Quantum algorithm for the Vlasov equation

Abstract

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 ) for a velocity grid with Nv grid points and simulation time T , our quantum algorithm scales as O(polylog(Nv)T /δ) where δ 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/δ cost factor to extract an accurate result remains a difficulty. This work provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.

Koopman–von Neumann approach to quantum simulation of nonlinear classical dynamics

“…Notably, Ref. [11] devised a quantum algorithm for solving the linearized Vlasov-Maxwell system that claims to allow the simulation of Landau damping with exponentially reduced computational work.…”

Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers

“…Quantum algorithms have also been put forward to solve linear systems of equations [9][10][11][12][13] and some of these have been used to efficiently solve systems of linear classical (non-quantum) physical systems [14][15][16][17][18][19].…”

Koopman wavefunctions and Clebsch variables in Vlasov–Maxwell kinetic theory

“…Ongoing discussions (Shenkel et al. 2018) on the potential of quantum information and computation in plasma physics have recently led to exploiting Hilbert-space approaches in the numerical simulation of magnetized plasmas (Dodin & Startsev 2020; Engel, Smith & Parker 2019, 2021; Joseph 2020), of the Navier–Stokes equations (Gaitan 2020), and of arbitrary non-Hamiltonian systems of equations (Joseph 2020; Liu et al. 2020).…”

“…Ongoing discussions (Shenkel et al 2018) on the potential of quantum information and computation in plasma physics have recently led to exploiting Hilbert-space approaches in the numerical simulation of magnetized plasmas (Dodin & Startsev 2020;Engel, Smith & Parker 2019Joseph 2020), of the Navier-Stokes equations (Gaitan 2020), and of arbitrary non-Hamiltonian systems of equations (Joseph 2020;Liu et al 2020). In particular, recent work (Joseph 2020) has emphasized the role of Koopman wavefunctions in classical dynamics while their usage in describing hybrid quantum-classical systems was presented in Bondar, Gay-Balmaz & Tronci (2019), Boucher & Traschen (1988), , , Sudarshan (1976) and Jauslin & Sugny (2010).…”