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Quantum computers can be used to simulate nonlinear non-Hamiltonian classicaldynamics on phase space by using the generalized Koopman-von Neumannformulation of classical mechanics. The Koopman-von Neumann formulation impliesthat the conservation of the probability distribution function on phase space,as expressed by the Liouville equation, can be recast as an equivalentSchr\"odinger equation on Hilbert space with a Hermitian Hamiltonian operatorand a unitary propagator. This Schr\"odinger equation is linear in the momentabecause it derives from a constrained Hamiltonian system with twice theclassical phase space dimension. A quantum computer with finite resources canbe used to simulate a finite-dimensional approximation of this unitaryevolution operator. Quantum simulation of classical dynamics is exponentiallymore efficient than a deterministic Eulerian discretization of the Liouvilleequation if the Koopman-von Neumann Hamiltonian is sparse. Utilizing quantumwalk techniques for state preparation and amplitude estimation for thecalculation of observables leads to a quadratic improvement over classicalprobabilistic Monte Carlo algorithms.

Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics