Comparison of the cost metrics through investigation of the relation between optimal NCV and optimal NCT three-qubit reversible circuits

A breadth-first search method for determining optimal three-qubit circuits composed of quantum NOT, CNOT, controlled-V and controlled-V+ (NCV) gates is introduced. Results are presented for simple gate count and for technology-motivated cost metrics. The optimal NCV circuits are also compared with NCV circuits derived from optimal NOT, CNOT and Toffoli (NCT) gate circuits. This work provides basic results and motivation for continued study of the direct synthesis of NCV circuits, and establishes relations between function realizations in different circuit cost metrics.