Large quantum Fourier transforms are never exactly realized by braiding conformal blocks

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 universal topological quantum computer is based on the Fibonacci 2+1 -topological quantum field theory TQFT , where the standard quantum circuits are replaced by unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms FN and the discrete sine or cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that an approximation is unavoidable in the implementation of Fourier transforms by braiding conformal blocks.