Reducing the Cost of Clifford+T Quantum Gates

Currently, the most used universal quantum gate set is the augmented Clifford+T set of gates. Most commonly, we use the cost reduction in terms of the T-depth or T-count to implement logic primitives such as the Toffoli gate. While a lot of research has been done in this area, the Clifford+T set of gates is difficult to use for the synthesis of larger multi-controlled NOT gates (CⁿNOT) or simply larger circuits. To address this issue, in this work, we examine the composition of the Toffoli gate in relation to the T-gates. We adopt an approach based on the logic formalization of the T-gate placement and deduce three main results. First, if the trailing phase is not a concern, we can efficiently reduce the number of T gates. The second result is that to construct CⁿNOT gates for n>2 the T gates require smaller angles of rotation (analogous to the original scaling problem in the Barenco et al. Toffoli implementation). Lastly, we demonstrate that it is feasible to substitute all T gates with RZ rotations, ensuring the correct logic operation and a constant phase across all outputs.