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arXiv:1402.7359 · Physical Review A

Quantum inference on Bayesian networks

Guang Hao Low, Theodore J. Yoder, Isaac L. Chuang

Published Jun 13, 2014• 109 citations
arXivHTMLPDF

Abstract

Performing exact inference on Bayesian networks is known to be $#P$-hard. Typically approximate inference techniques are used instead to sample from the distribution on query variables given the values $e$ of evidence variables. Classically, a single unbiased sample is obtained from a Bayesian network on $n$ variables with at most $m$ parents per node in time $O(nmP{(e)}^{\ensuremath{-}1})$, depending critically on $P(e)$, the probability that the evidence might occur in the first place. By implementing a quantum version of rejection sampling, we obtain a square-root speedup, taking $O(n{2}^{m}P{(e)}^{\ensuremath{-}\frac{1}{2}})$ time per sample. We exploit the Bayesian network's graph structure to efficiently construct a quantum state, a q-sample, representing the intended classical distribution, and also to efficiently apply amplitude amplification, the source of our speedup. Thus, our speedup is notable as it is unrelativized---we count primitive operations and require no blackbox oracle queries.

DOI
10.1103/physreva.89.062315

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Performing exact inference on Bayesian networks is known to be #P-hard. Typically approximate inference techniques are used instead to sample from the distribution on query variables given the values eee of evidence variables. Classically, a single unbiased sample is obtained from a Bayesian network on nnn variables with at most mmm parents per node in time O(nmP(e)\ensuremath−1)O(nmP{(e)}^{\ensuremath{-}1})O(nmP(e)\ensuremath−1), depending critically on P(e)P(e)P(e), the probability that the evidence might occur in the first place. By implementing a quantum version of rejection sampling, we obtain a square-root speedup, taking O(n2mP(e)\ensuremath−12)O(n{2}^{m}P{(e)}^{\ensuremath{-}\frac{1}{2}})O(n2mP(e)\ensuremath−21​) time per sample. We exploit the Bayesian network's graph structure to efficiently construct a quantum state, a q-sample, representing the intended classical distribution, and also to efficiently apply amplitude amplification, the source of our speedup. Thus, our speedup is notable as it is unrelativized---we count primitive operations and require no blackbox oracle queries.
Quantum machine learning promises great speedups over classical algorithms, but it often requires repeated computations to achieve a desired level of accuracy for its point estimates. Bayesian learning focuses more on sampling from posterior distributions than on point estimation, thus it might be more forgiving in the face of additional quantum noise. We propose a quantum algorithm for Bayesian neural network inference, drawing on recent advances in quantum deep learning, and simulate its empirical performance on several tasks. We find that already for small numbers of qubits, our algorithm approximates the true posterior well, while it does not require any repeated computations and thus fully realizes the quantum speedups.
This thesis explores the intersection of Bayesian inference and quantum computing, with a particular focus on developing and analyzing quantum approaches to probabilistic modeling. The motivation arises from the limitations of classical inference methods, which often become intractable when applied to high-dimensional distributions and complex Bayesian networks. By leveraging quantum algorithms, the work investigates how probabilistic reasoning can benefit from computational speedups, scalability, and more efficient handling of uncertainty.The implementation targets canonical benchmark problems—the Sprinkler Network and the Asia Network—where both classical and quantum Bayesian inference techniques are applied and compared. Using Qiskit, the thesis demonstrates how quantum circuits can encode probabilistic models through parameterized rotations and controlled operations, and how posterior distributions can be computed by sampling and amplitude amplification. Comparative experiments systematically evaluate the accuracy, variance, and scalability of quantum Bayesian inference against classical baselines, highlighting trade-offs between precision and shot counts while exploring the potential for quadratic speedups.The results show that although present-day hardware constraints, such as limited qubit counts and noise, restrict large-scale deployment, quantum implementations already display strong promise for complex probabilistic reasoning tasks. Importantly, the conclusions underscore the role of quantum Bayesian networks as a pathway toward practical quantum-enhanced machine learning.
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