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

TL;DR

This paper introduces a novel unitary discretization method for the Koopman-von Neumann equation, enabling quantum simulation of fluid and plasma dynamics with exact unitarity and high accuracy.

Contribution

It develops a Weyl-ordered KvN generator with summation-by-parts discretization, ensuring exact unitarity and efficient boundary management for quantum simulations.

Findings

Validated on three test cases with dissipative and Hamiltonian regimes.

Achieved fourth-order convergence and machine-precision unitarity.

Operates directly in physical amplitude space without phase-space doubling.

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.