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arXiv:2510.03596 · International Journal for Numerical Methods in Engineering

Quantum Algorithm for Electromagnetic Field Analysis

Hiroyuki Tezuka, Yuki Sato

Published May 15, 2026• 0 citations
arXivHTMLPDF

Abstract

ABSTRACT Partial differential equations (PDEs) form the mathematical foundation of computational electromagnetics (CEM) and photonic device design, but classical solvers face high costs for large or complex structures. Quantum Hamiltonian simulation provides a rigorous framework for mapping PDEs into unitary time evolution, offering a scalable approach to electromagnetic analysis. In this work, we formulate Maxwell's equations in the potential representation and embed governing equations, boundary conditions, and observables consistently into a Hamiltonian form. A major challenge arises from the exponential growth of Hamiltonian terms for complex geometries. We examine this issue and demonstrate that logical compression can significantly reduce the number of required terms, particularly in periodic or symmetric configurations. As a proof of concept, we simulate optical wave propagation through a metalens and illustrate that the proposed method can capture wavefront shaping and focusing behavior, suggesting its applicability to optical design optimization. This work highlights the feasibility of Hamiltonian‐based quantum simulation for electromagnetic and photonic systems, and identifies structural conditions favorable for efficient execution.

DOI
10.1002/nme.70344

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ABSTRACT Partial differential equations (PDEs) form the mathematical foundation of computational electromagnetics (CEM) and photonic device design, but classical solvers face high costs for large or complex structures. Quantum Hamiltonian simulation provides a rigorous framework for mapping PDEs into unitary time evolution, offering a scalable approach to electromagnetic analysis. In this work, we formulate Maxwell's equations in the potential representation and embed governing equations, boundary conditions, and observables consistently into a Hamiltonian form. A major challenge arises from the exponential growth of Hamiltonian terms for complex geometries. We examine this issue and demonstrate that logical compression can significantly reduce the number of required terms, particularly in periodic or symmetric configurations. As a proof of concept, we simulate optical wave propagation through a metalens and illustrate that the proposed method can capture wavefront shaping and focusing behavior, suggesting its applicability to optical design optimization. This work highlights the feasibility of Hamiltonian‐based quantum simulation for electromagnetic and photonic systems, and identifies structural conditions favorable for efficient execution.
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.
This work presents a novel approach for simulating electromagnetic (EM) wave propagation by leveraging quantum computing techniques. By reformulating Maxwell's equations through the Riemann-Silberstein formalism, we cast the problem into a Schrödinger-like framework, enabling the application of Hamiltonian simulation methods. In particular, we employ the Trotter-Suzuki decomposition to approximate the time evolution operator, facilitating efficient quantum simulation of EM wave generated by a Gaussian pulse propagating within a two-dimensional metallic cavity. Our results illustrate the potential of quantum algorithms to address complex electrodynamics problems, offering promising avenues for future advancements in electromagnetics engineering.
This work presents a novel approach for simulating electromagnetic (EM) wave propagation by leveraging quantum computing techniques. By reformulating Maxwell's equations through the Riemann-Silberstein formalism, we cast the problem into a Schrödinger-like framework, enabling the application of Hamiltonian simulation methods. In particular, we employ the Trotter-Suzuki decomposition to approximate the time evolution operator, facilitating efficient quantum simulation of EM wave generated by a Gaussian pulse propagating within a two-dimensional metallic cavity. Our results illustrate the potential of quantum algorithms to address complex electrodynamics problems, offering promising avenues for future advancements in electromagnetics engineering.
We present quantum algorithms for electromagnetic fields governed by Maxwell’s equations. The algorithms are based on the Schrödingerisation approach, which transforms any linear PDEs and ODEs with non-unitary dynamics into a system evolving under unitary dynamics, via a warped phase transformation that maps the equation into one higher dimension. In this paper, our quantum algorithms are based on either a direct approximation of Maxwell’s equations combined with Yee’s algorithm, or a matrix representation in terms of Riemann–Silberstein vectors combined with a spectral approach and an upwind scheme. We implement these algorithms with physical boundary conditions, including perfect conductor and impedance boundaries. We also solve Maxwell’s equations for a linear inhomogeneous medium, specifically the interface problem. Several numerical experiments are performed to demonstrate the validity of this approach. In addition, instead of qubits, the quantum algorithms can also be formulated in the continuous variable quantum framework, which allows the quantum simulation of Maxwell’s equations in analog quantum simulation.
We present quantum algorithms for electromagnetic fields governed by Maxwell's equations. The algorithms are based on the Schrödingersation approach, which transforms any linear PDEs and ODEs with non-unitary dynamics into a system evolving under unitary dynamics, via a warped phase transformation that maps the equation into one higher dimension. In this paper, our quantum algorithms are based on either a direct approximation of Maxwell's equations combined with Yee's algorithm, or a matrix representation in terms of Riemann-Silberstein vectors combined with a spectral approach and an upwind scheme. We implement these algorithms with physical boundary conditions, including perfect conductor and impedance boundaries. We also solve Maxwell's equations for a linear inhomogeneous medium, specifically the interface problem. Several numerical experiments are performed to demonstrate the validity of this approach. In addition, instead of qubits, the quantum algorithms can also be formulated in the continuous variable quantum framework, which allows the quantum simulation of Maxwell's equations in analog quantum simulation.
Transport phenomena play a key role in a variety of application domains, and efficient simulation of these dynamics remains an outstanding challenge. While quantum computers offer potential for significant speedups, existing algorithms either lack rigorous theoretical guarantees or demand substantial quantum resources, preventing scalable and efficient validation on realistic quantum hardware. To address this gap, we develop a comprehensive framework for simulating classes of transport equations, offering both rigorous theoretical guarantees -- including exponential speedups in specific cases -- and a systematic, hardware-efficient implementation. Central to our approach is the Hamiltonian embedding technique, a white-box approach for end-to-end simulation of sparse Hamiltonians that avoids abstract query models and retains near-optimal asymptotic complexity. Empirical resource estimates indicate that our approach can yield an order-of-magnitude (e.g., 42×42\times42×) reduction in circuit depth given favorable problem structures. We then apply our framework to solve linear and nonlinear transport PDEs, including the first experimental demonstration of a 2D advection equation on a trapped-ion quantum computer.