Quantum Computing promises accelerated simulation of certain classes of problems, in particular in plasma physics. The goal of this document is to provide a comprehensive list of citations for those developing and applying these approaches to experimental or theoretical analyses. As a living document, it will be updated as often as possible to incorporate the latest developments. Suggestions are most welcome.
The purpose of this note is to collect references for quantum algorithms already relevant to plasma physics. A minimal number of categories is chosen in order to be as useful as possible. Note that papers may be referenced in more than one category.
To facilitate search, if clearly appropriate, the following tags are applied
noisy-intermediate scale quantum computing
fault-tolerant quantum computing
quantum annealing
quantum-inspired
Color-filled tags indicate the type of content. Since most papers contain some form of theoretical analysis, we use
theoretical tag solely for the papers with analytical results, and no considerable numerical or experimental results
marks papers with numerical simulations, but no experimental results run on quantum devices.
marks papers with displayed experimental results. We may ommit this tag if the paper is referenced and tagged in a subsequent subsection.
We may omit tags if the paper is referenced and tagged in a subsequent subsection, or the paper is an overview of some topic, and covers many different approaches.
The fact that a paper is listed in this document does not endorse or validate its content - that is for the community (and for peer-review) to decide. Furthermore, the classification here is a best attempt and may have flaws - please let us know if (a) we have missed a paper you think should be included, (b) a paper has been misclassified or wrongly tagged, or (c) a citation for a paper is not correct or if the journal information is now available.
In order to be as useful as possible, this document will continue to evolve so please check back before you write your next paper. You can simply download the .bib file to get all of the latest references. Please consider citing Ref. AC23 when referring to this living review.
Modern reviews
Below are links to (static) general and specialized reviews.
System of linear equations HBR21 Bra+20 Xu+21 HHL09 CJS13 CKS17 WX22 SSO19 BL22 SM21
Many problems in plasma physics may be formulated, either exactly or approximately, as a problem of the form Ax = b where A and b encode the information about the system (including its initial conditions, if applicable), and the goal is to compute x , which encodes the desired data.
System of nonlinear equations DS21 XWG21 Xue+22
Nonlinear equations depend nonlinearly on x , and they are generally much harder to solve. As quantum mechanics is inherently linear, many techniques rely on mapping the original nonlinear problem to a (usually approximate) linear one, that is easier to solve.
System of polynomial equations Cha+19
System where each equation is a polynomial.
Ordinary differential equations Zan+21
Many plasma systems can be described by ordinary differential equations. Some techniques attempt to solve them directly, while others map the ODE to (larger) systems of linear equations, and solve those.
Linear Ber14 Ber+17 CL20 FLT22 JLY22a JLY22b Zan+21
As quantum mechanics is inherently linear, linear ODEs are often more straight-forward to solve with quantum computers than nonlinear ones.
Second-order SS19
The highest derivative appearing in these ODEs is the second derivative.
Nonlinear KPE21 Shi+21 LO08 DS21 JLY22b Liu+20 SGS22 SGS24 Zan+21
There is no general reliable procedure to solve nonlinear ODEs, but some methods have been proposed.
Partial differential equations (PDEs) GRG22 CLO21 Kro22 Gar21
In general, plasma systems are described by partial differential equations. Although some of the equations presented here do not commonly describe plasma systems, the techniques employed to solve them are often general enough that they could be adapted to tackle plasma PDEs.
For ease of search, PDEs arising from stochastic processes are indicated both here and in the following section.
Linear OMa+22, JLY22a JLY22b CS22
Non-homogeneous Bra+20 Arr+19 Ric+22
Vlasov Cap+23 ESP19 NDS24 TYH23 Ame+23 YL23
The Vlasov equation models the evolution of the distribution functions of (charged) particles in a plasma system, including their long-range Coulomb interactions. It can be written in the general form
\[\dfrac{\partial }{\partial t} f(x,p,t) + \dfrac{d r}{d t} \dfrac{\partial }{\partial r} f + \dfrac{d p}{d t} \dfrac{\partial }{\partial p} f = \mathcal{C}\]where C is a collision term. The dp/dt term can be coupled the (Electromagnetic) fields either through to the Poisson or the set of Maxwell’s equations.
Poisson Bra+20 Sat+21 AK22 Sah+22 Lub+20 Cao+13 Arr+19 Ric+22 Liu+21 Wan+20
The Poisson equation is an elliptic PDE that relates a charge/mass density or velocity field with the electrical/gravitational potential or pressure field it originates, respectively. It takes the form
\[\nabla^2 \phi(x,y,z) = f(x,y,z)\]This equation has significant applications in fluid dynamics (see Navier–Stokes equations).
Semi-classical Schrödinger JLL22
The semi-classical regime of the Schrödinger equation corresponds to the case when ħ ≪ 1. Possible applications include: quantum chemistry, including Born-Oppenheimer molecular dynamics and Ehrenfest dynamics.
