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arXiv:quant-ph/0004028

Quantum Computer as a Probabilistic Inference Engine

Robert R. Tucci

Published Apr 6, 2000• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0004028

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We propose a new family of quantum computing algorithms which generalize the Deutsch-Jozsa, Simon and Shor ones. The goal of our algorithms is to estimate conditional probability distributions. Such estimates are useful in applications of Decision Theory and Artificial Intelligence, where inferences are made based on uncertain knowledge. The family of algorithms that we propose is based on a construction method that generalizes a Fredkin-Toffoli (FT) construction method used in the field of classical reversible computing. FT 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) which can perform the same calculations as the classical one, but in a (piecewise) reversible manner. Thus, we extend the FT construction method so that it can be applied to any stochastic circuit, not just binary deterministic ones.
The astounding acceleration in Artificial Intelligence and Quantum Computing advances naturally gives rise to a line of research, which unrolls the potential advantages of quantum computing on classical Machine Learning tasks, known as Quantum Machine Learning or Quantum Machine Intelligence. The typical objectives are either (1) exploring the potential quantum advantages on classical learning tasks or (2) levering well-established classical ML algorithms to tackle quantum-related problems on Noisy Intermediate-Scale Quantum (NISQ) devices. Along the second research direction, we study Quantum Neural Networks (QNNs) to accomplish the purpose of Bayesian learning. By observing a wide range of studies on QNNs, in which the sole training method is based on frequentist training, we find that Bayesian learning benefits QNNs from two aspects. First, Bayesian-trained models enjoy a high level of generalization due to the prior and posterior distribution usage compared to frequentist training, which will be justified by this paper’s theoretical study of model capacity. Second, Bayesian Inference offers epistemic uncertainty estimation, which merits the decision-making process. It is worth mentioning that frequentist-trained QNNs generally lack this desirable property. Under the Bayesian training procedure, our derived models can be considered a new class of QNNs (called BayesianQNNs) which possesses both desirable properties of Bayesian Inference while maintaining comparable predictive performance as frequentist counterparts. The proposed Bayesian Quantum Neural Networks is justified by empirical evidence from numerical experiments.
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 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.
Abstract In multi-agent systems, Bayesian networks are pivotal for inference and decision-making under uncertainty, yet they face significant challenges, such as high computational complexity and slow decision speed. Quantum computing, leveraging superposition and entanglement, offers potential advantages for solving certain complex problems. Therefore, exploring Bayesian networks within the quantum multi-agent framework promises enhanced inference and decision-making capabilities. While quantum Bayesian networks for inference have been extensively studied, quantum dynamic Bayesian networks for inference and quantum decision networks for decision-making remain underexplored. In the noisy intermediate-scale quantum (NISQ) era, quantum computers struggle to process the long-term temporal structures of dynamic Bayesian networks due to limited resources. To address this, we propose a recursive quantum-classical hybrid Bayesian network inference method, which decomposes dynamic Bayesian networks into smaller subnetworks along the time series. Each subnetwork is represented by a reduced-scale quantum circuit, and the forward and backward operators are computed recursively, enabling efficient filtering and smoothing inference. In addition, we propose an optimal decision-making method based on quantum decision networks, which maps utility values to quantum probabilities and identifies the optimal action by determining the quantum state corresponding to the maximum expected probability. We validate the effectiveness of these algorithms using the IonQ quantum simulator and compare their performance with classical methods. The results demonstrate that our proposed quantum algorithms can effectively perform inference and decision-making tasks well on NISQ devices. The proposed methods provide a foundation for collaborative inference and decision-making within the quantum multi-agent framework.
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.