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.

The propagation and scattering of electromagnetic waves in magnetized plasmas in a state where a global mode has been established or is in turbulence, are of theoretical and experimental interest in thermonuclear fusion research. Interpreting experimental results, as well as predicting plasma behavior requires the numerical solutions of the underlying physics, that is, the numerical solution of Maxwell equations under various initial conditions and, under the circumstances, complex boundary conditions. Casting, the underlying equations in a coordinate free form that exploits the symmetries and the conserved quantities in a form that can easily encompass a variety of initial and boundary conditions is of tantamount importance. Pursuing this task we utilize the advantages the Clifford Algebras can possibly provide. For simplicity we deal with a cold multi-species lossless magnetized plasma. The formulation renders a Dirac type evolution equation for am augmented state that consists of the electric and magnetic field bivectors as well as the polarizations and their associated currents for each species. This evolution equation can be dealt with a general spatial lattice disretization scheme. The evolution operator that dictates the temporal advancement of the state is Hermitian. This formulation is computationally simpler whatever the application could be. However, small wavelength capabilities (on the Debye length scale) for spatially large systems (magnetic confinement devices) is questionable even for conventional super-computers. However, the formulation provided in this work it is entirely suitable and it can be directly transferred in a quantum computer. It is shown that the simplified problem in the present work could be suitable for contemporary rudimentary quantum computers.

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.

We propose a new dilation method for implementing non-unitary operators on a quantum computer using the biorthogonal framework. By selecting an appropriate biorthogonal basis for the non-unitary operator, a unitary counterpart can be constructed in the biorthogonal representation, enabling its implementation in the orthonormal computational basis. When compared to other dilation and decomposition methods, the proposed method is particularly efficient for non-contraction, non-unitary operators. In contrast to the Linear Combination of Unitaries (LCU) method, the efficiency of the biorthogonal dilation technique is not constrained by the number of unitary summands but instead by the dimensionality of the non-unitary operator. The proposed method complements the LCU method for implementing general non-unitary operators that arise in positive-only open quantum systems, pseudo-Hermitian, and general non-Hermitian systems.

The dissipative character of an electromagnetic medium breaks the unitary evolution structure that is present in lossless, dispersive optical media. In dispersive media, dissipation appears in the Schrödinger representation of classical Maxwell equations as a sparse diagonal operator occupying an r-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution operator enables us to isolate the non-unitary operators (associated with dissipation) from the unitary operators (associated with lossless media). The unitary operators can be implemented through qubit lattice algorithm (QLA) on n qubits, based on the discretization and the dimensionality of the pertinent fields. However, the non-unitary-dissipative part poses a challenge both physically and computationally on how it should be implemented on a quantum computer. In this paper, two probabilistic dilation algorithms are considered for handling the dissipative operators. The first algorithm is based on treating the classical dissipation as a linear amplitude damping-type completely positive trace preserving (CPTP) quantum channel where an unspecified environment interacts with the system of interest and produces the non-unitary evolution. Therefore, the combined system-environment is now closed, and must undergo unitary evolution in the dilated space. The unspecified environment can be modeled by just one ancillary qubit, resulting in an implementation scaling of O(2n−1n2) elementary gates for the total system-environment unitary evolution operator. The second algorithm approximates the non-unitary operators by the Linear Combination of Unitaries (LCU). On exploiting the diagonal structure of the dissipation, we obtain an optimized representation of the non-unitary part, which requires O(2n) elementary gates. Applying the LCU method for a simple dielectric medium with homogeneous dissipation rate, the implementation scaling can be further reduced into O[poly(n)] basic gates. For the particular case of weak dissipation we show that our proposed post-selective dilation algorithms can efficiently delve into the transient evolution dynamics of dissipative systems by calculating the respective implementation circuit depth. A connection of our results with the non-linear-in-normalization-only (NINO) quantum channels is also presented.

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.

