Estimation of the depth of reversible circuits consisting of NOT, CNOT and 2-CNOT gates

Abstract

The paper discusses the asymptotic depth of a reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The reversible circuit depth function D(n, q) is introduced for a circuit implementing a mapping f: Z 2 → Z 2 as a function of n and the number q of additional inputs. It is proved that for the case of implementation of a permutation from A(Z 2 ) with a reversible circuit having no additional inputs the depth is bounded as D(n, 0) ≳ 2 n /(3log2 n). It is also proved that for the case of transformation f: Z 2 → Z 2 with a reversible circuit having q 0 ~ 2 n additional inputs the depth is bounded as D(n,q 0) ≲ 3n.