In 1994 Peter Shor [65] published a factoring algorithm for a quantum computer that finds the prime factors of a composite integer N more efficiently than is possible with the known algorithms for a classical com puter. Since the difficulty of the factoring problem is crucial for the se curity of a public key encryption system, interest (and funding) in quan tum computing and quantum computation suddenly blossomed. Quan tum computing had arrived. The study of the role of quantum mechanics in the theory of computa tion seems to have begun in the early 1980s with the publications of Paul Benioff [6]' [7] who considered a quantum mechanical model of computers and the computation process. A related question was discussed shortly thereafter by Richard Feynman [35] who began from a different perspec tive by asking what kind of computer should be used to simulate physics. His analysis led him to the belief that with a suitable class of "quantum machines" one could imitate any quantum system.
In 1994 Peter Shor [65] published a factoring algorithm for a quantum computer that finds the prime factors of a composite integer N more efficiently than is possible with the known algorithms for a classical com puter. Since the difficulty of the factoring problem is crucial for the se curity of a public key encryption system, interest (and funding) in quan tum computing and quantum computation suddenly blossomed. Quan tum computing had arrived. The study of the role of quantum mechanics in the theory of computa tion seems to have begun in the early 1980s with the publications of Paul Benioff [6]' [7] who considered a quantum mechanical model of computers and the computation process. A related question was discussed shortly thereafter by Richard Feynman [35] who began from a different perspec tive by asking what kind of computer should be used to simulate physics. His analysis led him to the belief that with a suitable class of "quantum machines" one could imitate any quantum system.
In the twenty-fi rst century, many signifi cant developments in science and engineering have been achieved through interdisciplinary research. Quantum computing (QC) is about computing with quantum systems, called quantum computers. A quantum computer is a computational device that makes use of the quantum mechanics (QM) phenomena such as entanglement, superposition and interference to perform operations on data. Quantum computers are different from classical binary digital computers since they work with quantum bits. The theoretical model is the quantum turing machine (QTM) which is a normal turing machine (TM) with quantum parallelism. Richard Feynman introduced this fi eld for the fi rst time in the 80’s (Feynman 1982) since he observed that certain quantum-mechanical effects cannot be simulated in an effi cient way using a classical computer. Such observation let to speculate that using quantum effects may become more effi cient computing. The interest grew up in 1994 when Peter Shor published the paper entitled “Algorithms for Quantum Computation : Discrete Logarithms and Factoring” (Shor 1994) where he described polynomial time quantum algorithms for fi nding discrete logarithms and factoring integers.
Introduction In an ordinary computer, information is stored in a collection of tiny circuits each of which is designed to have two stable and easily distinguishable configurations: each represents a bit . In our study of quantum cryptography, we have seen how it can be useful to express information not in ordinary bits but in qubits . Whereas a bit can have only two values, say 0 and 1, a qubit can be in any quantum superposition of ∣0〉 and ∣1〉. Moreover, a qubit can be entangled with other qubits. Thus one might wonder whether a quantum computer, in which the basic elements for storing and processing information are qubits, can outperform an ordinary (classical) computer in certain ways. This question was addressed by researchers starting in the 1980s. In terms of practical consequences, perhaps the most dramatic answer has been given by Peter Shor in his 1994 factoring algorithm for a quantum computer, an algorithm that is exponentially faster than any known classical algorithm. As we have seen in Chapter 1, the difficulty of factoring a product of two large primes is the basis of the security of the RSA cryptosystem. So if one could build a large enough quantum computer – and there is no reason in principle why this could not be done – the RSA system would be rendered ineffective. In this chapter we present the basics of quantum computation and then focus on Shor's factoring algorithm.
It is difficult nowadays to graduate from a department of natural sciences and not hear anything about quantum computing, most likely about the fascinating prospects of it. Quantum computing by its origin is a rather abstract discipline, a branch of math or information science. It has emerged from a persistent search for more efficient ways to process information when quantum mechanics made it to the scope of the search. Being an abstract discipline, quantum computing approximates a physical quantum system with a number of axioms and explores the consequences of these axioms, precisely as conventional math has been. These activities begun in the 1970s, and for a long time it was not obvious why quantum calculational schemes have to be any better than common computer algorithms. This extensive work was rewarded with a break-through in 1994, when Peter Shor discovered a remarkable quantum algorithm for the factorization of large numbers into prime ones. This sounds quite abstract, but many public key cryptosystems will become obsolete if Shor's algorithm is ever implemented in a practical quantum computer. Modern communication security is based on the fact that the factorization is a tough problem for a classical computer: it takes too long to crack a code. The proposed algorithm speeds up the factorization enormously. The discovery by Shor has brought quantum computing to the scientific and even public attention. This progress has motivated a massive research effort towards the manipulation of individual quantum systems, the practical realization of quantum computing schemes being one of the most attractive goals. This chapter describes this goal, the quantum transport systems suitable for it, and the achievements made so far.
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.
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.