A Quantum Spectral Framework for Solving PDEs

TL;DR

This paper introduces a quantum spectral method for efficiently solving second-order linear PDEs by leveraging Fourier space properties and quantum block encoding, aiming to overcome classical computational limitations.

Contribution

It presents a novel quantum subroutine that exploits structural properties in Fourier space, improving upon standard quantum matrix inversion techniques for PDE solutions.

Findings

Validated the quantum method against classical solutions.

Demonstrated potential for extending to nonlinear PDEs and advanced quantum algorithms.

Provides a foundation for quantum PDE solvers using Fourier and wavelet analysis.

Abstract

Partial differential equations (PDEs) are fundamental across numerous scientific fields. As these problems scale to high dimensions, classical numerical schemes introduce severe computational bottlenecks, known as the curse of dimensionality. Attempts to solve this problem typically rely on either classical sparsity and low-rank decompositions, or neural network surrogate models. On the other hand, Quantum Computing offers a promising alternative, as it allows us to operate in significantly larger spaces while demanding far fewer resources. In this work, we present a quantum subroutine to solve second-order linear PDEs by exploiting the structural properties of the filter in Fourier space using Quantum Block Encoding (QBE) with quantum reversible arithmetic. This approach serves as a specialized alternative to standard quantum matrix inversion, which typically relies solely on Quantum Singular Value Transformation (QSVT) without exploiting the inherent structural properties of the matrix. We validate the proposed methodology against its classical counterpart to prove its correctness. This framework provides a foundation for extending these methods toward quantum group Fourier transforms, wavelet-based analysis, and equivariant quantum neural networks (EQNNs), offering a promising path toward solving broader classes of problems, including nonlinear PDEs.