Quantum computing is gaining increased attention as a potential way to speed up simulations of physical systems, and it is also of interest to apply it to simulations of classical plasmas. However, quantum information science is traditionally aimed at modeling linear Hamiltonian systems of a particular form that is found in quantum mechanics, so extending the existing results to plasma applications remains a challenge. Here, we report a preliminary exploration of the long-term opportunities and likely obstacles in this area. First, we show that many plasma-wave problems are naturally representable in a quantumlike form and thus are naturally fit for quantum computers. Second, we consider more general plasma problems that include non-Hermitian dynamics (instabilities, irreversible dissipation) and nonlinearities. We show that by extending the configuration space, such systems can also be represented in a quantumlike form and thus can be simulated with quantum computers too, albeit that requires more computational resources compared to the first case. Third, we outline potential applications of hybrid quantum–classical computers, which include analysis of global eigenmodes and also an alternative approach to nonlinear simulations.
References
1.
A.
Montanaro
, “Quantum algorithms: An overview
,” NPJ Quantum Inf.
2
, 15023
(2016
).2.
S.
Lloyd
, “Universal quantum simulators
,” Science
273
, 1073
(1996
).3.
A. M.
Childs
, D.
Maslov
, Y.
Nam
, N. J.
Ross
, and Y.
Su
, “Toward the first quantum simulation with quantum speedup
,” PNAS
115
, 9456
(2018
).4.
A.
Gilyén
, Y.
Su
, G. H.
Low
, and N.
Wiebe
, “Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics
,” arXiv:1806.01838 (2018
).5.
A. W.
Harrow
, A.
Hassidim
, and S.
Lloyd
, “Quantum algorithm for linear systems of equations
,” Phys. Rev. Lett.
103
, 150502
(2009
).6.
D. W.
Berry
, “High-order quantum algorithm for solving linear differential equations
,” J. Phys. A: Math. Theor.
47
, 105301
(2014
).7.
A. M.
Childs
, R.
Kothari
, and R. D.
Somma
, “Quantum algorithm for systems of linear equations with exponentially improved dependence on precision
,” SIAM J. Comput.
46
, 1920
(2017
).8.
F.
Arute
, K.
Arya
, R.
Babbush
, D.
Bacon
, J. C.
Bardin
, R.
Barends
, R.
Biswas
, S.
Boixo
, F. G. S. L.
Brandao
, D. A.
Buell
et al, “Quantum supremacy using a programmable superconducting processor
,” Nature
574
, 505
(2019
).9.
A.
Scherer
, B.
Valiron
, S.-C.
Mau
, S.
Alexander
, E.
van den Berg
, and T. E.
Chapuran
, “Concrete resource analysis of the quantum linear-system algorithm used to compute the electromagnetic scattering cross section of a 2D target
,” Quantum Inf. Process.
16
, 60
(2017
).10.
A.
Montanaro
and S.
Pallister
, “Quantum algorithms and the finite element method
,” Phys. Rev. A
93
, 032324
(2016
).11.
M.
Motta
, C.
Sun
, A. T. K.
Tan
, M. J.
O'Rourke
, E.
Ye
, A. J.
Minnich
, F. G. S. L.
Brandão
, and G. K.-L.
Chan
, “Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution
,” Nat. Phys.
16
, 205
(2019
).12.
R.
Di Candia
, J. S.
Pedernales
, A.
del Campo
, E.
Solano
, and J.
Casanova
, “Quantum simulation of dissipative processes without reservoir engineering
,” Sci. Rep.
5
, 9981
(2015
).13.
S. K.
Leyton
and T. J.
Osborne
, “A quantum algorithm to solve nonlinear differential equations
,” arXiv:0812.4423 (2008
).14.
A.
Engel
, G.
Smith
, and S. E.
Parker
, “Quantum algorithm for the Vlasov equation
,” Phys. Rev. A
100
, 062315
(2019
).15.
Y.
Shi
, A. R.
Castelli
, X.
Wu
, I.
Joseph
, V.
Geyko
, F. R.
Graziani
, S. B.
Libby
, J. B.
Parker
, Y. J.
Rosen
, L. A.
Martinez
, and J. L.
DuBois
, “Simulating non-native cubic interactions on noisy quantum machines
,” Phys. Rev. A
103
, 062608
(2021
).16.
I. M.
Georgescu
, S.
