Elementary gates for ternary quantum logic circuit

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In this article the elementary gates for ternary quantum logic circuit are studied. We propose the ternary controlled X (TCX) gate or ternary controlled Z (TCZ) gate as two-qutrit elementary gate, which is universal when assisted by arbitrary one-qutrit gates. It is primitive, efficient and easy to implement. Based on Cartan decomposition, we also give the one-qutrit elementary gates. Then the synthesis of some important ternary gates is investigated and the scheme of physical implementation for these ternary gates is discussed. Finally we extend these elementary gates to more general qudit case, so it provides a unified description for the synthesis of the binary and multi-valued quantum circuits.