Five Two-Qubit Gates Are Necessary for Implementing Toffoli Gate

TL;DR

This paper proves that at least five two-qubit gates are required to implement a Toffoli gate, establishing the minimal gate count and advancing understanding of quantum gate synthesis.

Contribution

It provides a rigorous proof that five two-qubit gates are necessary for the Toffoli gate, resolving a long-standing open problem.

Findings

Five two-qubit gates are necessary for Toffoli gate implementation

The proof settles the minimal gate count for Toffoli gate synthesis

Method can be applied to optimal simulation of three-qubit control phase

Abstract

In this paper, we settle the long-standing open problem of the minimum cost of two-qubit gates for simulating a Toffoli gate. More precisely, we show that five two-qubit gates are necessary. Before our work, it is known that five gates are sufficient and only numerical evidences have been gathered, indicating that the five-gate implementation is necessary. The idea introduced here can also be used to solve the problem of optimal simulation of three-qubit control phase introduced by Deutsch in 1989.