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.

An initial value algorithm is utilized to examine the time dependent evolution of the electromagnetic fields arising from oblique scattering of bounded pulses from an infinite planar dielectric interface. Since the qubit lattice algorithm (QLA) is almost fully unitary, one finds excellent conservation of electromagnetic energy. Various Gaussian envelope pulses are considered in regimes where the incident angle is below that needed for total internal reflection. While the reflected pulse retains its overall Gaussian shape, the transmitted pulse exhibits a combination of a Gaussian envelope along with Huygen-like emitted wave fronts from the collision point of the initial pulse with the infinite dielectric interface. The strength of these Huygen wavefronts depends on the width of the incident pulse.

Qubit lattice algorithm (QLA) simulations are performed for a two-dimensional spatially bounded pulse propagating onto a plane interface between two dielectric slabs. QLA is an initial value scheme that consists of a sequence of unitary collision and streaming operators, with appropriate potential operators, that recover Maxwell equations in inhomogeneous dielectric media to the second order in the lattice discreteness. For the case of total internal reflection, there is transient energy transfer into the second medium due to the evanescent fields as the Poynting unit vector of the pulse is rotated from its incident to reflected direction. Because of the finite spatial extent of the pulse, a self-consistent Goos–Hanchen-type displacement along the interface is found without imposing any explicit interface boundary conditions on the fields. For normal incidence, the standard Fresnel coefficients are recovered for appropriately averaged QLA fields. Energy is conserved at all times to seven significant figures.

Simulating nonlinear classical dynamics on a quantum computer is an inherently challenging task due to the linear operator formulation of quantum mechanics. In this work, we provide a systematic approach to alleviate this difficulty by developing an explicit quantum algorithm that implements the time evolution of a second-order time-discretized version of the Lorenz model. The Lorenz model is a celebrated system of nonlinear ordinary differential equations that has been extensively studied in the contexts of climate science, fluid dynamics, and chaos theory. Our algorithm possesses a recursive structure and requires only a linear number of copies of the initial state with respect to the number of integration time-steps. This provides a significant improvement over previous approaches, while preserving the characteristic quantum speed-up in terms of the dimensionality of the underlying differential equations system, which similar time-marching quantum algorithms have previously demonstrated. Notably, by classically implementing the proposed algorithm, we showcase that it accurately captures the structural characteristics of the Lorenz system, reproducing both regular attractors–limit cycles–and the chaotic attractor within the chosen parameter regime.

Using the Madelung transformation on a generalized scalar Gross-Pitaevski equation, a nonlinear continuum fluid equations are derived for a classical fluid. A unitary quantum lattice algorithm is then determined as a second order discrete representation of this Gross-Pitaevski equation and the simulations are compared to those using classical fluid dynamic techniques.

Based on the Dirac representation of Maxwell equations we present an explicit, discrete space-time, quantum walk-inspired algorithm suitable for simulating the electromagnetic wave propagation and scattering from inhomogeneities within magnetized plasmas. The quantum walk is implemented on a lattice with an internal space of n_q=4–qubits, used to encode the classical field amplitudes. Unitary rotation gates operate within this internal space to generate the non-trivial dynamics of the free plasma-Dirac equation. To incorporate the contributions from the cyclotron and plasma density terms–manifesting as inhomogeneous potential terms–in the plasma-Dirac equation, the walk process is complemented with unitary potential operators. This leads to a unitary qubit lattice sequence that recovers the plasma-Dirac equation under a second-order accurate discretization scheme. The proposed algorithm is explicit and demonstrates, in the worst case, a polynomial quantum advantage compared to the Finite Difference Time Domain (FDTD) classical method in terms of resource requirements and error complexity. In addition, we extend the algorithm to include dissipative effects by introducing a phenomenological collision frequency between plasma species. Then, a post-selective time-marching implementation scheme is delineated, featuring a non-vanishing overall success probability and, subsequently, eliminating the need for amplitude amplification of the output state while preserving the quantum advantage.

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.

It is well known that Maxwell equations can be expressed in a unitary Schrodinger-Dirac representation for homogeneous media. However, difficulties arise when considering inhomogeneous media. A Dyson map points to a unitary field qubit basis, but the standard qubit lattice algorithm of interleaved unitary collision-stream operators must be augmented by some sparse non-unitary potential operators that recover the derivatives on the refractive indices. The effect of the steepness of these derivatives on two dimensional scattering is examined with simulations showing quite complex wavefronts emitted due to transmissions/reflections within the dielectric objects. Maxwell equations are extended to handle dissipation using Kraus operators. Then, our theoretical algorithms are extended to these open quantum systems. A quantum circuit diagram is presented as well as estimates on the required number of quantum gates for implementation on a quantum computer.

