Efficient Toffoli Gates Using Qudits

TL;DR

This paper demonstrates that using a third state (qutrit) in quantum systems can significantly reduce the controlled-sign gates needed for Toffoli gates, enhancing efficiency in optical quantum computing.

Contribution

Introducing a method to implement Toffoli gates with fewer controlled-sign gates by leveraging accessible third states in quantum systems.

Findings

Toffoli gate implementation requires only three controlled-sign gates with a qutrit.

Using qutrits increases success probabilities in linear optical circuits.

The approach is applicable to systems like atoms and photonic modes.

Abstract

The simplest decomposition of a Toffoli gate acting on three qubits requires {\em five} 2-qubit gates. If we restrict ourselves to controlled-sign (or controlled-NOT) gates this number climbs to six. We show that the number of controlled-sign gates required to implement a Toffoli gate can be reduced to just {\em three} if one of the three quantum systems has a third state that is accessible during the computation, i.e. is actually a qutrit. Such a requirement is not unreasonable or even atypical since we often artificially enforce a qubit structure on multilevel quantums systems (eg. atoms, photonic polarization and spatial modes). We explore the implementation of these techniques in optical quantum processing and show that linear optical circuits could operate with much higher probabilities of success.