On Synthesis of Reversible Circuits with Small Number of Additional Inputs Consisting of NOT, CNOT and 2-CNOT Gates

TL;DR

This paper analyzes the gate complexity of reversible circuits with few additional inputs, specifically using NOT, CNOT, and 2-CNOT gates, and establishes bounds for the Shannon's gate complexity function in this context.

Contribution

It provides a theoretical bound on the gate complexity of reversible circuits with limited additional inputs, expanding understanding of their efficiency.

Findings

Proves a general bound for Shannon's gate complexity function $L(n, q)$ with $q o O(n^2)$

Shows that $L(n,q) hickapprox n2^n / \log_2 n$ for small $q$

Analyzes the impact of additional inputs on circuit complexity

Abstract

The paper discusses the gate complexity of reversible circuits with the small number of additional inputs consisting of NOT, CNOT and 2-CNOT gates. We study Shannon's gate complexity function $L(n, q)$ for a reversible circuit implementing a Boolean transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ with $q \leqslant O(n^2)$ additional inputs. The general bound $L(n,q) \asymp n2^n \mathop / \log_2 n$ is proved for this case.