A Classical-Quantum Hybrid Architecture for Physics-Informed Neural Networks

TL;DR

This work addresses the challenge of solving ODEs with physics-informed models by marrying classical PINNs with quantum neural networks in a multiplicative/additive coupling scheme (QPINN-MAC). The authors prove a universal approximation property for the hybrid model in $L_{\mathbb{R}^M}^p(\mathcal{K})$ spaces and derive gradient bounds that prevent barren plateaus, showing $\|\nabla_{\vec{\Xi}} \mathfrak{L}\|_{p,M} = O(1/\sqrt{\mathcal{N}N})$ and a trainability condition $\mathcal{N}N \lesssim O(1/\epsilon_{grad}^2)$. The theory rests on two new lemmas: a Quantum Universal Approximation Lemma for QNodes and a Classical Universal Approximation Lemma for MLPs,combined in a main theorem that yields an overall approximation error $\|y_{MAC} - y\|_{p,M} \le \epsilon_{\mathcal{Q}} + \epsilon_{\mathcal{CQ}}$ with $\epsilon_{\mathcal{Q}} = O(1/\sqrt{\mathcal{N}N})$ and $\epsilon_{\mathcal{CQ}} = O(1/(\mathcal{N}N))$. This framework thus delivers both expressivity and trainability, enabling robust quantum-enhanced solving of ODEs and offering a scalable blueprint for principled quantum-classical hybrids in scientific computing.

Abstract

In this work, we introduce the Quantum-Classical Hybrid Physics-Informed Neural Network with Multiplicative and Additive Couplings (QPINN-MAC): a novel hybrid architecture that integrates the framework of Physics-Informed Neural Networks (PINNs) with that of Quantum Neural Networks (QNNs). Specifically, we prove that through strategic couplings between classical and quantum components, the QPINN-MAC retains the universal approximation property, ensuring its theoretical capacity to represent complex solutions of ordinary differential equations (ODEs). Simultaneously, we demonstrate that the hybrid QPINN-MAC architecture actively mitigates the barren plateau problem, regions in parameter space where cost-function gradients decay exponentially with circuit depth, a fundamental obstacle in QNNs that hinders optimization during training. Furthermore, we prove that these couplings prevent gradient collapse, ensuring trainability even in high-dimensional regimes. Thus, our results establish a new pathway for constructing quantum-classical hybrid models with theoretical convergence guarantees, which are essential for the practical application of QPINNs.

Figures (4)

• Figure 1: Architecture of the Multilayer Perceptron (MLP) used in the Physics-Informed Neural Networks (PINNs) framework. The classical neural network supports multiple hidden layers (minimum of one) with flexible layer sizes, where the number of neurons in each layer can be chosen independently without architectural constraints.
• Figure 2: Architecture of a quantum neural network node (QNode) composed of $\mathcal{N}$ variational layers. Each layer consists of: (i) rotation gates $R_{Y}\left( \theta_{k}^{j} \right)$ applied to each qubit, (ii) a Hadamard gate ($H$) on the first qubit, (iii) a conditional phase gate $CP(\phi)$ with $\phi = \pi$ (equivalent to a controlled $Z$ gate) acting on the $N$-th qubit conditioned on the others, and (iv) measurements at the end of the circuit. The number of qubits and layers is configurable.
• Figure 3: Scheme of the Quantum-Classical Hybrid Architecture with $MAC$ (multiplicative/additive) coupling modes between classical ($Out$) and quantum ($\braket{\hat{O}}_{\vec{\Theta}}$) outputs. The classical neural network admits multiple hidden layers (minimum of one) and an arbitrary number of neurons per layer, independent of the configuration of the other layers.
• Figure 4: Proposed architecture of the QPINN-MAC model (Quantum Physics-Informed Neural Network with Multiplicative/Additive Coupling), illustrating the integration of $M$ quantum nodes (QNodes) coupled to the $M$ outputs of the classical neural network via $MAC$ modes. The classical network has a flexible architecture: multiple hidden layers (minimum of one) and a variable number of neurons per layer.

Theorems & Definitions (4)

• Lemma 1: Universal Approximation of a QNode in $L_{\mathbb{R}}^{p}(\mathcal{K})$
• Lemma 2: Universal Approximation
• Theorem 1: Universal Approximation of the Hybrid QPINN-MAC Model in $L_{\mathbb{R}^{M}}^{p}(\mathcal{K})$
• Corollary 1: Mitigation of Barren Plateaus and Trainability Condition