A Comparison of Quantum and Traditional Fourier Transform Computations

The quantum Fourier transform (QFT) can calculate the Fourier transform of a vector of size $N$N with time complexity $\mathcal {O}(\log ^2~N)$O(log2N) as compared to the classical complexity of $\mathcal {O}(N \;\log N)$O(NlogN). However, if one wanted to measure the full output state, then the QFT complexity becomes $\mathcal {O}(N \;\log ^2~N)$O(Nlog2N), thus losing its apparent advantage, indicating that the advantage is fully exploited for algorithms when only a limited number of samples is required from the output vector, as is the case in many quantum algorithms. Moreover, the computational complexity worsens if one considers the complexity of constructing the initial state. In this article, this issue is better illustrated by providing a concrete implementation of these algorithms and discussing their complexities as well as the complexity of the simulation of the QFT in matlab.