On Gate Complexity of Reversible Circuits Consisting of NOT, CNOT and 2-CNOT Gates

TL;DR

This paper establishes bounds on the gate complexity of reversible circuits with NOT, CNOT, and 2-CNOT gates, providing insights into their efficiency and limitations for implementing Boolean transformations.

Contribution

It derives new lower and upper bounds on the gate complexity of reversible circuits, advancing understanding of their computational efficiency.

Findings

Lower bound: $L(n,q) \\geq \\frac{2^n(n-2)}{3\\log_2(n+q)} - \\frac{n}{3}$

Upper bound without additional inputs: $L(n,0) \\leq 3n2^{n+4}(1+o(1))/\\log_2 n$

Upper bound with many additional inputs: $L(n,q_0) \\lesssim 2^n$ for $q_0 \\sim n2^{n-o(n)}$

Abstract

The paper discusses the gate complexity of reversible circuits consisting of NOT, CNOT and 2-CNOT gates. The Shannon gate complexity function $L(n, q)$ for a reversible circuit, implementing a Boolean transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$, is defined as a function of $n$ and the number of additional inputs $q$. The general lower bound $L(n,q) \geq \frac{2^n(n-2)}{3\log_2(n+q)} - \frac{n}{3}$ for the gate complexity of a reversible circuit is proved. An upper bound $L(n,0) \leqslant 3n2^{n+4}(1+o(1)) \mathop / \log_2n$ for the gate complexity of a reversible circuit without additional inputs is proved. An upper bound $L(n,q_0) \lesssim 2^n$ for the gate complexity of a reversible circuit with $q_0 \sim n2^{n-o(n)}$ additional inputs is proved.