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.
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.
We develop a hybrid classical-quantum method for solving the Lorenz system. We use the forward Euler method to discretize the system in time, transforming it into a system of equations. This set of equations is solved by using the Variational Quantum Linear Solver (VQLS) algorithm. We present numerical results comparing the hybrid method with the classical approach for solving the Lorenz system. The simulation results demonstrate that the VQLS method can effectively compute solutions comparable to classical methods. The method is easily extended to solving similar nonlinear differential equations.
Computational electromagnetics plays a central role in many industrial applications but often requires substantial computational resources, particularly when fine spatial discretizations are needed. While classical approaches remain the standard, quantum computing offers the potential to accelerate large-scale simulations by encoding them with a limited number of qubits.
Here, we investigate the performance and resource scaling of the Harrow-Hassidim-Lloyd (HHL) and Quantum Singular Value Transformation (QSVT) algorithms for solving linear systems generated by the finite-difference time-domain (FDTD) method, a widely adopted numerical scheme for discretizing Maxwell's equations. We benchmark their performance across representative industrial use cases, including radar propagation, lens simulations, and beamforming processes. Our results demonstrate the validity of the approaches, achieving state infidelities smaller than 2⋅10−3 with success probabilities greater than 10−3, compatible with practical quantum state sampling. Overall, we observe that the QSVT method consistently delivers higher accuracy.
We further observe that the condition number of the linear matrix, a key factor governing the performance of quantum solvers, saturates as the number of spatial lattice points increases. This implies that the spatial grid can be scaled to realistic industrial dimensions without increasing the HHL or QSVT circuit depth due to ill-conditioned matrices.
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.
Electromagnetic analysis based on the finite element method is widely used in antenna design. At higher frequencies, however, the required mesh refinement makes solving large sparse linear systems the main contributor to computational cost. Quantum linear system solvers offer a potential means of accelerating this step, but their output is a superposition state: reconstructing all components of the classical solution vector would forfeit the quantum speedup. Prior studies have proposed methods for obtaining physical quantities such as scattering cross sections and scattering parameters from the solution state. To support antenna design using fault-tolerant quantum computers, we formulate the total radiated power, the radiated power within a prescribed solid angle, and the dielectric and conductive losses as observables for efficient evaluation from a quantum solution state. These observables avoid the measurement cost of reconstructing all components of the output quantum state. We develop an end-to-end hybrid quantum--classical workflow that combines finite-element electromagnetic analysis, matrix preconditioning, a quantum linear system solver, and the evaluation of antenna design metrics. To validate the workflow, we performed numerical experiments evaluating these observables for an axisymmetric coaxially fed monopole antenna. Under idealized conditions, the relative errors in the total radiated power and the dissipated power with respect to the classical finite-element solution were on the order of 10−3, confirming the consistency of the workflow. The proposed method is expected to provide a basis for using quantum linear system solvers to evaluate antenna performance and, in the future, to design high-frequency wireless devices.
Quantum methods for solving partial differential equations (PDEs) generally fall into a few general categories, each leveraging different quantum mechanisms to achieve potential speedups. One well-known approach is the quantum linear system solver, epitomized by the Harrow-Hassidim-Lloyd (HHL) algorithm, which uses phase estimation and eigenvalue inversion to solve large, sparse linear systems arising from discretized linear PDEs [1]. This method can yield exponential improvements over classical solvers under certain conditions (mostly when the matrix is well-conditioned and sparse). More recent developments, such as Quantum Singular Value Transformation (QSVT), provide a framework for matrix inversion in a more flexible way, often improving on the cost and error dependence of earlier solvers [2]. Time-dependent PDEs can be tackled via Hamiltonian simulations. The Trotter-Suzuki method decomposes the exponential of a Hamiltonian into products of exponentials of simpler sub-Hamiltonians, allowing a piecewise approximation of continuous dynamics [3]. Continuous-time quantum walks, which treat a PDE operator as a Hamiltonian that governs the walk, similarly simulate processes like diffusion or wave propagation on discretized grids or graphs [4]. Beyond direct simulation, variational quantum algorithms have emerged as a promising route in the noisy intermediate-scale quantum era. These approaches encode the PDE solution as a parameterized quantum state and optimize the parameters via classical feedback, thereby minimizing a cost function tied to the PDE residual or energy functional [5]. Quantum annealing or adiabatic methods, meanwhile, can solve optimization problems derived from PDE constraints by adiabatically evolving a simple initial Hamiltonian to the final problem Hamiltonian and staying in the system's ground state [3]. In our study, we implement a new parameterized quantum model, specifically designed to solve large linear systems on quantum hardware. This model is less deep and computationally faster than the proposed one in [5], reducing the complexity and speeding up convergence times. Such improvements are particularly valuable in cases where classical algorithms may fail due to the problem's size or complexity. Ongoing research is focusing on generalized quantum models that can be adapted to practical PDE problems for solving large electromagnetic problems where classical computers can fail.
Numerical solvers for Partial Differential Equations (PDEs) are of great interest in various domains, e.g., in aerodynamics or for transport equations in lectrochemistry [1], and the need for fine-grained solutions of on-linear PDEs is growing. While subspace methods allow for a dimensionality reduction, non-linear PDEs require linearization schemes, such as the Carleman-linearization [2], resulting in linear systems of exponential dimensionality, operating on the limits of classical methods. Motivated by Krylov-subspace methods [3], which find approximate solutions in iteratively growing subspaces, the aim of this talk is to investigate the potential of two existing methods from quantum computing to compose a NISQ-era hybrid quantum-classical algorithm. Firstly, non-linear quantum computing (QNPUs) [4,5] and secondly Quantum Subspace Expansion (QSE) [6], both promising tools towards more scalable computations. The use of QNPUs enables linearization in a tensor-product-subspace using ancilla qubits, while offering efficient gate-based implementations. On the other hand, QSE measures high-dimensional overlaps on a quantum computer. The combination of these two methods yields the possibility of encapsulating the high-dimensional steps of linearization and subspace projection on a quantum device. As a result, only a lower-dimensional subspace problem remains to be solved on classical hardware.