The quantum FFT can be classically simulated

In this note we describe a simple and intriguing observation: the quantum Fourier transform (QFT) over $Z_q$, which is considered the most ``quantum'' part of Shor's algorithm, can in fact be simulated efficiently by classical computers. More precisely, we observe that the QFT can be performed by a circuit of poly-logarithmic path-width, if the circuit is allowed to apply not only unitary gates but also general linear gates. Recalling the results of Markov and Shi [MaSh] and Jozsa [Jo] which provided classical simulations of such circuits in time exponential in the tree-width, this implies the result stated in the title. Classical simulations of the FFT are of course meaningless when applied to classical input strings on which their result is already known; Our observation might be interesting only in the context in which the QFT is used as a subroutine and applied to more interesting superpositions. We discuss the reasons why this idea seems to fail to provide an efficient classical simulation of the entire factoring algorithm. In the course of proving our observation, we provide two alternative proofs of the results of [MaSh,Jo] which we use. One proof is very similar in spirit to that of [MaSh] but is more visual, and is based on a graph parameter which we call the ``bubble width'', tightly related to the path- and tree-width. The other proof is based on connections to the Jones polynomial; It is very short, if one is willing to rely on several known results.