Time-dependent Schrödinger Jou22 Oll+23 JLL22 LMJ18
Many problems can be mapped to the Schrödinger equation
\[\mathrm{i}\hbar \partial_t \vert{\psi(t)}\rangle = H\vert{\psi(t)}\rangle\]described by some Hamiltonian H. Solving the Schrödinger equation can often be done much more efficiently with quantum computers.
SFQED Hamiltonian HD24
Parabolic Pat+22
Heat/Convection LEK22 Alb+22 LMS22 JLY22a JLY22d JLY22b
The prototypical parabolic linear PDE. This equation describes the diffusion of heat in a material
\[\partial_t u(t,x) = \alpha ~\Delta u\]where α is the thermal diffusivity. Its applications are of fundamental importance in most branches of physics and engineering.
Black-Scholes FJO21 MK22 JLY22b
The Black–Scholes equation is a PDE which describes the price of the option V (S, t) over time t and price of underlying asset S(t), r is the ”force of interest, μ is the annualized drift rate of S, and σ is the standard deviation of the stock’s returns.
\[\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}+r S \frac{\partial V}{\partial S}-r V=0\]The Fokker-Planck equation for the probability density p(x, t) can be written as
\[\dfrac{\partial }{\partial t} p(x,t) = - \dfrac{\partial }{\partial x} \left( \mu(x,t)~p(x,t) \right) + \dfrac{\partial^2 }{\partial x^2} \left( D(x,t)~p(x,t) \right)\]where μ and D are the drift and diffusion coefficients (which may be time-dependent).
Pat+22Hyperbolic/Wave-related JLY22d JL22
The prototypical hyperbolic equation in physics, it describes oscillatory and propagating perturbations in a medium
\[\partial_{tt} u(t,x) = c^2 ~\nabla^2 u\]The Maxwell equations are a set of coupled PDEs that are foundational to the modeling of electromagnetic phenomena, which important applications in fundamental physics, classical optics and electric circuits.
\[\nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_0}, \nabla \cdot \mathbf{B}=0, \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, \nabla \times \mathbf{B}=\mu_0\left(\mathbf{J}+\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)\]Klein-Gordon CJO19
The Klein-Gordon relativistic wave equation is a 2nd order equation both in space and time
\[\left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2+\frac{m^2 c^2}{\hbar^2}\right) \psi(t, \mathbf{x})=0\]Helmholtz Ewe+22
The Helmholtz equation is the eigenvalue problem for the Laplace operator.
\[\nabla^2 f = - k^2~ f\]Important applications include wave-like phenomena and diffusion processes.
Nonlinear JL22
Evolution equation LEK22
General class of PDEs with time-domain.
Vlasov-Poisson YL22
The coupled Vlasov-Poisson system describes the self-consistent evolution of charges and their (electrical) potentials. Because of this, the system is nonlinear.
Schrödinger-Poisson MS21
Similar to the Vlasov-Poisson system, the Schrödinger-Poisson equation attempts to describe the self-consistent evolution of a wave-function and a potential.
Nonlinear-Schrödinger Lub+20
| Similar to the Schrödinger-Poisson system, however the potential is created by the square absolute value of the wave-function V= | ψ | ^2 which induces a cubic nonlinearity in the equations. |
Burger’s Oz+21
This nonlinear PDE with a quadratic nonlinearity is of fundamental importance to fluid dynamics and in particular in plasma physics. It is also a prototypical equation in turbulence related studies, and can be written in the form
\[\dfrac{\partial }{\partial t} u(x,t) +u \dfrac{\partial u}{\partial x} = \nu \dfrac{\partial^2 u}{\partial x^2}\]Reaction-diffusion LEK22 Dem+22, An+22
Reaction–diffusion equations describe transport/diffusion of substances and their transformation into other substances, for example to model the concentration of chemical components.
where D is the diffusion coefficient and R is the local reaction rate.
Navier-Stokes Gai20
The Navier-Stokes equations describe the motion of viscous fluids. They describe conservation of mass and momentum and often require equations of state for pressure, temperature and density to close the system of equations.
\[\partial_t \rho (x,t) + \nabla \cdot (\rho u) = 0, \frac{\partial}{\partial t}(\rho \mathbf{u})+\nabla \cdot(\rho \mathbf{u} \otimes \mathbf{u})=-\nabla p+\mu \nabla^2 \mathbf{u}+\frac{1}{3} \mu \nabla(\nabla \cdot \mathbf{u})+\rho \mathbf{g}\]A particular case of the Hamilton-Jacobi-Bellman. The Hamilton-Jacobi is an alternative formulation of classical mechanics, where given a Hamiltonian H(q, p, t) of a mechanical system, the Hamilton–Jacobi equation is a first-order, non-linear PDE for the Hamilton’s principal function S. One of its advantages is in efficiently identifying conserved quantities of mechanical systems.
