Two Approaches to Quantum Simulation of Classical Dynamics*

There are two approaches to quantum simulation of nonlinear classical dynamics: (i) quantize the classical Hamiltonian and (ii) use the Koopman-von Neumann approach to reformulate the conservation of probability, the Liouville equation, as an equivalent Schrodinger equation with a unitary evolution operator. The latter approach tracks the exact evolution of probability on phase space but requires simulating a system of twice the dimensionality. The former approach tracks the dynamics of the classical probability distribution until the Heisenberg time, at which point the quantum dynamics departs from that of the classical system. Simulating dissipative processes can be accomplished by embedding the N-dimensional system within a larger Hilbert space of size N², which is similar in cost to doubling the phase space dimension. For either approach, using a quantum computer to simulate these systems is exponentially more efficient than simulating the Eulerian discretization of the Liouville equation when the Hamiltonian is sparse. Using quantum walks to generate desired initial conditions and using amplitude estimation to measure observables is up to quadratically more efficient than time-dependent Monte Carlo techniques.