On asymptotic gate complexity and depth of reversible circuits without additional memory

In the paper we study reversible logic circuits, which consist of NOT, CNOT and C2NOT gates. We consider a set F(n,q) of all transformations Bn→Bn that can be realized by reversible circuits with (n+q) inputs. We define the Shannon gate complexity function L(n,q) and the depth function D(n,q) as functions of n and the number of additional inputs q. We prove general lower bounds for these functions. We introduce a new group-theory-based synthesis algorithm that can produce a reversible circuit S without additional inputs and with the gate complexity L(S)≤3n2n+4(1+o(1))/log2⁡n in the worst case.