Quantum algorithm for the linear Vlasov equation with collisions

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 paper, 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. For a given simulation time, 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 (ODEs) which can be solved with well-developed quantum algorithms: a 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.

Figure 1

The norm of the electric potential $\left|\phi \right|$ as a function of time $t$. The blue solid line is obtained by numerically solving Eqs. (11) and (12) with $k=0.4\sqrt{2}$, and $\alpha =2/k^2$. The red dashed line is an $e^{-\gamma t}$ envelope, where $\gamma =0.066237$. The parameters used here match those used in Ref. [34].

Figure 2

Workflow diagram of the quantum algorithm in both the collisionless and collisional cases. Here, $N$ is the system size, $T$ is the simulation time, and $N_s$ refers to the number of samples required at different times to calculate the Landau damping rate.

Figure 3

Numerically obtained norm of the system matrix $\parallel H\parallel$ [Eq. (28)] as a function of the system size $N=M+1$, where $M$ is the number of Hermite moments. The orange dashed line is a $\sqrt{N}$ scaling.

Figure 4

The relative error $\epsilon$ as a function of the number of Hermite moments $M$. The blue solid line is the error obtained from solving Eq. (25) using an ODE solver with $\alpha =\nu =0$, $k=2$, and $T=2.5$, and the dashed orange line is an $e^{-2M}$ scaling.

Figure 5

The relative error in integrating ${\int }_{-\infty }^{\infty }{e}^{-v^2}dv$ as a function of the number of velocity space grid points $N_v$ with $v_{\text{max}}=3$. The blue solid line is the total error, as shown in Eq. (45), the dashed-dotted orange line is the Riemann sum error, as shown by the second term on the right-hand side of Eq. (45), and the dashed yellow line is an $N_v^{-1}$ scaling.