A Quantum Algorithm for Nonlinear Electromagnetic Fluid Dynamics via Koopman-von Neumann Linearization

TL;DR

The paper tackles the computational bottlenecks of nonlinear plasma simulations by mapping electromagnetic fluid dynamics to a linear quantum dynamics via Koopman‑von Neumann (KvN) linearization and solving it with quantum singular value transformation (QSVT). The KvN framework yields a sparse, truncated Hamiltonian that can be simulated on a quantum computer with polylogarithmic dependence on grid points and exponential savings in spatial dimension, achieving a polynomial speedup in the number of grid points per dimension and an exponential advantage in the number of spatial dimensions. Numerical experiments, including 1D linear oscillations, 1D nonlinear advection, and 2D Kelvin–Helmholtz instability, demonstrate accurate behavior and reveal practical guidelines for truncation parameters $m$ and $R$ in real‑world problems. The results indicate that quantum computing can provide a viable pathway to overcome multiscale bottlenecks in nonlinear plasma modeling, with potential scalability to large‑scale simulations given adequate quantum resources.

Abstract

To simulate plasma phenomena, large-scale computational resources have been employed in developing high-precision and high-resolution plasma simulations. One of the main obstacles in plasma simulations is the requirement of computational resources that scale polynomially with the number of spatial grids, which poses a significant challenge for large-scale modeling. To address this issue, this study presents a quantum algorithm for simulating the nonlinear electromagnetic fluid dynamics that govern space plasmas. We map it, by applying Koopman-von Neumann linearization, to the Schrödinger equation and evolve the system using Hamiltonian simulation via quantum singular value transformation. Our algorithm scales $O \left(s N_x \, \mathrm{polylog} \left( N_x \right) T \right)$ in time complexity with $s$, $N_x$, and $T$ being the spatial dimension, the number of spatial grid points per dimension, and the evolution time, respectively. Comparing the scaling $O \left( s N_x^s \left(T^{5/4}+T N_x\right) \right)$ for the classical method with the finite volume scheme, this algorithm achieves polynomial speedup in $N_x$. The space complexity of this algorithm is exponentially reduced from $O\left( s N_x^s \right)$ to $O\left( s \, \mathrm{polylog} \left( N_x \right) \right)$. Numerical experiments validate that accurate solutions are attainable with smaller $m$ than theoretically anticipated and with practical values of $m$ and $R$, underscoring the feasibility of the approach. As a practical demonstration, the method accurately reproduces the Kelvin-Helmholtz instability, underscoring its capability to tackle more intricate nonlinear dynamics. These results suggest that quantum computing can offer a viable pathway to overcome the computational barriers of multiscale plasma modeling.

Figures (6)

• Figure 1: The quantum circuit $U_{\text{exp}}$ for an $(1, a+2, \epsilon)$-block encoding of the time evolution operator $\frac{1}{2}\exp(-i\tilde{H}T)$. Here $U$ is an $(\alpha,a,\epsilon/|2T|)$-block encoding of Hamiltonian $\tilde{H}$. See also Ref. Toyoizumi2024.
• Figure 2: Time evolution of the fluid velocity $\tilde{u}$ and electric field $\tilde{E}$ at a point of $x=0$. The solid lines are the discretized KvN‑QSVT result and the dot lines are the discretized KvN‑expm. Time scales with plasma frequency $\omega_{p.e}$
• Figure 3: Time evolution of the deviation errors between the $L^2$ norm of variable vector $\mathbf{x}$ and the initial $L^2$ norm by the KvN‑QSVT and KvN‑expm in the Case B.
• Figure 4: Time evolution of the deviation errors between the $L^2$ norm of variable vector $\mathbf{x}$ and the initial $L^2$ norm by the KvN‑QSVT and KvN‑expm in the Case C.
• Figure 5: The velocity vector fields of the 2D Kelvin-Helmholtz instability obtained from KvN‑QSVT simulations. (a): $t = 0$, (b): $t = 1.07$ ($=0.5T_{\text{eign}}$), and (c): $t = 2.15$ ($=1.0T_{\text{eign}}$). The color bar represents the magnitude of the velocity vector.

Theorems & Definitions (8)

• Definition 1: Definition 2 in Tanaka2024
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3: Hermitian block-encoding of sparse Hamiltonians camps2024explicit
• Lemma 4: alternative version of hermitian block-encoding of sparse Hamiltonians
• proof