Figure 28.

Four different quantum circuit implementations of the same balanced function, computing the majority $MAJ$ (see Figure 24) of three bits. (A) Straightforward construction using four $\tilde{T}\left(3\right)$ gates. (B) The majority is computed in place in the query register, the answer is copied out to the answer register, and the intermediate step is then uncomputed by applying $MA{J}^{†}$ to the query register. (C) is an optimization of A, tuned for as few gates as possible, and (D) is another optimized for few Toffolis followed by few $CNOT$s, generated as part of the list of all 72 possible functions in Appendix A.