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Photonics & Electromagnetics Research Symposium (PIERS)

Quantum Computing Method to Solve the Finite Element Equation in EM Problems

Tian Jian Peng, Jianan Zhang, Jian Wei You, Tie Jun Cui

Published Apr 21, 2024• 1 citation
Publisher

Abstract

Quantum computing, or quantum computation, has garnered increasing attention within the computational community in recent years, owing to its remarkable efficacy in addressing large-scale problems. In specific scenarios, quantum algorithms exhibit exponential enhancements over their classical counterparts executed on classical computing platforms. Quantum computing introduces myriad opportunities for simulating intricate physical systems, including but not limited to electromagnetic (EM) structures. Despite the extensive research devoted to quantum computing in contemporary literature, the exploration of quantum simulation techniques for EM structures remains relatively limited. In this paper, we will describe a quantum computing method for solving EM problems through finite element equations. The performance of this method will be verified by an EM example.

DOI
10.1109/piers62282.2024.10617947

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Quantum computing, or quantum computation, has garnered increasing attention within the computational community in recent years, owing to its remarkable efficacy in addressing large-scale problems. In specific scenarios, quantum algorithms exhibit exponential enhancements over their classical counterparts executed on classical computing platforms. Quantum computing introduces myriad opportunities for simulating intricate physical systems, including but not limited to electromagnetic (EM) structures. Despite the extensive research devoted to quantum computing in contemporary literature, the exploration of quantum simulation techniques for EM structures remains relatively limited. In this paper, we will describe a quantum computing method for solving EM problems through finite element equations. The performance of this method will be verified by an EM example.
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.
Applying quantum computation to solve electromagnetic (EM) problems is still at an early age. Recently, an initial study on applying quantum computation to solve the finite element method (FEM) equations in the EM domain has been made. This article makes a further development beyond the initial study. Specifically, we first develop an approach to systematically prepare the quantum state to represent the right-hand side (RHS) vector of the finite element equation in EM problems. Then, to reduce the number of gates needed in the quantum state preparation process, we propose a quantum-gate-reduction method, which explores the fact that the FEM cells in the input port are a small portion of the total cells in the 3-D EM structure and that the number of gates needed for state preparation depends on the RHS vector’s sparsity pattern. Based on the proposed quantum-gate-reduction method, we further derive the upper and lower bounds analytically for the number of gates needed in the quantum state preparation circuit. Furthermore, to deal with the large condition number of the finite element matrix in EM, we leverage a matrix preconditioner to modify the original linear equations, so as to reduce the number of qubits required in using quantum computation to solve EM problems. Two EM examples are used to illustrate how the proposed quantum computing method can be used to find solutions to EM problems.
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.
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.
Quantum approach to address electromagnetic (EM) computing presents a promising problems. Recently, preliminary conducted on the application of the Harrow/Hassidim/Lloyd (HHL) quantum method to solve finite-element-method (FEM) equations in the EM domain. Since quantum measurements are typically limited to measuring the magnitude rather than the phase, only the magnitude of the electric field can be calculated by existing quantum computing methods. However, without access to phase information of the electric field, the final S-parameters cannot be solved in a traditional computing way. This paper proposes a new quantum method to calculate magnitudes of the two-port S-parameters of lossless waveguides. Novel formulations are derived to leverage two inner products to develop the complete calculation. The correctness and accuracy of the proposed method are demonstrated using a waveguide example.
Numerical simulations of physical systems are found in many industries, as they currently play a crucial role in product development. There are many numerical methods for solving differential equations that describe the underlying physics behind the mathematical models in the simulation, among which, the finite element method (FEM) is one of the most commonly used. Although in many applications the FEM seems to provide an acceptable solution to the problem, there are still many complex real-life processes that can be challenging to simulate numerically due to their complexity and large size. Recently, there has been a shift in research towards efficiently applying quantum algorithms in finite element analysis (FEA), as the potential and speedup that they could offer have been shown, but little to no effort has been made towards the applicability and cost efficiency of these algorithms in real-world quantum devices. In this paper, we propose a cost-efficient method for applying quantum algorithms in FEA for industrial problems post-processed by classical algorithms in order to address the limitations of available quantum hardware and their cost when accessing them through different cloud-based services. We carry this out by approximating the solution of the initially large system with a suitable quantum algorithm and using the obtained solutions to generate a set of reduced-order models (ROMs) that are much smaller in complexity and size than the original model. This allows the simulation of the original model with different parameter sets and excitations to be run efficiently on classical computers without having the need to access quantum subroutines again. This way, we have reduced the usage of quantum hardware (and thus the development cost) while still taking advantage of its quantum speedup.