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IEEE MTT-S International Microwave Symposium digest

Quantum Method for Scaling the Finite Element Based Quantum Solutions of Electromagnetic Problems

Louis Zhang, Qi‐Jun Zhang

Published Jun 19, 2022• 6 citations
Publisher

Abstract

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.

DOI
10.1109/ims37962.2022.9865495

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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 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 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.
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.
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.