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arXiv:0803.1355

Parallelism of quantum computations from prequantum classical statistical field theory (PCSFT)

Andrei Yu. Khrennikov

Published Mar 10, 2008• 1 citation
arXivHTMLPDF

Abstract

This paper is devoted to such a fundamental problem of quantum computing as quantum parallelism. It is well known that quantum parallelism is the basis of the ability of quantum computer to perform in polynomial time computations performed by classical computers for exponential time. Therefore better understanding of quantum parallelism is important both for theoretical and applied research, cf. e.g. David Deutsch \cite{DD}. We present a realistic interpretation based on recently developed prequantum classical statistical field theory (PCSFT). In the PCSFT-approach to QM quantum states (mixed as well as pure) are labels of special ensembles of classical fields. Thus e.g. a single (!) ``electron in the pure state'' $ψ$ can be identified with a special `` electron random field,'' say $Φ_ψ(ϕ).$ Quantum computer operates with such random fields. By one computational step for e.g. a Boolean function $f(x_1,...,x_n)$ the initial random field $Φ_{ψ_0}(ϕ)$ is transformed into the final random field $Φ_{ψ_f}(ϕ)$ ``containing all values'' of $f.$ This is the objective of quantum computer's ability to operate quickly with huge amounts of information -- in fact, with classical random fields.

DOI
10.48550/arxiv.0803.1355

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This article analyzes mathematical models of quantum parallelism based on the transfer of a quantum system into a superposition state. The principles of quantum parallelism and its mathematical foundations are substantiated. To demonstrate the quantum advantage, mathematical models of fundamental quantum algorithms Deutsch-Joža and Grover are analyzed, which illustrate the efficiency of quantum computing compared to classical methods. The Deutsch-Joshy algorithm is considered, which generalizes the problem of determining the type of a binary function by extending it to a set of arguments written in a quantum register. This algorithm allows determining whether a function is constant or balanced in a single quantum call, while in classical calculations this requires exponential time. It is noted that although the Deutsch-Joshy algorithm demonstrates the possibilities of parallel quantum computing, it is of mainly theoretical importance. Grover’s algorithm implements quantum parallel computing for the practical problem of searching for an element in an unordered database. It is based on iterative amplification of the amplitude of the state corresponding to the searched element and provides quadratic speedup compared to classical algorithms. In addition to database search, Grover’s algorithm can be adapted to solve other optimization problems. The work of the considered algorithms is illustrated by numerical examples, which simplifies the understanding of their principles and contributes to the further methodological development of parallel quantum computing. The article also outlines the advantages of quantum parallelism over classical computing and identifies the prospects for the application of parallel quantum algorithms.
We first propose herein a novel parallel computation, even though today's algorithm methodology for quantum computing, for all of the combinations of values in variables of a logical function. Our concern so far has been to obtain an attribute of some function. In fact such a task is only for one task problem solving. However, we could treat positively the plural evaluations of some logic function in parallel instead of testing the function for finding out its attribute. In fact, these evaluations of the function are naturally included and evaluated, in parallel, in normal quantum computing discussing a function in a Boolean algebra stemmed from atoms in it. As is naturally understandable with mathematics, quantum computing with qubit systems naturally is included and exemplified by a Boolean algebra, which treats only both 0 and 1. Namely, the theory in this paper is quite natural in logical sense even though physics domain. Therefore, quantum computing has an ability to solve some mathematical problems described in a Boolean algebra. The reason why we positively introduce a Boolean algebra here is because we have multiple evaluations of a function in quantum computing general.
Quantum parallelism, a fundamental feature of quantum computing, enables the simultaneous evaluation of multiple inputs in a quantum superposition state. This paper explores the theoretical foundation, practical implementation challenges, and algorithmic implications of quantum parallelism. We examine how it empowers quantum algorithms such as the Deutsch-Jozsa and Grover's algorithms to achieve exponential speedups over classical counterparts. The paper concludes with a discussion on the current limitations and future potential of quantum parallelism in real-world quantum processors.
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In this paper we raise questions about the reality of computational quantum parallelism. Such questions are important because while quantum theory is rigorously established, the hypothesis that it supports a more powerful model of computation remains speculative. More specifically, we suggest the possibility that the seeming computational parallelism offered by quantum superpositions is actually effected by gate-level parallelism in the reversible implementation of the quantum operator. In other words, when the total number of logic operations is analyzed, quantum computing may not be more powerful than classical. This fact has significant public policy implications with regard to the relative levels of effort that are appropriate for the development of quantumparallel algorithms and associated hardware (i.e., qubit-based) versus quantum-scale classical hardware.
Albert Einstein did not believe in completeness of QM. He dreamed of creation of prequantum classical statistical mechanics such that QM will be reproduced as its approximation. He also dreamed of total exclusion of corpuscules from the future model. Reality of Einstein's dream was pure fields' reality. Recently I made his dream come true in the form of so called prequantum classical statistical field theory (PCSFT). In this approach quantum systems are described by classical random fields, e.g., electromagnetic field (instead of photon), electron field or neutron field. In this paper we generalize PCSFT to composite quantum system. It is well known that in QM, unlike classical mechanics, the state of a composite system is described by the tensor product of state spaces for its subsystems. In PCSFT one can still use Cartesian product, but state spaces are spaces of classical fields (not particles). In particular, entanglement is nothing else than correlation of classical random fields, cf. again Einstein. Thus entanglement was finally demystified.