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Keisan Rikigaku Koenkai koen ronbunshu/Keisan Rikigaku Kouenkai kouen rombunshuu

A quantum algorithm for nonlinear electromagnetic fluid dynamics using Koopman-von Neumann linearization

Hayato HIGUCHI, Yuki ITO, Kazuki Sakamoto, Keisuke Fujii, Akimasa Yoshikawa

Published Jan 1, 2025• 0 citations
Publisher

Abstract

To predict and simulate plasma phenomena, large-scale computational resources have been employed to develop high-precision, high-resolution plasma simulations. However, multi-scale plasma simulations require computational resources that scale polynomially with the number of spatial grids, posing a significant challenge for large-scale modeling. In this study, we present a quantum algorithm for simulating the nonlinear electromagnetic fluid dynamics systems that govern space plasmas. By applying Koopman–von Neumann (KvN) linearization, we map the nonlinear electromagnetic fluid dynamics systems to a Schrödinger equation and evolve it using Hamiltonian simulation via quantum singular value transformation (QSVT). The resulting algorithm reduces the computational complexity from O(Nx4) scaling of classical finite volume schemes to O(Nx log Nx), where Nx is the number of spatial grid points per dimension. And numerical experiments quantify behaviors of combined errors of discretization, KvN linearization, and QSVT-Hamiltonian simulation. As a practical demonstration, the method accurately reproduces the Kelvin-Helmholtz instability, underscoring its capability to tackle intricate nonlinear dynamics. These results suggest that quantum computing can offer a viable pathway to overcome the computational barriers of traditional multiscale plasma modeling.

