Canonical multi-target Toffoli circuits

Abstract

In this paper, we study reversible circuits as cascades of multi-target Toffoli gates. This new type of gates allows to shift parts of a gate to the preceding gate within a circuit provided that a certain independence condition holds. It turns out that shifts decrease the so-called waiting degree such that shifting as long as possible always terminates and yields shift-reduced circuits. As the main result, we show that shift-reduced circuits are unique canonical representatives of their shift equivalence classes. Canonical circuits are optimal with respect to maximal and as-early-as-possible parallelism of targets within gates.