The quantum Fourier transform is another fundamental quantum algorithm. The section begins with a simple phase-kick circuit and expands to quantum phase estimation before detailing the quantum Fourier transform itself. A short section on arithmetic in the quantum domain introduces techniques that are used in a final detailed section on Shor’s famous algorithm for number factorization.
The quantum Fourier transform is another fundamental quantum algorithm. The section begins with a simple phase-kick circuit and expands to quantum phase estimation before detailing the quantum Fourier transform itself. A short section on arithmetic in the quantum domain introduces techniques that are used in a final detailed section on Shor’s famous algorithm for number factorization.
Abstract Expanding on the factoring and periodicity in Chapter 14, the Shor algorithm is explained in detail, with a warning that this is one of the toughest chapters in the book. While the steps of Shor algorithm may look just like the RSA in reverse, the quantum computer is needed to find the periodicity of the factors and thereby the prime factors themselves. Fourier analysis is the tool for this. The quantum Fourier transform is described, including the quantum circuit, after first showing what the classical Fourier transform accomplishes. Extensive examples of all the steps are given.
This chapter details many of the fundamental quantum algorithms with full mathematical derivations and code. It discusses the quantum Fourier transform (QFT), arithmetic in the Fourier domain quantum phase estimation,Shor’s famous factorization algorithm and its quantum order-finding component, Grover’s search algorithm, amplitude amplification, and quantum counting, as well as quantum random walks. From the field of quantum simulation, the chapter discusses the variational quantum eigensolver, measurement in the Pauli bases, the quantum approximate optimization algorithm (QAOA), the NP-complete graph Max-Cut problem, and the related Subset Sum problem. The chapter concludes with an in-depth discussion of the elegant Solovay–Kitaev algorithm for gate approximation.
In this chapter, we will study the quantum Fourier transform and its application in different quantum algorithms. Problems such as factoring an integer into prime numbers or period finding are computationally intractable problems for a classical computer because of the exponentially large number of operations involved. Integer factoring and period finding can be efficiently solved using the quantum phase estimation algorithm that is heavily based on the quantum Fourier transform. Alternately, since quantum phase estimation aims to find the eigenvalue corresponding to an eigenvector of a unitary operator, it is backbone of important algorithms in optimization such as the HHL algorithm (named for Hassim, Harrow, and Lloyd), which serves as the matrix inversion routine in quantum computing. We start this chapter by revising our concepts of the Fourier transform and its discrete counterpart, the discrete Fourier transform , and then move on to the exciting domain of the quantum Fourier transform and the quantum phase estimation algorithm. We follow this up with a discussion and implementation of the few quantum Fourier transform–related algorithms such as factoring a number and period finding. At the end of the chapter, we briefly introduce the basics of group theory with an attempt to explain the hidden subgroup problem and how it relates to several of the Fourier transform–based algorithms.
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.
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.