Optimization of Quantum Boolean Circuits by Relative-Phase Toffoli Gates

To realize quantum Boolean circuits, Toffoli gates are often used as logic primitives. Then Toffoli gates are decomposed to physically realizable gates, i.e., CNOT, H and T gates when we consider fault-tolerant implementation. The realization cost of a T gate is huge compared to the other gates, and thus we often consider the number of T gates. We need seven T gates to decompose a Toffoli gate. However, if we allow to add some relative phases to some output quantum states, we can implement a Toffoli gate by only four T gates. Such an approximate Toffoli gate is called a relative-phase Toffoli gate (RTOF). This paper proposes an optimization method of quantum circuit by using RTOFs. When we optimize a circuit by replacing a Toffoli gate with a RTOF, some relative phase errors are added. Our method tries to correct such relative phases by using S gates.