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

Abstract: 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 $D(n,0) \geq 2^n(1-o(1)) \mathop / (3\log_2 n)$.