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Theoretical and Natural Science

Analysis of application of FFT in quantum field

Zhaoyang Xu

Published Nov 16, 2023• 0 citations
Publisher

Abstract

The algorithm concept of fast Fourier transform, Heisenberg uncertainty, and the combination of the two are expounded. Fast Fourier Transform (FFT) is not a new discrete Fourier Transform, but a fast algorithm for Discrete Fourier Transform (DFT). For a long time, the DFT was not really used because of the large amount of computation, even if it was used by computers, it was difficult to deal with problems in real time. It was not until when Cooley and Key first proposed a fast algorithm for DFT operations, and later, when fast algorithms by G. Sullivan and. Key appeared in succession, that fundamental changes took place. People began to realize some inherent laws of DFT operation, and thus quickly developed and improved a set of high-speed and effective operation methods, which is now commonly known as the fast Fourier transform FF11 algorithm. Specifically, the position of a particle and its momentum cannot be determined simultaneously in a quantum mechanical system. It focuses on the proof and application of the Heisenberg inequality by TTF. Due to the different ways and methods of practical application, the practical significance and practical value of the application are for readers' reference. Besides, the space is limited and my knowledge is limited. Finally, the development direction of FFT in quantum physics is summarized.

DOI
10.54254/2753-8818/10/20230363

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The fast Fourier transform (FFT) is the unsung digital workhorse of modern life.It's a clever mathematical shortcut that makes possible the many signals in our device-connected world.Every minute of every video stream, for instance, entails computing some hundreds of FFTs.The FFT's importance to practically every data-processing application in the digital age explains why some researchers have begun exploring how quantum computing can run the FFT algorithm more efficiently still."The fast Fourier transform is an important algorithm that's had lots of applications in the classical world," says Ian Walmsley, physicist at Imperial College London."It also has many applications in the quantum domain.[So] it's important to figure out effective ways to be able to implement it."The first proposed killer app for quantum computers-finding a number's prime factors-was discovered by mathematician Peter Shor at AT&T Bell Laboratories in 1994.Shor's algorithm scales up its factorization of numbers more efficiently and rapidly than any classical computer anyone could ever design.And at the heart of Shor's phenomenal quantum engine is a subroutine called-you guessed it-the quantum Fourier transform (QFT).Here is where the terminology gets a little out of hand.There is the QFT at the center of Shor's algorithm, and then there is the QFFT-the quantum fast Fourier transform.They represent different calculations that produce different results, although both are based on the same core mathematical concept, known as the discrete Fourier transform.

IEEE Spectrum•Dec 30, 2020•0 citations

Quantum Fourier transform revisited

Camps, D, Van Beeumen, R, Yang, C

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.

•Jan 1, 2021•25 citations

A quantum search algorithm of two entangled registers to realize quantum discrete Fourier transform of signal processing

Chao-Yang Pang, Ben-Qiong Hu

The discrete Fourier transform (DFT) is the base of modern signal processing. 1-dimensional fast Fourier transform (ID FFT) and 2D FFT have time complexity O ( N log N ) and O ( N 2 log N ) respectively. Since 1965, there has been no more essential breakthrough for the design of fast DFT algorithm. DFT has two properties. One property is that DFT is energy conservation transform. The other property is that many DFT coefficients are close to zero. The basic idea of this paper is that the generalized Grover's iteration can perform the computation of DFT which acts on the entangled states to search the big DFT coefficients until these big coefficients contain nearly all energy. One-dimensional quantum DFT (ID QDFT) and two-dimensional quantum DFT (2D QDFT) are presented in this paper. The quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, ID and 2D QDFT have time complexity O(√ N ) and O ( N ) respectively. QDFT and quantum convolution demonstrate that quantum computation to process classical signal is possible.

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Chao-Yang Pang, Ben-Qiong Hu

Discrete Fourier transform (DFT) is the base of modern signal or information processing. 1-Dimensional fast Fourier transform (1D FFT) and 2D FFT have time complexity O(NlogN) and O(N^2logN) respectively. Quantum 1D and 2D DFT algorithms with classical output (1D QDFT and 2D QDFT) are presented in this paper. And quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, QDFT has two advantages at least. One of advantages is that 1D and 2D QDFT has time complexity O(sqrt(N)) and O(N) respectively. The other advantage is that QDFT can process very long signal sequence at a time. QDFT and quantum convolution demonstrate that quantum signal processing with classical output is possible.

