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2021 International Applied Computational Electromagnetics Society Symposium (ACES)

Quantum Algorithms for the Solution of Matrix Equations in Electromagnetics

Christopher Phillips, Vladimir Okhmatovski

Published Aug 1, 2021• 5 citations
Publisher

Abstract

We present the Harrow/Hassidim/Lloyd (HHL) algorithm-a fundamental technique in quantum computing for matrix equations-and describe its application to matrix equations in electromagnetics. A method of moments (MoM) matrix equation is analyzed, and we compare the results of a simulated HHL solution with a classical solution.
DOI
10.1109/aces53325.2021.00103

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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 is a potential approach for addressing computational challenges in solving electromagnetic problems. A quantum method, called the Harrow/Hassidim /Lloyd (HHL) algorithm has recently been applied to solve the equations from finite-element-method (FEM) formulation of electromagnetic problems. Here we address a further step in this direction. Since quantum state vectors always have unit norm, the solution vector from HHL is only proportional to the final electromagnetic solutions. The magnitude of the final electromagnetic solution vector remains unavailable from HHL In this paper, a new quantum formulation is proposed to find the magnitude of the final electromagnetic solutions. The proposed method can be used to scale the HHL results to the correct magnitude of electromagnetic solutions. In this way, the proposed method is a further advance in making the quantum computing more effective in solving electromagnetic problems.
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.
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Abstract This study explores the application of the Harrow–Hassidim–Lloyd (HHL) algorithm for solving linear systems within the context of information security and quantum computing. Linear systems are fundamental in scientific, engineering, financial, and economic applications, impacting tasks such as differential equation solving, regression analysis, and Fourier transforms. In the quantum domain, effective implementation of the HHL algorithm relies on proper utilization of Pauli matrices and the choice of computational bases, including the computational basis and Hadamard basis. This work highlights how basis selection influences algorithmic efficiency and solution accuracy, offering insights for optimizing quantum computation workflows. By examining the HHL algorithm, we demonstrate its potential for accelerating linear system solutions and enhancing information processing capabilities, which may contribute to more secure and efficient computational methodologies in diverse application areas.