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

TL;DR

This work addresses designing binary‑input, multivalued‑output optical cascades for reversible and quantum technologies using group‑theoretic decomposition. It introduces a canonical cascade framework based on dihedral groups $D_n$, with Shannon‑type expansions and Walsh transform mappings that yield exponents encoding the cascade, and demonstrates for 3, 5, and 7 valued outputs. Seven local transformations are presented to simplify cascades, and a rigorous upper bound on the gate count is derived for arbitrary $n$‑variable, $k$‑valued outputs, yielding practical scalability in optical hardware using NOT, SWAP, and Fredkin gates. The approach directly supports optical implementations of reversible and quantum logic with multivalued outputs and provides concrete examples including modulo adders, highlighting potential impact on energy‑efficient photonic computation and quantum information processing.

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.

Figures (57)

• Figure 1: Applicability of groups of order two through fourteen
• Figure 2: Group Maps for elements $a,g$
• Figure 3: $D_3$ elements $I, g, a$
• Figure 4: Element $a^2$
• Figure 5: Elements of $D_3$

Theorems & Definitions (1)

• proof