Asymptotic bounds of depth for a reversible circuit consisting of NOT, CNOT and 2-CNOT gates

TL;DR

This paper establishes asymptotic bounds on the depth of reversible circuits with NOT, CNOT, and 2-CNOT gates, depending on the number of additional inputs, providing insights into their efficiency for implementing permutations and transformations.

Contribution

It introduces bounds on the depth of reversible circuits with specific gates, considering the impact of additional inputs, advancing understanding of circuit complexity.

Findings

Depth for permutation implementation without extra inputs is at least exponential in n.

With approximately 2^n additional inputs, the depth is bounded above by a linear function of n.

The results clarify how additional inputs influence the circuit depth for reversible transformations.

Abstract

The paper discusses the asymptotic depth of a reversible circuit consisting of NOT, CNOT and 2-CNOT gates. Reversible circuit depth function $D(n, q)$ for a circuit implementing a transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ is introduced as a function of $n$ and the number of additional inputs $q$. It is proved that for the case of implementing a permutation from $A(\mathbb Z_2^n)$ with a reversible circuit having no additional inputs the depth is bounded as $D(n, 0) \gtrsim 2^n / (3\log_2 n)$. It is proved that for the case of implementing a transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ with a reversible circuit having $q_0 \sim 2^n$ additional inputs the depth is bounded as $D(n, q_0) \lesssim 3n$.