On the dependence of the complexity and depth of reversible circuits consisting of NOT, CNOT, and 2-CNOT gates on the number of additional inputs

Abstract

Abstract The paper is concerned with the complexity and depth of reversible circuits consisting of NOT, CNOT, and 2-CNOT gates under constraints on the number of additional inputs. We study the Shannon functions for the complexity L(n, q) and depth D(n, q) of a reversible circuit implementing a map f : ℤ 2 n → ℤ 2 n $f\colon \mathbb{Z}_2^n \to \mathbb{Z}_2^n$ under the condition that the number of additional inputs q is in the range 8 n < q ≲ n 2 n − ⌈ n / ϕ ( n ) ⌉ $8n < q \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} n{2^{n - \left\lceil {n{\rm{ }}/\phi (n)} \right\rceil }}$ , where ϕ(n) → ∞ and n / ϕ(n) − log2 n → ∞ as n → ∞. We establish the upper estimates L ( n , q ) ≲ 2 n + 8 n 2 n / ( log 2 ( q − 4 n ) − log 2 n − 2 ) $L(n,q) \lesssim 2^n + 8n2^n \mathop / (\log_2 (q-4n) - \log_2 n - 2)$ and D ( n , q ) ≲ 2 n + 1 ( 2 , 5 + log 2 n − log 2 ( log 2 ( q − 4 n ) − log 2 n − 2 ) ) $D(n,q) \lesssim 2^{n+1}(2,5 + \log_2 n - \log_2 (\log_2 (q - 4n) - \log_2 n - 2))$ for this range of q. The asymptotics L ( n , q ) ≍ n 2 n / log 2 q $L(n,q) \asymp n2^n \mathop / \log_2 q$ is established for q such that n 2 ≲ q ≲ n 2 n − ⌈ n / ϕ ( n ) ⌉ ${n^2} \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} q \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} n{2^{n - \left\lceil {n{\rm{ }}/\phi (n)} \right\rceil }}$ , where ϕ(n) → ∞ and n / ϕ(n) − log2 n → ∞ as n → ∞.