Synthesis of binary-input multi-valued output optical cascades

This paper extends the decomposition from the group theory based methods of Sasao and Saraivanov to design binary input multivalued output quantum cascades realized with optical NOT, SWAP, and Fredkin Gates. We present this method for 3, 5, and 7-valued outputs, but in general it can be used for odd prime-valued outputs. The method can be extended to realize hybrid functions with different valued outputs. A class of local transformations is presented that can simplify the final cascade circuits. Using these simplifying transformations, we present an upper bound on the maximum number of gates in an arbitrary $n$-variable input and $k$-valued output function.