Quantum Orthogonal Separable Physics-Informed Neural Networks

TL;DR

This paper presents Quantum Orthogonal Separable Physics-Informed Neural Networks (QO-SPINNs), integrating quantum computing to efficiently solve PDEs and develop a novel uncertainty quantification method, significantly improving computational complexity over classical approaches.

Contribution

The paper introduces a quantum-enhanced architecture for PINNs that accelerates matrix operations and provides a new uncertainty quantification technique leveraging quantum orthogonality.

Findings

Quantum algorithms reduce matrix multiplication complexity to $ ilde{O}(d ext{log}d/ ext{epsilon}^2)$.

QO-SPINNs effectively solve forward and inverse PDE problems.

The proposed UQ method eliminates the need for spectral normalization.

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 $\mathcal O(d\log d/\epsilon^2)$ complexity, a significant improvement over the classical $\mathcal O(d^2)$ complexity, where $d$ is the dimension of the matrix, $\epsilon$ 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.