We focus on the electromagnetic fluid dynamics governing plasmas and develop a quantum algorithm for nonlinear partial differential equations. Specifically, we apply Koopman-von Neumann (KvN) linearization to transform the electromagnetic fluid dynamics into a Schrödinger equation. The time evolution of this system is then computed using Hamiltonian simulation via Quantum Singular Value Transformation (QSVT). The proposed quantum algorithm has the potential to significantly reduce computational complexity compared to an ideal classical high-precision scheme, which scales as$O\left(N_{x}^{4}\right)$where$N_{x}$is the number of grid points per direction. By leveraging quantum computation, this complexity is reduced to$O\left(N_{x} \log N_{x}\right)$offering a potential advantage in large-scale simulations. To assess the feasibility and performance of quantum computing in plasma simulations, we implement the proposed quantum algorithm and conduct numerical validations. Specifically, we present results for two test cases: one with a linear solution and the other with a nonlinear solution. As a result, the behavior of composite numerical errors arising from discretization, KvN, and QSVT is elucidated, demonstrating the effectiveness of the proposed quantum algorithm. The novelty of this study lies in the application of a promising quantum algorithm for nonlinear partial differential equations to plasma simulations, along with a concrete numerical evaluation of its computational performance. Furthermore, this work presents a novel direction by identifying a nonlinear, high- degree-of-freedom physical system that can be efficiently solved by a quantum algorithm under theoretical guarantees.
We focus on the electromagnetic fluid dynamics governing plasmas and develop a quantum algorithm for nonlinear partial differential equations. Specifically, we apply Koopman-von Neumann (KvN) linearization to transform the electromagnetic fluid dynamics into a Schrödinger equation. The time evolution of this system is then computed using Hamiltonian simulation via Quantum Singular Value Transformation (QSVT). The proposed quantum algorithm has the potential to significantly reduce computational complexity compared to an ideal classical high-precision scheme, which scales asO(Nx4)whereNxis the number of grid points per direction. By leveraging quantum computation, this complexity is reduced toO(NxlogNx)offering a potential advantage in large-scale simulations. To assess the feasibility and performance of quantum computing in plasma simulations, we implement the proposed quantum algorithm and conduct numerical validations. Specifically, we present results for two test cases: one with a linear solution and the other with a nonlinear solution. As a result, the behavior of composite numerical errors arising from discretization, KvN, and QSVT is elucidated, demonstrating the effectiveness of the proposed quantum algorithm. The novelty of this study lies in the application of a promising quantum algorithm for nonlinear partial differential equations to plasma simulations, along with a concrete numerical evaluation of its computational performance. Furthermore, this work presents a novel direction by identifying a nonlinear, high- degree-of-freedom physical system that can be efficiently solved by a quantum algorithm under theoretical guarantees.
To predict and simulate plasma phenomena, large-scale computational resources have been employed to develop high-precision, high-resolution plasma simulations. However, multi-scale plasma simulations require computational resources that scale polynomially with the number of spatial grids, posing a significant challenge for large-scale modeling. In this study, we present a quantum algorithm for simulating the nonlinear electromagnetic fluid dynamics systems that govern space plasmas. By applying Koopman–von Neumann (KvN) linearization, we map the nonlinear electromagnetic fluid dynamics systems to a Schrödinger equation and evolve it using Hamiltonian simulation via quantum singular value transformation (QSVT). The resulting algorithm reduces the computational complexity from O(Nx4) scaling of classical finite volume schemes to O(Nx log Nx), where Nx is the number of spatial grid points per dimension. And numerical experiments quantify behaviors of combined errors of discretization, KvN linearization, and QSVT-Hamiltonian simulation. As a practical demonstration, the method accurately reproduces the Kelvin-Helmholtz instability, underscoring its capability to tackle intricate nonlinear dynamics. These results suggest that quantum computing can offer a viable pathway to overcome the computational barriers of traditional multiscale plasma modeling.
The Koopman-von Neumann (KvN) formalism recasts classical mechanics in a Hilbert space framework using complex wavefunctions and linear operators, akin to quantum mechanics. Instead of evolving probability densities in phase space (as in Liouville's equation), KvN uses a Schrödinger-like equation for a classical wavefunction, with commuting position and momentum operators. Mapped to quantum computing, KvN offers a promising route to simulate classical dynamical systems using quantum algorithms by leveraging unitary evolution and quantum linear algebra tools, potentially enabling efficient classical-to-quantum mappings without invoking full quantum uncertainty. In this work, we specifically explore the implementation of the KvN approach on continuous-variable photonic quantum computing architectures, with the goals of leveraging quantum simulation for both sampling and computing intractable nonlinear dynamics. We will demonstrate its implementation and feasibility with two problems: the harmonic oscillator and a 1D partial differential equation governing nonlinear dynamics.
