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Journal of Physics Condensed Matter

Quantum computing and probability

David K. Ferry

Published Nov 5, 2009• 8 citations
Publisher

Abstract

Over the past two decades, quantum computing has become a popular and promising approach to trying to solve computationally difficult problems. Missing in many descriptions of quantum computing is just how probability enters into the process. Here, we discuss some simple examples of how uncertainty and probability enter, and how this and the ideas of quantum computing challenge our interpretations of quantum mechanics. It is found that this uncertainty can lead to intrinsic decoherence, and this raises challenges for error correction.

DOI
10.1088/0953-8984/21/47/474201

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Over the past two decades, quantum computing has become a popular and promising approach to trying to solve computationally difficult problems. Missing in many descriptions of quantum computing is just how probability enters into the process. Here, we discuss some simple examples of how uncertainty and probability enter, and how this and the ideas of quantum computing challenge our interpretations of quantum mechanics. It is found that this uncertainty can lead to intrinsic decoherence, and this raises challenges for error correction.
The quantum computing theoretically serves as a path to many high-level complex problems, but suffers some profound challenges which become hurdles in its path to a practical scenario. It has deep rooted inexplicable and unsettled challenges like Quantum Error Correction (QEC), qubit instability and issues of quantum decoherence. One of the main obstacles in quantum computing is tenuousness of quantum information processing. The time limit and decoherence problem have been a long-term scientific barrier in the theory of quantum mechanics. The quantum behavior of a quantum computer is influenced and affected by the surroundings creating a ‘race against time’ causing errors in the quantum computations making later unreliable. The error correction requires resource overhead further causing the data processing complexities and so remains as a monumental issue. This paper discusses some major unresolved limitations and deficiencies of quantum computing to eliminate the hurdles, for a need to create a transformative potential technology in the era of quantum computing.
Quantum computing is necessary to be described as a powerful tool is the new generation of computational power that works based on the principles of quantum mechanics, allowing the solving of problems that are intractable for classical computers. Due to the utilization of such phenomena as superposition, quantum entanglement and quantum interference, quantum computers are said to be able to solve problems in such areas as cryptography, artificial intelligence, optimization problems and material sciences, among others. This paper attempts to examine the basic concepts of quantum computing while focusing on their different aspects and their potential impacts. This study its experience and potential in its current form and in future applications through examples, illustrations, and theoretical discussion. Specific problems, including error correction, hardware scalability, and requirements for resources to address, make the paper a comprehensive overview of the quantum computing potential and weaknesses. Moving towards practical examples, the discussion covers all aspects of the book, providing readers with a holistic understanding of how quantum computing is going to change technological paradigms in the forthcoming decades.
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.
In a ground breaking discovery in 1994, Shor has shown that quantum computers, if built, can factor numbers efficiently. Since then quantum computing has become a burgeoning field of research, attracting theoreticians and experimentalists alike, and regrouping researchers from fields like computer science, physics, mathematics and engineering. Quantum information is very fragile and prone to decoherence. Yet by the middle of 1996 it has been shown that fault-tolerant quantum computation is possible. We give a simple description of the elements of quantum error-correction and quantum fault-tolerance. After characterizing quantum errors we present several error correction schemes and outline the elements of a full fledged fault-tolerant computation, which works error-free even though all of its components can be faulty. We also mention alternative approaches to error-correction, so called error-avoiding or decoherence-free schemes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Despite high hopes for quantum computation in the 1990s, progress in the past decade has been slow; we still cannot perform computation with more than about three qubits and are no closer to solving problems of real interest than a decade ago. Separately, recent experiments in fluid mechanics have demonstrated the emergence of a full range of quantum phenomena from completely classical motion. We present two specific hypotheses. First, Kuramoto theory may give a basis for geometrical thinking about entanglement. Second, we consider a recent soliton model of the electron, in which the quantum-mechanical wave function is a phase modulation of a carrier wave. Both models are consistent with one another and with observation. Both models suggest how entanglement and decoherence may be related to device geometry. Both models predict that it will be difficult to maintain phase coherence of more than three qubits in the plane, or four qubits in a three-dimensional structure. The soliton model also shows that the experimental work which appeared to demonstrate a violation of Bell's inequalities might not actually do so; regardless of whether it is a correct description of the world, it exposes a flaw in the logic of the Bell tests. Thus the case for the security of EPR-based quantum cryptography has just not been made. We propose experiments in quantum computation to test this. Finally, we examine two possible interpretations of such soliton models: one is consistent with the transactional interpretation of quantum mechanics, while the other is an entirely classical model in which we do not have to abandon the idea of a single world where action is local and causal.