Abstract
In this paper, we study reversible circuits as cascades of multi-target Toffoli gates. This 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. Further, we discuss how successive equal subgates can be removed in order to reduce the waiting degree even more. Interestingly after applying shifts and removals as long as possible again a unique normal form is obtained.
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