  We provide a simple framework for the synthesis of quantum circuits based on
a numerical optimization algorithm. This algorithm is used in the context of
the trapped-ions technology. We derive theoretical lower bounds for the number
of quantum gates required to implement any quantum algorithm. Then we present
numerical experiments with random quantum operators where we compute the
optimal parameters of the circuits and we illustrate the correctness of the
theoretical lower bounds. We finally discuss the scalability of the method with
the number of qubits.
