CRC Press eBooks

Quantum Fourier Transform and Applications

Eduardo R. Mucciolo

Published Aug 6, 2025• 0 citations

Abstract

Because of quantum interference, quantum computers are very efficient at finding and enhancing periodic structures in data. In 1994, Peter Shor introduced the first quantum algorithm that explored this property to solve a mathematical problem of great practical importance: factoring semi-prime numbers. To understand the algorithm (warning: it is not simple!), we need first to review some math.

DOI: 10.1201/9781003485124-8

Related papers

Shor Factoring Algorithm and the Quantum Fourier Transform

Alice Flarend, Robert Hilborn

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.

Jul 9, 20260 citations

Mathematical Analysis and Simulation of Shor's Algorithm and the Quantum Fourier Transform

Marcus Ahlström, Peda Dizdarevic, Patrik Henelius

In 1994, Peter Shor presented an algorithm for integer factorization that used exponentiallyless operations than the most efficient known algorithm. His algorithm requiresthe use of a quantum computer, a theoretical computational device using quantum mechanicaleffects not utilized in contemporary computers. In this paper we have analysedthe mathematics behind Shor’s algorithm and the quantum circuits on which it operates.We have also studied the Quantum Fourier Transform, a central component of Shor’sAlgorithm. Furthermore we have written a program in C++ to simulate a quantum circuitperforming Shor’s algorithm and the quantum Fourier transform. We were able tounderstand the critical parts of Shor’s algorithm that contribute with the great increasein efficiency compared to classical algorithms.

Jan 1, 20121 citation

News

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 SpectrumDec 30, 20200 citations

Simulation of a Quantum Prime Factoring Algorithm

Elizabeth Ellen Parsons

The intent of this thesis is to elucidate the quantum computing algorithm developed by Peter Shor called Shor's algorithm. We will provide a detailed description and simulation of the algorithm using MATLAB. Precursory information regarding quantum phenomena such as superposition, entanglement, and Dirac notation, will be described in great detail so that the reader may have a better understanding of the operations in Shor's algorithm. Quantum computers store and transport information quite differently than their classical counterparts. We will provide a quick overview of these differences to high- light the benefit of utilizing quantum phenomena in a computer in order to create massive parallel computations. Thus, reducing the complexity time for classical algorithms used to solve problems such as the prime factorization problem and the period finding problem. The Quantum Fourier Transform is a principle component in Shor's algorithm. We will explicitly define the Quantum Fourier Transform and show that it is a unitary transformation. We will also show how the Quantum Fourier Transform, as well as another unitary transformation called the Hadamard transform, functions in Shor's algorithm. One of the initial parameters in Shor's algorithm is to select a random variable. We will examine the erratic effects of this random variable as well as how it effects the probability of us successfully reducing an integer into a product of two primes. We will provide a thorough analysis of the randomness in Shor's algorithm. We will also show how measuring the state of our quantum system as well as selecting a suitable random variable impacts finding the period of the Quantum Fourier Transform which in turn will either give a high or low probability of obtaining a factor of some integer.

Jan 1, 20160 citations

Shor Algorithm

Ray LaPierre

In 1994, a mathematician named Peter Shor surprised the world by publishing an efficient quantum algorithm for finding the prime factors of very large numbers, which is the basis for today’s data encryption. This discovery spurred the further development of quantum computing, making Shor’s algorithm one of the most important algorithms.

The Materials Research Society seriesOct 24, 20250 citations

Quantum Fourier Transform and Related Algorithms

Santanu Pattanayak

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.

Apress eBooksJan 1, 20210 citations