# General Upper Bounds for Gate Complexity and Depth of Reversible Circuits Consisting of NOT, CNOT and 2-CNOT Gates

## TL;DR

This paper establishes new upper bounds on the gate complexity and depth of reversible circuits with NOT, CNOT, and 2-CNOT gates when the number of additional inputs is limited, advancing understanding of their efficiency.

## Contribution

It provides the first general upper bounds for gate complexity and depth of reversible circuits with limited additional inputs, specifically for functions with a large number of inputs.

## Key findings

- Upper bounds for gate complexity $L(n,q)$ derived.
- Upper bounds for circuit depth $D(n,q)$ established.
- Results applicable for $8n < q
oughly n2^{n-o(n)}$ additional inputs.

## Abstract

The paper discusses the gate complexity and the depth of reversible circuits consisting of NOT, CNOT and 2-CNOT gates in the case, when the number of additional inputs is limited. We study Shannon's gate complexity function $L(n, q)$ and depth function $D(n, q)$ for a reversible circuit implementing a Boolean transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ with $8n < q \lesssim n2^{n-o(n)}$ additional inputs. The general upper bounds $L(n,q) \lesssim 2^n + 8n2^n \mathop / (\log_2 (q-4n) - \log_2 n - 2)$ and $D(n,q) \lesssim 2^{n+1}(2,5 + \log_2 n - \log_2 (\log_2 (q - 4n) - \log_2 n - 2))$ are proved for this case.