Black-Scholes-Barenblatt Pat+22
The Black-Scholes-Barenblatt equation is a nonlinear extension to the Black-Scholes equation, which models uncertain volatility and interest rates derived from the Black-Scholes equation.
Stefan problems Sar22.
Stefan problems are a particular kind of boundary value problems for a system of PDEs in which the boundary between the phases can move with time.
Stochastic/difusive processes Kub+20 Alg+22 An+21
(Integro-)differential equations in which one or more of the terms is a stochastic process, leading to a solution which is stochastic in nature. Stochastic Differential Equations (SDEs) can be used to model physical systems subject to thermal fluctuations.
Through the Feynman-Kac formula, many common SDEs can be reduced to solving a PDE for the probability density of interest, as is the case for the -Planck equation.
For ease of search, PDEs arising from stochastic processes are indicated both here and in the previous section.
The Fokker-Planck equation for the probability density p(x, t) can be written as
\[\dfrac{\partial }{\partial t} p(x,t) = - \dfrac{\partial }{\partial x} \left( \mu(x,t)~p(x,t) \right) + \dfrac{\partial^2 }{\partial x^2} \left( D(x,t)~p(x,t) \right)\]where μ and D are the drift and diffusion coefficients (which may be time-dependent).
Linear Boltzmann/Rate equation JLY22b IKK24
The linear Boltzmann or Rate equation is a stochastic integro-differential equation for the probability density p(x, t) can be written as
\[\dfrac{d }{d t} p(x,t) = \int p(x',t) W(x,x') dx' - p(x,t) \int W(x',x) dx'\]where W(x,x’) is the probability rate of transition from state x’ to state x, where dp/dt can include partial derivatives of p(x, t).
Other techniques:
Linear embedding of nonlinear dynamical systems ESP21 JLY22c Liu+20
Several linear embedding of nonlinear dynamical systems have been developed to extend the class of problems that can be tackled by quantum computers. These include Koopman–von Neumann formulation, Quantum nonlinear Schrödinger lineariza- tion formulation and Carleman linearization, amongst others.
Koopman–von Neumann formulation Jos20 JLY22c TF23 Lin+22
Koopman–von Neumann mechanics is a description of classical mechanics embedded in a Hilbert space. The dynamical equation can be written as
\[i \partial_t \psi = H_\mathrm{KvN} \psi\]where the operator
\[H_\mathrm{KvN} = - i \sum_j \left(F_j \frac{\partial}{\partial x_j} + \frac{1}{2} \frac{\partial F_j}{\partial x_j} \right)\]| Furthermore, the probability density is interpreted as ρ = | ψ | ^2 . Applications may include the Vlasov-Maxwell coupled system of equations. |
Quantum nonlinear Schrödinger linearization formulation Llo+20
Formulation of ODEs/PDEs of the type dx/dt + f(x) x = b(t) with f = x^(†⊗m) F x^(⊗m) , as nonlinear Schrödinger equations. Potential applications of the method may include the Navier-Stokes equation, plasma hydrodynamics, epidemiology.
Madelung transform for nonlinear relativistic fluids Hat+19 Zyl+22
The Madelung equations are an alternative formulation to the Schrödinger equation, written in terms of hydrodynamical variables, with the addition of the Bohm quantum potential Q
\[\partial_t \rho_m+\nabla \cdot\left(\rho_m \mathbf{u}\right)=0, ~\frac{d \mathbf{u}}{d t}=\partial_t \mathbf{u}+\mathbf{u} \cdot \nabla \mathbf{u}=-\frac{1}{m} \nabla(Q+V)\]Applications include modeling of shocks in plasmas.
Finite element method MP16
This approach relies on dividing a system/domain into smaller regions called finite elements. The discretization process applied to a boundary value problem leads to a system of algebraic equations, which can be less computationally expensive to resolve than the original PDE. Important applications include fluid flow, heat transfer, and electromagnetic potentials.
Lattice Boltzmann algorithms Lju22
Originally developed as a classical algorithm, this approach can be used to simulate fluid dynamics without having to solve the Navier–Stokes equations directly. The fluid density is represented on a lattice and evolves in time with streaming and collision processes. One of the advantages of the method is its efficiency/scalability in parallel architectures.
Quantum lattice algorithms (QLA) And+22 Kou+22 Oga+18 Ram+21 Vah+20a Vah+20b Vah+21a Vah+21b Vah+22 VSV20 VVS20 Vah+21c Vah+20c Vah+19 Vah+20d Yep02 Yep05 Yep16 VYV03 Vah+11 Vah+10 Oga+16a Oga+16b Oga+15 Shi+18 And+23 Kou+23
Highly parallelizable approach amenable to classical supercomputers, allowing the study of (Klein-Gordon-)Maxwell’s equations, the Gross-Pitaevski equation, the nonlinear Schrödinger equation, and the KdV equation. In some cases, the method may also be suitable for fault-tolerant quantum computers