Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2408.11550 · Journal of Plasma Physics

Quantum calculation for two-stream instability and advection test of Vlasov–Maxwell equations: numerical evaluation of Hamiltonian simulation

Hayato Higuchi, Juan W. Pedersen, Kiichiro Toyoizumi, Kohji Yoshikawa, Chusei Kiumi, Akimasa Yoshikawa

Published Aug 1, 2025• 2 citations
arXivHTMLPDF

Abstract

The Vlasov–Maxwell equations provide kinetic simulations of collisionless plasmas, but numerically solving them on classical computers is often impractical. This is due to the computational resource constraints imposed by the time evolution in the six-dimensional phase space, which requires broad spatial and temporal scales. The novelty of this study is to implement a quantum–classical hybrid Vlasov–Maxwell solver and the rigorous numerical scheme evaluation by numerical simulations. Specifically, the Vlasov solver implements the Hamiltonian simulation based on quantum singular value transformation, coupled with a classical Maxwell solver. We perform numerical simulation of a one-dimensional advection test and a one-spatial-dimension, one-velocity-dimension two-stream instability test on the Qiskit-Aer-GPU quantum circuit emulator with an A100 GPU. The computational complexity of our quantum algorithm can potentially be reduced from the classical $\mathcal{O}(N^6T^2/\epsilon )$ to $\mathcal{O}\left (\text{poly}(\log {N})\left (NT+T\log \left (T/\epsilon \right )\right )\right )$ for the $N$ grid system, simulation time $T$ and error tolerance $\epsilon$ in the limit where the number of queries is large enough and the error is small enough. Furthermore, the numerical analysis reveals that our quantum algorithm is robust under larger time steps compared with classical algorithms with the constraint of Courant–Friedrichs–Lewy condition.

DOI
10.1017/s0022377825100500

Related papers

Quantum algorithm for the Vlasov equation

Alexander Engel, Graeme Smith, Scott Parker

The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $O({N}_{v}t)$ for a velocity grid with ${N}_{v}$ grid points and simulation time $t$, our quantum algorithm scales as $O[\text{polylog}({N}_{v})t/\ensuremath{\delta}]$ where $\ensuremath{\delta}$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/\ensuremath{\delta}$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.

arXiv:1907.09418•Physical Review A•Dec 11, 2019•66 citations

Quantum Calculation of Classical Kinetic Equations: A Novel Approach for Numerical Analysis of 6D Boltzmann-Maxwell Equations in Collisionless Plasmas Using Quantum Computing

Hayato Higuchi, Juan W. Pedersen, Akimasa Yoshikawa

A novel quantum algorithm for solving the Boltzmann-Maxwell equations of the 6D collisionless plasma is proposed. The equation describes the kinetic behavior of plasma particles in electromagnetic fields and is known for the classical first-principles equations in various domains, from space to laboratory plasmas. We have constructed a quantum algorithm for a future fault-tolerant large-scale quantum computer to accelerate its costly computation. This algorithm consists mainly of two routines: the Boltzmann solver and the Maxwell solver. Quantum algorithms undertake these dual procedures, while classical algorithms facilitate their interplay. Each solver has a similar structure consisting of three steps: Encoding, Propagation, and Integration. We conducted a preliminary implementation of the quantum algorithm and performed a parallel validation against a comparable classical approach. IBM Qiskit was used to implement all quantum circuits.

arXiv:2306.05967•Jun 23, 2023•1 citation

A Quantum Algorithm for Nonlinear Electromagnetic Fluid Dynamics via Koopman-Von Neumann Linearization: 1D Numerical Evaluation Using Qulacs

Hayato Higuchi, Yuki Ito, Kazuki Sakamoto +3 more

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.

IEEE International Conference on Quantum Software (QSW)•Jul 7, 2025•0 citations

Quantum Algorithm for the Linear Vlasov Equation with Collisions

Abtin Ameri, Erika Ye, Paola Cappellaro +2 more

The Vlasov equation is a nonlinear partial differential equation that provides a first-principles description of the dynamics of plasmas. Its linear limit is routinely used in plasma physics to investigate plasma oscillations and stability. In this paper, we present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions, in the one-dimensional electrostatic limit. Rather than solving this equation in its native spatial and velocity phase space, we adopt an efficient representation in the dual space yielded by a Fourier-Hermite expansion. For a given simulation time, the Fourier-Hermite representation is exponentially more compact, thus yielding a classical algorithm that can match the performance of a previously proposed quantum algorithm for this problem. This representation results in a system of linear ordinary differential equations (ODEs) which can be solved with well-developed quantum algorithms: a Hamiltonian simulation in the collisionless case, and quantum ODE solvers in the collisional case. In particular, we demonstrate that a quadratic speedup in system size is attainable.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 17, 2023•2 citations

