The role of quantum computing in advancing plasma physics simulations for fusion energy and high-energy

Its complexity constrains advancements in fusion energy and high energy applications driven by plasma physics, multiscale phenomena beyond classical computing limits. These transformative solutions, especially in plasma simulations, for which exponential speedup is possible, represent significant promise toward breakthroughs in sustainable energy and extreme state studies. In this review, Quantum Computing (QC) is explored as a means to drive plasma physics simulations forward by providing applications such as fusion energy and high-energy systems. This includes computational methods for simulating turbulence, wave-particle interactions, and Magnetohydrodynamic (MHD) instabilities that have near-quantum efficiency. We show that by integrating QC into plasma research, one can solve large-scale linear equations, compute eigenvalues, and optimize complex systems, performing better than classical methods. This discussion examines the potential of quantum computing for plasma physics, highlighting its current limitations, including hardware constraints and the need for specialized algorithms tailored to model complex plasma phenomena accurately. These challenges notwithstanding, QC has the potential to dramatically change plasma modeling and expedite the development of fusion reactors. QC represents a new approach to engineer away computational bottlenecks, providing unprecedented views of plasma behavior needed for sustainable energy breakthroughs. The results from this work underscore the continued importance of looking outside of plasma physics to realize QC’s full potential in advancing high-energy science.

1 Introduction

Plasma physics is at the forefront of scientific inquiry, providing profound insight into the fundamental mechanisms that underlie the universe along with transformative technological and application developments at the surface of the Earth. Plasmas are commonly referred to, or often called, “the fourth state of matter,” describing particles, such as electrons and ions, with charged particles whose unique behavior is dependent on electromagnetic interactions [1]. Plasmas are distinctly different from solids, liquids, and gases because, unlike solids, liquids, and gases, they are highly dynamic systems and possess properties such as conductivity, magnetic containment, and high energy reaction-sustaining capability [2]. The properties of plasmas make them of key importance in a variety of scientific fields, including astrophysics, industrial processes, and energy generation.

Plasma physics plays a pivotal role in the pursuit of sustainable energy because fusion promises access to nuclear power. Stars create the fusion that powers their mass, and fusion holds the potential to serve as an immaculate energy source on Earth [3]. Fusion produces no long-lived radioactive waste, and unlike fossil fuels or even nuclear fission, the fuel available is relatively abundant: isotopes of hydrogen (deuterium and tritium). Harnessing fusion energy success might ease worldwide energy needs whilst reducing the adverse impacts on our environment—both desirable scientific and technological goals [4].

Plasma must be heated to temperatures beyond tens of millions of degrees Celsius, conditions well beyond what will be found on the fleet of fusion reactors that will launch sometime in the second half of the 21st Century [5]. Such plasma stability and dynamics need to be precisely controlled in order to confine it in a reactor: a tokamak like a stellarator. Whilst plasma behavior is unstable, plasma turbulence, MHD instabilities, and complex wave-particle collisions may lead to reactor deterioration [6]. To power these reactions for long enough that they are sustainable and to maintain the plasma’s energy balance to these, the phenomena that occur must be understood and controlled.

These challenges must both be overcome; however, simulating the way plasma behaves in a fusion reactor is key. By computational models, we can predict plasma behavior, design an effective confinement system, and optimize reactor performance. However, the complexity of plasma systems makes it challenging to use traditional computational methods [7]. Plasma dynamics are by nature nonlinear and extend from the nanosecond interaction of the charged particles to the macroscopic behavior of the reactor. Solving these spectra simultaneously requires substantial computational resources and novel algorithms [8].

Plasma physics is key to understanding high energy systems and extreme states of matter, as well as beyond fusion energy. In astrophysics, plasmas govern phenomena from solar flares and cosmic rays to interstellar media. High-energy plasmas are essential at the forefront of particle accelerators, laser-matter interaction studies, and advanced manufacturing processes on Earth [9]. Both these applications rely on accurate plasma interaction simulations and performance optimization. Even the most potent class of classical computers find plasma physics simulations computationally intractable. The calculation of the vast number of variables involved in solving coupled equations describing plasma behavior, e.g., the Vlasov-Maxwell or Boltzmann equations, requires a significant amount of calculation [10]. The modeling of plasma turbulence (a primary factor in energy loss from fusion reactors) requires a resolution of interactions over nine-dimensional phase space (three spatial dimensions, three velocity dimensions, and time) [7]. However, classical computing resources are far too complex to cope with this quickly, which requires new approaches to computation.

In recent years, quantum computing (QC) has become a transformative technology, potentially being able to change the way the plasma simulations in plasma physics are organized. Unlike classical computers that store information as a binary bit (0 or 1), quantum computers operate using quantum bits or qubits, which are governed by the very essence of quantum mechanics [11]. By doing this, quantum computers can perform certain types of calculations in an exponentially faster time than their classical counterparts. Large-scale linear equations, eigenvalues of large matrices, and optimization of complex systems are basic to plasma physics simulations and can be efficiently solved by QC [12].

Simulating quantum systems is one of the most promising applications for QC. Since plasma behavior results from the quantum scale quantum interactions, wave-particle dynamics, and quantum tunneling, quantum computers are inherently suited to modeling it [11]. QC could provide the precision solutions to these problems that would provide an unparalleled understanding of plasma behavior and lead to fundamental breakthroughs in fusion energy research and high energy physics.

In this review, we discuss how QC is expanding the realm of simulations in plasma physics, particularly in the case of fusion energy and high-energy applications. The review examines the intersection of these cutting-edge fields with the aim of learning about the state of the art, challenges, and prospects in the future. In this review, we aim to make a qualitative and nuanced examination of how QC is helping to advance the conduct of simulations in plasma physics. The review aims to bridge these disciplines, stimulating further research and innovation to realize sustainable fusion energy and lead high-energy science. Due to the potential of QC to change our understanding of plasmas, it is set to help drive breakthroughs that will revolutionize energy production, open routes to space exploration, and enhance our knowledge of extreme states of matter.

