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Photonics & Electromagnetics Research Symposium - Fall (PIERS-Fall)

Investigation of Quantum Algorithms for the Solution of EFIE-Based Matrix Equation System for 3D PEC Scatterers

Rui Chen, Teng-Yang Ma, Menghan Dou, Chao-Fu Wang

Published Nov 4, 2025• 2 citations
Publisher

Abstract

In this work, we investigate the Harrow-Hassidim-Lloyd (HHL) quantum algorithm for the solution of electric field integral equation (EFIE)-based matrix equation system for analyzing electromagnetic scattering from three-dimensional (3D) perfect electrically conducting (PEC) objects. The HHL algorithm is one of the representative quantum algorithms for the solution of linear equation systems. Numerical experiments are presented to investigate the computational cost and accuracy of the HHL algorithm for solving the EFIE matrix equation system executed on the virtual simulator on the classical computers.

DOI
10.23919/piers-fall62445.2025.11394208

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In this work, we investigate the Harrow-Hassidim-Lloyd (HHL) quantum algorithm for the solution of electric field integral equation (EFIE)-based matrix equation system for analyzing electromagnetic scattering from three-dimensional (3D) perfect electrically conducting (PEC) objects. The HHL algorithm is one of the representative quantum algorithms for the solution of linear equation systems. Numerical experiments are presented to investigate the computational cost and accuracy of the HHL algorithm for solving the EFIE matrix equation system executed on the virtual simulator on the classical computers.
Conventional classical solvers are commonly used for solving matrix equation systems resulting from the discretization of SIEs in computational electromagnetics (CEM). However, the memory requirement would become a bottleneck for classical computing as the electromagentic problems become much larger. As an alternative, quantum computing has a natural "parallelization" advantage with much lower storage complexity due to the superposition and entanglement in quantum mechanics. Even though several quantum algorithms have been applied for the SIEs-based methods in the literature, the size of the matrix equation systems solvable using them is still limited. In this work, we use a hybrid quantum-classical scheme to solve the EFIE for analyzing electromagentic scattering from three-dimensional (3D) perfect electrically conducting objects with arbitrary shapes in CEM for the first time. Instead of directly solving the original EFIE matrix equation system using the quantum algorithms, the hybrid scheme first designs the preconditioned linear system and then uses a double-layer iterative strategy for its solution, where the external iteration layer builds subspace matrix equation systems with smaller dimension and the internal iteration layer solves the smaller systems using the quantum algorithms. Two representative quantum algorithms, HHL and VQLS, are considered in this work, which are executed on the quantum simulator and quantum computer platforms. We present the theoretical time complexity analysis of the hybrid quantum-classical scheme and perform numerical experiments to investigate the accuracy and efficiency of the hybrid scheme. The results show that the computational complexity of the hybrid VQLS-classical scheme is lower than the conventional fast solvers in classical computing, which indicates the hybrid scheme is more promising for analyzing large-scale electromagnetic problems.
Quantum simulation of electromagnetic (EM) structures is still in its infancy. In this paper, we investigate the possibility of applying quantum computing to solve the finite element equations for EM problems. Specifically, we propose to leverage a milestone in quantum computing, i.e., the Harrow/Hassidim/Lloyd (HHL) algorithm, to solve the finite element equations in the EM domain. To do this, we first present a systematic method to reformulate the finite element equations into quantum computation format and apply the HHL algorithm to solve the new equations. Taking advantage of the special properties of finite element matrices in EM problems, we then devise a stepwise algorithm to efficiently determine the suitable values for the hyperparameters of HHL. The proposed method can in theory solve the EM finite element equations in logarithmic time relative to the size of the finite element matrix. This is exponentially faster than that achievable by classical computation as the matrix size increases. A two-dimensional EM example is used to illustrate how the proposed method can be used to find the solutions to EM problems through quantum computation.
Quantum computing in the electromagnetic (EM) field represents a transformative frontier with significant potential for advancement. Recent studies have demonstrated the potential of quantum computing for solving waveguide mode problems, solving finite element method (FEM) equations for EM problems, and scaling the FEM solution. Among these applications, the Harrow–Hassidim–Lloyd (HHL) algorithm is an algorithm with large quantum advantages and has attracted attention. However, existing quantum methods remain constrained by the problem that the quantum-based FEM solution only obtains electric field magnitude and lacks phase information. The lack of phase information results in the inability to solve an important characteristic of EM problems: scattering parameters (S-parameters). To address the above issue, this work proposes a novel quantum method of calculating multiport S-parameter magnitudes based on FEM and HHL algorithm for microwave components. The proposed method introduces novel mathematical formulations optimized for quantum architectures. To streamline the quantum computation process and avoid the need for multiple quantum circuit simulations for each S-parameter, we design an integrated quantum circuit capable of computing magnitudes of all multiport S-parameters within a single execution. Furthermore, a specific matrix-scaling method is also developed to simplify the proposed circuits. The specific matrix-scaling method is used to improve the condition of the FEM matrix and reduce the complexity of the designed quantum circuits. The proposed method is validated using two microwave structures, demonstrating how our proposed quantum method computes S-parameter magnitudes in practical EM applications.
The Poisson equation is a fundamental equation of mathematical physics that describes the potential distribution in static fields. Solving the Poisson equation on a grid is computationally intensive and can be challenging for large grids. In recent years, quantum computing has emerged as a potential approach to solving the Poisson equation more efficiently. This article uses quantum algorithms, particularly the Harrow–Hassidim–Lloyd (HHL) algorithm, to solve the 2D Poisson equation. This algorithm can solve systems of equations faster than classical algorithms when the matrix A is sparse. The main idea is to use a quantum algorithm to transform the state vector encoding the solution of a system of equations into a superposition of states corresponding to the significant components of this solution. This superposition is measured to obtain the solution of the system of equations. The article also presents the materials and methods used to solve the Poisson equation using the HHL algorithm and provides a quantum circuit diagram. The results demonstrate the low error rate of the quantum algorithm when solving the Poisson equation.
Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed “NP-completeness” as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm’s potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.