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Iterated dynamical maps on a quantum computer

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I will discuss an alternative 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 labelled 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.

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