2 Fundamentals of plasma physics and computational challenges

2.1 Overview of plasma physics

Plasma behavior is highly complex, including phenomena from turbulence, wave-particle interactions, and nonlinear instabilities, which are crucial to understanding and controlling plasma environments [13].

Plasma physics has one of the most significant applications, especially fusion reactions that happen within plasma at very high temperatures at very high pressure. In these reactions, the light atomic nuclei (usually hydrogen isotopes) come together to form heavier nuclei, emitting gigantic amounts of energy upon absorption [14]. Fusion happens between hydrogen nuclei to form helium and release energy described by Einstein’s mass-energy equivalence formula $E=m{c}^2$ the generator of the Sun, as well as the rest of stars, is fusion. One of the defining challenges of fusion energy research is replicating these conditions on Earth in a controlled manner. To be practical for fusion power generation, the researchers want to confine a high-temperature plasma (more than 100 million degrees Celsius) long enough for fusion reactions to occur [15]. The two most commonly explored approaches are magnetic confinement devices — tokamaks and stellarators — and inertial confinement devices — laser-driven fusion.

Fusion energy is but one of the utilities of plasma physics; plasma physics is also essential in high-energy physics and astrophysics. Plasmas are the dominant form of matter in these fields in a number of extreme environments, including interstellar medium, the solar wind, and the accretion disks around black holes. So, it is vital to understand plasma behavior in such high-energy environments to learn something about how the universe evolved, how galaxies have formed, and the dynamics of cosmic phenomena. Particle accelerators use controlled plasmas as plasma as well; charged particles are accelerated nearly to the speed of light. Plasma-based technologies are increasingly being investigated for application to space propulsion and materials processing.

2.2 Computational needs in plasma physics

Accurate and efficient simulation of plasma behavior is necessary to advance both theoretical and experimental plasma research. Plasma phenomena occur on an extensive range of spatial and temporal scales, from fundamental motions of individual particles to the large-scale behavior of the complete plasma system [16]. Thus, they need to address complex interactions and feedback mechanisms at different scales, and they necessarily must do so with some level of accuracy. For each one of these, the designer depends on the successful completion of a simulation that predicts or has predicted plasma confinement, energy losses, or stability, which directly determines the efficiency and even the feasibility of the fusion power reactor. For astrophysical purposes, accurate simulations are also required to model the evolution of plasma systems in space.

Given the problems in predicting plasma behavior, stability, and energy yield, this need for simulations is even more pressing. To study or design practical fusion reactors, they need to know what plasma will do when exposed to different settings, like varying magnetic fields or plasma heating rates [17]. To do so, we need to solve very complex plasma dynamics equations (Vlasov-Maxwell or Boltzmann) or macroscopic plasma equations (MHD equation). Such models are often coupled and nonlinear and usually complex to solve on a long time or large scale [18].

The distribution function of the plasma particles in phase space, as described by the Vlasov equation, is one of the most fundamental equations of plasma physics. The Vlasov equation is given by Equation 1:

$\frac{\partial f}{\partial t}+v.\nabla f+F.{\nabla }_{v}f=0(1)$

where $f=f\left(r,v,t\right)$ Is the distribution function for particles in phase space, v is the velocity of particles, F is the force acting on the particles (for example, electrical and magnetic forces) ${\nabla }_v$. It is the velocity gradient. This equation must be solved in conjunction with the Maxwell equations, which describe the electric and magnetic fields.

These equations are computationally expensive due to their complexity and high dimensionality, and hence we solve them. For many plasma systems, there are millions or even billions of particles interacting with each other through electromagnetic forces. As the system size increases, traditional methods become intractable for large-scale simulations due to rapidly growing computational power requirements. Plasma phenomena involve the interaction of thinly excited particles on vastly different timescales (e.g., electrons and fluid bulk plasma flow), which makes it difficult to simultaneously simulate these interactions in a single simulation [19].

Modeling plasma turbulence, which is a significant factor in the energy losses of fusion reactors, is an important computational challenge in plasma simulations. The turbulence of plasma is characterized by small-scale, chaotic fluctuations that can affect the large plasma structures, increasing the energy dissipation and decreasing the plasma confinement [20]. Given this, the modeling of these turbulent dynamics requires the resolution of multiple scales simultaneously, from microscopic instabilities to macroscopic flows. The catches are accurate descriptions of these scales in a computable model without high computational cost.

Plasma simulations produce large amounts of data, another problem with traditional computing thinking. Let’s say that in a plasma simulation, we are solving for the state of millions of particles, each with multiple velocity components per time step [11]. Because the resulting data sets can be enormous, their storage and processing have become sophisticated. These large data sets are often interpreted by identifying the key physical phenomena (turbulence or instabilities) of which little can be discerned without extensive data processing, as shown in Table 1.

Table 1

Table 1. Summarizes the various challenges associated with plasma simulations

2.3 Advances in computational techniques

Due to the complexity and challenges outlined, traditional computational methods based on classical computing architectures are not sufficient for plasma systems of large, realistic size to be simulated. These limitations have been overcome in various ways, though, such as the use of particle-in-cell (PIC) simulations, MHD models, and hybrid models, which incorporate both fluid and kinetic descriptions of plasma behavior [21]. While these methods have yielded beneficial insights into the physics of inertial confinement fusion plasmas, they still have a significant number of obstacles, particularly as plasma systems become progressively more extensive and more complex.

