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arXiv:2207.03104 · Proceedings of the National Academy of Sciences

Quantum advantage in variational Bayes inference

Hideyuki Miyahara, Vwani Roychowdhury

Published Jul 25, 2023• 5 citations

Abstract

Variational Bayes (VB) inference algorithm is used widely to estimate both the parameters and the unobserved hidden variables in generative statistical models. The algorithm—inspired by variational methods used in computational physics—is iterative and can get easily stuck in local minima, even when classical techniques, such as deterministic annealing (DA), are used. We study a VB inference algorithm based on a nontraditional quantum annealing approach—referred to as quantum annealing variational Bayes (QAVB) inference—and show that there is indeed a quantum advantage to QAVB over its classical counterparts. In particular, we show that such better performance is rooted in key quantum mechanics concepts: i) The ground state of the Hamiltonian of a quantum system—defined from the given data—corresponds to an optimal solution for the minimization problem of the variational free energy at very low temperatures; ii) such a ground state can be achieved by a technique paralleling the quantum annealing process; and iii) starting from this ground state, the optimal solution to the VB problem can be achieved by increasing the heat bath temperature to unity, and thereby avoiding local minima introduced by spontaneous symmetry breaking observed in classical physics based VB algorithms. We also show that the update equations of QAVB can be potentially implemented using ⌈log K ⌉ qubits and 𝒪( K ) operations per step, where K is the number of values hidden categorical variables can take. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.

DOI: 10.1073/pnas.2212660120

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Variational Bayes (VB) inference algorithm is used widely to estimate both the parameters and the unobserved hidden variables in generative statistical models. The algorithm—inspired by variational methods used in computational physics—is iterative and can get easily stuck in local minima, even when classical techniques, such as deterministic annealing (DA), are used. We study a VB inference algorithm based on a nontraditional quantum annealing approach—referred to as quantum annealing variational Bayes (QAVB) inference—and show that there is indeed a quantum advantage to QAVB over its classical counterparts. In particular, we show that such better performance is rooted in key quantum mechanics concepts: i) The ground state of the Hamiltonian of a quantum system—defined from the given data—corresponds to an optimal solution for the minimization problem of the variational free energy at very low temperatures; ii) such a ground state can be achieved by a technique paralleling the quantum annealing process; and iii) starting from this ground state, the optimal solution to the VB problem can be achieved by increasing the heat bath temperature to unity, and thereby avoiding local minima introduced by spontaneous symmetry breaking observed in classical physics based VB algorithms. We also show that the update equations of QAVB can be potentially implemented using ⌈log K ⌉ qubits and 𝒪( K ) operations per step, where K is the number of values hidden categorical variables can take. Thus, QAVB can match the time complexity of existing VB algorithms, while delivering higher performance.

Variational Bayes (VB) inference is one of the most important algorithms in machine learning and is widely used in engineering and industry. However, VB is known to suffer from the problem of local optima. In this paper, we generalize VB by using quantum mechanics and propose an algorithm, which we call quantum annealing variational Bayes (QAVB) inference. We then show that QAVB drastically improves the performance of VB in a clustering problem described by a Gaussian mixture model, which is essentially important from the viewpoint of optimization. Finally, we discuss an intuitive understanding of how QAVB works well.

Variational Bayes (VB) is a critical method in machine learning and statistics, underpinning the recent success of Bayesian deep learning. The natural gradient is an essential component of efficient VB estimation, but it is prohibitively computationally expensive in high dimensions. We propose a hybrid quantum-classical algorithm to improve the scaling properties of natural gradient computation and make VB a truly computationally efficient method for Bayesian inference in highdimensional settings. The algorithm leverages matrix inversion from the linear systems algorithm by Harrow, Hassidim, and Lloyd [Phys. Rev. Lett. 103, 15 (2009)] (HHL). We demonstrate that the matrix to be inverted is sparse and the classical-quantum-classical handoffs are sufficiently economical to preserve computational efficiency, making the problem of natural gradient for VB an ideal application of HHL. We prove that, under standard conditions, the VB algorithm with quantum natural gradient is guaranteed to converge. Our regression-based natural gradient formulation is also highly useful for classical VB.

Variational Bayes (VB) has become a widely-used tool for Bayesian inference in statistics and machine learning. VB algorithms are generally restricted to the case where the variational parameter space is Euclidean. Recently, the VB framework has been extended to the case where the variational parameter space is a Riemannian manifold. Such a manifold-based algorithm, by utilizing the natural gradient, can exploit the geometric structures to derive an efficient VB procedure. However, natural gradient in high dimension can be prohibitively computationally expensive to compute. This work proposes using a quantum natural gradient within the manifold VB framework. We demonstrate that the quantum natural gradient does not suffer from the well-known quantum-classical readout computational bottleneck, and hence delivers a highly computational efficiency. The resulting quantum manifold VB algorithm is provably convergent.

The pursuit of massive foundation models is hindered by the limitations of Maximum Likelihood Estimation (MLE), which struggles with uncertainty quantification and out-of-distribution generalization. The Bayesian framework resolves these deficiencies but is crippled by the "marginalisation bottleneck"—the computational intractability of integrating over high-dimensional parameter spaces. Classical approximations like MCMC and Variational Inference (VI) are fundamentally limited by slow mixing times or sub-optimal local minima. Quantum computing offers a paradigm-shifting resolution, leveraging superposition and entanglement to execute high-dimensional integrals with theoretical exponential speedups by mapping statistical probabilities to quantum amplitude vectors. This review systematically details foundational quantum mapping and state-of-the-art Mamun, SM (2026) ………………………………….1 Quantum Computing for Bayesian Inference: Algorithms, Speedups, Challenges, and Synergies with Scalable Machine Learning (Vol-III) K S C algorithms, including Quantum Bayesian Computation (QBC), Quantum Annealing Variational Bayes (QAVB), and Quantum Gaussian Processes (Q-GPs). A key insight is the deep synergy where inherent quantum noise (decoherence) functionally serves as a native Bayesian regularizer, analogous to classical dropout. Also available on Academia.edu: https://www.academia.edu/attachments/132601227/download_file?s=portfolio