Studying the Cost of n-qubit Toffoli Gates

There are several Toffoli gate designs for quantum computers in the literature. Each of these designs is focused on a specific technology or on optimising one or several metrics (T-count, number of qubits, etc.), and therefore has its advantages and disadvantages. While there is some consensus in the state of the art on the best implementations for the Toffoli gate, scaling this gate for use with three or more control qubits is not trivial. In this paper, we analyse the known techniques for constructing an n-qubit Toffoli gate, as well as the existing state-of-the-art designs for the 2-qubit version, which is an indispensable building block for the larger gates. In particular, we are interested in a construction of the temporary logical-AND gate with more than two control qubits. This gate is widely used in the literature due to the T-count and qubit reduction it provides. However, its use with more than two control qubits has not been analysed in detail in any work. The resulting information is offered in the form of comparative tables that will facilitate its consultation for researchers and people interested in the subject, so that they can easily choose the design that best suits their interests. As part of this work, the studied implementations have been reproduced and tested on both quantum simulators and real quantum devices.

Automated summary

This paper analyses the known techniques for constructing multi-qubit Toffoli gates and the latest designs for the 2-qubit version. It focuses on the construction of a temporary logical-AND gate with more than two control qubits. The results are presented in comparative tables to facilitate researchers and interested individuals in choosing the design that suits their needs.