PIC methods manifest plasma as discrete particles and solve the evolution of their motion over time in a self-consistent electromagnetic field. These methods have been successfully applied to simulate small-scale plasma phenomena such as beam-plasma interactions and wave-particle dynamics [22]. Unfortunately, scaling these simulations to model large fusion plasmas is challenging due to the size of the number of particles and the complexity of the electromagnetic fields involved.

On the contrary, MHD models treat the plasma as a continuous fluid, and we solve for its macroscopic behavior, including the plasma’s motion and the development of magnetic fields. MHD models are computationally more efficient but do not take into account the individual particle interactions, which are essential for fine features in plasma dynamics, such as turbulence [23]. To reconcile computational efficiency with accuracy, hybrid models that meld PIC and MHD approaches have been developed.

Recently, QC has become a promising alternative to classical methods of computation. This quantum processing in superposition states allows quantum computers to solve specific problems much faster than classical computers. Thus, QC promises to enhance significantly plasma physics simulation capability greatly, mainly for solving large-scale, nonlinear issues that have long been computationally intractable [24]. However, quantum algorithms, including a quantum version of Monte Carlo, have the potential to fundamentally change the way we model complex plasma phenomena and reveal new aspects of how plasmas behave in extreme conditions.

3 Quantum computing: principles and potential

3.1 Basics of quantum computing

QC is a revolutionary computational paradigm based upon principles of quantum mechanics; information is processed using this paradigm in ways qualitatively different from classical computing. The foundation is based on quantum bits or qubits, themselves the essential element of quantum information. Qubits are unlike classical bits, as they exist in a superposition of 0 and 1 [25]. Quantum computers have unprecedented computational potential because this property enables qubits to represent and process much more information simultaneously than classical computers.

Mathematically, the state of a qubit can be described as in Equation 2:

${\left|\psi \right.〉\rangle }^2=\alpha \left|0〉\rangle +\beta \right|0〉\rangle ,(2)$

where $\left|\psi \right.〉$ represents the quantum state, $\left|0〉\right.$ Are the basis states and $\left|1〉\right.$ Are complex coefficients satisfying the normalization condition as shown in Equation 3:

${\left|\alpha \right|}^2+{\left|\beta \right|}^2=1(3)$

This superposition enables quantum computers to perform parallel computations on a scale unattainable by classical systems.

Another key property of quantum systems is entanglement, a phenomenon where the states of multiple qubits become interdependent, regardless of the distance between them. In a two-qubit entangled state, as shown in Equation 4:

$\left|\psi 〉\rangle =\frac{1}{\sqrt{2}}\left(\left|00〉\rangle \left.+\right|11\rangle 〉\right.\right)\right.(4)$

Therefore, the state of one qubit is measured immediately and determines the state of the other. This means that entanglement is a critical resource in QC for solving complex problems.

One must remember that quantum computation is based on quantum gates, which are, in some sense, quantum counterparts of classical logic gates. These gates are realized as unitary matrices and perform unitary transformations that maintain the normalization of the quantum state [26]. The Hadamard gate (H), Pauli-X gate, and controlled-NOT (CNOT) gate are all standard gates that can be used to build quantum circuits [27].

Specializing in key algorithms, quantum computation clearly shows its power in scientific fields. The Quantum Phase Estimation (QPE) algorithm provides an example in which, given a unitary operator, it is very fast to determine its eigenvalues, a pivotal problem within many physical simulations, such as plasma physics [28]. The Variational Quantum Eigensolver (VQE), another critical algorithm, is employed to compute the ground state energy of quantum systems through the optimization of a parameterized quantum circuit [29]. In particular, these algorithms find application in plasma physics since they allow for the simulation of quantum phenomena as well as the resolution of the complex mathematics of the field.

3.2 Advantages of classical computing

QC achieves transformative advantages of classical computing by solving high-complexity and huge computational-scale problems. In traditional plasma physics models, one typically has to solve systems of partial differential, large-scale linear, and eigenvalue problems of charged particle behavior under electromagnetic forces [30]. The computations of these interactions, turbulence, and multiscale phenomena are often very costly.

Perhaps the most significant advantage of QC is that it can solve large-scale linear systems with exponential speedup. Though classical methods like Gaussian elimination or iterative solvers are tractable fr input size n caping $O\left(n^3\right)$ or $O\left(n^2\right)$ operations, respectively), each method scales poorly as the system size increases n, meaning one tends to need $O\left(n^3\right)$ operations to solve directly or $O\left(n^2\right)$ operations per iteration for an iterative approach. HHL algorithm is an algorithm that provides exponential speed up for quantum algorithms under some conditions; it simplifies the complexity to $\left(Olog\left(n\right)\right)$ [31]. This could be a functional capability in plasma physics for solving the Maxwell-Vlasov equations, a set that describes the dynamics of charged particles in electromagnetic fields.

HHL algorithm solves linear equations of the form (Equation 5):

$Ax=b(5)$

where $A$ is a Hermitian matrix, $x$ is the solution vector, and $b$ is the known vector. The quantum approach involves encoding $b$ as a quantum state, applying QPE to approximate the eigenvalues of $A$, and then using these eigenvalues to construct the solution $x$ [32]. While the algorithm’s performance depends on the condition number of $A$ and the availability of efficient state preparation techniques, it represents a significant leap forward in computational efficiency.

Like QC, plasma physics is interested in solving eigenvalue problems, which QC excels in. The QPE algorithm efficiently extracts eigenvalues from Hermitian operators and, hence, accurately simulates wave-particle interactions and MHD instabilities [33]. These phenomena are essential to understanding plasma behavior in fusion reactors and optimizing confinement systems.

