Asymptotic bounds of depth for a reversible circuit consisting of NOT, CNOT and 2-CNOT gates

The paper discusses the asymptotic depth of a reversible circuit consisting of NOT, CNOT and 2-CNOT gates. Reversible circuit depth function $D(n,q)$ for a circuit implementing a transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$isintroducedasafunctionof$n$andthenumberofadditionalinputs$q$\mathrm{.}Itisprovedthatforthecaseofimplementingapermutationfrom$A(\mathbb Z_2^n)$ with a reversible circuit having no additional inputs the depth is bounded as $D(n,0)\gtrsim 2^n\mathrm{/}(3{\log }_{2}n)$. It is proved that for the case of implementing a transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$withareversiblecircuithaving$q_0 \sim 2^n$ additional inputs the depth is bounded as $D(n,q_0)\lesssim 3n$.