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arXiv:quant-ph/9706039 · International Journal of Modern Physics B

QUANTUM BAYESIAN NETS

Robert R. Tucci

Published Jan 30, 1995• 70 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217979295000148

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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 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 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.
We present a new implementation of quantum computation that treats quantum computers as a special type of Bayesian Network called a QuDot Net. QuDot Nets allow for the efficient representation of some qubit systems. Single qubit quantum gates can be implemented as edge transformations on QuDot Nets. The X, H, R(k), M and SWAP gates are discussed in detail and results show linear scaling as the number of qubits are increased. We show that measurement and semi-quantum control gates can be efficiently implemented using QuDot Nets and present results from a QuDot Net implementation of the terminal Quantum Fourier Transform. We show how QuDot Nets can implement coherent control gates using multi-digraphs by labelling parallel edges. Lastly, we discuss implications to quantum foundations if a classical implementation of quantum computation is realized.
Quantum Bayesian networks provide a mathematical formalism to describe causal relations, to analyse correlations, and to predict the probabilities of measurement outcomes, in systems involving both classical and quantum data. They generalize Pearl's Bayesian networks -- prominent graphical models for classical probabilistic reasoning and inference. The goal of this paper is to bring compositional principles and a typing discipline into this setting. A key feature of our compositional semantics is that when all causes are classical, it coincides with the standard factor-based semantics of Bayesian networks, while in the purely quantum case it reduces to tensor networks. We then propose a typed formalism based on linear logic proof-nets, where types ensure well-behaved composition of systems, and which we prove sound and complete with respect to quantum Bayesian networks.