🤖 AI Summary
Quantum machine learning faces two major bottlenecks: barren plateaus and noise sensitivity, compounded by the absence of a unified theoretical framework. This work proposes a novel paradigm based on Lie-algebraic generator dynamics, modeling parameterized quantum circuits as Lie subalgebras of $\mathfrak{u}(2^n)$ and characterizing trainability and expressivity through the induced Riemannian manifold geometry. The key innovation is structured Lie algebra truncation (LieTrunc), which contracts the manifold to circumvent concentration of measure while preserving non-vanishing gradients. We establish the first “geometric capacity–plateau” principle, proving that expressivity is governed by the span of generators rather than parameter count, and identify trainable regions where gradient variance decays polynomially. Experiments on 2–6 qubit systems demonstrate that LieTrunc-QNN maintains stable gradients and high effective dimensionality, fully preserving the Fubini–Study metric rank (rank=16 at $n=6$) and significantly outperforming random truncation, thereby validating the scaling law between gradient variance and effective dimension.
📝 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.