  We provide numerical evidence that the quantum Fourier transform can be
efficiently represented in a matrix product operator with a size growing
relatively slowly with the number of qubits. Additionally, we numerically show
that the tensors in the operator converge to a common tensor as the number of
qubits in the transform increases. Together these results imply that the
application of the quantum Fourier transform to a matrix product state with $n$
qubits of maximum Schmidt rank $\chi$ can be simulated in $O(n (log(n))^2
\chi^2)$ time. We perform such simulations and quantify the error involved in
representing the transform as a matrix product operator and simulating the
quantum Fourier transform of periodic states.
