This paper studies the factorization in classical computation,the concept and properties of quantum computing and quantum computers.The authors focus on the theoretical principles of the Shor quantum Algorithm to show the obvious advantages and the great potential of this method.
Quantum Information and Quantum Computation are rapidly evolving domains in their theoretical foundations, implementations and applications. The main scope of the present book is to give the basic ideas of this field to undergraduate students in Physics. A rigorous approach is used to provide the key concepts and methods. Many examples are discussed. To be self-contained, the principles of Quantum Mechanics are reviewed. The discussion is aimed to the specific, but paradigmatic, factorization problem of integers, dealt by the Quantum Shor’s Algorithm. Thus, notions of the Number Theory are given in the first Chapter, while the exposition of the algorithm and its implementation is reserved to the last Section.
Current technology is beginning to allow us to manipulate rather than just observe individual quantum phenomena. This opens up the possibility of exploiting quantum effects to perform computations beyond the scope of any classical computer. Recently Peter Shor discovered an efficient algorithm for factoring whole numbers, which uses characteristically quantum effects. The algorithm illustrates the potential power of quantum computation, as there is no known efficient classical method for solving this problem. The authors give an exposition of Shor's algorithm together with an introduction to quantum computation and complexity theory. They discuss experiments that may contribute to its practical implementation. [S0034-6861(96)00303-0]
The quantum algorithms covered in this chapter, i.e., the Deutsch, Deutsch-Jozsa and Bernstein-Vazirani algorithms, have been chosen for their relative simplicity and theoretical interest. They do not address problems of real practical value, their purpose being to demonstrate, using simple examples, the great potential power of quantum computation relative to the digital case. The most significant breakthrough in quantum computing is Shor's integer factorisation algorithm. Shor's algorithm is beyond the scope of this introductory text, but its significance lies in the fact that it demonstrates the potential to factor large integers efficiently. No efficient digital algorithms for factoring large integers exist, and many current encryption algorithms rely on this being the case. Shor's algorithm therefore threatens current encryption methods, and is responsible for much of the recent accelerated interest in quantum computation.
Quantum computing, based on the fundamental principles of quantum mechanics, has become a breakthrough that has the potential to revolutionize computing. At the forefront of this quantum revolution are quantum algorithms that use the origin of qubits to solve computational problems much faster than classical algorithms. This article provides a review of quantum algorithms to introduce their principles, examine famous examples such as the Grover and Shor algorithms, and delve into their applications, including quantum machine learning. Exploring the fundamentals of quantum computing begins with an overview that highlights the unique features of quantum computing compared to classical computing. We then examine several different algorithmic paradigms to highlight the unique advantages that quantum parallelism and interference bring to computing. The best knowledge of quantum algorithms focuses on Grover's blind search, Shor's efficient factorization algorithm, and the quantum phase approach for solving quantum chemistry and eigenvalue problems. As well as celebrating the achievements of quantum algorithms, this research also looks at the challenges and limitations that hinder their widespread use. Topics such as decoherence, error correction, and finding errors in quantum computing are discussed in the context of overcoming obstacles to the use of practical quantum algorithms. In addition, the latest developments are clarified by showing the successes of the experiments and their implications for the future. As we stand at the forefront of a new era in computing, this article not only provides a snapshot of the current state of quantum algorithms but also speculates on future directions. The potential applications of quantum algorithms in many fields and the ongoing quest to overcome current limitations offer exciting opportunities for further research. This research aims to contribute to the ongoing debate about quantum algorithms, gaining a deeper understanding of their impact on the future of computing.
Basic concepts of quantum theory of information, principles of quantum calculations and the possibility of creation on this basis unique on calculation power and functioning principle device, named quantum computer, are briefly reviewed. The main blocks of quantum logic, schemes of implementation of quantum calculations, as well as some known today effective quantum algorithms, called to realize advantages of quantum calculations upon classical, are concerned. Among them special place is taken by Shor's algorithm of number factorization, Grover's algorithm of unsorted database search and, finally, the most promising in application methods of quantum phenomena simulation, particularly quantum chaos. The most perspective methods of experimental realization of quantum computer, namely nuclear-magnetic resonance and trapped ions realizations, are discussed. Phenomena of decoherence, its influence on quantum computer stability and methods of quantum error correction are described.