Fourier transform is an essential ingredient in Shor’s factoring algorithm. In the standard quantum circuit model with the gate set U 2 , controlled-NOT , the discrete Fourier transforms FN= ij N N, i , j=0,1 ,... ,N −1, =e2 i/N, can be realized exactly by quantum circuits of size O n2 , n=ln N, and so can the discrete sine or cosine transforms. In topological quantum computing, the simplest un...