On Asymptotic Gate Complexity and Depth of Reversible Circuits With Additional Memory

TL;DR

This paper analyzes the asymptotic gate complexity and depth of reversible circuits with additional memory, providing bounds for their synthesis using an adapted Lupanov method.

Contribution

It introduces bounds for the complexity and depth of reversible circuits with extra inputs, extending classical synthesis methods to this context.

Findings

Upper bounds for gate complexity: $L(n,q_0) ot o 2^n$ with sufficient additional inputs.

Upper bounds for circuit depth: $D(n,q_1) ot o 3n$ with enough extra memory.

Application of Lupanov's method to reversible logic with additional inputs.

Abstract

The reversible logic can be used in various research areas, e.g. quantum computation, cryptography and signal processing. In the paper we study reversible logic circuits with additional inputs, which consist of NOT, CNOT and C\textsuperscript{2}NOT gates. We consider a set $F(n,q)$ of all transformations $\mathbb B^n \to \mathbb B^n$ that can be realized by reversible circuits with $(n+q)$ inputs. An analogue of Lupanov's method for the synthesis of reversible logic circuits with additional inputs is described. We prove upper asymptotic bounds for the Shannon gate complexity function $L(n,q)$ and the depth function $D(n,q)$ in case of $q > 0$: $L(n,q_0) \lesssim 2^n$ if $q_0 \sim n 2^{n-o(n)}$ and $D(n,q_1) \lesssim 3n$ if $q_1 \sim 2^n$.