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