The quantum FFT can be classically simulated

TL;DR

This paper demonstrates that the quantum Fourier transform, a key component of Shor's algorithm, can be efficiently simulated classically by circuits with poly-logarithmic path-width, challenging assumptions about its quantum uniqueness.

Contribution

It shows that the quantum Fourier transform can be classically simulated efficiently using circuits with poly-logarithmic path-width, extending previous classical simulation results.

Findings

Classical simulation of the QFT is possible with poly-logarithmic path-width circuits.

Provides two alternative proofs of classical simulation results, one graph-theoretic and one based on the Jones polynomial.

Discusses limitations of this simulation approach for full factoring algorithms.

Abstract

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.