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The Quantum Fourier Transform for Continuous Variables

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The quantum Fourier transform for discrete variable (dvQFT) is an efficient algorithm for several applications. It is usually considered for the processing of quantum bits (qubits) and its efficient implementation is obtained with two elementary components: the Hadamard gate and the controlled--phase gate. In this paper, the quantum Fourier transform operating with continuous variables (cvQFT) is considered. Thus, the environment becomes the Hilbert space, where the natural definition of the cvQFT will be related to rotation operators, which in the $N$--mode are completely specified by unitary matrices of order $N$. Then the cvQFT is defined as the rotation operator whose rotation matrix is given by the discrete Fourier transform (DFT) matrix. For the implementation of rotation operators with primitive components (single--mode rotations and beam splitters), we follow the well known Murnaghan procedure, with appropriate modifications. Moreover, algorithms related to the fast Fourier transform (FFT) are applied to reduce drastically the implementation complexity. The final part is concerned with the application of the cvQFT to general Gaussian states. In particular, we show that cvQFT has the simple effect of transforming the displacement vector by a one-dimensional DFT, the squeeze matrix by a two-dimensional DFT, and the rotation matrix by a Fourier-like similarity transform.

quant-pharXiv:2512.12771•Dec 16, 2025•0 citations

FOURIER TRANSFORM ANALYSIS USING QUANTUM COMPUTATION METHOD

Surya P, U. Pavan Kumar, Ch. Arun Prakash +2 more

signals into frequency-domain forms is the Fourier Transform (FT). It is extensively utilized in several fields, such as spectrum analysis, audio and video processing, and 5G networks. Effective frequency analysis is essential for channel optimization, signal reliability, and noise reduction in 5G networks. The FFT is appropriate for real-time applications since it greatly increases processing speed. However, the Quantum Fourier Transform makes use of quantum entanglement and superposition, providing exciting new developments for secure data transfer and high-speed communication in the future. Furthermore, both the Inverse Fast Fourier Transform (IFFT) and the Inverse Quantum Fourier Transform are used to correctly recreate the signal. Implementation of Quantum's real valued computing of quantum-based signal processing is also covered in the survey.

