Quantum Simulation of Nonlinear Classical Dynamics*

Nonlinear classical dynamics can be simulated by a quantum computer with enough resources to approach the semiclassical limit. There is an exact embedding of a classical system of N ordinary differential equations (ODEs) within an enlarged quantum mechanical system with 2N degrees of freedom. Any set of ODEs can be derived from a classical Hamiltonian that is a sum over a set of N constraints, thereby yielding 2N equations of motion. Quantizing the constrained system leads to a Schrodinger equation that is equal to the classical Liouville equation which ensures conservation of phase space density for the original set of ODEs. Heisenberg’s uncertainty principle is satisfied by each variable and its canonically conjugate momentum, the Lagrange multiplier, on the extended phase space. Hence, there is no uncertainty in a simultaneous measurement of any of the variables of the original ODE. An appropriate choice of Planck’s constant can be used to reduce the uncertainty in the degrees of freedom of interest to a width on the order of the level spacing. Thus, excellent fidelity to the classical system can be achieved.