Many computational fluid dynamics (CFD) algorithms solve partial differential equations by discretization, resulting in large and sparse systems of linear equations. While iterative methods are widely used to solve these systems classically, most existing quantum linear system solvers target the solution through matrix inversion rather than approximating it using an iterative procedure. In this work, we develop a quantum implementation of the Jacobi method based on quantum singular value transformation (QSVT). By reformulating the Jacobi iteration as a polynomial transformation of a block-encoded operator, the algorithm requires only a constant ancilla overhead with respect to the number of iterations while maintaining a circuit depth that scales linearly with the iteration number. We demonstrate the algorithm for one- and two-dimensional Poisson problems, including the pressure Poisson equation arising in Chorin's projection method for the lid-driven cavity flow. The proposed algorithm provides a promising building block for future quantum implementations of multigrid methods and preconditioning techniques, bringing quantum algorithms closer to established CFD solution strategies.
Computational fluid dynamics lies at the heart of many issues in science and engineering, but solving the associated partial differential equations remains computationally demanding. With the rise of quantum computing, new approaches have emerged to address these challenges. In this work, we investigate the potential of quantum algorithms for solving the shallow water equations, which are, for example, used to model tsunami dynamics. By extending a linearization scheme previously developed in [Phys. Rev. Research 7, 013036 (2025)] for the Navier-Stokes equations, we create a mapping from the nonlinear shallow water equation to a linear system of equations, which, in principle, can be solved exponentially faster on a quantum device than on a classical computer. To validate our approach, we compare its results to an analytical solution and benchmark its dependence on key parameters. Additionally, we implement a quantum linear system solver based on quantum singular value transformation and study its performance in connection to our mapping. Our results demonstrate the potential of applying quantum algorithms to fluid dynamics problems and highlight necessary considerations for future developments.
We present a comprehensive review of quantum approaches for solving partial differential equations (PDEs) arising in computational fluid dynamics (CFD). We examine fully quantum approaches, including quantum linear system algorithms (QLSAs), ranging from the Harrow--Hassidim--Lloyd (HHL) algorithm to quantum singular value transformation (QSVT), Hamiltonian simulation, and quantum lattice Boltzmann methods (QLBMs), while emphasizing hybrid quantum--classical approaches, including quantum physics-informed neural networks (QPINNs) and amplitude-encoded variational PDE solvers. We focus on hardware-agnostic algorithms compatible with present noisy processors and emerging fault-tolerant architectures. For each framework, we analyze the mathematical formulation, algorithmic structure, and principal limitations. We also examine tensor-network (TN) representations, since CFD fields, differential operators, and geometrical information can often be encoded efficiently in low-rank form. The TN formalism bridges CFD discretizations and quantum states, operators, and circuits, enabling compact representations to be translated into tensor-programmable variational quantum algorithms (TP-VQAs). We further review benchmark problems, including Poisson, reaction, diffusion, and nonlinear model equations, and assess how well quantum algorithms capture key features of fluid dynamics. Our analysis highlights that potential quantum advantage is highly problem dependent and governed by condition number, representational complexity, state preparation, and measurement constraints. We outline capabilities, limitations, and challenges toward scalable quantum algorithms for CFD.
Computational fluid dynamics (CFD) is a specialized branch of fluid mechanics that utilizes numerical methods and algorithms to solve and analyze fluid-flow problems. One promising avenue to enhance CFD is the use of quantum computing, which has the potential to resolve nonlinear differential equations more efficiently than classical computers. Here, we try to answer the question of which regimes of nonlinear partial differential equations for fluid dynamics can have an efficient quantum algorithm. We propose a connection between the numerical parameter R , which guarantees efficiency in the truncation of the Carleman linearization, and the physical parameters that describe the fluid flow. This link can be made thanks to the Kolmogorov scale, which determines the minimum size of the grid needed to properly resolve the energy cascade induced by the nonlinear term. Additionally, we introduce the formalism for vector field simulation in different spatial dimensions, providing the discretization of the operators and the boundary conditions.