Quantum algorithm for the Vlasov equation
Abstract
Typically, the VlasovMaxwell 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 VlasovMaxwell 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 errorcorrected universal quantum computer. Although the classical simulation has costs that scale as $$\mathcal{O}(N_v t)$$ for a velocity grid with $$N_v$$ grid points and simulation time $t$, our quantum algorithm scales as $$\mathcal{O}(\text{polylog}(N_v) t/\delta)$$ where $$\delta$$ 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 highresolution, sixdimensional phasespace grid, but the $$1/\delta$$ cost factor to extract an accurate result remains a difficulty. Our work provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.
 Authors:

 Univ. of Colorado, Boulder, CO (United States)
 Publication Date:
 Research Org.:
 Univ. of Colorado, Boulder, CO (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Fusion Energy Sciences (FES)
 OSTI Identifier:
 1579923
 Grant/Contract Number:
 SC0020393
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review A
 Additional Journal Information:
 Journal Volume: 100; Journal Issue: 6; Journal ID: ISSN 24699926
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Citation Formats
Engel, Alexander, Smith, Graeme, and Parker, Scott E. Quantum algorithm for the Vlasov equation. United States: N. p., 2019.
Web. doi:10.1103/PhysRevA.100.062315.
Engel, Alexander, Smith, Graeme, & Parker, Scott E. Quantum algorithm for the Vlasov equation. United States. doi:https://doi.org/10.1103/PhysRevA.100.062315
Engel, Alexander, Smith, Graeme, and Parker, Scott E. Wed .
"Quantum algorithm for the Vlasov equation". United States. doi:https://doi.org/10.1103/PhysRevA.100.062315. https://www.osti.gov/servlets/purl/1579923.
@article{osti_1579923,
title = {Quantum algorithm for the Vlasov equation},
author = {Engel, Alexander and Smith, Graeme and Parker, Scott E.},
abstractNote = {Typically, the VlasovMaxwell 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 VlasovMaxwell 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 errorcorrected universal quantum computer. Although the classical simulation has costs that scale as $\mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $\mathcal{O}(\text{polylog}(N_v) t/\delta)$ where $\delta$ 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 highresolution, sixdimensional phasespace grid, but the $1/\delta$ cost factor to extract an accurate result remains a difficulty. Our work provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.},
doi = {10.1103/PhysRevA.100.062315},
journal = {Physical Review A},
number = 6,
volume = 100,
place = {United States},
year = {2019},
month = {12}
}
Web of Science
Works referenced in this record:
Optimal Hamiltonian Simulation by Quantum Signal Processing
journal, January 2017
 Low, Guang Hao; Chuang, Isaac L.
 Physical Review Letters, Vol. 118, Issue 1
A fast lowtohigh confinement mode bifurcation dynamics in the boundaryplasma gyrokinetic code XGC1
journal, May 2018
 Ku, S.; Chang, C. S.; Hager, R.
 Physics of Plasmas, Vol. 25, Issue 5
On the Relationship Between Continuous and DiscreteTime Quantum Walk
journal, October 2009
 Childs, Andrew M.
 Communications in Mathematical Physics, Vol. 294, Issue 2
Multiscale gyrokinetic simulation of tokamak plasmas: enhanced heat loss due to crossscale coupling of plasma turbulence
journal, December 2015
 Howard, N. T.; Holland, C.; White, A. E.
 Nuclear Fusion, Vol. 56, Issue 1
Design for U.S. exascale computer takes shape
journal, February 2018
 Service, Robert F.
 Science, Vol. 359, Issue 6376
PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
journal, October 1997
 Shor, Peter W.
 SIAM Journal on Computing, Vol. 26, Issue 5
Quantum Algorithm for Linear Systems of Equations
journal, October 2009
 Harrow, Aram W.; Hassidim, Avinatan; Lloyd, Seth
 Physical Review Letters, Vol. 103, Issue 15
Secondary reconnection sites in reconnectiongenerated flux ropes and reconnection fronts
journal, July 2015
 Lapenta, Giovanni; Markidis, Stefano; Goldman, Martin V.
 Nature Physics, Vol. 11, Issue 8
LowDepth Quantum Simulation of Materials
journal, March 2018
 Babbush, Ryan; Wiebe, Nathan; McClean, Jarrod
 Physical Review X, Vol. 8, Issue 1
Hamiltonian Simulation by Qubitization
journal, July 2019
 Low, Guang Hao; Chuang, Isaac L.
 Quantum, Vol. 3
Simulating physics with computers
journal, June 1982
 Feynman, Richard P.
 International Journal of Theoretical Physics, Vol. 21, Issue 67
Methodology of Resonant Equiangular Composite Quantum Gates
journal, December 2016
 Low, Guang Hao; Yoder, Theodore J.; Chuang, Isaac L.
 Physical Review X, Vol. 6, Issue 4
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
journal, December 2006
 Berry, Dominic W.; Ahokas, Graeme; Cleve, Richard
 Communications in Mathematical Physics, Vol. 270, Issue 2
Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision
journal, January 2017
 Childs, Andrew M.; Kothari, Robin; Somma, Rolando D.
 SIAM Journal on Computing, Vol. 46, Issue 6
Efficient state preparation for a register of quantum bits
journal, January 2006
 Soklakov, Andrei N.; Schack, Rüdiger
 Physical Review A, Vol. 73, Issue 1