Synthesis of Binary-Input Multi-Valued Output Optical Cascades for Reversible and Quantum Technologies

TL;DR

This paper introduces a group theory-based method for designing binary-input, multi-valued output optical cascades for reversible and quantum computing, extending to odd prime valued outputs and hybrid functions.

Contribution

It extends existing decomposition methods to multi-valued outputs using optical gates and provides an upper bound on circuit complexity with simplification techniques.

Findings

Method applicable to 3, 5, 7 valued outputs

Can be extended to hybrid functions with different outputs

Provides an upper bound on the number of gates in the cascade

Abstract

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.