The presence of turbulence in the form of large density fluctuations and coherent filamentary structures in the edge region of fusion plasmas has been well documented. Radio frequency waves, launched from structures near the wall of a tokamak, have to propagate through this turbulent plasma before reaching the core. These density fluctuations can reflect, refract, and diffract the electromagnetic waves, thereby modifying the flow of energy and momentum to the core plasma. Conversely, the radiation pressure of the radio frequency waves can modify the turbulence, whether it is in the edge region or in the core. This article examines some consequences of the radiation force induced by electron cyclotron waves in plasmas. The effect of waves on two different representations of density fluctuations are studied. In the first representation, suitable for both edge and core plasmas, it is assumed that a planar interface separates two different density regimes. The physics basis for the radiation force on an interface separating two different scalar dielectric media was first elucidated by Poynting in 1905 [J. H. Poynting, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 9 , 393-406 (1905)]. Poynting’s results are explained within the context of Snell’s law and Fresnel equations, and, subsequently, extended to magnetized plasmas. The analysis shows that electron cyclotron waves lead to peaking of the density profile – the interface is pushed towards the region of higher density. The planar interface approximation is the basis of Kirchhoff theory [P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech, Massachusetts, 1987) Chapter 3] used to study wave scattering by turbulent media. In the second representation, appropriate for coherent structures in edge plasmas, the radiation force on a cylindrical filament embedded in a background plasma is determined using the Maxwell stress tensor. A detailed study reveals that the radiation force has a different effect on filaments – those with densities higher than the background density are pulled in towards the source launching the waves, while the lower density filaments are pushed away. The reaction on a filament is large enough to be observed experimentally.

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.

The scattering of lower hybrid (LH) waves due to scrape-off layer filaments is investigated. It is revealed that scattering can account for the LH spectral gap without any ad hoc modification to the wave-spectrum. This is shown using a multiscale simulation approach which allows, for the first time, the inclusion of full-wave scattering physics in ray-tracing/Fokker–Planck calculations. In this approach, full-wave scattering probabilities are calculated for a wave interacting with a statistical ensemble of filaments. These probabilities are coupled to ray-tracing equations using radiative transfer theory. This allows the modeling of scattering along the entire ray-trajectory, which can be important in the multi-pass regime. Simulations are conducted for LH current drive (LHCD) in Alcator C-Mod, resulting in excellent agreement with experimental current and hard x-ray profiles. A region in filament parameter space is identified in which the impact of scattering on LHCD is saturated. Such a state coincides with experimental LHCD measurements, suggesting saturation indeed occurs in C-Mod, and therefore the exact statistical properties of the filaments are not important.

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 reformulation of Maxwell equations for an inhomogeneous, anisotropic, passive and non-dispersive medium results in a quantum-like Dirac equation that admits unitary time evolution. In contrast to other approaches, there is no a-priori introduction of the Riemann-Silberstein-Weber (RSW) vector but the Maxwell equations are considered in their standard fields, with given constitutive relations. From the electromagnetic conservation quantities a pseudo-Hermitian dynamics is found together with a Dyson map that recovers the full Hermicity of the dynamics in an extended Hilbert space that describes the physical notion of unitary evolution. As an example, a uniaxial tensor dielectric medium is considered, with the explicit Dyson map yielding an optimal representation in a set of generalized RSW vectors. In this newly discovered form, a Quantum Computing (QC) implementation for simulation of electromagnetic wave propagation in complex media can be made, with further extension into plasmas.

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.

Detailed simulations are presented on a quantum lattice algorithm (QLA) for the solution of pulse propagation in a two dimensional (2D) refractive index medium. In particular, the initial value problem of an electromagnetic pulse scattering from a localized dielectric object is considered. The properties of the scattered waves depend on the scale length of the transition layer separating the vacuum from the core of the dielectric. The QLA perturbatively recovers the Maxwell equations to second order in a perturbation parameter ε that is just the normalized initial speed of light. QLA simulations show that the Maxwell equations are still recovered when ε = 1.