This chapter deals with some of the most prominent successes of quantum computing. The most well-known quantum computing algorithm, Shor’s algorithm for factoring a number in its prime factors, is discussed in details. The key to Shor’s algorithm is the quantum Fourier transform that is explained with the help of simple examples. The role of quantum entanglement is also discussed. The next important quantum computing algorithm is Grover’s algorithm that helps in searching an item in an unsorted database. This algorithm is motivated by first discussing a quantum shell game in which a pea hidden under one of the four shells is found in one measurement with certainty each time. This amazing result is then generalized to an arbitrary number of objects and Grover’s algorithm.
This chapter deals with some of the most prominent successes of quantum computing. The most well-known quantum computing algorithm, Shor’s algorithm for factoring a number in its prime factors, is discussed in details. The key to Shor’s algorithm is the quantum Fourier transform that is explained with the help of simple examples. The role of quantum entanglement is also discussed. The next important quantum computing algorithm is Grover’s algorithm that helps in searching an item in an unsorted database. This algorithm is motivated by first discussing a quantum shell game in which a pea hidden under one of the four shells is found in one measurement with certainty each time. This amazing result is then generalized to an arbitrary number of objects and Grover’s algorithm.
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.
The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.
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.
Quantum computing exploits quantum mechanical properties to perform computations. It enables quantum parallelism and provides much more powerful data processing capabilities than classical computers. With a quantum computer, one can exponentially accelerate quantum system simulation and accelerate some important classical algorithms. Traditional quantum algorithms use unitary evolution to process information. A quantum computing process is the product of a series of unitary operators. In 1994, Shor invented a quantum prime factorization algorithm that exponentially accelerates the factorization of an integer. In 1996, Grover proposed a quantum search algorithm that accelerates the search of an unsorted database using square-root steps. Afterwards, the development of quantum algorithms slowed down, leading to a subsequent query by Shor in 2003 regarding why no more significant quantum algorithms have been found. Since 2009, many important new quantum algorithms have been proposed, such as a quantum algorithm for solving linear equations with the capability of exponential acceleration, a quantum algorithm for sparse Hamiltonian simulation using linear combinations of unitary operators, and a novel algorithm for Hamiltonian simulation, which provides exponential improvements in precision. In this paper, we first describe the basic principles of quantum computation. We then describe the Shor algorithm and Grover/Long search algorithm. These quantum algorithms are the general quantum algorithms that use unitary operations in their computational processes. Next, we introduce the basic principles of duality quantum computing, which was proposed in 2002. In contrast to traditional quantum computing, duality quantum computing allows the use of linear combinations of unitary operators in the computation process, meaning the multiplication, division, addition, and subtraction of unitary operators are all possible in duality quantum computing. Thus, duality quantum computing provides more flexibility for constructing quantum algorithms and the techniques used in classical algorithm design can be directly used to construct quantum algorithms. We then review recent work regarding newer quantum algorithms that enable the linear combination of unitary operators, which are actually duality quantum algorithms. Additionally, a duality quantum simulation algorithm for open quantum systems is introduced. This algorithm not only reduces computational complexity, but also improves accuracy exponentially. Finally, a summary and the future prospects of quantum algorithms are provided.
“An Introduction to Quantum Algorithms” provides an introduction to quantum algorithms, starting with the fundamentals of quantum computing. It covers the basics of quantum mechanics, the potential for quantum speedup, and the challenges and limitations of quantum computing. The chapter also highlights the key differences between classical and quantum computing, focusing on concepts such as bits and qubits, quantum measurement, superposition, and qubit properties, with a detailed explanation of qubit states on the Bloch sphere. Additionally, it explores quantum gates and algorithms, including single-qubit and multiple-qubit gates, the Quantum Fourier Transform (QFT), and important quantum algorithms such as Shor's algorithm for factoring, Grover's algorithm for searching, and Simon's algorithm for collision finding.
In this diploma paper we present the Shor’s algorithm for factorising numbers, which runs on quantum computers.
Initially, we explain the basics of quantum computing. Afterwards, we present a general review of the Fourier and the discrete Fourier transform, from which we derive the quantum Fourier transform.
In the last part, we first give a short introduction to modular arithmetics. Later on, we describe the concept of factorization that is used in Shor’s algorithm. Diploma thesis ends with a detailed review of how the algorithm works on quantum computers and a simple example.