Koopman–von Neumann approach to quantum simulation of nonlinear classical dynamics

Abstract

Quantum computers can be used to simulate nonlinear non-Hamiltonian classical dynamics on phase space by using the generalized Koopman-von Neumann formulation of classical mechanics. The Koopman-von Neumann formulation implies that the conservation of the probability distribution function on phase space, as expressed by the Liouville equation, can be recast as an equivalent Schrödinger equation on Hilbert space with a Hermitian Hamiltonian operator and a unitary propagator. This Schrödinger equation is linear in the momenta because it derives from a constrained Hamiltonian system with twice the classical phase-space dimension. A quantum computer with finite resources can be used to simulate a finite-dimensional approximation of this unitary evolution operator. Quantum simulation of classical dynamics is exponentially more efficient than a deterministic Eulerian discretization of the Liouville equation if the Koopman-von Neumann Hamiltonian is sparse. Utilizing quantum walk techniques for state preparation and amplitude estimation for the calculation of observables leads to a quadratic improvement over classical probabilistic Monte Carlo algorithms.

Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

“…As an interesting alternative direction, we point out that the inherent dynamics of quantum systems could be used for solving nonlinear differential equations via the Koopman-von Neumann formulation of classical mechanics [19].…”

Koopman wavefunctions and Clebsch variables in Vlasov–Maxwell kinetic theory

“…As discussed in Bondar et al. (2019) and Joseph (2020), this Hamiltonian structure readily identifies a Hamiltonian functional given by This is accompanied by the usual Poisson bracket from standard quantum mechanics (Chernoff & Marsden 1976; Foskett, Holm & Tronci 2019) where we have introduced the double-bracket notation to distinguish from the canonical Poisson bracket. Then, the Madelung transform expresses the Koopman wavefunction in terms of the density and the phase , thereby leading to the celebrated Clebsch representation of the real-valued Vlasov density (Holm & Kupershmidt 1983; Marsden & Weinstein 1983; Morrison 1981), that is Here, the classical Liouville equation follows by combining the Jacobi identity with the relations which arise from (1.1).…”

“…Second, the Clebsch representation (2.4) does not generally identify a probability density, since whenever and are differentiable continuous functions. In Joseph (2020), a solution to this apparent issue was provided by selecting a singular phase ensuring the following consistency condition Due to the equations of motion (2.5 a , b ), if this relation holds true as an initial condition, then it will hold for all time. In this case, the expression of the phase is necessarily singular and this is strictly necessary to produce the boundary terms generally required for the Clebsch representation (2.4) to hold true.…”

“…Ongoing discussions (Shenkel et al. 2018) on the potential of quantum information and computation in plasma physics have recently led to exploiting Hilbert-space approaches in the numerical simulation of magnetized plasmas (Dodin & Startsev 2020; Engel, Smith & Parker 2019, 2021; Joseph 2020), of the Navier–Stokes equations (Gaitan 2020), and of arbitrary non-Hamiltonian systems of equations (Joseph 2020; Liu et al. 2020).…”

“…2020). In particular, recent work (Joseph 2020) has emphasized the role of Koopman wavefunctions in classical dynamics while their usage in describing hybrid quantum-classical systems was presented in Bondar, Gay-Balmaz & Tronci (2019), Boucher & Traschen (1988), Gay-Balmaz & Tronci (2019), Tronci & Gay-Balmaz (2021), Sudarshan (1976) and Jauslin & Sugny (2010). In this paper, we want to show how these enter in the variational and Hamiltonian formulation of Vlasov kinetic theory.…”