The driving force in the pursuit for quantum computation is the exciting possibility that quantum algorithms can be more efficient than their classical analogues. Research on the subject has unraveled several aspects of how that can happen. Clever quantum algorithms have been discovered in recent years, although not systematically, and the field remains under active investigation. Richard Feynman was one of the pioneers who foresaw the power of quantum computers. In this issue dedicated to him, I give an introduction to how particle and wave aspects contribute to the power of quantum computers. Shor's and Grover's algorithms are analysed as examples.
The driving force in the pursuit for quantum computation is the exciting possibility that quantum algorithms can be more efficient than their classical analogues. Research on the subject has unraveled several aspects of how that can happen. Clever quantum algorithms have been discovered in recent years, although not systematically, and the field remains under active investigation. Richard Feynman was one of the pioneers who foresaw the power of quantum computers. In this issue dedicated to him, I give an introduction to how particle and wave aspects contribute to the power of quantum computers. Shor's and Grover's algorithms are analysed as examples.
In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation. The review begins with background on theoretical computer science, Turing machines and Boolean circuits. In light of these models, I define quantum computers, and discuss the issue of universal quantum gates. Quantum algorithms, including Shor's factorization algorithm and Grover's algorithm for searching databases, are explained. I will devote much attention to understanding what the origins of the quantum computational power are, and what the limits of this power are. Finally, I describe the recent theoretical results which show that quantum computers maintain their complexity power even in the presence of noise, inaccuracies and finite precision. I tried to put all results in their context, asking what the implications to other issues in computer science and physics are. In the end of this review I make these connections explicit, discussing the possible implications of quantum computation on fundamental physical questions, such as the transition from quantum to classical physics.
Richard Feynman's observation that quantum mechanical effects could not be simulated efficiently on a computer led to speculation that computation in general could be done more efficiently if it used quantum effects. This speculation appeared justified when Peter Shor described a polynomial time quantum algorithm for factoring integers. In quantum systems, the computational space increases exponentially with the size of the system which enables exponential parallelism. This parallelism could lead to exponentially faster quantum algorithms than possible classically. The catch is that accessing the results, which requires measurement, proves tricky and requires new non-traditional programming techniques. The aim of this paper is to guide computer scientists and other non-physicists through the conceptual and notational barriers that separate quantum computing from conventional computing. We introduce basic principles of quantum mechanics to explain where the power of quantum computers comes from and why it is difficult to harness. We describe quantum cryptography, teleportation, and dense coding. Various approaches to harnessing the power of quantum parallelism are explained, including Shor's algorithm, Grover's algorithm, and Hogg's algorithms. We conclude with a discussion of quantum error correction.
Quantum computation has teased scientific imagination ever since Richard Feynman dreamed in 1982 of quantum-mechanical computers, which could solve problems that are effectively out of reach of classical computation thanks to their ability to use superposition and entanglement. This critical survey recalls the field’s journey from theoretical advances to present-day engineering challenges, highlighting the yawning gulf between quantum computing’s lofty promises and its actual realizations. By examining three decades of developments, from Shor’s integer factorization and Grover’s search algorithms to recent demonstrations of quantum supremacy, this paper shows how idealized mathematical models interact with the noisy, resource-constrained realities of physical quantum systems. The author presents a literature review through critically analyzing key achievements in quantum computing research.. This paper concludes that quantum advantage will not arise from brute-force universality, but through application specific co-design, combining noise-adjusted algorithms to quantum hardware restrictions whilst mixing classical and quantum processing modes and fundamentally changing the way we think of error correction through the physics of qubits. This shift in paradigm from theoretical maximalism to engineering pragmatism embodies a realistic roadmap for practical 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.
Richard Feynman's observation that certain quantum mechanical effects cannot be simulated efficiently on a computer led to speculation that computation in general could be done more efficiently if it used these quantum effects. This speculation proved justified when Peter Shor described a polynomial time quantum algorithm for factoring intergers. In quantum systems, the computational space increases exponentially with the size of the system, which enables exponential parallelism. This parallelism could lead to exponentially faster quantum algorithms than possible classically. The catch is that accessing the results, which requires measurement, proves tricky and requires new nontraditional programming techniques. The aim of this paper is to guide computer scientists through the barriers that separate quantum computing from conventional computing. We introduce basic principles of quantum mechanics to explain where the power of quantum computers comes from and why it is difficult to harness. We describe quantum cryptography, teleportation, and dense coding. Various approaches to exploiting the power of quantum parallelism are explained. We conclude with a discussion of quantum error correction.
A challenging question in the overlap of computer science, mathematics, and physics, is the exploration of potential capabilities of quantum computers. Milestones were the factoring algorithm of Shor (1994) and the search algorithm of Grover (1996). So far, major research was concentrated on discrete and algebraic problems. Much less was known about computational problems of analysis, including such a prominent example as high dimensional numerical integration, which is well-studied in the classical settings. We seek to understand how efficiently this and related problems can be solved in the quantum model of computation (i.e., on a quantum computer) and how the outcome compares to the efficiency of deterministic or randomized algorithms on a classical (i.e. non-quantum) computer. In this paper we give a survey of the state of the art in this field, including also a brief introduction to the general ideas of quantum computing. Introduction A quantum computer is a computing device based on quantum mechanical laws of the (sub)atomic world. The idea of such a computer was developed by Feynman [8] in 1982. He emphasized that simulating quantum mechanics on a classical computer is extremely hard, probably infeasible. So why not try to simulate quantum mechanics using quantum devices themselves. (Thoughts in this direction were also expressed by Manin [21] in 1980, see also [22].) In 1985 Deutsch [6] presented a formal model of computation for quantum computing.