I will discuss an alternative approach to simulating Hamiltonian flows with a quantum computer. A Hamiltonian system is a continuous time dynamical system represented as a flow of points in phase space. An
alternative dynamical system, first introduced by Poincare, is defined
in terms of an area preserving map. The dynamics is not continuous but discrete and successive dynamical states are labeled by integers rather than a continuous time variable. Discrete unitary maps are
naturally adapted to the quantum computing paradigm. Grover's
algorithm, for example, is an iterated unitary map. In this talk I
will discuss examples of nonlinear dynamical maps which are well adapted to simple ion trap quantum computers, including a transverse field Ising map, a non linear rotor map and a Jahn-Teller map. I will show how a good understanding of the quantum phase transitions
and entanglement exhibited in these models can be gained by first
describing the classical bifurcation structure of fixed points.