A Quantum Algorithm for Nonlinear Electromagnetic Fluid Dynamics via Koopman-Von Neumann Linearization: 1D Numerical Evaluation Using Qulacs

We focus on the electromagnetic fluid dynamics governing plasmas and develop a quantum algorithm for nonlinear partial differential equations. Specifically, we apply Koopman-von Neumann (KvN) linearization to transform the electromagnetic fluid dynamics into a Schrödinger equation. The time evolution of this system is then computed using Hamiltonian simulation via Quantum Singular Value Transformation (QSVT). The proposed quantum algorithm has the potential to significantly reduce computational complexity compared to an ideal classical high-precision scheme, which scales as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O\left(N_{x}^{4}\right)$</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N_{x}$</tex> is the number of grid points per direction. By leveraging quantum computation, this complexity is reduced to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O\left(N_{x} \log N_{x}\right)$</tex> offering a potential advantage in large-scale simulations. To assess the feasibility and performance of quantum computing in plasma simulations, we implement the proposed quantum algorithm and conduct numerical validations. Specifically, we present results for two test cases: one with a linear solution and the other with a nonlinear solution. As a result, the behavior of composite numerical errors arising from discretization, KvN, and QSVT is elucidated, demonstrating the effectiveness of the proposed quantum algorithm. The novelty of this study lies in the application of a promising quantum algorithm for nonlinear partial differential equations to plasma simulations, along with a concrete numerical evaluation of its computational performance. Furthermore, this work presents a novel direction by identifying a nonlinear, high- degree-of-freedom physical system that can be efficiently solved by a quantum algorithm under theoretical guarantees.