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

Haijian Shao; Dalong Zhao; Xing Deng; Wenzheng Zhu; Yingtao Jiang

cs.LGApr 3, 2026

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

Organizations: School of Computer, Jiangsu University of Science and Technology, Zhenjiang 212003, China · Department of Electrical and Computer Engineering, University of Nevada, Las Vegas, USA

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: (1) a trainability lower bound for LieTrunc-QNN, and (2) that the Fubini-Study metric rank is bounded by the algebraic span of generators, showing expressivity is governed by structure rather than parameter count. Compact Lie subalgebras also provide inherent robustness to perturbations. Importantly, we establish a polynomial trainability regime where gradient variance decays polynomially instead of exponentially. Experiments (n=2-6) validate the theory: LieTrunc-QNN maintains stable gradients and high effective dimension, while random truncation leads to metric rank collapse. At n=6, full metric rank is preserved (rank=16). Results support a scaling law between gradient variance and effective dimension. This work provides a unified geometric framework for QNN design, linking Lie algebra, manifold geometry, and optimization.

Figures & tables

\mathfrak{g}_{\text{circ}}\xrightarrow{\text{Lie reduction}}\mathfrak{g}_{\text{trunc}}\;\Longleftrightarrow\;\mathcal{M}\xrightarrow{\text{manifold contraction}}\mathcal{M}_{\text{trunc}}\;\Longleftrightarrow\;T\mathcal{M}\xrightarrow{\text{tangent reduction}}T\mathcal{M}_{\text{trunc}}

Table 1

Method	$d_{\mathrm{eff}}$	$\mathrm{rank}(G)$	Gradient Variance
Full	12.41	16	$2.00\times 10^{0}$
RandomTrunc	1.99	2	$2.75\times 10^{-1}$
LieTrunc	13.36	16	$1.16\times 10^{0}$

Table 1: Geometric properties for

n=6

qubits.

Figure 1: Left: gradient variance vs. qubit number; middle: effective dimension $d_{\text{eff}}$ ; right: variance–effective dimension product. LieTrunc stabilizes gradients while preserving expressivity.

Figure 2: Fubini–Study metric eigenvalue spectra across architectures. RandomTrunc shows spectral collapse; LieTrunc maintains full structure.

Figure 3: Task loss across qubit counts. LieTrunc-QNN achieves stable, competitive performance.

Figure 4: Fubini–Study metric spectrum at $n=6$ . RandomTrunc collapses; LieTrunc remains full-rank.

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Beyond the Expressivity-Trainability Paradox: A Dynamical Lie Algebra Perspective on Navigating Barren Plateaus in Quantum Machine Learning

Kung-Ming Lan, Edward Huang

Department of Industrial Engineering and Enterprise Information, Tunghai University, Taichung 40704, Taiwan · Department of Industrial and Systems Engineering, 3306 Shelby Center, Auburn University, Auburn, AL, USA

Quantum Machine Learning Barren Plateaus Variational Quantum Circuits Quantum Neural Networks

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As Quantum Machine Learning (QML) transitions toward practical implementation, the field faces a critical architectural bottleneck that challenges the fundamental assumptions of classical statistical learning theory. In classical deep learning, increasing model capacity typically risks overfitting. However, this study advances a counter-intuitive paradigm: unstructured contemporary QML architectures suffer from a profound state of quantum underfitting, driven by the "expressivity-trainability paradox." We demonstrate that the vast Hilbert space capacity of Parameterized Quantum Circuits (PQCs)-traditionally chased as the source of quantum advantage is the direct mathematical cause of Barren Plateaus (BPs), where gradient landscapes become exponentially flat. By synthesizing recent breakthroughs in Dynamical Lie Algebras (DLAs) and Geometric QML, we establish a comprehensive framework linking the algebraic dimension of circuit generators to their optimization dynamics. Furthermore, we empirically validate this framework on a non-linear binary classification task, illuminating a uniquely quantum manifestation of the bias-variance tradeoff: while unstructured architectures achieve near-perfect training accuracy via unscalable parameterization (quantum overfitting), embedding group-theoretic geometric priors acts as a structural regularizer. By restricting the DLA growth to a polynomial regime, our symmetry-preserving approach sacrifices raw memorization capacity to guarantee scalable, gradient-rich training landscapes, offering a robust roadmap for "Trainability-by-Design" in scalable quantum neural networks.

Abstract

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