On synthesis of reversible circuits consisting of NOT, CNOT, 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 a circuit implementing a map f:Z2n→Z2n, $f\colon \mathbb Z_2^n \to \mathbb Z_2^n,$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/log2n. $L(n,q) \asymp n2^n \mathop / \log_2 n.$We show that L(n,q)≍n2n/log2(n+q) $L(n,q) \asymp n2^n \mathop / \log_2 (n+q)$for all q≲n2n−⌈n/ϕ(n)⌉, $q \lesssim n2^{n-\lceil n \mathop / \phi(n)\rceil},$where ϕ(n)→∞andn/ϕ(n)−log2n→∞asn→∞. $\phi(n) \to \infty {\text {and}} \,n \mathop / \phi(n) - \log_2 n \to \infty \,{\text {as}}\, n \to \infty.$