DOI
10.1299/jsmecmd.2025.38.os6-8

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To predict and simulate plasma phenomena, large-scale computational resources have been employed to develop high-precision, high-resolution plasma simulations. However, multi-scale plasma simulations require computational resources that scale polynomially with the number of spatial grids, posing a significant challenge for large-scale modeling. In this study, we present a quantum algorithm for simulating the nonlinear electromagnetic fluid dynamics systems that govern space plasmas. By applying Koopman–von Neumann (KvN) linearization, we map the nonlinear electromagnetic fluid dynamics systems to a Schrödinger equation and evolve it using Hamiltonian simulation via quantum singular value transformation (QSVT). The resulting algorithm reduces the computational complexity from O(Nx4) scaling of classical finite volume schemes to O(Nx log Nx), where Nx is the number of spatial grid points per dimension. And numerical experiments quantify behaviors of combined errors of discretization, KvN linearization, and QSVT-Hamiltonian simulation. As a practical demonstration, the method accurately reproduces the Kelvin-Helmholtz instability, underscoring its capability to tackle intricate nonlinear dynamics. These results suggest that quantum computing can offer a viable pathway to overcome the computational barriers of traditional multiscale plasma modeling.
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The Koopman-von Neumann (KvN) formalism recasts classical mechanics in a Hilbert space framework using complex wavefunctions and linear operators, akin to quantum mechanics. Instead of evolving probability densities in phase space (as in Liouville's equation), KvN uses a Schrödinger-like equation for a classical wavefunction, with commuting position and momentum operators. Mapped to quantum computing, KvN offers a promising route to simulate classical dynamical systems using quantum algorithms by leveraging unitary evolution and quantum linear algebra tools, potentially enabling efficient classical-to-quantum mappings without invoking full quantum uncertainty. In this work, we specifically explore the implementation of the KvN approach on continuous-variable photonic quantum computing architectures, with the goals of leveraging quantum simulation for both sampling and computing intractable nonlinear dynamics. We will demonstrate its implementation and feasibility with two problems: the harmonic oscillator and a 1D partial differential equation governing nonlinear dynamics.
Quantum computers have the potential to efficiently solve a system of nonlinear ordinary differential equations (ODEs), which play a crucial role in various industries and scientific fields. However, it remains unclear which system of nonlinear ODEs, and under what assumptions, can achieve exponential speedup using quantum computers. In this work, we introduce a class of systems of nonlinear ODEs that can be efficiently solved on quantum computers, where the efficiency is defined as solving the system with computational complexity of O(Tlog(N)polylog(1/ε))O(T {\rm log}(N) {\rm polylog}(1/ε))O(Tlog(N)polylog(1/ε)), where TTT is the evolution time, εεε is the allowed error, and NNN is the number of variables in the system. Specifically, we employ the Koopman-von Neumann linearization to map the system of nonlinear ODEs to Hamiltonian dynamics and find conditions where the norm of the mapped Hamiltonian is preserved and the Hamiltonian is sparse. This allows us to use the optimal Hamiltonian simulation technique for solving the nonlinear ODEs with O(log(N))O({\rm log}(N))O(log(N)) overhead. Furthermore, we show that the nonlinear ODEs include a wide range of systems of nonlinear ODEs, such as the nonlinear harmonic oscillators and the short-range Kuramoto model. This is the first concrete example of solving systems of nonlinear ODEs with exponential quantum speedup by the Koopman-von Neumann linearization, although it is noted that this assumes efficient preparation of the initial state and computation of the output. These findings contribute significantly to the application of quantum computers in solving nonlinear problems.
The Koopman--von Neumann (KvN) formulation of spectrally truncated fluid and plasma dynamics is considered as a potential approach for quantum computation. The KvN framework embeds the Liouville equation into a Hilbert space with norm-preserving, unitary evolution. Here, we propose a Weyl-ordered KvN generator along with a summation-by-parts discretization, which ensures that the resulting operators are exactly unitary as required for quantum computers. The Weyl-ordered KvN generator is derived as the unique anti-Hermitian operator symmetrization for real velocity fields. The formulation operates directly in the physical amplitude space without phase-space doubling, so the Heisenberg uncertainty principle does not constrain the grid resolution during evolution. This limitation re-enters only at the measurement stage on a quantum computer. Exact discrete unitarity is proved as a purely algebraic identity that holds regardless of grid resolution or stencil order. To manage boundaries, a split-step Kraus absorbing layer is introduced via a Stinespring dilation requiring only one ancilla qubit. Validation on three test cases spanning dissipative and Hamiltonian regimes (a viscous Navier--Stokes triad, an incompressible Euler triad, and a Hasegawa--Mima drift-wave triad) confirms fourth-order convergence and machine-precision unitarity.
The Vlasov-Maxwell 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 Vlasov-Maxwell 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 error-corrected universal quantum computer. While the classical simulation has costs that scale as O(Nvt)O({N}_{v}t)O(Nv​t) for a velocity grid with Nv{N}_{v}Nv​ grid points and simulation time ttt, our quantum algorithm scales as $O[\text{polylog}({N}_{v})t/\ensuremath{\delta}]$ where $\ensuremath{\delta}$ 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 high-resolution, six-dimensional phase-space grid, but the $1/\ensuremath{\delta}$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.
We propose an explicit algorithm based on the Linear Combination of Hamiltonian Simulations technique to simulate both the advection-diffusion equation and a nonunitary discretized version of the Koopman--von Neumann formulation of nonlinear dynamics. By including dissipation into the model, through an upwind discretization of the advection operator, we avoid spurious parasitic oscillations which usually accompany standard finite difference discretizations of the advection equation. In contrast to prior works on quantum simulation of nonlinear problems, we explain in detail how different components of the algorithm can be implemented by using the Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) methods. In addition, we discuss the general method for implementing the block-encoding (BE) required for QSP and QSVT circuits and provide explicit implementations of the BE oracles tailored to our specific test cases. We simulate the resulting circuit on a digital emulator of quantum fault-tolerant computers and investigate its complexity and success probability. The proposed algorithm is universal and can be used for modeling a broad class of linear and nonlinear differential equations including the KvN and Carleman embeddings of nonlinear systems, the semiclassical Koopman-van Hove (KvH) equation, as well as the advection and Liouville equations.