This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum computation, and quantum algorithms. Then, we explore the topic of using quantum computers to simulate both linear and nonlinear dynamics in greater detail. Because quantum computers can only efficiently perform linear operations on the quantum state, it is challenging to perform nonlinear operations that are generically required to describe the nonlinear differential equations of interest. In this work, we extend previous results on embedding nonlinear systems within linear systems by explicitly deriving the connection between the Koopman evolution operator, the Perron–Frobenius evolution operator, and the Koopman–von Neumann evolution (KvN) operator. We also explicitly derive the connection between the Koopman and Carleman approaches to embedding. Extension of the KvN framework to the complex-analytic setting relevant to Carleman embedding, and the proof that different choices of complex analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert space metric are covered in the appendixes. Finally, we conclude with a review of recent quantum hardware implementations of algorithms on present-day quantum hardware platforms that may one day be accelerated through Hamiltonian simulation. We discuss the simulation of toy models of wave–particle interactions through the simulation of quantum maps and of wave–wave interactions important in nonlinear plasma dynamics.
This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum computation, and quantum algorithms. Then, we explore the topic of using quantum computers to simulate both linear and nonlinear dynamics in greater detail. Because quantum computers can only efficiently perform linear operations on the quantum state, it is challenging to perform nonlinear operations that are generically required to describe the nonlinear differential equations of interest. In this work, we extend previous results on embedding nonlinear systems within linear systems by explicitly deriving the connection between the Koopman evolution operator, the Perron–Frobenius evolution operator, and the Koopman–von Neumann evolution (KvN) operator. We also explicitly derive the connection between the Koopman and Carleman approaches to embedding. Extension of the KvN framework to the complex-analytic setting relevant to Carleman embedding, and the proof that different choices of complex analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert space metric are covered in the appendixes. Finally, we conclude with a review of recent quantum hardware implementations of algorithms on present-day quantum hardware platforms that may one day be accelerated through Hamiltonian simulation. We discuss the simulation of toy models of wave–particle interactions through the simulation of quantum maps and of wave–wave interactions important in nonlinear plasma dynamics.
A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear representations, such as the Koopman representation and Koopman von Neumann mechanics, have regained attention from the dynamical-systems research community. Here, we aim to present a unified theoretical framework, currently missing in the literature, with which one can compare and relate existing methods, their conceptual basis, and their representations. We also aim to show that, despite the fact that quantum simulation of nonlinear classical systems may be possible with such linear representations, a necessary projection into a feasible finite-dimensional space will in practice eventually induce numerical artifacts which can be hard to eliminate or even control. As a result, a practical, reliable and accurate way to use quantum computation for solving general nonlinear dynamical systems is still an open problem.
Plasma simulations for magnetic-confinement fusion (MCF) involve high-dimensional kinetic dynamics, strong nonlinear coupling, complex geometries, and multiscale spatiotemporal behavior, imposing severe computational bottlenecks. While quantum computing offers a potential alternative for accelerating plasma simulation problems, its practical advantage for fusion research has not yet been demonstrated. We review existing quantum computing studies across kinetic dynamics, wave interactions, equilibrium, and macroscopic stability, evaluating their hardware implementations and operational bottlenecks. Current studies remain largely restricted to linearized equations, reduced models, or isolated computational subroutines, leaving a substantial gap from realistic fusion simulations. We summarize key problem characteristics, including high dimensionality, structured dynamics, and measurement requirements, under which plasma applications are most likely to yield a quantum advantage. Guided by these considerations, we highlight potential approaches and present a roadmap toward practical quantum advantage of simulating plasma in MCF.
Important nonlinear dynamics, such as those found in plasma and fluid systems, are typically hard to simulate on classical computers. Thus, if fault-tolerant quantum computers could efficiently solve such nonlinear problems, it would be a transformative change for many industries. In a recent breakthrough [Liu et al., PNAS 2021], the first efficient quantum algorithm for solving nonlinear differential equations was constructed, based on a single condition R<1, where R characterizes the ratio of nonlinearity to dissipation. This result, however, is limited to the class of purely dissipative systems with negative log-norm, which excludes application to many important problems. In this work, we correct technical issues with this and other prior analysis, and substantially extend the scope of nonlinear dynamical systems that can be efficiently simulated on a quantum computer in a number of ways. Firstly, we extend the existing results from purely dissipative systems to a much broader class of stable systems, and show that every quadratic Lyapunov function for the linearized system corresponds to an independent R-number criterion for the convergence of the Carleman scheme. Secondly, we extend our stable system results to physically relevant settings where conserved polynomial quantities exist. Finally, we provide extensive results for the class of non-resonant systems. With this, we are able to show that efficient quantum algorithms exist for a much wider class of nonlinear systems than previously known, and prove the BQP-completeness of nonlinear oscillator problems of exponential size. In our analysis, we also obtain several results related to the Poincaré-Dulac theorem and diagonalization of the Carleman matrix, which could be of independent interest.
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.
The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally extreme. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. We introduce a framework for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this framework that correspond to previously-studied mappings. Additionally, we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). A quantum computer could simulate these finite linear systems using a number of qubits that scales only logarithmically with the number of variables of the original nonlinear system. Computational efficiency of the three detailed embedding strategies is discussed.
Solving high-dimensional dynamical systems by encoding their state in the amplitudes of a quantum state is expected to be a core application of fault-tolerant quantum computers. However, the conditions under which this paradigm yields quantum advantage can be restrictive, motivating complementary approaches. Here, we introduce Quantum Koopman Algorithms, where quantum amplitudes encode approximately closed sets of observables for a linear or nonlinear dynamics. These amplitudes evolve under the Koopman operator, and we establish conditions for efficient quantum algorithms addressing both dynamical and spectral problems. With this approach, we show that classes of open-system dynamics involving exponentially large fermionic systems can be efficiently simulated on a quantum computer. For nonlinear dynamics, we introduce a novel nonlinear interaction picture that enables quantum simulation beyond existing weak-nonlinearity conditions, and extract late-time spectral data where quantum phase estimation is unavailable.