Quantum algorithm for partial differential equations of non-conservative systems with spatially varying parameters

TL;DR

This paper introduces a quantum algorithm for efficiently solving complex PDEs with spatially varying parameters, utilizing Hamiltonian simulation, logic minimization, and tensor-network techniques to address computational challenges in engineering models.

Contribution

It presents a novel quantum algorithm that transforms PDEs into qubit operators, efficiently constructs operators for spatially varying parameters, and implements scalable quantum circuits using tensor networks.

Findings

Validated on acoustic and heat equations with spatially varying parameters.

Reduces circuit depth through logic minimization of operator terms.

Provides a detailed quantum circuit construction recipe for PDEs.

Abstract

Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large computational models is essential for improving product performance and reducing development costs. However, solving large-scale PDEs, particularly for systems with spatially varying material properties, poses significant computational challenges. In this paper, we propose a quantum algorithm for solving second-order linear PDEs of non-conservative systems with spatially varying parameters, using the linear combination of Hamiltonian simulation (LCHS) method. Our approach transforms those PDEs into ordinary differential equations represented by qubit operators, through spatial discretization using the finite difference method. Then, we provide an algorithm that efficiently constructs the operator corresponding to the spatially varying parameters of PDEs via a logic minimization technique, which reduces the number of terms and subsequently the circuit depth. We also develop a scalable method for realizing a quantum circuit for LCHS, using a tensor-network-based technique, specifically a matrix product state (MPS). We validate our method with applications to the acoustic equation with spatially varying parameters and the dissipative heat equation. Our approach includes a detailed recipe for constructing quantum circuits for PDEs, leveraging efficient encoding of spatially varying parameters of PDEs and scalable implementation of LCHS, which we believe marks a significant step towards advancing quantum computing's role in solving practical engineering problems.