Another domain where QC offers promise is for reducing the scalability of plasma physics models. Plasmas are multiscale, involving interactions from microscopic particle dynamics to second life-like behavior, and traditional simulations suffer from resolving this multiscale character. However, quantum computers are able to do so compactly, bypassing these limitations [34]. Quantum tensor networks can efficiently represent the high dimensional phase space of plasma turbulence, providing more accurate and scalable simulations. Table 2 provides a comparison of classical and quantum approaches to everyday computational tasks in plasma physics.

Table 2

Table 2. Comparison of classical and quantum methods for computational tasks.

3.3 Challenges and future directions

QC presents an unrivaled opportunity for progress in plasma physics simulations but is not trivial. Current quantum hardware is still limited by the number of qubits, gate fidelity, and noise, limiting the size of quantum simulations that can be achieved with current qubits. To overcome these limitations, error correction, qubit design, and algorithm optimization need to be advanced [35]. At the same time, these challenges notwithstanding, the pace of QC research appears to be progressing exceedingly quickly. A practical route to quantum advantage leveraging hybrid approaches, coupling quantum algorithms with classical techniques, is also explored. Hybrid quantum-classical frameworks solve complex problems, such as selecting eigenvalues using quantum computers but require classical resources for larger volumes [36].

QC in plasma physics requires interdisciplinary collaboration between physicists, computer scientists, and engineers to fully realize its potential [37]. Designing quantum algorithms specifically for plasma simulations opens up new avenues for understanding and controlling plasmas and leads to new discoveries in fusion energy and high-energy science.

A QC paradigm shift in scientific computation with consequences for plasma physics and beyond. Superposition, entanglement, and quantum gates can be leveraged to transformically enable QC to produce solutions for large-scale linear equations, eigenvalue problems, and multiscale models [38]. These capabilities are particularly important to the computationally and intensively complex field of plasma physics, where standard methods frequently fail.

The integration of quantum hardware and algorithms into plasma physics research is poised to accelerate the development of fusion energy systems, gain insight into high-energy plasmas, and drive innovation in many scientific and industrial applications. However, as researchers continue to iterate and work together, QC will redefine what is computationally possible, ushering in a new era of discovery and technological advancement.

3.4 Applications of plasma physics and principles of quantum computing

The field of plasma physics operates as a basic science that delivers solutions for various possible scientific and technical applications. Plasma serves as an essential element for astrophysicists who study solar flares alongside research of interstellar medium collaborations and black hole accretion disk activity [13]. Research examines plasma propulsion systems as an alternative for deep-space missions since they generate superior thrust efficiency than traditional chemical rockets [24].

Plasma technology finds its main application in industrial manufacturing because semiconductor manufacturers need plasma etching to make microchips. Plasma-based processes are actively used in materials science because they assist scientists with surface transformations and coating techniques, as well as nanomaterial formation [20]. The controlled generation of plasma reactions represents a necessary step toward developing sustainable nuclear fusion energy, which can provide both unlimited clean power and sustainable energy capabilities. The computational problems that classical systems find difficult to solve efficiently receive breakthrough solutions through QC. The quantum computing element stands in contrast to classical bits, which exclusively work between 0 or 1 states because quantum bits (qubits) apply superposition principles to represent multiple states together [7]. The implementation of quantum entanglement and quantum parallelism leads to drastic speed improvement for sophisticated calculations.

Plasma physics researchers use it to address large linear equations, optimize plasma confinement models, and accurately replicate turbulence effects and wave-particle interactions. These technologies’ operational capabilities enable advanced research on sustainable energy solutions and leading-edge technologies for high-energy physics applications and industrial plasma operations [31].

4 Quantum computing in plasma physics simulations

Plasma physics is technology-focused at the crossroads of energy research, astrophysics, and the newest generation of laboratory experiments. It is a key cornerstone of scientific and technological progress. In all its applications, such as fusion energy, space physics, and particle acceleration, accurate, large-scale simulations are needed to understand highly nonlinear and multiscale dynamics [39]. However, simulations of plasma behavior are inherently complicated by the presence of collective effects, turbulence, and wave-particle interactions, making solutions computationally challenging. Although their advances have outperformed traditional computational tools, the latter rarely manage to capture the wide-ranging scope of plasma dynamics in high dimensions or extreme conditions [11].

QC, a nascent yet rapidly developing technology, offers a revolutionary solution to these challenges. A quantum computer uses superposition, entanglement, and quantum parallelism to calculate problems that classical computers can’t solve [40].

4.1 Current applications of quantum computing in plasma physics

The simulation of plasma systems remains one of the most computationally expensive tasks in modern physics. More specifically, plasmas are comprised of charged particles with long-range interaction driven by electromagnetic forces, whose characteristics still contain rich turbulence, wave-particle interactions, and instabilities occurring at large spatial and temporal scales [41]. These features consist of high-dimensional phase space and nonlinear coupled partial differential equations, the solution of which needs to be found. And just as traditional computational methods were very good for a few scenarios, they suffer from scalability and precision [42]. However, QC does have some options now, and, now quantum parallelism and quantum entanglement are being applied in new and potentially promising ways to solve these new problems.

A classical problem in plasma physics, turbulence modeling is a primary instance of application. High energy and particle transport in fusion devices and astrophysical plasmas is dominated by turbulence [7]. Statistical properties of turbulent systems are particularly amenable to quantum MC. The results of QMC methods are employed to calculate turbulence-related eigenstates and spectra more precisely than previous classical approaches [43]. VQEs have been shown to be able to solve energy eigenvalues of systems with strong turbulence, giving us insight into the dynamics of turbulent cascades [44].

