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arXiv:2605.19187

Unitary discretization of the Koopman-von Neumann equation for quantum simulation of fluid and plasma dynamics

Aleksandar Jemcov, Scott C. Morris

physics.flu-dynquant-phSubmitted May 18, 2026• Announced May 20, 2026• 0 citations
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Abstract

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.

Comments
18 pages, 7 figures. Submitted to Physical Review Research

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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.
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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.
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The simulation of nonlinear ordinary differential equations on quantum computers is inherently challenging, as quantum gates are linear operators on qubit states. In this paper, we put forth a Koopman-von Neumann (KvN) operator based algorithm for solving nonlinear ordinary differential equations on a quantum computer which overcomes the innate limitations of quantum operations. In this approach, a Liouville probability density is embedded into a wavefunction, the evolution of which is governed by an operator dual to the Koopman operator. The trajectories corresponding to the nonlinear differential equations are reconstructed from the average value of Koopman observables evaluated through quantum measurements. We specifically evaluate the computational limitations of solving nonlinear equations within this Liouville embedding framework. An Ehrenfest-type estimate is derived that relates the reconstruction error to the covariance of the transported density, highlighting the competition between linear stretching and Hessian-induced folding.This leads to a key stability criterion, which depends on the local Ehrenfest-Reynolds number. We discuss the effects of measurement-induced errors including those due to Hadamard-test sampling, amplitude estimation, and bias due to covariance in probabilistic Grover-type inference on the evaluation of trajectories. The limiting bounds on the accuracy of quantum computations are numerically verified for the Lotka-Volterra system and for the quartic oscillator. We observe a rapid growth in computational errors when the Ehrenfest-Reynolds number approaches the predicted threshold. The bounds resulting from our analysis provide key guidelines for selecting the width of the initial Gaussian probability distribution associated with a system of nonlinear ordinary differential equations.
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.