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Computation with quantum systems

David J. Wineland

Published Jun 1, 2010• 0 citations
Publisher

Abstract

Summary form only given. In 1994, Peter Shor showed that a computer based on the rules that govern quantum systems could efficiently factorize large numbers. Because of the implications of this idea on the security of data encryption, funding for the development of such a device increased significantly and sparked research for other applications of quantum information processing (QIP). Since then, the elementary logic operations and simple algorithms for such a device have been demonstrated, but building a useful quantum computer is an extremely daunting task due to the necessity of overcoming decoherence of the inherent large entangled quantum superposition states. Nevertheless, in the near term, the principles of QIP are finding applications in metrology (such as for atomic clocks) and may also provide a way to efficiently simulate other quantum systems of interest, a motivation that intrigued Richard Feynman in the early 1980's. A number of physical systems are currently considered for building a quantum computer; this talk will focus on the use of registers of atomic ions, but connections to other possible physical implementations are rather direct.

DOI
10.1109/drc.2010.5551926

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Summary form only given. In 1994, Peter Shor showed that a computer based on the rules that govern quantum systems could efficiently factorize large numbers. Because of the implications of this idea on the security of data encryption, funding for the development of such a device increased significantly and sparked research for other applications of quantum information processing (QIP). Since then, the elementary logic operations and simple algorithms for such a device have been demonstrated, but building a useful quantum computer is an extremely daunting task due to the necessity of overcoming decoherence of the inherent large entangled quantum superposition states. Nevertheless, in the near term, the principles of QIP are finding applications in metrology (such as for atomic clocks) and may also provide a way to efficiently simulate other quantum systems of interest, a motivation that intrigued Richard Feynman in the early 1980's. A number of physical systems are currently considered for building a quantum computer; this talk will focus on the use of registers of atomic ions, but connections to other possible physical implementations are rather direct.
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.
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.
After some remarks on the fundamental physical nature of information, Bennett and Fredkin’s ideas reversible computation are introduced. This leads to the suggestions of Benioff and Feynman as to the possibility of a new type of essentially ‘quantum computer’. If we happen to build such devices, Deutsch scientists showed that‘quantum parallelism’ leads to new algorithms and new complexity classes. This is dramatically illustrated by Shor's quantum algorithm for factorization which is polynomial in time in contrast to algorithms for factorization on a classical Turing computer. This discovery has potentially important implications for the security of many modern cryptographic systems. The fundamentals of quantum computing are then introduced - reversible logic gates, qubits and quantum registers. The key quantum property of ‘entanglement’ is described, with due homage to Einstein and Bell. As an illustration of a quantum program, Grover's database search algorithm is described in some detail. After all this theory, the status of experimental attempts to build a quantum computer is reviewed: it will become evident that we have a long way to go before we can factorize even small numbers. Finally, we end with some thoughts about the process of ‘quantum compilation’ - translating a quantum algorithm into actual physical operations on a quantum system - and some comments on prospects for future progress.
Building a useful quantum computer is a grand science and engineering challenge, currently pursued intensely by teams around the world. In the 1980s, Richard Feynman and Yuri Manin observed independently that computers based on quantum mechanics might enable better simulations of quantum phenomena. Their vision remained an intellectual curiosity until Peter Shor published his famous quantum algorithm for integer factoring, and shortly thereafter a proof that errors in quantum computations can be corrected. Since then, quantum computing R&D has progressed rapidly, from small-scale experiments in university physics laboratories to well-funded industrial efforts and prototypes. Hype notwithstanding, quantum computers have yet to solve scientifically or practically important problems -- a target often called quantum utility. In this article, we describe the capabilities of contemporary quantum computers, compare them to the requirements of quantum utility, and illustrate how to track progress from today to utility. We highlight key science and engineering challenges on the road to quantum utility, touching on relevant aspects of our own research.
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.
Quantum computers operate on the principles of quantum logic, a fundamentally distinct paradigm from classical Boolean logic. This disparity leads to quantum computation's enhanced efficiency compared to classical computing. In this comprehensive review, we demystify the fundamental concepts of quantum computation, covering the creation of elementary gates and networks. We highlight the capabilities of quantum algorithms by examining the simple Deutsch problem and, in straightforward terms, dissect the renowned Shor algorithm for factoring large numbers into primes. Furthermore, we delve into the realm of physical quantum computer implementations, with a particular focus on tlo he linear ion trap approach. Here, we shed light on the primary challenge hindering the realization of practical quantum computers: the issue of decoherence. Nonetheless, we demonstrate that this hurdle can be overcome through the application of quantum error correction methods