Another cornerstone of plasma dynamics is wave-particle interactions, and this benefits tremendously from QC. Plasma configuration changes have to be modeled since instabilities and plasma heating have to be modeled as depending on particle behavior simultaneously as being collective and individual [45]. We have shown that the decomposition-based Suzuki method is promising for the quantum simulation of the time evolution of wave-particle systems in complex electromagnetic fields [46]. A more satisfactory description of the wave dispersion and particle acceleration in plasmas has been realized through quantum lattice-based research, which has allowed for greater control of plasma processes in experiments [47].

QC also improves the modeling of plasma instabilities in high-confinement fusion scenarios. Using algorithms such as the Quantum Approximate Optimization Algorithm (QAOA), we optimize stability criteria in tokamak plasmas. These approaches improve our understanding of confinement dynamics and better control strategies for sustaining stable plasma states [48].

QC applications in plasma physics are built upon quantum algorithms. Some of the key algorithms from Table 3 along with their applications, are summarized in Table 3.

Table 3

Table 3. Applications and advantages of quantum algorithms in turbulence and plasma physics modeling.

4.2 Turbulence in plasma systems

Turbulence is the dominant controlling mechanism in energy and particle transport in plasmas, primarily fusion device tokamaks and stellarators. Due to computational cost and the fact they cannot resolve all turbulent scales, classical solvers, such as those based on direct numerical simulation (DNS), become prohibitive. QMC methods and VQEs have been promising in this domain [49].

Quantum parallelism is exploited by QMC algorithms to reduce the complexity of sample turbulence spectra with an efficiency scale decoupled from the fine-scale resolution [50]. The energy cascade in plasma turbulence, modeled, for example, by the Kolmogorov scaling law given by Equation 6, is considered.

$E\left(k\right)\propto k^{-5/3}(6)$

with $E\left(k\right)$ energy spectrum as a function of wave number $k$. It has been shown that quantum algorithms can perform this computation more efficiently and can thereby simulate turbulent cascades with good agreement of the results with experimental data.

4.3 Wave-particle interactions

Wave-particle interactions govern the phenomena of plasma heating, energy transport, and instability generation. The Vlasov-Maxwell system in Equation 7 then forms a model for these interactions.

$\frac{\partial f}{\partial t}+v.\nabla f+\frac{q}{m}\left(E+v\times B\right).\nabla vF=0(7)$

Distribution function, particle velocity, electric field, and magnetic field are denoted by $f$, $v$, $E$ and $B$ respectively here. On the other hand, classical solvers are required to discretize phase space onto a computationally expensive grid. The system’s time evolution is simulated using quantum approaches, for instance, the Suzuki-Trotter decomposition, by approximating the exponential of the Hamiltonian [51] using Equation 8.

$e^{-iHt}\approx \prod _j{e}^{i{H}_i∆t}(8)$

In terms of the Hamiltonian decomposed into terms $H_i$ For the quantum gates. This decomposition allows quantum computers to more accurately simulate the time evolution and subsequent wave function collapse of wave-particle systems, giving insight into how resonant wave interaction can manifest itself in Landau damping.

4.4 Plasma instabilities

Kink modes and ballooning modes are important plasma instabilities that are critical in understanding the confinement dynamics in fusion devices. MHD describes plasmas as a conducting fluid [52] and governs these instabilities. The MHD Equations 9–11 include:

$\frac{\partial p}{\partial t}+\nabla .\left(pv\right)=0(9)$

$\frac{\partial v}{\partial t}+\left(v.\nabla \right)v=\frac{1}{p}\nabla p+\frac{J\times B}{p}(10)$

$\frac{\partial B}{\partial t}=\nabla \times \left(v\times B\right)-\nabla \times \left(\eta J\right)(11)$

$J$, the current density, and $\eta$, resistivity, it is where $p$ is mass density, $v$ is velocity, and it is also when the field strength, the electric field strength, and the electrical resistivity are. We apply quantum optimization techniques, the QAOA, to find configurations that minimize instability growth rates, resulting in improved plasma stability for fusion experiments.

5 Fusion energy simulations

A significant application is plasma physics, as the fusion energy source is a sustainable and clean energy source. In fusion research, the modeling of MHD is usually central to the effort. QC provides innovative solutions for breaking out computational bottlenecks that have previously limited classical algorithms used to solve for simulating these systems [34]. Understanding plasma, described through MHD, as a fluid subjected to magnetic fields is necessary for tokamak and stellarator studies of plasma confinement. These MHD equations are very nonlinear; thus, they are really hard to solve [53]. Quantum algorithms, like QFTs, were shown to be more efficient at simulating MHD wave dynamics. Alfven Waves, one of the most critical to plasma stability’s delicate structure, can be resolved via QFTs, which resolve delicate structures far too large to figure out on a conventional scale [54].

Confinement systems for stabilizing plasma under extreme conditions have been demonstrated to be optimized using quantum computing. Hybrid quantum-classical algorithms were used to optimize magnetic field configurations in tokamaks [55] to solve the high-order energy efficiency and stability problems. Advanced simulations of nonlinear plasma-material interactions that are of exceptionally high importance for understanding reactor boundary behavior are explored [56]. However, the application of these methods in practice is challenged by limitations to quantum hardware immaturity, scaling of algorithms to large systems, and the noise and error rates that prevent precise simulation of complex plasma phenomena.

5.1 Magnetohydrodynamics in fusion

Magnetic confinement in devices such as tokamaks cannot be understood without MHD. In contrast, QC-based simulations use QFTs to efficiently solve MHD wave equations [57]. Equation 12 models Alfwen waves, which are important for stability in the frequency domain of tokamaks.

$\frac{{\partial }^2\varphi }{{\partial t}^2}+v_A^2{\nabla }^2\varphi (12)$

where $v_A=B/\sqrt{{\mu }_{o}p}$ It is the Alfven velocity. Quantum methods on scalable qubit systems solve these equations for real-time wave propagation and damping analysis in fusion plasmas.

