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Optimal synthesis of multivalued quantum circuit

Although many of works have been done in multivalued quantum logic synthesis, the question whether multivalued quantum circuits are more efficient than the conventional binary quantum circuits is still open. In this article we devote to the optimization of generic multivalued quantum circuits. The multivalued quantum Shannon decompositions (QSD) are improved so that the circuits obtained are asymptotically optimal for all dimensionality d. The syntheses of uniformly multifold controlled $R_y$ rotations are also optimized to make the circuits further simplified. Moreover, the theoretical lower bound of complexity for multivalued quantum circuits is investigated, and a quantity known as efficiency index is proposed to evaluate the efficiency of synthesis of various quantum circuits. The algorithm for qudit circuits given here is an efficient synthesis routine which produces best known results for all dimensionality d, and for both cases the number of qudit n is small and that is asymptotic. The multivalued quantum circuits are indeed more efficient than the binary quantum circuits. The facts, the leading factor of the lower bound of complexity for qudit circuits is small by a factor of d-1 in comparison to that for qubit circuits and the asymptotic efficiency index is increased with the increase of dimensionality d, reveal the potential advantage of qudit circuits over generic qubit circuits. The generic n-qudit circuits with $d\geq5$ and generic two-ququart circuits synthesized by the algorithm given here are practical circuits which are more efficient than the most efficient qubit circuits.