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International Journal of Parallel Emergent and Distributed Systems

Quantum computing: beyond the limits of conventional computation†

Marius Nagy, Selim G. Akl

Published Mar 30, 2007• 22 citations
Publisher

Abstract

The quantum model of computation not only offers entirely new ways to manipulate information, but also allows information processing tasks to be formulated in unconventional, genuine quantum mechanical terms. We show that the task of distinguishing among entangled quantum states combines entanglement and non-determinism in a way that makes the quantum solution impossible to simulate on any classical machine (even one equipped with the same measurement capabilities as the quantum computational device). A new class of information processing tasks is thus uncovered whose members are readily carried out by a quantum computer, yet are impossible to perform on any classical machine (whether deterministic or probabilistic). In the broad, unconventional context created by quantum mechanics, the computational power of a quantum computer is therefore strictly greater than that of a classical computer.

DOI
10.1080/13547500600899209

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The quantum model of computation not only offers entirely new ways to manipulate information, but also allows information processing tasks to be formulated in unconventional, genuine quantum mechanical terms. We show that the task of distinguishing among entangled quantum states combines entanglement and non-determinism in a way that makes the quantum solution impossible to simulate on any classical machine (even one equipped with the same measurement capabilities as the quantum computational device). A new class of information processing tasks is thus uncovered whose members are readily carried out by a quantum computer, yet are impossible to perform on any classical machine (whether deterministic or probabilistic). In the broad, unconventional context created by quantum mechanics, the computational power of a quantum computer is therefore strictly greater than that of a classical computer.
The physics and the mathematics of computation are examined to provide a foundation and perspective for the investigation of the quantum mechanics of computation. Our purpose is to explore the fundamental limits and constraints imposed on computation by Nature through the laws of physics and the mathematics of computational complexity. Inasmuch as information storage and transmission are an integral part of computation, their physical bounds are considered. The computer is viewed both physically and mathematically as a dynamical system, and is depicted in terms of the basic Turing machine paradigm. Three fundamental classes of the Turing machine are defined; the deterministic, stochastic and quantum Turing machines. Hamiltonian models and physical realizations of quantum computing are described. Quantum computers can perform some tasks which have no classical analogue, but they cannot compute functions that are non-computable by classical means. Some classically intractable problems can be solved with quantum computers.
In this chapter we shall deal with the most recent and challenging development of quantum theory, and also one of the most important ones for interpretational, foundational, and even technological issues. This field finds its roots in the observation that quantum states can be viewed as bricks of information in a way that is intrinsically different from classical information. As a consequence, the ability to manipulate quantum states translates immediately into a new form of information processing and exchange. What has been discovered during this conceptual passage is that this type of information processing is in many respects much richer than its classical analogue. This has contributed to the understanding of quantum states as an extension of the classical concept of state and not as a defective reality (see also Subsec. 2.3.4 and Sec. 15.5). The impossibility of knowing the value of all observables at the same time that had been seen as a strong limitation in the early days of quantum mechanics, now turns out to be a manifestation of a different – but not necessarily poorer – resource. On the contrary, we have increasingly discovered that superposition and entanglement are additional informational resources. These resources, for instance, allow for certain particular computations that are much faster on a quantum device than on its classical counterpart, and therefore the former is able to solve problems that cannot be practically solved using classical means.
Quantum entanglement, a cornerstone of quantum mechanics, holds immense potential for revolutionizing computation. However, the inherent fragility of entangled states poses significant challenges to building practical and scalable quantum computers. This paper explores the concept of quantum entanglement's non-symmetry, proposing a novel approach – the incorporation of non-symmetry into quantum computations – to enhance stability and unlock new computational capabilities. We investigate how this approach mitigates the limitations of single-state entanglement and paves the way for fundamentally different algorithms. The core claim is that strategically introducing non-symmetry into quantum algorithms can lead to improved resilience and performance, offering a pathway to more robust and capable quantum systems. This work details a mechanism for encoding and manipulating non-symmetric entangled states, illustrating its potential impact on quantum computation. We present a preliminary analysis of this approach and outline future research directions, emphasizing the importance of understanding and harnessing the power of non-symmetry in quantum information processing.
We consider a quantum computational algorithm that can be used to determine (probabilistically) how close a given signal is to one of a set of previously observed signals stored in the state of a quantum neurocomputional machine. The realization of a new quantum algorithm for factorization of integers by Shor and its implication to cryptography has created a rapidly growing field of investigation. Although no physical realization of a quantum computer is available, a number of software systems simulating a quantum computation process exist. In light of the rapidly increasing power of desktop computers and their ability to carry out these simulations, it is worthwhile to investigate possible advantages as well as realizations of quantum algorithms in signal processing applications. The algorithm presented offers a glimpse of the potential of this approach. Neural networks (NN) provide a natural paradigm for parallel and distributed processing of a wide class of signals. Neural networks within the context of classical computation have been used for approximation and classification tasks with some success. We propose a model for quantum neurocomputation (QN) and explore some of its properties and potential applications to signal processing in an information theoretic context.
Information is something that can be encoded in the state of a physical system, and a computation is a task that can be performed with a physically realizable device. Therefore, since the physical world is fundamentally quantum mechanical, the foundations of information theory and computer science should be sought in quantum physics. In fact, quantum information has weird properties that contrast sharply with the familiar properties of classical information. A quantum computer -- a new type of machine that exploits the quantum properties of infomation -- could perform certain typ$s of calculations far more efficiently than any foreseeable classical computer. To build a functional quantum computer will be an enormous technical challenge. New methods for quantum error correction are being developed that can help to prevent a quantum computer from crashing.
This paper presents a novel framework for constructing quantum networks specifically designed for non-classical information processing. The core argument is that leveraging quantum entanglement and quantum superposition allows us to build networks capable of exceeding the computational limitations inherent in classical systems. We propose a mechanism utilizing these quantum phenomena to create a fundamentally new type of quantum network. This work explores the potential of quantum networks to perform tasks fundamentally impossible for classical networks, marking a significant step towards harnessing the power of non-classical information processing. The architecture and key principles are outlined, emphasizing the transition from classical information transmission to the manipulation of quantum states within the network. This research lays the groundwork for future development in quantum communication and computation.