5.2 Plasma-material interactions

Phenomena, such as erosion and impurity generation, arise in plasma interactions with materials at the boundary of fusion reactors. Electron bombardment and surface modification are on a quantum scale. Quantum simulations using density functional theory (DFT) model these processes with great precision.

5.3 High-energy physics applications

Astrophysics systems such as supernovae, accretion disks, and even solar flares are all ruled with behavior by plasmas, see Table 4. To simulate these phenomena, these equations solving the quantum and relativistic effects are required [58]. Quantum simulations of relativistic plasmas have been used to simulate tensor networks and, hence, understand particle acceleration and magnetic reconnection processes [59]. These simulations are essential for interpreting observations from these space-based telescopes and improving our understanding of cosmic events.

Table 4

Table 4. Key contributions of QC to high-energy physics applications.

5.4 Laser-plasma interactions

Intense laser fields generate plasmas that exhibit nonlinear QED effects, modeled by the Klein-Gordon Equation 13.

$\left(⊡+m^2\right)\psi =0(13)$

where $⊡$ is the d’Alembertian operator, and $\psi$ is the wave function. QC-based solutions provide insights into pair production and radiation under extreme fields, aiding in the development of next-generation particle accelerators.

QC is addressing computational challenges in plasma physics, including turbulence, wave-particle interactions, MHD, and QED phenomena. The integration of QC into plasma research, combined with advancements in quantum hardware and algorithms, will revolutionize our ability to perform actuator control in fusion energy and high-energy physics.

6 Quantum algorithms and techniques for plasma physics

Tackling computational challenges in plasma physics is an exciting next step for the application of QC. Systems in plasma physics, including those critical to fusion energy and high-energy applications, exhibit complex behavior governed by nonlinear dynamics and multiscale interactions. Traditional approaches to these phenomena are too large and too complex, and therefore, novel techniques are needed [60]. Another option is for a QC to solve a problem that our old-fashioned processors can’t solve.

6.1 Quantum algorithms for plasma physics

Advantages gained by quantum algorithms stem from the quantum mechanical properties when the associated problems are spacelike nonlinear interactions and stochastic dynamics in large dimensions [61]. QMC, VQE, and Tensor Networks are all algorithms that have much promise in their application to plasma physics.

6.2 Quantum monte carlo (QMC)

Solving stochastic differential equations is an important issue in plasma physics. Monte Carlo methods are one of the main tools available for applications such as particle interaction simulation and turbulence modeling. Although classical Monte Carlo is tremendously versatile see Table 5, these methods suffer from certain computational costs that scale unfavorably with system size and which render systems requiring high dimensional integration or long simulation time daunting even with the state-of-the-art Metropolis algorithms [62].

Table 5

Table 5. Advantages of QMC in plasma physics.

Similar to most other experiments, QMC methods use quantum parallelism to sample many states at once, accelerating the convergence, diminishing the efficiency overhead [63]. QMC algorithms, allow efficient computation of partition functions and energy distributions in magnetized plasmas that deepen our understanding of plasma stability and confinement properties important for fusion energy research [37].

The improvements suggest QMC as a choice to explore plasma turbulence, wave particle interactions, and energy transfer mechanisms in fusion systems [64].

6.3 Variational quantum eigensolvers (VQE)

The VQE is a hybrid quantum classical algorithm for approximating eigenvalues of Hamiltonian operators. It is particularly useful in the solution of quantum mechanical problems of plasma interactions (wave particle resonances and instabilities).

The VQE algorithm minimizes the expectation value of the Hamiltonian, H, using parameterized quantum circuits given in Equation 14.

$E\left(\stackrel{\rightharpoonup }{\theta }\right)=\mathrm{\psi }\left(\stackrel{\rightharpoonup }{\theta }\right)\left|H\right|\psi \left(\stackrel{\rightharpoonup }{\theta }\right)〉(14)$

$\left|\psi \right.\left(\stackrel{\rightharpoonup }{\theta }\right)〉$ represents here the quantum state that is determined by the set of parameters $\stackrel{\rightharpoonup }{\theta }$. Efficient computation of low energy states in plasma systems is enabled by iteratively updating $\stackrel{\rightharpoonup }{\theta }$ in classical optimization algorithms that minimize E.

Hamiltonians describing MHD stability or collisional plasma dynamics are approximated by eigenvalues of the VQE for plasma physics applications. This tool is a promising one for current and near-term QC applications in this area because it can be adapted to Noisy Intermediate Scale Quantum (NISQ) devices [65].

6.4 Tensor networks

The system has been adapted to QC for modeling strongly correlated systems; tensor networks, mathematical structures originally developed for condensed matter physics, have been adapted to QC. Means similar to representation of quantum states as tensors connected through entanglement and correlations offer efficient encoding of entanglement and correlations, necessary for the modelling of nonlinear plasma dynamics [66]. Tensor networks are used in plasma physics to understand nonlinear wave interactions and extreme plasma states in astrophysical environments and quantum turbulence. This ability to integrate with quantum circuit improves the utility of QUDs in hybrid computational frameworks.

6.5 Hybrid computational techniques

In the current NISQ era there are limited combinations of qubit count and coherence times in a quantum device. Hybrid quantum classical methods are made straightforward by these devices, allowing for simple pathways to exploit the devices well as quantum bounded ancilla qubits for eigenvalue computations and classical machines for large scale simulation.

Modeling the behavior of plasmas via solving the Vlasov Maxwell equations is one of the most notable hybrid techniques—a central problem in plasma physics [67]. These equations describe the evolution of the plasma distribution function $f\left(x,v,t\right)$ in phase space as given in Equation 15.

