Reversible Circuits Synthesis from Functional Decision Diagrams by using Node Dependency Matrices

Decision diagrams are a data structure suitable for reversible circuit synthesis. Functional decision diagrams (FDDs) are particularly convenient in synthesis with Toffoli gates, since the functional expressions for decomposition rules used in them are similar to the functional expressions of Toffoli gates. The main drawback of reversible circuit synthesis based on decision diagrams is the usually large number of ancilla lines. This paper presents two methods for the reduction of the number of ancilla lines in reversible circuits derived from FDDs by selecting the order of implementation of nodes. In the first method, nodes are implemented by levels, starting from the bottom level to the top. The method uses appropriately defined level dependency matrices for choosing the optimal order of implementation of nodes at the same level. In this way, the optimization is performed level by level. The second method uses a diagram dependency matrix expressing mutual dependencies among all the nodes in the diagram. This method is computationally more demanding than the first method, but the reductions of both the number of lines and the Quantum cost of the circuits are larger.