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arXiv:2512.13887 · AIAA SciTech Forum

Implementing the Koopman-von Neumann approach on continuous-variable photonic quantum computers

Xinfeng Gao, Olivier Pfister, Stefan Bekiranov

quant-phSubmitted Dec 15, 2025• Announced Dec 17, 2025• 0 citations
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Abstract

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.

Comments
9 pages, 3 figures, submitted for publication

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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.
Solving Ordinary Differential Equations (ODE) is important for a broad range of domains, such as engineering, weather forecast, and finance. Most quantum algorithms proposed to solve them revolve around transforming ODEs into a system of linear equations, in order to benefit from the exponential advantage promised by the HHL algorithm. However, there are known limitations to this subroutine such as the linear scaling with condition number. In particular, for HHL-based ODE solvers, this dependency yields a complexity that generally grows exponentially with the integration time. In this paper, we present a scheme that does not rely on the HHL subroutine. We use the Koopman Von Neumann (KvN) formalism that maps arbitrary nonlinear dynamics to a Hilbert space where the dynamics are unitary. This effectively reduces the ODE problem to a Hamiltonian time evolution, which is a well-known problem in quantum computing. However, this comes at a cost as the KvN formalism is expressed in an infinite-dimensional Hilbert space, and involves nonphysical states with infinite energies. Previous works have tackled this by truncating the Hilbert space, and by approximating all the relevant states and operations on qubit-based systems. Instead of mapping to qubit-based computations, in this paper, we investigate the direct use of continuous-variable quantum computers for this problem. We provide an algorithm to compile a sequence of Gaussian and non-Gaussian continuous variables gates to solve an arbitrary one-dimensional polynomial ODE. We analyze the algorithm and propose that it is intrinsically better suited for solving so-called initial distribution problems, rather than initial condition problems. We propose the first steps towards a comprehensive complexity and specify which steps need to be developed for a complete analysis.
Quantum computers have long been expected to efficiently solve complex classical differential equations. Most digital, fault-tolerant approaches use Carleman linearization to map nonlinear systems to linear ones and then apply quantum linear-system solvers. However, provable speedups typically require digital truncation and full fault tolerance, rendering such linearization approaches challenging to implement on current hardware. Here we present an analog, continuous-variable algorithm based on coupled bosonic modes with qubit-based adaptive measurements that avoids Hilbert-space digitization. This method encodes classical fields as coherent states and, via Kraus-channel composition derived from the Koopman-von Neumann (KvN) formalism, maps nonlinear evolution to linear dynamics. Unlike many analog schemes, the algorithm is provably efficient: advancing a first-order, LLL-grid point, ddd-dimensional, order-KKK spatial-derivative, degree-rrr polynomial-nonlinearity, strongly dissipative partial differential equations (PDEs) for TTT time steps costs O(T(log⁡L+drlog⁡K))\mathcal{O}\left(T(\log L + d r \log K)\right)O(T(logL+drlogK)). The capability of the scheme is demonstrated by using it to simulate the one-dimensional Burgers' equation and two-dimensional Fisher-KPP equation. The resilience of the method to photon loss is shown under strong-dissipation conditions and an analytic counterterm is derived that systematically cancels the dominant, experimentally calibrated noise. This work establishes a continuous-variable framework for simulating nonlinear systems and identifies a viable pathway toward practical quantum speedup on near-term analog hardware.
The Koopman–von Neumann (KvN) formulation recasts deterministic classical mechanics in a Hilbert-space operator framework analogous to quantum mechanics. In this approach, the Liouville equation is expressed as a Schrödinger-type evolution equation generated by a Hermitian operator, yielding unitary propagation for phase-space-volume-preserving dynamics. We first clarify the relationship among the Koopman, Perron–Frobenius, and Liouville operators through an explicit example. Wilczek's construction shows that the KvN formalism is not only restricted to canonical Hamiltonian mechanics but this can be generalized any first-order classical dynamical system. Wilczek shows how a general autonomous first-order system can be generated by a Hermitian temporal-evolution operator after the introduction of auxiliary conjugate variables. We develop the KvN description of systems driven by Hamiltonian curl forces–position-dependent, non-conservative forces that are non-dissipative and preserve phase-space volume. Finally, we extend the KvN formalism to non-Hamiltonian systems using the Jacobi Last Multiplier and examine its connection to the Boltzmann H-theorem.