A Dyson map explicitly determines the appropriate basis of electromagnetic fields which yields a unitary representation of the Maxwell equations in an inhomogeneous medium. A qubit lattice algorithm (QLA) is then developed perturbatively to solve this representation of Maxwell equations. A QLA consists of an interleaved unitary sequence of collision operators (that entangle on lattice-site qubits) and streaming operators (that move this entanglement throughout the lattice). External potential operators are introduced to handle gradients in the refractive indices, and these operators are typically non-unitary but sparse matrices. By also interleaving the external potential operators with the unitary collide-stream operators, one achieves a QLA which conserves energy to high accuracy. Some two dimensional simulations results are presented for the scattering of a one-dimensional (1D) pulse off a localized anisotropic dielectric object.

A qubit lattice algorithm (QLA), which consists of a set of interleaved unitary collision-streaming operators, is developed for electromagnetic wave propagation in tensor dielectric media. External potential operators are required to handle gradients in the refractive indices, and these operators are typically non-unitary but sparse. A similar problem arises in the QLA for the Korteweg–de Vries equation, as the potential operator that models the KdV nonlinear term is also non-unitary. Several QLAs are presented here that avoid the need of this non-unitary potential operator by perturbing the collision operator. These QLAs are fully unitary.

The effect of the thickness of the dielectric boundary layer that connects a material of refractive index n_1 to another of index n_2 is considered for the propagation of an electromagnetic pulse. For very thin boundary layer the scattering properties of the pulse mimics that found from the Fresnel jump conditions for a plane wave - except that the transmission to incident amplitudes are augmented by a factor of √(n_2/n_1). As the boundary layer becomes thicker one finds deviations away from the Fresnel conditions and eventually one approaches WKB propagation. However there is found a small but unusual dip in part of the transmitted pulse that persists in time. The quantum lattice algorithm (QLA) used recovers the Maxwell equations to second order in a small parameter – but QLA still recovers Maxwell equations when this parameter is unity. The expansion parameter is the speed of the pulse in medium n_1.

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.

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.

A quantum lattice representation (QLA) is devised for the initial value problem of one-dimensional (1D) propagation of an electromagnetic disturbance in a scalar dielectric medium satisfying directly only the two curl equations of Maxwell. It si found that only 4 qubits/node are required. The collision, streaming, and potential operators are determined so as to recover the two curl equations to second order. Both polarizations are considered.

Long-time quantum lattice algorithm (QLA) simulations are performed for the multiple reflection–transmission of an initial electromagnetic pulse propagating normally to a boundary layer region joining two media of different refractive index. For these one-dimensional (1D) simulations, there is excellent agreement between x-, y- and z-representations, as well as very good agreement with nearly all the standard plane wave boundary condition results for reflection and transmission off a dielectric discontinuity. In the QLA simulation, no boundary conditions are imposed at the continuous, but sharply increasing, dielectric boundary layers. Two-dimensional (2D) QLA scattering simulations in the x–z plane are performed for an electromagnetic pulse interacting with a conical dielectric obstacle for the 8–16 qubit model.

There is considerable interest in the application of quantum information science to advance computations in plasma physics. Many of the topics in fusion plasma physics are classical in nature. In order to implement them on quantum computers it will require couching a classical problem in the language of quantum mechanics. Electromagnetic waves are routinely used in fusion experiments. The propagation of electromagnetic waves is described by Maxwell equations with an appropriate description of the plasma as a dielectric medium. Before advancing to the tensor dielectric of a magnetized plasma, this paper considers wave propagation in a one-dimensional inhomogeneous scalar dielectric. The classic theory of scattering of plane electromagnetic waves at a planar interface, separating two different dielectric media, leads to Fresnel equations for reflection and transmission coefficients. In contrast to plane waves, this paper is on the reflection and transmission of a spatially confined electromagnetic pulse. An analytical formulation for the scattering of a Gaussian pulse shows that the maximum transmission coefficient is modified by the square-root of the ratio of the two permittivities. The theory is complemented by numerical simulations using a quantum lattice algorithm for Maxwell equations. The algorithm, based on the Riemann-Silberstein-Weber representation of the electromagnetic fields and expressed in term of qubits, is an interleaved sequence of entangling operators at each lattice site and unitary streaming operators which transmit information from one site to an adjacent lattice site. Besides substantiating results from the theory for Gaussian pulses, numerical simulations show their validity for non-Gaussian pulses.