Quantum Algorithms Using Quantum Fourier Transform

Moghareh Abed Zohreh, Mohsen Ashorian, Kiumars Ghowsi

In this paper, we introduce quantum fourier transform as a key ingredient for many useful algorithms. These algorithms make a solution for problems which is considered to be intractable problems on a classical computer. Quantum Fourier transform is propounded as a key for quantum phase estimation algorithm. In this paper our aim is the implementation of period-finding algorithm.Quantum computer solves this problem, exponentially faster than classical one. Quantum phase estimation algorithm is the key for the period-finding problem .Therefore, by means of simulating quantum Fourier transform, we are able to implement the period-finding algorithm. In this paper, the simulation of quantum Fourier transform is carried out by Matlab software.

Umesh V. Vazirani

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.

S. S. Zhou, T. Loke, Josh Izaac +1 more

Jamie Smith, Michele Mosca

Robert Hundt

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.