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

Abstract

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 $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ is introduced as a function of $n$ and the number of additional inputs $q$. It is proved that for the case of implementing a permutation from $A(\mathbb Z_2^n)$ with a reversible circuit having no additional inputs the depth is bounded as $D(n, 0) \gtrsim 2^n / (3\log_2 n)$. It is proved that for the case of implementing a transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ with a reversible circuit having $q_0 \sim 2^n$ additional inputs the depth is bounded as $D(n, q_0) \lesssim 3n$.