$\frac{\partial f}{\partial t}+v.\nabla f+\frac{q}{m}\left(E+v\times B\right).\nabla vF=C\left(f\right)(15)$

where $E$ and $B$ are the electric and magnetic fields, and $C\left(f\right)$ represents collisional effects. This equation is too computationally intensive to be handled by quantum processors, and so they do integrations and visualizations, while classical systems handle the parts of the equation requiring nonlinear wave particle interaction.

Magnetic confinement of fusion reactors has also been explained by hybrid approaches. Finally, by optimizing control parameters such as magnetic field strength and heating rates, the QAOA increases plasma stability and increases the efficiency of the confinement of ions [68].

7 Software frameworks for quantum computing in plasma physics

The success of QC towards solving plasma physics problems requires robust software tools. In particular, Qiskit, Cirq and Pennylane have been enabling researchers to design, implement and test quantum algorithms through these open-source frameworks.

7.1 Qiskit

Qiskit is a complete development, optimization and simulation platform for quantum algorithms built by IBM. The modules that accompany Qiskit — such as “Qiskit Aer” for simulation, and “Qiskit Aqua” for running applications in optimization or quantum chemistry — make Qiskit very adaptable for plasma physics research [69].

Qiskit enables computer scientists to implement or simulate QMC algorithms or simulate Hamiltonians for exploring plasma dynamics such in stability and turbulence. It provides for experiment validation of the theoretical models for its integration with IBM quantum hardware.

7.2 Cirq

Cirq is designed for the creation of quantum circuits optimized for NISQ devices. Its noise modeling and error mitigation tools are particularly useful for plasma simulations, which are inherently sensitive to accuracy. Cirq is used to implement hybrid algorithms like VQE and QAOA and they are used as plasma control and optimization studies.

7.3 Pennylane

Pennylane is a hybrid quantum classical technique which is supported. Since it can readily interface with a wide range of platforms that enable quantum hardware, it provides an attractive platform for plasma physics applications like diagnostics and control system optimization [70].

8 Challenges and limitations

Many practical implementation challenges still lie ahead despite the potential QC can revolutionize plasma physics simulation. These limitations come from the interplay of the nascent nature of quantum hardware with plasma physics itself and its peculiarities. Therefore, collaboration between the QC experts and plasma physicists and also computational scientists is needed to overcome these challenges.

8.1 Technical barriers

Most modern quantum systems, build by IBM, Google and other leading technology companies, are in the tens to hundreds of qubit size [71]. Although impressive this scale is insufficient, appropriate in the sense that it falls well short of the requirements for realistic plasma physics simulations that often involve millions of (or billions of) interacting particles and their associated parameters [72].

The ability to move plasma simulations to QC has been hampered by the challenges inherent to scaling qubits, achieving precision, and improving their quality. Quantum states are environmental noise, or decoherence, degraded, with errors inciting errors that tarnish computational reliability [73]. Little in the way of error correction methods currently exists, and any that have emerged require a great and unmanageably large increase in physical qubits to realize fault tolerant logical qubits, which already severely restricts the already wildly constrained computational capacity of existing quantum hardware. Hurdle number two is scalability. Modeling interactions across a multiple of scales ranging from microscopic particle dynamic to macroscopic system behavior is an impetus for plasma simulations. To get this, we will need progress in qubit connectivity and system architecture as well as coherence time. Plasma simulations are very sensitive to precise physical parameters such as temperature gradients and magnetic field strength [72]. Quantum algorithms, usually stochastic, fail to reach the same numerical precision as classical simulations. This limitation limits the capabilities of quantum computers to monitor phenomenon like plasma turbulence and instabilities with the required levels of precision of plasma physics.

8.2 Algorithmic constraints

Plasma systems are inherently nonlinear and multiscale and are, therefore, hard to describe using the common linear operators and matrix form used by quantum computing. In equations like the Vlasov-Maxwell and MHD equations, the interaction between particles and fields is primarily described by nonlinear terms, coupled terms. A critical task is to translate these equations into computationally efficient, but complex, quantum representations. Low resource quantum algorithms often perform iterative tasks such as finding ground or solving linear systems [34]. They are very sensitive to parameters, including circuit design and optimization methods. In these plasma systems, evaluations in an extremely large solution space make identifying optimal configurations extremely difficult. Furthermore, there is a strong need for quantum algorithms for plasma simulations to be scalable and to require a large number of quantum gates, which increase computational cost and hardware error sensitivity. However, hybrid quantum classical approaches are typically practical, but their complexity in merging the two computational paradigms limits their potential success. Adapting algorithms for stochastic processes in plasma physics, such as turbulence and energy dissipation, are still a major open challenge in research due to the fact that classical Monte Carlo methods need quantum specific adaptations.

8.3 Field-specific challenges

Quantum computing beyond technical and algorithmic constraints has special difficulties in the plasma physics, simulating plasma turbulence. Plasmas are also intrinsically nonlinearity, resulting from the complex electromagnetic interactions between the particles which constitute the plasma. Inherent in plasma behavior are phenomena such as turbulence, instabilities, and self-organization, and all are inherently nonlinear. Despite their potential, these nonlinear systems are, however, challenging to simulate with standard computational methods nor are they well represented in a quantum setting. Plasma behavior extends from particle dynamics to global system-wide behavior over many spatial and temporal scales. Therefore, quantum simulations must be able to seamlessly transition between the fine grain particle level detail and macroscopically system level dynamical scales [11]. The lack of full integration is due to the limited qubit count and coherence times of present quantum hardware.

QC simulations for plasma physics are an empirically driven field, and thus any QC simulation must agree with experimental results in order to be accepted. To achieve this, we need high quality experimental data and robust quantum algorithms [56]. Integration of QC to plasma physics is complicated by their interdisciplinary nature. Quantum algorithms must simultaneously do justice to the various numerical requirements that arise, from fluid dynamics traversed through statistical mechanics and electromagnetic theory, involving collaborations between disciplines on quantum solutions.

