The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank‐1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix‐2 QFT decomposition to a radix‐d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.

中文翻译：

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再谈量子傅立叶变换

快速傅立叶变换（FFT）是20世纪最成功的数值算法之一，并且在计算科学和工程学的许多分支中都有大量应用。FFT算法可以从离散傅里叶变换（DFT）矩阵的特定矩阵分解中得出。在本文中，我们表明可以通过将FFT矩阵分解的对角线因子进一步分解为具有Kronecker乘积结构的矩阵的乘积来导出量子傅立叶变换（QFT）。我们分析了这种Kronecker乘积结构对经典计算机上秩1张量的离散傅里叶变换的影响。我们还将解释为什么这种结构可以利用重要的量子计算机功能，该功能使QFT算法在量子计算机上的性能比经典计算机上的FFT算法快。此外，在DFT矩阵的矩阵分解和量子电路之间进行连接。我们还讨论了将基数2 QFT分解自然扩展到基数d QFT分解。不需要了解量子计算的先验知识即可理解本文介绍的内容。但是，我们认为本文可以帮助读者从矩阵计算的角度对量子计算的性质有一些基本的了解。