Ashhab
, and F.
Nori
, “Quantum simulation
,” Rev. Mod. Phys.
86
, 153
(2014
).17.
S. J.
Devitt
, W. J.
Munro
, and K.
Nemoto
, “Quantum error correction for beginners
,” Rep. Prog. Phys.
76
, 076001
(2013
).18.
S.
Bravyi
and A.
Kitaev
, “Universal quantum computation with ideal Clifford gates and noisy ancillas
,” Phys. Rev. A
71
, 022316
(2005
).19.
B. D.
Clader
, B. C.
Jacobs
, and C. R.
Sprouse
, “Preconditioned quantum linear system algorithm
,” Phys. Rev. Lett.
110
, 250504
(2013
).20.
E. R.
Tracy
, A. J.
Brizard
, A. S.
Richardson
, and A. N.
Kaufman
, Ray Tracing and beyond: Phase Space Methods in Plasma Wave Theory
(Cambridge University Press
, New York
, 2014
).21.
22.
I.
Bialynicki-Birula
, in Progress in Optics
, edited by E.
Wolf
(Elsevier
, Amsterdam
, 1996
), p. 245
, Vol. XXXVI
.23.
P. C. S.
Costa
, S.
Jordan
, and A.
Ostrander
, “Quantum algorithm for simulating the wave equation
,” Phys. Rev. A
99
, 012323
(2019
).24.
A.
Suau
, G.
Staffelbach
, and H.
Calandra
, “Practical quantum computing: Solving the wave equation using a quantum approach
,” ACM Trans. Quantum Comput.
2
, 1
(2021
).25.
D. E.
Ruiz
and I. Y.
Dodin
, “First-principles variational formulation of polarization effects in geometrical optics
,” Phys. Rev. A
92
, 043805
(2015
).26.
G.
Vahala
, L.
Vahala
, M.
Soe
, and A. K.
Ram
, “Unitary quantum lattice simulations for Maxwell equations in vacuum and in dielectric media
,” J. Plasma Phys.
86
, 905860518
(2020
).27.
I. Y.
Dodin
, “Geometric view on noneikonal waves
,” Phys. Lett. A
378
, 1598
(2014
).28.
I. Y.
Dodin
, D. E.
Ruiz
, K.
Yanagihara
, Y.
Zhou
, and S.
Kubo
, “Quasioptical modeling of wave beams with and without mode conversion. I. Basic theory
,” Phys. Plasmas
26
, 072110
(2019
).29.
K.
Yanagihara
, I. Y.
Dodin
, and S.
Kubo
, “Quasioptical modeling of wave beams with and without mode conversion. II. Numerical simulations of single-mode beams
,” Phys. Plasmas
26
, 072111
(2019
).30.
K.
Yanagihara
, I. Y.
Dodin
, and S.
Kubo
, “Quasioptical modeling of wave beams with and without mode conversion. III. Numerical simulations of mode-converting beams
,” Phys. Plasmas
26
, 072112
(2019
).31.
L.
Friedland
, “Gyroresonant absorption from congruent reduction of an anisotropic pressure fluid model
,” Phys. Fluids
31
, 2615
(1988
).32.
D. E.
Ruiz
and I. Y.
Dodin
, “Extending geometrical optics: A Lagrangian theory for vector waves
,” Phys. Plasmas
24
, 055704
(2017
).33.
D. E.
Ruiz
, “Geometric theory of waves and its applications to plasma physics
,” Ph.D. thesis (Princeton University
, 2017
).34.
F.
Fillion-Gourdeau
, S.
MacLean
, and R.
Laflamme
, “Algorithm for the solution of the Dirac equation on digital quantum computers
,” Phys. Rev. A
95
, 042343
(2017
).35.
C. S.
Gardner
, “Bound on the energy available from a plasma
,” Phys. Fluids
6
, 839
(1963
).36.
I. B.
Bernstein
, “Waves in a plasma in a magnetic field
,” Phys. Rev.
109
, 10
(1958
).37.
I. Y.
Dodin
and N. J.
Fisch
, “Variational formulation of the Gardner's restacking algorithm
,” Phys. Lett. A
341
, 187
(2005
).38.
P.
Helander
, “Available energy and ground states of collisionless plasmas
,” J. Plasma Phys.
83
, 715830401
(2017
).39.
J.
