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arXiv:2604.26099 · Springer series on atomic, optical, and plasma physics

Mathematical Foundation for Quantum Computing of Electromagnetic Wave Propagation in Dielectric Media

Abhay K. Ram, Efstratios Koukoutsis, George Vahala, Kyriakos Hizanidis

quant-phSubmitted Apr 28, 2026• Announced Apr 30, 2026• 0 citations
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Abstract

Can quantum computers effectively simulate the propagation and scattering of electromagnetic waves in a classical plasma? This chapter introduces some of the basic concepts in mathematics and physics essential to answering that question. The numerical simulations of Maxwell equations for wave propagation in dielectrics are constrained by technological limitations of the present-day computers. In contrast, there has been ample fanfare around quantum computers and their potential to far exceed the performance of traditional computers. Whether the enhanced capabilities of a quantum computer can be put to use for simulating topics in classical physics is a source of intrigue and curiosity.

DOI
10.1007/978-3-031-84245-0
Comments
This chapter is part of the book: Emerging Applications of Ions and Plasmas (Editors: Samar K. Guharay & Motoi Wada)

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Can quantum computers effectively simulate the propagation and scattering of electromagnetic waves in a classical plasma? This chapter introduces some of the basic concepts in mathematics and physics essential to answering that question. The numerical simulations of Maxwell equations for wave propagation in dielectrics are constrained by technological limitations of the present-day computers. In contrast, there has been ample fanfare around quantum computers and their potential to far exceed the performance of traditional computers. Whether the enhanced capabilities of a quantum computer can be put to use for simulating topics in classical physics is a source of intrigue and curiosity.
Electromagnetic waves are an inherent part of all plasmas—laboratory fusion plasmas or astrophysical plasmas. The conventional methods for studying properties of electromagnetic waves rely on discretization of Maxwell equations suitable for implementing on classical, present day, computers. The traditional methodology is not efficient for quantum computing implementation—a future computational source offering a tantalizing possibility of enormous speed up and a significant reduction in computational cost. This paper addresses two topics relevant to implementing Maxwell equations on a quantum computer. The first is on formulating a quantum Schrödinger representation of Maxwell equations for wave propagation in a cold, inhomogeneous, and magnetized plasma. This representation admits unitary, energy preserving, evolution and conveniently lends itself to appropriate discretization for a quantum computer. Riding on the coattails of these results, the second topic is on developing a sequence of unitary operators which form the basis for a qubit lattice algorithm (QLA). The QLA, suitable for quantum computers, can be implemented and tested on existing classical computers for accuracy as well as scaling of computational time with the number of available processors. In order to illustrate the QLA for Maxwell equations, results are presented from a time evolving, full wave simulation of propagation and scattering of an electromagnetic wave packet by non-dispersive dielectric medium localized in space.
The prospect that quantum computers could be exponentially faster than conventional computers has motivated our research on the application of quantum information science to plasma physics. Quantum computers leverage quantum parallelization, entanglement, and superposition to speed up computations. However, our usual approach to computations on topics in classical plasma physics is not directly suited to take advantage of quantum computers. In this presentation, we will put forth some ideas on formulating algorithms for electromagnetic wave propagation in plasmas which could be implemented on quantum computers.
The presence of waves in laboratory and space plasma is due to collective processes which are inherently classical in nature. The linear propagation of electromagnetic waves follows from Maxwell equations in which the constitutive relations are obtained using either a kinetic or a fluid description of the plasma. Even though these descriptions require a large number of charged particles inside a Debye sphere, the interparticle distance is long compared to the de Broglie wavelength of any electron or ion in the plasma. As such, quantum effects are negligible and classical treatments have been successful in providing insight into plasma behavior in fusion, terrestrial, and space plasma. However, this does not preclude the exploitation of quantum computers for simulating classical plasma. It requires us to formulate appropriate theoretical models that appear to be “quantum-like” while maintaining the integrity of the classical description of electromagnetic wave propagation in plasma.
Quantum computers are ideally set up to solve linear systems which are of a form similar to the Schrodinger/Dirac equation of quantum mechanics. In the framework of linear response theory, the propagation and scattering of electromagnetic waves in a dielectric medium are described by Maxwell equations. The qubit lattice algorithm consists of a series of alternating unitary streaming and entanglement operators acting on qubit amplitudes constructed from the electric and magnetic fields. It is not a direct discretization of Maxwell equations, but recovers the desired equations to second order in lattice grid spacing. The resulting algorithm is implemented on a present-day supercomputer and is the basis of studying scattering of electromagnetic waves by an elliptical dielectric. As opposed to the steady state description of Mie scattering in frequency domain, the temporal evolution provides insights into transient scattering. The QLA simulations, reveal that a spatially localized wave packet propagating past an elliptic dielectric, embedded in vacuum, leads to several reflections generated by wave fields trapped within the dielectric. The physics insight brought forth by these simulations is not apparent from frequency domain studies of scattering. A complimentary simulation on transient scattering of a wave packet by an elliptical vacuum bubble inserted in a uniform dielectric demonstrates a stark contrast with respect to scattering off an elliptical dielectric in vacuum. Essentially, there is only a single internal reflection in which the field amplitudes are significantly smaller than those for side and forward scattering. A simple model based on the Kirchhoff tangent plane approximation helps explain the differences between these two scattering examples.
Quantum computers, through quantum entanglement and parallelization, offer an intriguing prospect of exponential speedup relative to classical computers for performing numerical simulations. Our interest is in the application of quantum information science to plasma physics in order to develop algorithms which can implemented on quantum computers. Concurrently, we want to test these algorithms on present conventional (classical) computers as large scale, error-correcting quantum computers with long coherence times are not yet forthcoming. Even though the constituents of laboratory and space plasmas are electrons and ions, the physics of these plasmas is dominated by collective processes. Quantum effects that are prominent at sub-atomic scales are averaged out in the statistical description of a plasma. Consequently, we need to render classical plasma physics in the language of quantum mechanics in preparation for quantum computers.
We investigate quantum information preserving computational electromagnetics. To do so, we propose two different methods which are (1) numerical canonical quantization and (2) quantum finite-difference time-domain scheme. The proposed algorithms find numerical solutions of quantum Maxwell's equations involved in arbitrary inhomogeneous dielectric media based on the macroscopic theory on quantum electrodynamics; hence, they are useful to study the quantum nature of electromagnetic fields in arbitrary passive and lossless quantum optical instruments. We provide a numerical example associated with two photon interference occurring in a quantum beam splitter, which is the underlying principle of recent quantum technologies.