Synthesis of reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs

Abstract Reversible circuits consisting of NOT, CNOT and 2-CNOT gates with small number of additional inputs are considered. For such circuits implementing maps f : Z2n $\begin{array}{} \displaystyle \mathbb Z_2^n \end{array}$ → Z2n $\begin{array}{} \displaystyle \mathbb Z_2^n \end{array}$, we study the Shannon complexity function L(n, q) under the condition that the number of additional inputs is q = O(n2). For this range of q, it is shown that L(n, q) ≍ n2n / log2 n. The growth order L(n, q) ≍ n2n / log2 (n + q) for all q ≲ n2n−⌈n/ϕ(n)⌉, where ϕ(n) → ∞ and n / ϕ(n) − log2 n → ∞ as n → ∞, is evaluated.