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近畿大学理工学部研究報告

Spectral Decomposition of Quantum Fourier Transform

Takashi Aoki, Yusuke Maesaka, Mikio Nakahara

Published Sep 1, 2006• 0 citations
OpenAlex

Abstract

Their explicit forms are In this article we present a concrete form of the spectral decomposition of the quantum Fourier transform (QFT). QFT is one of the most useful mathematical tools in applied mathematics and mathematical sciences such as signal processing and quantum computing') . For example, in the famous quantum algorithm of factorization of a large integer due to Shor2}, QFT plays an essential role. Thus studying QFT is an important subject in various fields of applied mathematics. Spectral decomposition of QFT provides us with a very convenient way of theoretical treatment of QFT. As an application, we compute the logarithm of QFT.

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Their explicit forms are In this article we present a concrete form of the spectral decomposition of the quantum Fourier transform (QFT). QFT is one of the most useful mathematical tools in applied mathematics and mathematical sciences such as signal processing and quantum computing') . For example, in the famous quantum algorithm of factorization of a large integer due to Shor2}, QFT plays an essential role. Thus studying QFT is an important subject in various fields of applied mathematics. Spectral decomposition of QFT provides us with a very convenient way of theoretical treatment of QFT. As an application, we compute the logarithm of QFT.
This survey presents a comprehensive analysis of the quantum Fourier transform (QFT), a fundamental tool in quantum computing, in comparison to its classical counterpart, the fast Fourier transform (FFT). The study begins with an introduction to the classical Fourier transform, which is widely used in different disciplines such as signal and image processing. The article introduces the QFT as a quantum version of the Fourier transform, detailing how it leverages quantum parallelism and superposition to reduce the time complexity of the operation, and highlighting its crucial role in quantum algorithms like Shor’s algorithm for integer factorization. The analysis also addresses the mathematical foundations of the QFT, its implementation in quantum circuits, and the key advantages and challenges associated with its use, such as measurement precision and quantum decoherence. Finally, it concludes with an exploration of the current and potential applications of QFT in quantum computing.
Quantum Fourier transform (QFT) plays an eminent role in quantum computation.It creates a superposition of different quantum states, allowing simultaneous calculation, which would take many steps were the same program implemented on a classical computer.The computational speed of a quantum computer is thus boosted dramatically.In this article, we explain the application of QFT in Shor's algorithm, which was proposed by Peter Shor to factor large integers on quantum computers.Specifically, the principle and design of quantum Fourier transform are explained.We stress the subtle distinction between QFT and its inverse.Since former articles did not emphasize it, we hope it could be a supplement to former articles.Our next work is to verify the existing articles on executing Shor's algorithm by conducting two experiments on IBMQ of factoring N = 15 in two ways (a = 7 and a = 11).We find that the effect of quantum entanglement might be crucial to the speed boost of factoring large integers in Shor's algorithm.
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.
The quantum Fourier transform (QFT) is the important operation in quantum computing. It is used in many algorithms, including the Shor's quantum algorithm for finding the prime factors of integers. Color image encryption, processing and representation of quantum images are areas where the QFT is also used. In this paper, we discuss the quantum superpositions of the images and methods of calculation of the analogues of the quantum 2-D discrete Fourier transform of images. The quantum algorithms and circuits will be described, and examples of calculation of the 2-D QFT of images of 8×8 and 16×16 pixels will be given in detail.
Summary 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.