On a Central Limit Theorem and Sanov's principle for quantum neural networks

Abstract: In this work, we study the fluctuations of a Mixture of Experts (MoE) generated by a quantum neural network trained via gradient flow on supervised learning problems. Our main results establish the Central Limit Theorem (CLT), and Sanov's principle for an MoE as the number of experts diverges. We demonstrate that the fluctuations of the empirical measure of its parameters close to its corresponding limit probability measure solve a linear transport equation. As a byproduct, we show that the MoE converges to a limit function which solves an evolution equation governed by the neural tangent kernel associated with the quantum neural network.

• The paper establishes a central limit theorem for quantum neural mixtures by analyzing parameter fluctuations under gradient flow.
• It derives Sanov’s principle to quantify large deviations in the empirical distribution of quantum circuit parameters.
• The results connect quantum NTK evolution with classical dynamics, offering insights into finite-size effects in quantum learning models.

Central Limit Theorem and Sanov’s Principle for Quantum Neural Networks

Overview

This paper investigates the asymptotic behavior and fluctuation dynamics of mixtures of quantum neural networks (QNNs) in supervised learning, employing techniques from mean-field theory, stochastic processes, and quantum machine learning. The study focuses on mixtures of experts (MoE) generated by QNNs trained via gradient flow, establishing a Central Limit Theorem (CLT) and Sanov’s principle for the empirical distribution of parameters as the number of experts $N \to \infty$. The approach provides a rigorous characterization of stochastic fluctuations in quantum neural ensembles and elucidates connections to continuity equations, neural tangent kernels (NTK), and large deviation theory.

Problem Formulation and Related Work

The paper models each expert as a parametric quantum circuit, with parameters evolved under gradient flow to minimize the empirical quadratic loss. The MoE architecture forms the ensemble average

$$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$

where $f$ is the QNN model function. Previous works have demonstrated that wide quantum networks converge to Gaussian processes under certain scaling limits [girardi2025], and bounds on convergence rates in Wasserstein distance have been established [melchor2025quantitative]. Efficient classical computation schemes for QNN NTKs have also been introduced [hernandez2025efficientclassicalcomputationneural], showing limitations on quantum advantage in MoE settings.

The mean-field limit frames each expert as an interacting particle. In the infinite-width limit, the empirical measure over parameters converges to a probability evolution governed by a nonlinear continuity PDE, analogously to classical neural networks [sirignano2021meanfieldanalysisdeep, mei2019meanfieldtheorytwolayersneural]. This propagation-of-chaos result connects microscopic parameter dynamics to macroscopic functional evolution.

Main Results

Central Limit Theorem for Quantum Mixtures

The primary result is a CLT for the rescaled fluctuation

$\delta_t^N = \sqrt{N}(\mu_{\Theta_t^N} - \mu_t)$

of the empirical parameter measure $\mu_{\Theta_t^N}$ around its limit $\mu_t$, as $N \to \infty$. The fluctuation process converges in law to a limiting signed measure $\delta_t$ solving the linearized transport PDE:

$$\frac{\partial \delta_t}{\partial t} = -\nabla_\theta \cdot \big( \nabla \mathcal{V}(\theta, \mu_t)\, \delta_t + \mu_t \nabla \mathcal{G}(\theta, \delta_t) \big)$$

where $\mathcal{V}$ encodes mean-field vector fields and $$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$0 captures nonlinear interactions.

The rate of convergence $$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$1 surpasses previous Wasserstein-2 bounds, especially in regimes where parameter dimension $$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$2.

Sanov’s Principle and Large Deviations

An explicit large-deviation principle is established for empirical measure trajectories $$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$3, with rate function given by relative entropy. Sanov’s theorem provides exponential equivalence between the process with correlated parameters and a system of independent processes, with speed $$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$4 for $$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$5. This analysis exploits propagation-of-chaos and continuity equation frameworks.

Functional Limit for MoE Output

The fluctuation of the MoE output propagates the convergence rate:

$$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$6

which quantifies finite-size stochastic effects in the ensemble output.

Limit Dynamics via NTK

The limiting function $$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$7 evolves according to a kernel-based ODE driven by the quantum NTK:

$$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$8

with $$F_N(\Theta, x) = \frac{1}{N} \sum_{i=1}^N f(\theta^i, x)$$9 defined as expected gradient correlations over $f$0. This dynamics generalizes classical NTK theory to quantum architectures, revealing connections to quantum encoding and training mechanisms.

Numerical and Theoretical Findings

• CLT rate: The empirical parameter measure convergence is quantified as $f$1, superior to Wasserstein-2 rates for large $f$2.
• Gaussian process initialization: At $f$3, fluctuations converge to a Gaussian law in functional space.
• Linearized dynamics for fluctuations: The evolution of stochastic fluctuations differs from nonlinear limit dynamics, being governed by a deterministic linear transport PDE.
• Large deviations: The empirical measure process exhibits a large deviation principle with relative entropy rate, providing robustness to modeling errors in finite $f$4 settings.

Implications and Future Directions

The results rigorously characterize asymptotic fluctuations of quantum MoEs, establishing stochastic convergence rates and underlying transport PDEs. The findings enable:

• Quantitative analysis of stochastic generalization errors and convergence rates in quantum ensembles.
• Theoretical foundation for NTK-based optimization and kernel methods in QML settings.
• Application of large deviation theory to analyze rare event probabilities under quantum parameter dynamics.
• Insights into trainability and representation learning in overparameterized quantum architectures, beyond lazy training regimes.

Future investigations should focus on:

• Uniform-in-time convergence properties, relevant for late-stage training.
• Joint infinite-width and depth limits, integrating more complex quantum architectures and entanglement structures.
• Infinite-dimensional parameter limits, leveraging continuity equations in Hilbert spaces.
• Extension to quantum-classical hybrid models and MoE paradigms with adaptive expert selection.

Conclusion

The paper presents a rigorous probabilistic and PDE-theoretic analysis of quantum neural network ensembles, deriving a CLT, Sanov’s principle, and large deviation properties under mean-field dynamics. The stochastic behavior of quantum MoEs is characterized at both the parameter and functional levels, with explicit convergence rates and evolution equations. The approach bridges quantum machine learning, statistical physics, and stochastic analysis, advancing the theoretical understanding of quantum neural architectures and their asymptotic properties (2606.21721).