  It is shown how to exactly simulate many-body interactions and multi-qubit
gates by coupling finite dimensional systems, e.g., qubits with a continuous
variable. Cyclic evolution in the phase space of such a variable gives rise to
a geometric phase, depending on a product of commuting operators. The latter
allows to simulate many-body Hamiltonians and nonlinear Hamiltonians, and to
implement a big variety of multi-qubit quantum gates on both qubits and encoded
qubits. An application to the quantum amplitude amplification algorithm will be
discussed.
