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 Schr\"odinger equation on Hilbert space with a Hermitian Hamiltonian operator and a unitary propagator. This Schr\"odinger 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.
Highlights
The Koopman–von Neumann formulation implies that the classical phase-space dynamics expressed by the Liouville equation can be recast as an equivalent Schrödinger equation for the wave function on Hilbert space
The wave function completely specifies the probability distribution function and its dynamics is generated by a Hermitian Hamiltonian operator and a unitary evolution operator
A quantum computer with finite resources can be used to simulate a finite-dimensional approximation of this unitary evolution operator
Summary
Future error-corrected quantum computers have the power to simulate quantum mechanical systems exponentially more efficiently [1,2] than computers that are bound to satisfy the laws of classical physics. It is important to understand whether quantum computers can provide similar gains in efficiency for the simulation of classical dynamics. Classical dynamical systems are typically nonlinear and many important examples are not Hamiltonian. Since quantum computers can only perform linear unitary operations, it is not clear how nonlinear nonunitary simulations can be performed efficiently. While efficient quantum algorithms for linear ordinary differential equations (ODEs) are known [6,7], an attempt to simulate nonlinear dynamics by measuring the full state at each time step and feeding this information into the time step would require an exponential amount of resources. Requires an exponential amount of resources in the number of time steps and the polynomial degree of the nonlinearity
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