An Algorithm for Reversible Logic Circuit Synthesis Based on Tensor Decomposition

An algorithm for reversible logic synthesis is proposed. The task is, for a given $n$-bit substitution map $Pn: {0,1}ⁿ \rightarrow {0,1}^n$, to find a sequence of reversible logic gates that implements the map. The gate library adopted in this work consists of multiple-controlled Toffoli gates denoted by $C^m!X$, where $m$ is the number of control bits that ranges from 0 to $n-1$. Controlled gates with large $m \,\,(>2)$ are then further decomposed into $C^0!X$, $C^1!X$, and $C^2!X$ gates. A primary concern in designing the algorithm is to reduce the use of $C^2!X$ gate (also known as Toffoli gate) which is known to be universal. The main idea is to view an $n$-bit substitution map as a rank-$2n$ tensor and to transform it such that the resulting map can be written as a tensor product of a rank-($2n-2$) tensor and the $2\times 2$ identity matrix. Let $\mathcal{P}n$ be a set of all $n$-bit substitution maps. What we try to find is a size reduction map $\mathcal{A}{\rm red}: \mathcal{P}n \rightarrow {Pn: Pn = P{n-1} \otimes I2}$. %, where $Im$ is the $m\times m$ identity matrix. One can see that the output $P{n-1} \otimes I2$ acts nontrivially on $n-1$ bits only, meaning that the map to be synthesized becomes $P{n-1}$. The size reduction process is iteratively applied until it reaches tensor product of only $2 \times 2$ matrices.