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Quantum Information Processing

A review on quantum Fourier transform

Tomás Barros, Pablo Álvarez, Bárbara Vidal, Mauricio Solar

Published Jan 27, 2026• 2 citations
Publisher

Abstract

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.

DOI
10.1007/s11128-026-05067-7

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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.
Fourier Transform (FT) is an important signal processing technique that translates time-domain signals to frequency-domain forms. It is widely used in several applications, including 5G networks, image processing, & spectrum analysis. In 5G networks, efficient frequency analysis is critical for noise reduction, channel optimisation, and signal dependability.This study explores the implementation of Classical Fourier Transform, Fast Fourier Transform, and Quantum Fourier Transform using Python (NumPy, Pandas, Matplotlib) and Qiskit. The FFT significantly improves processing speed, making it suitable for real-time applications. On the other hand, QFT leverages quantum superposition and entanglement, offering promising advancements for future high-speed communication and secure data transmission. Additionally, the signal is successfully reconstructed using both the Inverse Fast Fourier Transform (IFFT) and the Inverse Quantum Fourier Transform (IQFT). While IFFT efficiently restores the original signal with minimal numerical artifacts, IQFT ensures exact reconstruction without errors. The study also includes execution on IBM Quantum’s real hardware, showcasing the practical feasibility of quantum-based signal processing.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.
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 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.
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.