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arXiv:2512.02323

Unlocking the Power of Boltzmann Machines by Parallelizable Sampler and Efficient Temperature Estimation

Kentaro Kubo, Hayato Goto

cs.LGquant-phstat.MLSubmitted Dec 2, 2025• Announced Dec 3, 2025
v1 · Dec 2, 2025v2 · Sep 15, 2026
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Abstract

Boltzmann machines (BMs) are powerful energy-based generative models, but their heavy training cost has largely confined practical use to Restricted BMs (RBMs) trained with an efficient learning method called contrastive divergence. More accurate learning typically requires Markov chain Monte Carlo (MCMC) Boltzmann sampling, but it is time-consuming due to the difficulty of parallelization for more expressive models. To address this limitation, we first propose a new Boltzmann sampler inspired by a quantum-inspired combinatorial optimization called simulated bifurcation (SB). This SB-inspired approach, which we name Langevin SB (LSB), enables parallelized sampling while maintaining accuracy comparable to MCMC. Furthermore, this is applicable not only to RBMs but also to BMs with general couplings. However, LSB cannot control the inverse temperature of the output Boltzmann distribution, which hinders learning and degrades performance. To overcome this limitation, we also developed an efficient method for estimating the inverse temperature during the learning process, which we call conditional expectation matching (CEM). By combining LSB and CEM, we establish an efficient learning framework for BMs with greater expressive power than RBMs. We refer to this framework as sampler-adaptive learning (SAL). SAL opens new avenues for energy-based generative modeling beyond RBMs.

DOI
10.1103/2vb5-2lt8
Comments
16 pages, 14 figures

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Boltzmann machines (BMs) are powerful energy-based generative models, but their heavy training cost has largely confined practical use to Restricted BMs (RBMs) trained with an efficient learning method called contrastive divergence. More accurate learning typically requires Markov chain Monte Carlo (MCMC) Boltzmann sampling, but it is time-consuming due to the difficulty of parallelization for more expressive models. To address this limitation, we first propose a new Boltzmann sampler inspired by a quantum-inspired combinatorial optimization called simulated bifurcation (SB). This SB-inspired approach, which we name Langevin SB (LSB), enables parallelized sampling while maintaining accuracy comparable to MCMC. Furthermore, this is applicable not only to RBMs but also to BMs with general couplings. However, LSB cannot control the inverse temperature of the output Boltzmann distribution, which hinders learning and degrades performance. To overcome this limitation, we also developed an efficient method for estimating the inverse temperature during the learning process, which we call conditional expectation matching (CEM). By combining LSB and CEM, we establish an efficient learning framework for BMs with greater expressive power than RBMs. We refer to this framework as sampler-adaptive learning (SAL). SAL opens new avenues for energy-based generative modeling beyond RBMs.
Energy-based generative models, such as restricted Boltzmann machines (RBMs), require unbiased Boltzmann samples for effective training. Classical Markov chain Monte Carlo methods, however, converge slowly and yield correlated samples, making large-scale training difficult. We address this bottleneck by applying the analytic relation between annealing schedules and effective inverse temperature in diabatic quantum annealing. By implementing this prescription on a quantum annealer, we obtain temperature-controlled Boltzmann samples that enable RBM training with faster convergence and lower validation error than classical sampling. We further identify a systematic temperature misalignment intrinsic to analog quantum computers and propose an analytical rescaling method that mitigates this hardware noise, thereby enhancing the practicality of quantum annealers as Boltzmann samplers. In our method, the model's connectivity is set directly by the qubit connectivity, transforming the computational complexity inherent in classical sampling into a requirement on quantum hardware. This shift allows the approach to extend naturally from RBMs to fully connected Boltzmann machines, opening opportunities inaccessible to classical training methods.
An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact in deep learning and other machine learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggests it will do so with an instance-dependent effective temperature, different from the physical temperature of the device. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this talk, we present a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a kind of restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep learning architectures. We also provide a comparison to k-step contrastive divergence (CD-k) with k up to 100. Although assuming a suitable fixed effective temperature also allows to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature we find a performance close to that of CD-100 for the case studied here. We discuss generalizations of the algorithm to other more expressive generative models, beyond restricted Boltzmann machines.
An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact on deep learning and other machine-learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggest it will do so with an {\it instance-dependent} effective temperature, different from its physical temperature. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a special class of a restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep-learning architectures. We also provide a comparison to kkk-step contrastive divergence (CD-kkk) with kkk up to 100. Although assuming a suitable fixed effective temperature also allows us to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature do we find a performance close to that of CD-100 for the case studied here.
An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact on deep learning and other machine-learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggest it will do so with an instance-dependent effective temperature, different from its physical temperature. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a special class of a restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep-learning architectures. We also provide a comparison to kkk-step contrastive divergence (CD-kkk) with kkk up to 100. Although assuming a suitable fixed effective temperature also allows us to outperform one-step contrastive divergence (CD-1), only when using an instance-dependent effective temperature do we find a performance close to that of CD-100 for the case studied here.
Restricted Boltzmann Machine (RBM) is an energy-based, undirected graphical model. It is commonly used for unsupervised and supervised machine learning. Typically, RBM is trained using contrastive divergence (CD). However, training with CD is slow and does not estimate the exact gradient of the log-likelihood cost function. In this work, the model expectation of gradient learning for RBM has been calculated using a quantum annealer (D-Wave 2000Q), where obtaining samples is faster than Markov chain Monte Carlo (MCMC) used in CD. Training and classification results of RBM trained using quantum annealing are compared with the CD-based method. The performance of the two approaches is compared with respect to the classification accuracies, image reconstruction, and log-likelihood results. The classification accuracy results indicate comparable performances of the two methods. Image reconstruction and log-likelihood results show improved performance of the CD-based method. It is shown that the samples obtained from quantum annealer can be used to train an RBM on a 64-bit “bars and stripes” dataset with classification performance similar to an RBM trained with CD. Though training based on CD showed improved learning performance, training using a quantum annealer could be useful as it eliminates computationally expensive MCMC steps of CD.
Abstract Boltzmann machines, a class of machine learning models, are the basis of several deep learning methods that have been successfully applied to both supervised and unsupervised machine learning tasks. These models assume that some given dataset is generated according to a Boltzmann distribution, and the goal of the training procedure is to learn the set of parameters that most closely match the input data distribution. Training such models is difficult due to the intractability of traditional sampling techniques, and proposals using quantum annealers for sampling hope to mitigate the cost associated with sampling. However, real physical devices will inevitably be coupled to the environment, and the strength of this coupling affects the effective temperature of the distributions from which a quantum annealer samples. To counteract this problem, error correction schemes that can effectively reduce the temperature are needed if there is to be some benefit in using quantum annealing for problems at a larger scale, where we might expect the effective temperature of the device to be too high. To this end, we have applied nested quantum annealing correction (NQAC) to do unsupervised learning with a small bars and stripes dataset, and to do supervised learning with a coarse-grained MNIST dataset, which consists of black-and-white images of hand-written integers. For both datasets we demonstrate improved training and a concomitant effective temperature reduction at higher noise levels relative to the unencoded case. We also find better performance overall with longer anneal times and offer an interpretation of the results based on a comparison to simulated quantum annealing and spin vector Monte Carlo. A counterintuitive aspect of our results is that the output distribution generally becomes less Gibbs-like with increasing nesting level and increasing anneal times, which shows that improved training performance can be achieved without equilibration to the target Gibbs distribution.