On the advantages of using relative phase Toffolis with an application to multiple control Toffoli optimization

TL;DR

This paper demonstrates that relative phase Toffoli gates can be optimized and integrated into larger quantum circuits, reducing resource counts and expanding their practical applicability in quantum computing.

Contribution

The paper introduces three circuit identities enabling the replacement of certain multiple control Toffoli configurations with their simpler relative phase versions, improving circuit efficiency.

Findings

Reduced CNOT-count in optimized circuits

Lower T-count achieved through new identities

Fewer ancillae needed in the optimized implementations

Abstract

Various implementations of the Toffoli gate up to a relative phase have been known for years. The advantage over regular Toffoli gate is their smaller circuit size. However, their use has been often limited to a demonstration of quantum control in designs such as those where the Toffoli gate is being applied last or otherwise for some specific reasons the relative phase does not matter. It was commonly believed that the relative phase deviations would prevent the relative phase Toffolis from being very helpful in practical large-scale designs. In this paper, we report three circuit identities that provide the means for replacing certain configurations of the multiple control Toffoli gates with their simpler relative phase implementations, up to a selectable unitary on certain qubits, and without changing the overall functionality. We illustrate the advantage via applying those identities to the optimization of the known circuits implementing multiple control Toffoli gates, and report the reductions in the CNOT-count, T-count, as well as the number of ancillae used. We suggest that a further study of the relative phase Toffoli implementations and their use may yield other optimizations.