Figure 44.

Shor’s algorithm with semiclassical inverse Fourier transform. Please note that $\times 2^2\equiv \times 7^2\equiv \times 8^2\equiv \times {13}^2\equiv \times 4^1$ and $\times 4^2\equiv \times {11}^2\equiv \times 1$ (mod 15), so that many of the controlled multiplications will be identities. Therefore, most rotations $R_{2^k}$ will never be applied (in the ideal situation), in fact only the very last $R_2$ operation can ever occur.