A quantum algorithm for the linear Vlasov equation with collisions

Abstract

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 work, 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. 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 which can be solved with well-developed quantum algorithms: 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.

References in corpus (31)

• Quantum algorithm for solving linear systems of equations
• Quantum Amplitude Amplification and Estimation
• Hamiltonian Simulation by Qubitization
• A quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors
• Optimal Hamiltonian Simulation by Quantum Signal Processing
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Efficient quantum algorithms for simulating sparse Hamiltonians
• Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
• Quantum Algorithms for Quantum Field Theories
• High-order quantum algorithm for solving linear differential equations
• Efficient quantum algorithm for dissipative nonlinear differential equations
• Quantum algorithm for linear differential equations with exponentially improved dependence on precision
• Solving nonlinear differential equations with differentiable quantum circuits
• Quantum algorithm and circuit design solving the Poisson equation
• Quantum Algorithm for Simulating the Wave Equation
• Quantum spectral methods for differential equations
• Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection and electron heating in low-beta plasmas
• Hamiltonian Simulation Using Linear Combinations of Unitary Operations
• Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics
• Improved quantum algorithms for linear and nonlinear differential equations
• Quantum Algorithm for the Vlasov Equation
• Eigenvector Approximation Leading to Exponential Speedup of Quantum Eigenvalue Calculation
• Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms
• Fluctuation-dissipation relations for a plasma-kinetic Langevin equation
• Measurement-based quantum phase estimation algorithm for finding eigenvalues of non-unitary matrices
• Quantum Computing for Fusion Energy Science Applications
• A solvable model of Vlasov-kinetic plasma turbulence in Fourier-Hermite phase space
• A Universal Quantum Circuit Scheme For Finding Complex Eigenvalues
• Simulating nonnative cubic interactions on noisy quantum machines
• Quantum Signal Processing for simulating cold plasma waves
• Quantum Lattice Representation for the Curl Equations of Maxwell Equations

Cited by in corpus (6)

• Quantum Computing Perspective for Electromagnetic Wave Propagation in Cold Magnetized Plasmas
• Quantum algorithm for the Vlasov simulation of the large-scale structure formation with massive neutrinos
• The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts
• Quantum Calculation for Two-Stream Instability and Advection Test of Vlasov-Maxwell Equations: Numerical Evaluation of Hamiltonian Simulation
• Post-Moore Technologies for Plasma Simulation: A Community Roadmap
• Solving the Nonlinear Vlasov Equation on a Quantum Computer