再论量子傅里叶变换
Quantum Fourier Transform Revisited
摘要：
快速傅里叶变换（FFT）是20世纪最成功的数值算法之一，在计算科学和工程的许多分支中有着广泛的应用。FFT算法可以从离散傅里叶变换（DFT）矩阵的特定矩阵分解导出。本文证明了量子傅里叶变换（QFT）可以通过将FFT矩阵分解的对角因子进一步分解为具有Kronecker乘积结构的矩阵的乘积而导出。我们分析了这种克罗内克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。我们还解释了为什么这样的结构可以利用一个重要的量子计算机功能，使QFT算法在量子计算机上实现指数加速超过经典计算机上的FFT算法。此外，DFT矩阵的矩阵分解和量子电路之间的联系。我们还讨论了一个基数2 QFT分解的一个基数d QFT分解的自然扩张。不需要量子计算的先验知识就可以理解本文所提出的内容。然而，我们相信这篇论文可以帮助读者从矩阵计算的角度对量子计算的本质有一些初步的了解。
英文摘要：
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.
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