A periodic function can be represented in the frequency space. The frequency is the number of occurrences of a repeating event per one unit of time. If something changes rapidly, then we say that it has a high frequency. If it does not change rapidly, i.e., it changes smoothly, we say that it has a low frequency. The discrete Fourier transform (DFT) converts discrete time-based or space-based data into the frequency domain.
A periodic function can be represented in the frequency space. The frequency is the number of occurrences of a repeating event per one unit of time. If something changes rapidly, then we say that it has a high frequency. If it does not change rapidly, i.e., it changes smoothly, we say that it has a low frequency. The discrete Fourier transform (DFT) converts discrete time-based or space-based data into the frequency domain.
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 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.
In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in Rn, that generalizes the n-th fold DFT of the integer lattice Zn to arbitrary lattices. This definition is not applicable for every lattice, but can be defined on lattices known as Systematic Normal Form (SysNF) introduced in \cite{ES16}. Systematic Normal Form lattices are sets of integer vectors that satisfy a single homogeneous modular equation, which itself satisfies a certain number-theoretic property. Such lattices form a dense set in the space of n-dimensional lattices, and can be used to approximate efficiently any lattice. This implies that for every lattice L a DFT can be computed efficiently on a lattice near L. Our proof of the statement above uses arguments from quantum computing, and as an application of our definition we show a quantum algorithm for sampling from discrete distributions on lattices, that extends our ability to sample efficiently from the discrete Gaussian distribution \cite{GPV08} to any distribution that is sufficiently "smooth". We conjecture that studying the eigenvectors of the newly-defined lattice DFT may provide new insights into the structure of lattices, especially regarding hard computational problems, like the shortest vector problem.
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.
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.
Eine philosophisch und mathematisch fundierte Darstellung der Fouriertransformation im Kontext der Quantenmechanik. Der Text verbindet historische Reflexion (Parmenides, Demokrit, Fourier, Dirac) mit einer präzisen Herleitung von Fourierreihe, Fouriertransformation und diskreter Fouriertransformation und zeigt, wie Orts‑ und Impulsdarstellung der Wellenfunktion durch Fouriertransformation ineinander übergehen. Besonderer Schwerpunkt liegt auf der ontologischen Bedeutung des Kontinuums und der Spektralanalyse für die Struktur der Quantenphysik. This work presents a concise and conceptually unified introduction to Fourier analysis in the context of quantum mechanics. It traces the historical development from classical harmonic decomposition to the modern Fourier transform and shows how periodic functions, spectral components, and exponential modes form the mathematical foundation of quantum theory. The transition from Fourier series to the continuous Fourier transform is derived rigorously, leading to the representation of quantum states in position and momentum space. Using Dirac notation, the text demonstrates that the wavefunction in momentum space is the Fourier transform of the position‑space wavefunction, revealing the spectral structure of quantum motion. The discrete Fourier transform (DFT) is introduced as a numerical analogue relevant for lattice models and quantum field theory. Overall, the text connects mathematical formalism, physical interpretation, and the ontology of the continuum in a compact and accessible way.