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Fourier transform is an essential ingredient in Shor's factoring algorithm.In the standard quantum circuit model with the gate set $\{\U(2),\textrm{CNOT}\}$, the discrete Fourier transforms $F_N=(\omega^{ij})_{N\timesN},i,j=0,1,..., N-1, \omega=e^{\frac{2\pi i}{N}}$, can be realized exactly byquantum circuits of size $O(n^2), n=\textrm{log}N$, and so can the discretesine/cosine transforms. In topological quantum computing, the simplestuniversal topological quantum computer is based on the Fibonacci(2+1)-topological quantum field theory (TQFT), where the standard quantumcircuits are replaced by unitary transformations realized by braiding conformalblocks. We report here that the large Fourier transforms $F_N$ and the discretesine/cosine transforms can never be realized exactly by braiding conformalblocks for a fixed TQFT. It follows that approximation is unavoidable toimplement the Fourier transforms by braiding conformal blocks

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