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.

中文翻译：

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量子与传统傅里叶变换计算的比较

量子傅里叶变换 (QFT) 可以计算大小为 $N$N 的向量的傅里叶变换，其时间复杂度为 $\mathcal {O}(\log ^2~N)$O(log2N) 与经典复杂度相比$\mathcal {O}(N \;\log N)$O(NlogN)。但是，如果要测量完整的输出状态，那么 QFT 复杂度变为 $\mathcal {O}(N \;\log ^2~N)$O(Nlog2N)，从而失去其明显的优势，表明优势当仅需要从输出向量中获取有限数量的样本时，就可以充分利用算法，就像在许多量子算法中一样。此外，如果考虑构建初始状态的复杂性，计算复杂性会恶化。在本文中，