNN: Lie Algebra Truncation and Quantum Expressivity Phase Transition from LiePrune to Provably Stable Quantum Neural Networks Authors:Haijian Shao, Dalong Zhao, Xing Deng, Wenzheng Zhu, Yingtao Jiang View a PDF of the paper titled LieTrunc-QNN: Lie Algebra Truncation and Quantum Expressivity Phase Transition from LiePrune to Provably Stable Quantum Neural Networks, by Haijian Shao and 4 other authors View PDF HTML (experimental) Abstract:Quantum Machine Learning (QML) is fundamentally limited by two challenges: barren plateaus (exponentially vanishing gradients) and the fragility of parameterized quantum circuits under noise. Despite extensive empirical studies, a unified theoretical framework remains lacking. We introduce LieTrunc-QNN, an algebraic-geometric framework that characterizes trainability via Lie-generated dynamics. Parameterized quantum circuits are modeled as Lie subalgebras of u(2^n), whose action induces a Riemannian manifold of reachable quantum states. Expressivity is reinterpreted as intrinsic manifold dimension and geometry. We establish a geometric capacity-plateau principle: increasing effective dimension leads to exponential gradient suppression due to concentration of measure. By restricting to structured Lie subalgebras (LieTrunc), the manifold is contracted, preventing concentration and preserving non-degenerate gradients. We prove two main results:...