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CRC Press eBooks

Quantum Integral Transforms

Mikio Nakahara

Published Mar 11, 2008• 0 citations
Publisher

Abstract

We demonstrated in the previous chapter that there are some quantum algorithms superior to their classical counterparts. It is, however, rather difficult to find any practical use of these algorithms. There are two quantum algorithms, known to date, which are potentially useful: Grover’s search algorithm and Shor’s prime number factorization algorithm. Both of them depend on quantum integral transforms, which will be introduced in the present chapter. We mainly follow [1] in our presentation.

DOI
10.1201/9781420012293-12

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We demonstrated in the previous chapter that there are some quantum algorithms superior to their classical counterparts. It is, however, rather difficult to find any practical use of these algorithms. There are two quantum algorithms, known to date, which are potentially useful: Grover’s search algorithm and Shor’s prime number factorization algorithm. Both of them depend on quantum integral transforms, which will be introduced in the present chapter. We mainly follow [1] in our presentation.
Abstract This chapter presents early quantum algorithms. We first describe Deutsch’s and Bernstein-Vazirani, some of the first quantum algorithms that showed a query advantage over their classical counterparts. We then present Grover’s algorithm, a quantum algorithm that finds an item in an unstructured database with a quadratic speed-up over classical algorithms. We explain the quantum Fourier transform, an important subroutine in various quantum algorithms. The chapter concludes with a detailed presentation of two of the most prominent applications of the quantum Fourier transform: the period finding algorithm and Shor’s algorithm for prime factorization.
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.
Shor’s factorization algorithm is one of the prime examples in which a quantum computer demonstrates enormous power surpassing its classical counterpart [1, 2]. Although the factorization algorithm may be carried out with a classical computer, it takes an exponentially longer time (i.e., practically impossible) compared to Shor’s quantum algorithm. Shor’s algorithm is almost identical with the classical one; it consists of a sequence of classical steps with one exception, which is replaced by a quantum algorithm. Our presentation here closely follows [3] and [4].
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.