Quantum algorithm for the linear Vlasov equation with collisions

Abtin Ameri, Erika Ye, Paola Cappellaro +2 more

The Vlasov equation is a nonlinear partial differential equation that provides a first-principles description of the dynamics of plasmas. Its linear limit is routinely used in plasma physics to investigate plasma oscillations and stability. In this paper, we present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions, in the one-dimensional electrostatic limit. Rather than solving this equation in its native spatial and velocity phase space, we adopt an efficient representation in the dual space yielded by a Fourier-Hermite expansion. For a given simulation time, the Fourier-Hermite representation is exponentially more compact, thus yielding a classical algorithm that can match the performance of a previously proposed quantum algorithm for this problem. This representation results in a system of linear ordinary differential equations (ODEs) which can be solved with well-developed quantum algorithms: a Hamiltonian simulation in the collisionless case, and quantum ODE solvers in the collisional case. In particular, we demonstrate that a quadratic speedup in system size is attainable.

arXiv:2303.03450•Physical Review A•Jun 12, 2023•11 citations

Efficient Quantum Simulation of Linearized Vlasov--Poisson Dynamics Using Trotter and THRIFT Hamiltonian Simulation Methods

Kartick Paul, Rahul V, S. Aravinda +1 more

The Vlasov--Poisson system provides the fundamental kinetic description of plasma and plays a central role in understanding collective phenomena such as Landau damping and wave--particle interactions. Efficient numerical simulation of these dynamics remains challenging because of the high dimensionality of phase space. In this work, we studied magnetized and non-magnetized plasma using a quantum simulation framework for the linearized Vlasov-Poisson equation by reformulating the discretized system as a Hermitian Hamiltonian suitable for gate-based quantum computation. The time evolution is implemented using first, second and fourth-order Trotter--Suzuki product formulas and the recently proposed Time-Resolved Interaction Framework (THRIFT). The performance of the different simulation methods is systematically evaluated through electric field evolution, state fidelity, convergence behavior, energy conservation, entanglement entropy and quantum resource requirements, including circuit depth and two-qubit gate complexity, for both magnetized and non-magnetized plasma models. To further reduce finite time step errors without increasing circuit depth, Richardson extrapolation is incorporated as a error-mitigation technique. The results provide a comprehensive comparison of Trotter and THRIFT approaches and establish practical guidelines for accurate and resource-efficient quantum simulation of plasma dynamics on gate-based quantum computers.

