On Asymptotic Gate Complexity and Depth of Reversible Circuits Without Additional Memory

Reversible computation is one of the most promising emerging technologies of the future. The usage of reversible circuits in computing devices can lead to a significantly lower power consumption. In this paper we study reversible logic circuits consisting of NOT, CNOT and 2-CNOT gates. We introduce a set $F(n,q)$ of all transformations $\mathbb Z_2^n \to \mathbb Z_2^n$ that can be implemented by reversible circuits with $(n+q)$ inputs. We define the Shannon gate complexity function $L(n,q)$ and the depth function $D(n,q)$ as functions of $n$ and the number of additional inputs $q$. First, we prove general lower bounds for functions $L(n,q)$ and $D(n,q)$. Second, we introduce a new group theory based synthesis algorithm, which can produce a circuit $\mathfrak S$ without additional inputs and with the gate complexity $L(\mathfrak S) \leq 3n 2^{n+4}(1+o(1)) \mathop / \log_2 n$. Using these bounds, we state that almost every reversible circuit with no additional inputs, consisting of NOT, CNOT and 2-CNOT gates, implements a transformation from $F(n,0)$ with the gate complexity $L(n,0) \asymp n 2^n \mathop / \log_2 n$ and with the depth $D(n,0) \geq 2^n(1-o(1)) \mathop / (3\log_2 n)$.