  We propose the generalized controlled X (GCX) gate as the two-qudit
elementary gate, and based on Cartan decomposition, we also give the one-qudit
elementary gates. Then we discuss the physical implementation of these
elementary gates and show that it is feasible with current technology. With
these elementary gates many important qudit quantum gates can be synthesized
conveniently. We provide efficient methods for the synthesis of various kinds
of controlled qudit gates and greatly simplify the synthesis of existing
generic multi-valued quantum circuits. Moreover, we generalize the quantum
Shannon decomposition (QSD), the most powerful technique for the synthesis of
generic qubit circuits, to the qudit case. A comparison of ququart (d=4)
circuits and qubit circuits reveals that using ququart circuits may have an
advantage over the qubit circuits in the synthesis of quantum circuits.