quant-pharXiv:2608.23127•Aug 25, 2026•0 citations
The Vlasov–Maxwell equations provide kinetic simulations of collisionless plasmas, but numerically solving them on classical computers is often impractical. This is due to the computational resource constraints imposed by the time evolution in the six-dimensional phase space, which requires broad spatial and temporal scales. The novelty of this study is to implement a quantum–classical hybrid Vlasov–Maxwell solver and the rigorous numerical scheme evaluation by numerical simulations. Specifically, the Vlasov solver implements the Hamiltonian simulation based on quantum singular value transformation, coupled with a classical Maxwell solver. We perform numerical simulation of a one-dimensional advection test and a one-spatial-dimension, one-velocity-dimension two-stream instability test on the Qiskit-Aer-GPU quantum circuit emulator with an A100 GPU. The computational complexity of our quantum algorithm can potentially be reduced from the classical O(N6T2/ϵ)\mathcal{O}(N^6T^2/\epsilon )O(N6T2/ϵ) to O(poly(log⁡N)(NT+Tlog⁡(T/ϵ)))\mathcal{O}\left (\text{poly}(\log {N})\left (NT+T\log \left (T/\epsilon \right )\right )\right )O(poly(logN)(NT+Tlog(T/ϵ))) for the NNN grid system, simulation time TTT and error tolerance ϵ\epsilonϵ in the limit where the number of queries is large enough and the error is small enough. Furthermore, the numerical analysis reveals that our quantum algorithm is robust under larger time steps compared with classical algorithms with the constraint of Courant–Friedrichs–Lewy condition.
The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as O(Nvt)O({N}_{v}t)O(Nv​t) for a velocity grid with Nv{N}_{v}Nv​ grid points and simulation time ttt, our quantum algorithm scales as $O[\text{polylog}({N}_{v})t/\ensuremath{\delta}]$ where $\ensuremath{\delta}$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/\ensuremath{\delta}$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.
A novel quantum algorithm for solving the Boltzmann-Maxwell equations of the 6D collisionless plasma is proposed. The equation describes the kinetic behavior of plasma particles in electromagnetic fields and is known for the classical first-principles equations in various domains, from space to laboratory plasmas. We have constructed a quantum algorithm for a future fault-tolerant large-scale quantum computer to accelerate its costly computation. This algorithm consists mainly of two routines: the Boltzmann solver and the Maxwell solver. Quantum algorithms undertake these dual procedures, while classical algorithms facilitate their interplay. Each solver has a similar structure consisting of three steps: Encoding, Propagation, and Integration. We conducted a preliminary implementation of the quantum algorithm and performed a parallel validation against a comparable classical approach. IBM Qiskit was used to implement all quantum circuits.
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)O\left(N_{x}^{4}\right)O(Nx4​)whereNxN_{x}Nx​is the number of grid points per direction. By leveraging quantum computation, this complexity is reduced toO(Nxlog⁡Nx)O\left(N_{x} \log N_{x}\right)O(Nx​logNx​)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.
The Vlasov equation is a nonlinear partial differential equation that provides a first-principles description of the dynamics of plasmas. Its linear limit is routinely used in plasma physics to investigate plasma oscillations and stability. In this paper, we present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions, in the one-dimensional electrostatic limit. Rather than solving this equation in its native spatial and velocity phase space, we adopt an efficient representation in the dual space yielded by a Fourier-Hermite expansion. For a given simulation time, the Fourier-Hermite representation is exponentially more compact, thus yielding a classical algorithm that can match the performance of a previously proposed quantum algorithm for this problem. This representation results in a system of linear ordinary differential equations (ODEs) which can be solved with well-developed quantum algorithms: a Hamiltonian simulation in the collisionless case, and quantum ODE solvers in the collisional case. In particular, we demonstrate that a quadratic speedup in system size is attainable.
The Vlasov equation is a nonlinear partial differential equation that provides a first-principles description of the dynamics of plasmas. Its linear limit is routinely used in plasma physics to investigate plasma oscillations and stability. In this paper, we present a quantum algorithm that simulates the linearized Vlasov equation with and without collisions, in the one-dimensional electrostatic limit. Rather than solving this equation in its native spatial and velocity phase space, we adopt an efficient representation in the dual space yielded by a Fourier-Hermite expansion. For a given simulation time, the Fourier-Hermite representation is exponentially more compact, thus yielding a classical algorithm that can match the performance of a previously proposed quantum algorithm for this problem. This representation results in a system of linear ordinary differential equations (ODEs) which can be solved with well-developed quantum algorithms: a Hamiltonian simulation in the collisionless case, and quantum ODE solvers in the collisional case. In particular, we demonstrate that a quadratic speedup in system size is attainable.
The Vlasov--Poisson system provides the fundamental kinetic description of plasma and plays a central role in understanding collective phenomena such as Landau damping and wave--particle interactions. Efficient numerical simulation of these dynamics remains challenging because of the high dimensionality of phase space. In this work, we studied magnetized and non-magnetized plasma using a quantum simulation framework for the linearized Vlasov-Poisson equation by reformulating the discretized system as a Hermitian Hamiltonian suitable for gate-based quantum computation. The time evolution is implemented using first, second and fourth-order Trotter--Suzuki product formulas and the recently proposed Time-Resolved Interaction Framework (THRIFT). The performance of the different simulation methods is systematically evaluated through electric field evolution, state fidelity, convergence behavior, energy conservation, entanglement entropy and quantum resource requirements, including circuit depth and two-qubit gate complexity, for both magnetized and non-magnetized plasma models. To further reduce finite time step errors without increasing circuit depth, Richardson extrapolation is incorporated as a error-mitigation technique. The results provide a comprehensive comparison of Trotter and THRIFT approaches and establish practical guidelines for accurate and resource-efficient quantum simulation of plasma dynamics on gate-based quantum computers.