Quantum Orthogonal Separable Physics-Informed Neural Networks

TL;DR

The paper addresses the high computational burden of solving high-dimensional PDEs with physics-informed neural networks by introducing Quantum Orthogonal SPINNs (QO-SPINNs), which use a Hamming-weight-preserving quantum circuit and unary data encoding to achieve a matrix-multiplication complexity of $O(d\log d/\epsilon^2)$. It integrates Separable PINNs (SPINNs) with Quantum Orthogonal MLPs and develops a novel uncertainty quantification framework based on Spectral Normalized Gaussian Processes adapted to SPINNs, leveraging the orthogonality to avoid expensive spectral normalization. The methodology is validated on forward and inverse PDE problems (advection-diffusion, Burgers, Sine-Gordon) and demonstrates favorable accuracy and parameter efficiency in higher dimensions, along with competitive UQ performance against Monte Carlo dropout. Hardware considerations are discussed, highlighting that the theoretical speedups require large-scale, high-fidelity quantum hardware, while the work provides a practical roadmap and publicly available code for reproduction.

Abstract

This paper introduces Quantum Orthogonal Separable Physics-Informed Neural Networks (QO-SPINNs), a novel architecture for solving Partial Differential Equations, integrating quantum computing principles to address the computational bottlenecks of classical methods. We leverage a quantum algorithm for accelerating matrix multiplication within each layer, achieving a $\mathcal O(d\log d/ε^2)$ complexity, a significant improvement over the classical $\mathcal O(d^2)$ complexity, where $d$ is the dimension of the matrix, $ε$ the accuracy level. This is accomplished by using a Hamming weight-preserving quantum circuit and a unary basis for data encoding, with a comprehensive theoretical analysis of the overall architecture provided. We demonstrate the practical utility of our model by applying it to solve both forward and inverse PDE problems. Furthermore, we exploit the inherent orthogonality of our quantum circuits (which guarantees a spectral norm of 1) to develop a novel uncertainty quantification method. Our approach adapts the Spectral Normalized Gaussian Process for SPINNs, eliminating the need for the computationally expensive spectral normalization step. By using a Quantum Orthogonal SPINN architecture based on stacking, we provide a robust and efficient framework for uncertainty quantification (UQ) which, to our knowledge, is the first UQ method specifically designed for Separable PINNs. Numerical results based on classical simulation of the quantum circuits, are presented to validate the theoretical claims and demonstrate the efficacy of the proposed method.

Figures (8)

• Figure 1: A decomposition of the $\mathrm{RBS}(\theta)$ gate.
• Figure 2: Encoding circuit for $d=4$.
• Figure 3: Pyramidal circuit for $d=4$.
• Figure 4: High level representation of the whole circuit.
• Figure 5: Forward problem for 2d AD equation: Analytical solution, QO-SPINN prediction and absolute error at different time slices.

Theorems & Definitions (2)

• Theorem 1: General SPINN Lipschitz Bound
• Theorem 2: Refined Lipschitz Bound for QO-SPINN