International Journal of Advanced Research•Feb 21, 2026•0 citations
The algorithm concept of fast Fourier transform, Heisenberg uncertainty, and the combination of the two are expounded. Fast Fourier Transform (FFT) is not a new discrete Fourier Transform, but a fast algorithm for Discrete Fourier Transform (DFT). For a long time, the DFT was not really used because of the large amount of computation, even if it was used by computers, it was difficult to deal with problems in real time. It was not until when Cooley and Key first proposed a fast algorithm for DFT operations, and later, when fast algorithms by G. Sullivan and. Key appeared in succession, that fundamental changes took place. People began to realize some inherent laws of DFT operation, and thus quickly developed and improved a set of high-speed and effective operation methods, which is now commonly known as the fast Fourier transform FF11 algorithm. Specifically, the position of a particle and its momentum cannot be determined simultaneously in a quantum mechanical system. It focuses on the proof and application of the Heisenberg inequality by TTF. Due to the different ways and methods of practical application, the practical significance and practical value of the application are for readers' reference. Besides, the space is limited and my knowledge is limited. Finally, the development direction of FFT in quantum physics is summarized.
The fast Fourier transform (FFT) is the unsung digital workhorse of modern life.It's a clever mathematical shortcut that makes possible the many signals in our device-connected world.Every minute of every video stream, for instance, entails computing some hundreds of FFTs.The FFT's importance to practically every data-processing application in the digital age explains why some researchers have begun exploring how quantum computing can run the FFT algorithm more efficiently still."The fast Fourier transform is an important algorithm that's had lots of applications in the classical world," says Ian Walmsley, physicist at Imperial College London."It also has many applications in the quantum domain.[So] it's important to figure out effective ways to be able to implement it."The first proposed killer app for quantum computers-finding a number's prime factors-was discovered by mathematician Peter Shor at AT&T Bell Laboratories in 1994.Shor's algorithm scales up its factorization of numbers more efficiently and rapidly than any classical computer anyone could ever design.And at the heart of Shor's phenomenal quantum engine is a subroutine called-you guessed it-the quantum Fourier transform (QFT).Here is where the terminology gets a little out of hand.There is the QFT at the center of Shor's algorithm, and then there is the QFFT-the quantum fast Fourier transform.They represent different calculations that produce different results, although both are based on the same core mathematical concept, known as the discrete Fourier transform.
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 discrete Fourier transform (DFT) is the base of modern signal processing. 1-dimensional fast Fourier transform (ID FFT) and 2D FFT have time complexity O ( N log N ) and O ( N 2 log N ) respectively. Since 1965, there has been no more essential breakthrough for the design of fast DFT algorithm. DFT has two properties. One property is that DFT is energy conservation transform. The other property is that many DFT coefficients are close to zero. The basic idea of this paper is that the generalized Grover's iteration can perform the computation of DFT which acts on the entangled states to search the big DFT coefficients until these big coefficients contain nearly all energy. One-dimensional quantum DFT (ID QDFT) and two-dimensional quantum DFT (2D QDFT) are presented in this paper. The quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, ID and 2D QDFT have time complexity O(√ N ) and O ( N ) respectively. QDFT and quantum convolution demonstrate that quantum computation to process classical signal is possible.
Discrete Fourier transform (DFT) is the base of modern signal or information processing. 1-Dimensional fast Fourier transform (1D FFT) and 2D FFT have time complexity O(NlogN) and O(N^2logN) respectively. Quantum 1D and 2D DFT algorithms with classical output (1D QDFT and 2D QDFT) are presented in this paper. And quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, QDFT has two advantages at least. One of advantages is that 1D and 2D QDFT has time complexity O(sqrt(N)) and O(N) respectively. The other advantage is that QDFT can process very long signal sequence at a time. QDFT and quantum convolution demonstrate that quantum signal processing with classical output is possible.
The quantum Fourier transform for discrete variable (dvQFT) is an efficient algorithm for several applications. It is usually considered for the processing of quantum bits (qubits) and its efficient implementation is obtained with two elementary components: the Hadamard gate and the controlled--phase gate. In this paper, the quantum Fourier transform operating with continuous variables (cvQFT) is considered. Thus, the environment becomes the Hilbert space, where the natural definition of the cvQFT will be related to rotation operators, which in the NNN--mode are completely specified by unitary matrices of order NNN. Then the cvQFT is defined as the rotation operator whose rotation matrix is given by the discrete Fourier transform (DFT) matrix. For the implementation of rotation operators with primitive components (single--mode rotations and beam splitters), we follow the well known Murnaghan procedure, with appropriate modifications. Moreover, algorithms related to the fast Fourier transform (FFT) are applied to reduce drastically the implementation complexity. The final part is concerned with the application of the cvQFT to general Gaussian states. In particular, we show that cvQFT has the simple effect of transforming the displacement vector by a one-dimensional DFT, the squeeze matrix by a two-dimensional DFT, and the rotation matrix by a Fourier-like similarity transform.
signals into frequency-domain forms is the Fourier Transform (FT). It is extensively utilized in several fields, such as spectrum analysis, audio and video processing, and 5G networks. Effective frequency analysis is essential for channel optimization, signal reliability, and noise reduction in 5G networks. The FFT is appropriate for real-time applications since it greatly increases processing speed. However, the Quantum Fourier Transform makes use of quantum entanglement and superposition, providing exciting new developments for secure data transfer and high-speed communication in the future. Furthermore, both the Inverse Fast Fourier Transform (IFFT) and the Inverse Quantum Fourier Transform are used to correctly recreate the signal. Implementation of Quantum's real valued computing of quantum-based signal processing is also covered in the survey.