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arXiv:1404.0055

Quantum Circuit For Discovering from Data the Structure of Classical Bayesian Networks

Robert R. Tucci

Published Mar 31, 2014• 2 citations
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Abstract

We give some quantum circuits for calculating the probability $P(G|D)$ of a graph $G$ given data $D$. $G$ together with a transition probability matrix for each node of the graph, constitutes a Classical Bayesian Network, or CB net for short. Bayesian methods for calculating $P(G|D)$ have been given before (the so called structural modular and ordered modular models), but these earlier methods were designed to work on a classical computer. The goal of this paper is to "quantum computerize" those earlier methods.

DOI
10.48550/arxiv.1404.0055

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We give some quantum circuits for calculating the probability P(G∣D)P(G|D)P(G∣D) of a graph GGG given data DDD. GGG together with a transition probability matrix for each node of the graph, constitutes a Classical Bayesian Network, or CB net for short. Bayesian methods for calculating P(G∣D)P(G|D)P(G∣D) have been given before (the so called structural modular and ordered modular models), but these earlier methods were designed to work on a classical computer. The goal of this paper is to "quantum computerize" those earlier methods.
We begin with a review of a well-known class of networks, Classical Bayesian (CB) nets (also called causal probabilistic nets by some). Given a situation which includes randomness, CB nets are used to calculate the probabilities of various hypotheses about the situation, conditioned on the available evidence. We introduce a new class of networks, which we call Quantum Bayesian (QB) nets, that generalize CB nets to the quantum mechanical regime. We explain how to use QB nets to calculate quantum mechanical conditional probabilities (in case of either sharp or fuzzy observations), and discuss the connection of QB nets to Feynman Path integrals. We give examples of QB nets that involve a single spin-[Formula: see text] particle passing through a configuration of two or three Stern—Gerlach magnets. For the examples given, we present the numerical values of various conditional probabilities, as calculated by a general computer program specially written for this purpose.
We begin with a review of a well known class of networks, Classical Bayesian (CB) nets (also called causal probabilistic nets by some). Given a situation which includes randomness, CB nets are used to calculate the probabilities of various hypotheses about the situation, conditioned on the available evidence. We introduce a new class of networks, which we call Quantum Bayesian (QB) nets, that generalize CB nets to the quantum mechanical regime. We explain how to use QB nets to calculate quantum mechanical conditional probabilities (in case of either sharp or fuzzy observations), and discuss the connection of QB nets to Feynman Path integrals. We give examples of QB nets that involve a single spin-half particle passing through a configuration of two or three Stern-Gerlach magnets. For the examples given, we present the numerical values of various conditional probabilities, as calculated by a general computer program especially written for this purpose.
We propose a new class of quantum computing algorithms which generalize many standard ones. The goal of our algorithms is to estimate probability distributions. Such estimates are useful in, for example, applications of Decision Theory and Artificial Intelligence, where inferences are made based on uncertain knowledge. The class of algorithms that we propose is based on a construction method that generalizes a Fredkin-Toffoli (F-T) construction method used in the field of classical reversible computing. F-T showed how, given any binary deterministic circuit, one can construct another binary deterministic circuit which does the same calculations in a reversible manner. We show how, given any classical stochastic network (classical Bayesian net), one can construct a quantum network (quantum Bayesian net). By running this quantum Bayesian net on a quantum computer, one can calculate any conditional probability that one would be interested in calculating for the original classical Bayesian net. Thus, we generalize the F-T construction method so that it can be applied to any classical stochastic circuit, not just binary deterministic ones. We also show that, in certain situations, our class of algorithms can be combined with Grover's algorithm to great advantage.
This paper explores the potential of quantum computing to accelerate Bayesian network inference. Traditional Bayesian network inference relies on iterative calculations of probability distributions, which can be computationally expensive, especially for complex networks. We propose a novel approach that leverages quantum circuits to simulate the transition probability matrices inherent in Bayesian networks. By mapping these matrices to quantum states and utilizing quantum parallelism, we aim to achieve significant speedups in inference processes. This work outlines the core concept, detailing the mapping strategy and the proposed quantum algorithm. While a full experimental implementation is not presented, this research lays a foundational framework for future investigations into quantum-enhanced Bayesian network inference. The core claim is to utilize quantum circuits to simulate Bayesian network probability transition matrices, leading to accelerated inference. The core mechanism involves mapping transition matrices to quantum states and employing quantum lines for parallel computation. This represents a new approach combining quantum computing and Bayesian networks for powerful probabilistic inference. ---
This project deals with the implementation of Bayesian networks, a type of artificial intelligence based on graph theory and conditional probability, in the quantum computing paradigm. Examples related to the world of sports will be used as input in order to test the developed model. Besides, a comparison between implementations in classical and quantum paradigms will be carried out, so their differences in performance and attributes can be appreciated. The work will conclude with a debate about quantum computing and its future as a basis for executing AI models.