8.4 Pathways forward

However, the prospective of QC to improve plasma physics simulations is still high. To overcome the limitations described above we need to make concerted advances in quantum hardware, developing tailored algorithms and building interdisciplinary collaborations. Looking ahead years, hardware development on the quantum computer space such as error correction, qubit scalability, and system coherence is expected to grow capabilities of quantum computers. There could also be a practical path to take advantage of quantum advantages while exploiting hardware constraints, using advances in hybrid quantum classical computing [35].

Unlike in many computational physics problems, the specific needs of plasma physics are very well known and researchers must algorithmically adaptifiable existing quantum methods or develop new quantum approaches to plasma physics dynamics that are nonlinear and multiscale. By exploiting machine learning techniques that accelerate classical plasma simulations, machine learning may have a role to play in optimizing quantum algorithms, and increasing their efficiency [11]. Stronger connections between the QC and plasma physics communities will be the final key to surmounting specific field barriers. Through collaborative research initiatives, workshops, and cross disciplinary training programs, the gap between these disciplines and innovation at their intersection can be bridged.

This review makes progress in high-energy physics by adopting quantum computing techniques to boost the simulation of complex plasmas and their phenomena like MHD, relativistic plasmas, and QED effects. QC-based approaches, including QFTs QMC and VQE solutions, help scientists achieve better computational performance in astrophysical plasma modeling as well as modeling of laser-plasma interactions and advanced particle accelerators through their integration.

9 Future directions and opportunities

The potential of QC to revolutionize plasma physics simulation is becoming apparent as QC develops. Qualitative and quantitative breakthroughs, with game changing impacts in energy, high energy physics, and space exploration, may be promised. Though there has been significant progress, there are still great opportunities for improvement in quantum hardware, algorithms, as well as interdisciplinary cooperation. Together, these activities will further our understanding of plasma dynamics and help us address critical human advance challenges. Current NISQ devices, while noisy and subject to finite times of coherence, are not yet suited to tackling the more complex (and useful) problems [74]. Topological qubits and error resilient architectures that use these novel approaches are touted as innovations to increase robustness and increase fidelity. The computational demands of plasma simulations require scaling quantum systems to thousands or millions of qubits. Refined error correction protocols along with advances in cryogenics, fabrication methods, and interconnects will be key enablers to performing fault-tolerant operations [75].

Algorithmic innovation is therefore vital parallel to hardware development. Because plasma physics is inherently nonlinear and multiscale, it requires specialized quantum algorithms. However, existing methods, such as the VQE, and QMC lie well behind where plasma needs to be for applications. As powerful tools, efficient hybrid quantum classical approaches are emerging in which quantum algorithms can be used to solve high complexity tasks, but other operations are based on classical computation [76]. But quantum machine learning has also been transformative for plasma simulations, leading to classical speeds in turbulence modeling, anomaly detection, and parameter optimization, based on analysis of high dimensional datasets.

QC has a need for robust interdisciplinary collaboration at the intersection with plasma physics. What physicists write out as equations and what phenomena to model, what computer scientist optimizes in algorithms and what quantum engineers struggle with to fit into hardware. QC also has potential beyond fusion for space exploration to accurately simulate solar wind interactions, magnetospheric dynamics and plasma propulsion systems [34]. In order to fully harness the transformative power of QC, sustained investment in research, hardware scalability, and algorithm development is required. As equally important are education and training programs that will train the next-generation to walk this interdisciplinary frontier. QC will create a culture of innovation and cooperation, giving plasma physics and its broader applications a new shape.

10 Conclusion

There is significant potential for revolutionizing our understanding of complex systems, fusion energy and high energy using the intersection of QC and plasma physics. The study of plasma physics, that is, the fourth state of matter, is central to progress in fusion energy, material science, astrophysics and industrial processes. Consequently, the simulation of plasma behavior with its nonlinearity, multiscale properties, and turbulence is still an unsolved challenge that goes beyond the scope of classical computers. Nevertheless, these challenges can be addressed in a new way using novel principals of QC such as superposition, entanglement, and parrallelism. Quantum algorithms (e.g. Quantum Monte Carlo, VQE, and QPE) have been shown to have potential in plasma simulation (including turbulence modeling, wave particle interactions, and MHD stability analysis). Processing giga data, large linear equations and complex matrix eigenvalues are essential to advance plasma research and these algorithms make these possible. Fusion energy, one of the most ambitious challenges of modern science, is especially transformed by the impact of QC. But quantum systems can also be used to simulate interactions at atomic and subatomic scales, allowing for the design and optimisation of fusion reactors. Controlled fusion depends upon understanding plasma dynamics under extreme conditions, and QC can provide a significant improvement in reactor design and operation. To fusion, QC also plays a role in high energy plasma applications in astrophysics and other sectors of industry. Through astronomical observations, it offers insights into cosmic plasmas in solar flares and in black holes, and could provide the impetus for some of the most powerful innovations in semiconductor manufacturing, materials science, and propulsion systems. While its promise is great, there’s still a way to go before QC is ready: noise, decoherence, and a small number of available qubits are some of the issues in the way. To overcome these challenges breakthroughs in quantum hardware, error correction, and tailored algorithms for plasma physics are needed. However, it will be necessary to continue the research and the collaboration to open the way for the full power of QC in this area. Plasma physics challenges hinge on solutions from QC which has the potential to solve problems that would drive energy and science forward.

Author contributions

YY: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Resources, Visualization, Writing–original draft, Writing–review and editing.

Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The authors declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

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