Quantum Computing Method for Solving Electromagnetic Problems Based on the Finite Element Method

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.