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ACIS International Conference on Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing (SNPD)

Quantum Mechanics of Theorem of Bayes Modeled by Machine Learning Principles

Huber Nieto–Chaupis

Published Dec 7, 2022• 0 citations
Publisher

Abstract

A theory consisting in quantum mechanics and theorem of Bayes, is presented. In essence, the Bayes probability has been built from two subspaces. While in one some quantum measurements are done, in the another it is seen that the probabilities acquire their highest values. Thus, the validity of a prior probability makes sense is there is a clear difference between the done measurement of probability amplitude. Thus, the principles of machine learning compacted in the criteria of Tom Mitchell have been employed. The simulations have shown that the size of space has direct impact on the prior probability that presumably would get low values of probability in a limited subspace. These values have turned out to be strongly correlated to the times in which measurements are done in a big space. Therefore, it is evident the prospective applicability of this novel approach in all those scenarios that require of a quantum measurement in separated times.

DOI
10.1109/snpd54884.2022.10051776

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A theory consisting in quantum mechanics and theorem of Bayes, is presented. In essence, the Bayes probability has been built from two subspaces. While in one some quantum measurements are done, in the another it is seen that the probabilities acquire their highest values. Thus, the validity of a prior probability makes sense is there is a clear difference between the done measurement of probability amplitude. Thus, the principles of machine learning compacted in the criteria of Tom Mitchell have been employed. The simulations have shown that the size of space has direct impact on the prior probability that presumably would get low values of probability in a limited subspace. These values have turned out to be strongly correlated to the times in which measurements are done in a big space. Therefore, it is evident the prospective applicability of this novel approach in all those scenarios that require of a quantum measurement in separated times.
This paper proposes a hybrid model of machine learning, based at the principles of quantum mechanics in conjunction to Bayes theorem. With such construction, it is tested onto a scenario dictated by the Mitchell criteria. By assuming stochastic system whose main dependence is through time, the proposed model exhibits a high experience probability just at the beginning of processes. The results of this paper would corroborate the hypothesis that stochastic system might be strongly dependent at the very beginning or initial conditions.
We present a theory of Machine Learning based entirely on the formalism of Quantum Mechanics from the fact that the diverse instances on the application of the algorithms would contain certain concepts linked to stochastic. In this manner, the probabilistic formalism of the Quantum Mechanics might be well applied. Thus, we implement the Mitchell's criteria with mathematical methodologies based on the Hilbert's space as well as the employment of quantum operators to describe the behavior of the experience in terms of probabilities. We illustrate the application of this theory through a quantitative analysis of the time evolution of the experience.
Abstract This paper reviews arguments for the claim that all probabilities arising in quantum mechanics are Bayesian degrees of belief. The starting point is the observation that there is a one-to-one map from quantum states to points on the probability simplex. Bayes’s rule remains valid in quantum theory, provided it is kept in mind that the outcomes of quantum measurements have no preassigned values. The probabilities of quantum measurement outcomes depend explicitly on a quantum prior whose general form will be discussed. Similar to the classical case, the quantum de Finetti theorem simplifies the analysis of repeated quantum trials if the prior is exchangeable. The Bayesian approach to quantum theory described here solves the problem of the ‘collapse of the wave function’. It also provides almost trivial explanations of a number of phenomena in quantum information theory, including the no-cloning theorem and quantum teleportation.
We consider the problem of gambling on a quantum experiment and enforce rational behaviour by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalised to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalised Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalisation (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers.
The goal of this paper is the presentation of the elementary procedures that normally are done in nonrelativistic Quantum Mechanics in terms of the principles of Machine Learning. In essence, this paper discusses Mitchell’s criteria, whose block fundamental dictates that the universal evolution of any system is composed by three fundamental steps: (i) Task, (ii) Performance and (iii) Experience. In this paper, the quantum mechanics formalism reflected on the usage of evolution operator and Green’s function are assumed to be part of mechanisms that are inherently engaged to the Machine Learning philosophy. The action for measuring observables through experiments and the intrinsic apparition of statistical or systematic errors are discussed in terms of “quantum learning”.
Abstract In the 1950s and 1960s it was causing interest among physicists that in the formalism of quantum mechanics (complex-valued) ‘probability amplitudes’ obeyed laws similar to the laws obeyed by probabilities in the ordinary probability calculus. But this did not then lead decisively to the claim that probabilities should be represented by complex numbers. It became fashionable instead to regard probability amplitudes as an abstract concept from which actual probabilities could be derived by taking the squared moduli of the amplitudes. In this monograph, however, we make another attempt to show how probability amplitudes might after all be identified with actual probabilities. To do this, probability itself is interpreted in a rational Bayesian manner (i.e. as an extension of logic) and a new (complex-valued) probability theory is formulated that incorporates the uncertainty principle (i.e. that takes account of the fact that acquisition of knowledge of a quantum mechanical process generally interferes with it). Taking this probability theory as the new logic of science, and assuming certain physical laws and properties of matter, an interpretation of non-relativistic quantum mechanics is built up. It is consistent with the usual quantum mechanical formalism but allows a clear distinction to be made between the physical world and our knowledge of it.