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.

• Received 12 June 2012

DOI:https://doi.org/10.1103/PhysRevA.87.012325