Simulation of many-body interactions by conditional geometric phases

It is shown how to exactly simulate many-body interactions and multiqubit 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 one to simulate many-body Hamiltonians and nonlinear Hamiltonians, and to implement a big variety of multiqubit quantum gates on both qubits and encoded qubits. An application to the quantum amplitude amplification algorithm will be discussed.