Reducing T-Count in Quantum Circuits Using Alternate Forms of the Relative Phase Toffoli Gate

Abstract

Toffoli gates are an important primitive in reversible Boolean logic. In quantum computation, these Toffoli gates are composed using other elementary gates, most notably the Clifford+T basis. However, in fault-tolerant implementations of quantum circuits, the T-gate incurs extra cost relative to Clifford gates like the S-gate and CNOT gate. Relative-phase Toffoli Gates (RTOFs) have been proposed as a way to reduce this T-count at the cost of incurring a relative phase that could skew the final quantum states. In this paper, we utilize an observation that the relative phase which RTOFs introduce can be canceled by the appropriate application of less expensive S-gates instead of T-gates. It leverages alternate forms of the RTOF including incorporating S-gates into it or moving around its input bits in order to simplify the logic to erase the relative phase. We find experimentally that our method has a clear advantage in most cases, and identify several types of circuits that it could be synergistic with.