De-quantisation of the quantum Fourier transform

The quantum Fourier transform (QFT) plays an important role in many known quantum algorithms such as Shor's algorithm for prime factorisation. In this paper we show that the QFT algorithm can, on a restricted set of input states, be de-quantised into a classical algorithm which is both more efficient and simpler than the quantum algorithm. By working directly with the algorithm instead of the circuit, we develop a simple classical version of the quantum basis-state algorithm. We formulate conditions for a separable state to remain separable after the QFT is performed, and use these conditions to extend the de-quantised algorithm to work on all such states without loss of efficiency. Our technique highlights the linearity of quantum mechanics as the fundamental feature accounting for the difference between quantum and de-quantised algorithms, and that it is this linearity which makes the QFT such a useful tool in quantum computation.