Larsson
, “Hermitian structure for the linearized Vlasov-Poisson and Vlasov-Maxwell equations
,” Phys. Rev. Lett.
66
, 1466
(1991
).40.
A.
Brizard
, “On the relation between pseudo-Hermiticity and dissipation
,” Phys. Lett. A
187
, 382
(1994
).41.
A.
Mostafazadeh
, “Pseudo-hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
,” J. Math. Phys.
43
, 205
(2002
).42.
A.
Brizard
, “Hermitian structure for linearized ideal MHD equations with equilibrium flows
,” Phys. Lett. A
168
, 357
(1992
).43.
H.
Zhu
and I. Y.
Dodin
, “Wave-kinetic approach to zonal-flow dynamics: Recent advances
,” Phys. Plasmas
28
, 032303
(2021
).44.
Y.
Zhou
, H.
Zhu
, and I. Y.
Dodin
, “Solitary zonal structures in subcritical drift waves: A minimum model
,” Plasma Phys. Controlled Fusion
62
, 045021
(2020
).45.
H.
Qin
, R.
Zhang
, A. S.
Glasser
, and J.
Xiao
, “Kelvin–Helmholtz instability is the result of parity-time symmetry breaking
,” Phys. Plasmas
26
, 032102
(2019
).46.
E.
Mengi
, E. A.
Yildirim
, and M.
Kiliç
, “Numerical optimization of eigenvalues of Hermitian matrix functions
,” SIAM J. Matrix Anal. Appl.
35
, 699
(2014
).47.
D. E.
Ruiz
, J. B.
Parker
, E. L.
Shi
, and I. Y.
Dodin
, “Zonal-flow dynamics from a phase-space perspective
,” Phys. Plasmas
23
, 122304
(2016
).48.
A quantumlike formulation of the general linearized Vlasov–Maxwell system is also possible39 but requires a more complicated definition of the state function, so we do not consider the general case in this preliminary study.
49.
G. H.
Low
and I. L.
Chuang
, “Optimal Hamiltonian simulation by quantum signal processing
,” Phys. Rev. Lett.
118
, 010501
(2017
).50.
G. H.
Low
and I. L.
Chuang
, “Hamiltonian simulation by qubitization
,” Quantum
3
, 163
(2019
).51.
G. H.
Low
and I. L.
Chuang
, “Hamiltonian simulation by uniform spectral amplification
,” arXiv:1707.05391 (2017
).52.
G. H.
Low
, T. J.
Yoder
, and I. L.
Chuang
, “Methodology of resonant equiangular composite quantum gates
,” Phys. Rev. X
6
, 041067
(2016
).53.
In this approach, the evolution operator is represented through a series of rotations whose angles are found numerically. The authors of Ref. 3 “were unable to compute (those angles) explicitly except in very small instances.”
54.
A. M.
Childs
, “On the relationship between continuous- and discrete-time quantum walk
,” Commun. Math. Phys.
294
, 581
(2009
).55.
T.
Loke
and J. B.
Wang
, “Efficient circuit implementation of quantum walks on non-degree-regular graphs
,” Phys. Rev. A
86
, 042338
(2012
).56.
M. A.
Nielsen
and I. L.
Chuang
, Quantum Computation and Quantum Information
(Cambridge University Press
, New York
, 2009
).57.
G.
Brassard
, P.
Hoyer
, M.
Mosca
, and A.
Tapp
, “Quantum amplitude amplification and estimation
,” arXiv:quant-ph/0005055 (2000
).58.
That said, this algorithm is advantageous in that the number of qubits it requires scales logarithmically with the number of degrees of freedom. We return to this in Sec. III E.
59.
D. E.
Ruiz
and I. Y.
Dodin
, “On the correspondence between quantum and classical variational principles
,” Phys. Lett. A
379
, 2623
(2015
).60.
R. L.
Seliger
and G. B.
Whitham
, “Variational principles in continuum mechanics
,” Proc. R. Soc. A
305
, 1
(1968
).61.
62.
F.
Gaitan
, “Finding flows of a Navier–Stokes fluid through quantum computing
,” NPJ Quantum Inf.
6
(1
), 61
(2020
).63.
R.
Steijl
, “Quantum algorithms for nonlinear equations in fluid mechanics
,” in Quantum Computing and Communications
(IntechOpen
, London
, 2020
).64.
