The increased effort in recent years towards methods for computer aided design of reversible logic circuits has also lead to research in algorithms for optimising the resulting circuits; both with higher-level data structures and directly on the reversible circuits. To obtain structural patterns that can be replaced by a cheaper realisation, many direct algorithms apply so-called moving rules; a simple form of rewrite rules that can only swap gate order.
In this paper we first describe the few basic rules that are needed to perform rewriting directly on reversible logic circuits made from general Toffoli circuits. We also show how to use these rules to derive more complex formulas. The major difference compared to existing approaches is the use of negative controls (white dots), which significantly increases the algebraic strength. We show how existing optimisation approaches can be adapted as problems based on our rewrite rules.
Finally, we outline a path to generalising the rewrite rules by showing their forms for reversible control-gates. This can be used to expand our method to other gates such as the controlled-swap gate or quantum gates.