Solution of the Electric Field Integral Equation Using a Hybrid Quantum-Classical Scheme: Investigation of Accuracy and Efficiency

TL;DR

The paper tackles the memory bottleneck of EFIE-based solvers in computational electromagnetics by proposing a hybrid quantum-classical approach that preconditions the system and solves smaller subspace problems via quantum algorithms. It combines a double-layer iteration—an exterior subspace construction and an interior quantum solve—with HHL or VQLS as the quantum engines, and provides time-complexity analyses for both hybrids. Numerical experiments on a statevector simulator and a real quantum computer show that the hybrid VQLS-classical scheme can outperform classical solvers in scaling while maintaining high accuracy, indicating potential for large-scale electromagnetic analyses. The work demonstrates a viable near-term pathway to leverage quantum computation in 3D PEC scattering problems, highlighting practical considerations such as preconditioning, subspace sizing, and hardware noise.

Abstract

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.

Figures (6)

• Figure 1: Comparison of the RCS results of the PEC unit sphere obtained after using the hybrid HHL- and VQLS-classical schemes with those obtained using the Mie series analytical solution with respect to $\theta =[0^{\mathrm{o}}, 180^{\mathrm{o}}]$ at 300 MHz.
• Figure 2: The surface current distributions of the PEC unit sphere obtained after using (a) the Mie series solution, (b) the hybrid HHL-classical scheme, (c) the difference between (a) and (b), (d) the hybrid VQLS-classical scheme, and (e) the difference between (a) and (d).
• Figure 3: Comparison of the time cost of the hybrid VQLS-classical scheme with its theoretical computational complexity presented in (13) with respect to the number of unknowns.
• Figure 4: The topology structure of the 72-qubit quantum chip.
• Figure 5: Comparison of the RCS results of the PEC flower-shaped scatterer obtained after using the hybrid VQLS-classical scheme with those obtained using the QR decomposition method with respect to $\theta =[0^{\mathrm{o}}, 180^{\mathrm{o}}]$ at 69.89 MHz.