I.
Joseph
, “Koopman–von Neumann approach to quantum simulation of nonlinear classical dynamics
,” Phys. Rev. Res.
2
, 043102
(2020
).65.
Since the first preprint of our paper was released, related ideas have also been proposed in Refs. 63 and 70–72. In all these methods, the underlying idea is to convert a nonlinear problem into a linear problem via phase-space extension. In particular, Ref. 63 proposes to digitize nonlinear equations and then perform quantum simulations in the computational basis.
66.
B. D.
Craven
, “Generalized functions for applications
,” ANZIAM J.
26
, 362
(1985
).67.
E. A.
Novikov
, “Functionals and the random-force methods in turbulence theory
,” Sov. Phys. JETP
20
, 1290
(1965
).68.
W. D.
McComb
, The Physics of Fluid Turbulence
(Oxford University Press
, New York
, 1990
), Appendix H and Sec. 6.2.169.
S. F.
Edwards
, “The statistical dynamics of homogeneous turbulence
,” J. Fluid Mech.
18
, 239
(1964
).70.
A.
Engel
, G.
Smith
, and S. E.
Parker
, “Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms
,” Phys. Plasmas
28
, 062305
(2021
).71.
J.-P.
Liu
, H. Ø.
Kolden
, H. K.
Krovi
, N. F.
Loureiro
, K.
Trivisa
, and A. M.
Childs
, “Efficient quantum algorithm for dissipative nonlinear differential equations
,” arXiv:2011.03185 (2020
).72.
S.
Lloyd
, G. D.
Palma
, C.
Gokler
, B.
Kiani
, Z.-W.
Liu
, M.
Marvian
, F.
Tennie
, and T.
Palmer
, “Quantum algorithm for nonlinear differential equations
,” arXiv:2011.06571 (2020
).73.
P.
Rebentrost
, M.
Schuld
, L.
Wossnig
, F.
Petruccione
, and S.
Lloyd
, “Quantum gradient descent and Newton's method for constrained polynomial optimization
,” arXiv:1612.01789 (2016
).74.
P.
Helander
, M.
Drevlak
, M.
Zarnstorff
, and S. C.
Cowley
, “Stellarators with permanent magnets
,” Phys. Rev. Lett.
124
, 095001
(2020
).75.
J. B.
Parker
and I.
Joseph
, “Quantum phase estimation for a class of generalized eigenvalue problems
,” Phys. Rev. A
102
, 022422
(2020
).76.
J. P.
Friedberg
, Ideal Magnetohydrodynamics
(Plenum Press
, New York
, 1987
).77.
Notably, there also exist quantum algorithms for calculating (complex) eigenvalues of non-Hermitian operators [
A.
Daskin
, A.
Grama
, and S.
Kais
, “A universal quantum circuit scheme for finding complex eigenvalues
,” Quantum Inf. Process.
13
, 333
(2013
)]. However, these algorithms are considerably less efficient.78.
D. S.
Abrams
and S.
Lloyd
, “Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors
,” Phys. Rev. Lett.
83
, 5162
(1999
).79.
A.
Peruzzo
, J.
McClean
, P.
Shadbolt
, M.-H.
Yung
, X.-Q.
Zhou
, P. J.
Love
, A.
Aspuru-Guzik
, and J. L.
O'Brien
, “A variational eigenvalue solver on a photonic quantum processor
,” Nat. Commun.
5
, 4213
(2014
).80.
J. R.
McClean
, J.
Romero
, R.
Babbush
, and A.
Aspuru-Guzik
, “The theory of variational hybrid quantum-classical algorithms
,” New J. Phys.
18
, 023023
(2016
).81.
M.
Lubasch
, J.
Joo
, P.
Moinier
, M.
Kiffner
, and D.
Jaksch
, “Variational quantum algorithms for nonlinear problems
,” Phys. Rev. A
101
, 010301(R)
(2020
).82.
H. E.
Haber
, “Useful relations among the generators in the defining and adjoint representations of SU(N)
,” arXiv:1912.13302 (2019
).83.
G. V.
Skrotski
, “The Landau–Lifshitz equation revisited
,” Sov. Phys. Usp.
27
, 977
(1984
).84.
L.
Schwartz
, Lectures on Modern Mathematics
(Wiley
, New York
, 1963
), Vol. I
, pp. 23
–58
.© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.