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arXiv:2306.13504 · Journal of Physics A Mathematical and Theoretical

Existence and uniqueness of solutions of the Koopman–von Neumann equation on bounded domains

Marian Stengl, Patrick Gelß, Stefan Klus, Sebastian Pokutta

Published Aug 14, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract The Koopman–von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman–von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set’s closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman–von Neumann framework to transport equations is highlighted.

DOI
10.1088/1751-8121/ad6f7d

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Abstract The Koopman–von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman–von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set’s closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman–von Neumann framework to transport equations is highlighted.
Koopman and von Neumann (KvN) extended the Liouville equation by introducing a phase space function S(K)(q,p,t)S^{(K)}(q,p,t)S(K)(q,p,t) whose physical meaning is unknown. We show that a different S(q, p, t), with well-defined physical meaning, may be introduced without destroying the attractive “quantum-like” mathematical features of the KvN theory. This new S(q, p, t) is the classical action expressed in phase space coordinates. It defines a mapping between observables and operators which preserves the Lie bracket structure. The new evolution equation reduces to Schrödinger’s equation if functions on phase space are reduced to functions on configuration space. This new kind of “quantization” does not only establish a correspondence between observables and operators, but provides in addition a derivation of quantum operators and evolution equations from corresponding classical entities. It is performed by replacing ∂∂p\frac{\partial }{\partial p}∂p∂​ by 0 and p by ℏı∂∂q\frac{\hbar }{\imath } \frac{\partial }{\partial q}ℏ​∂q∂​ , thus providing an explanation for the common quantization rules.
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.
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.
Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed classical variables into a quantum state. While the Carleman and Koopman-von Neumann embeddings have been investigated so far, the class of problems that can be mapped to the Schrödinger equation is not well understood even for linear differential equations. In this work, we investigate the conditions for linear differential equations to be mapped to the Schrödinger equation and solved on a quantum computer. Interestingly, we find that these conditions are identical for both Carleman and Koopman-von Neumann embeddings. We also compute the computational complexity associated with estimating the expected values of an observable. This is done by assuming a state preparation oracle, block encoding of the mapped Hamiltonian via either Carleman or Koopman-von Neumann embedding, and block encoding of the observable using O(log⁡M)O(\log M)O(logM) qubits with MMM is the mapped system size. Furthermore, we consider a general classical quadratic Hamiltonian dynamics and find a sufficient condition to map it into the Schrödinger equation. As a special case, this includes the coupled harmonic oscillator model [Babbush et al., \cite{babbush_exponential_2023}]. We also find a concrete example that cannot be described as the coupled harmonic oscillator but can be mapped to the Schrödinger equation in our framework. These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.
The so-called Koopman-von Neumann (KvN) formulation brings classical mechanics to Hilbert space, but many techniques familiar from quantum mechanics remain missing. We use the gauge freedom of the KvN equation to derive separable solutions of the KvN equation, and by extension the classical Liouville equation. The canonical ensemble solution of statistical mechanics is shown to arise from separable solutions. We discuss the Hermiticity of the Liouvillian operator, and treat a case where its spectrum is discrete. Finally, we analyze an uncertainty relation between the so-called dynamical time and the Liouvillian, and argue that such a relation is natural and can be understood entirely within a classical framework.
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.