# Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models

arXiv: https://arxiv.org/abs/1609.02542 · HTML: https://mlsanity.com/paper/1609.02542/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Sep 2016.

## Built on by (most cited first, 26 of 26)

- [Recent Advances for Quantum Neural Networks in Generative Learning](https://mlsanity.com/paper/2206.03066/) (2022)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [On the Sample Complexity of Quantum Boltzmann Machine Learning](https://mlsanity.com/paper/2306.14969/) (2023)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [High-quality Thermal Gibbs Sampling with Quantum Annealing Hardware](https://mlsanity.com/paper/2109.01690/) (2021)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Experimental quantum pattern recognition in IBMQ and diamond NVs](https://mlsanity.com/paper/2205.00561/) (2022)
- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)
- [Fermi-Dirac thermal measurements: A framework for quantum hypothesis testing and semidefinite optimization](https://mlsanity.com/paper/2603.04061/) (2026)
- [Assessment of image generation by quantum annealer](https://mlsanity.com/paper/2103.08373/) (2021)
- [Non-parametric Semi-Supervised Learning in Many-body Hilbert Space with Rescaled Logarithmic Fidelity](https://mlsanity.com/paper/2107.00195/) (2021)
- [Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems](https://mlsanity.com/paper/2508.09103/) (2025)
- [Training Multilayer Perceptrons by Sampling with Quantum Annealers](https://mlsanity.com/paper/2303.12352/) (2023)
- [Density Operator Expectation Maximization](https://mlsanity.com/paper/2507.22786/) (2025)
- [Comparison of D-Wave Quantum Annealing and Gibbs Monte Carlo for Sampling from a Probability Distribution of a Restricted Boltzmann Machine](https://mlsanity.com/paper/2508.10228/) (2025)
- [Quantum Reinforcement Learning in Non-Abelian Environments: Unveiling Novel Formulations and Quantum Advantage Exploration](https://mlsanity.com/paper/2406.06531/) (2024)
- [Robust Hybrid Classical-Quantum Transfer Learning Model for Text Classification Using GPT-Neo 125M with LoRA & SMOTE Enhancement](https://mlsanity.com/paper/2501.10435/) (2025)
- [Training Energy-Based Models with Non-MCMC Samplers and Efficient Temperature Estimation](https://mlsanity.com/paper/2512.02323/) (2025)
- [Generative modeling using evolved quantum Boltzmann machines](https://mlsanity.com/paper/2512.02721/) (2025)
- [Molecular Design beyond Training Data with Novel Extended Objective Functionals of Generative AI Models Driven by Quantum Annealing Computer](https://mlsanity.com/paper/2602.15451/) (2026)
- [Quantum Maximum Likelihood Prediction via Hilbert Space Embeddings](https://mlsanity.com/paper/2602.18364/) (2026)
- [Layer-wise QUBO-Based Training of CNN Classifiers for Quantum Annealing](https://mlsanity.com/paper/2603.02958/) (2026)
- [Toward Generative Quantum Utility via Correlation-Complexity Map](https://mlsanity.com/paper/2603.06440/) (2026)
- [Canonical quantization of neurons](https://mlsanity.com/paper/2607.05000/) (2026)
- [Unifying quantum measurement constructions via a relative-entropy minimum change principle](https://mlsanity.com/paper/2608.04055/) (2026)

## Abstract (arXiv)

Mainstream machine-learning techniques such as deep learning and probabilistic programming rely heavily on sampling from generally intractable probability distributions. There is increasing interest in the potential advantages of using quantum computing technologies as sampling engines to speed up these tasks or to make them more effective. However, some pressing challenges in state-of-the-art quantum annealers have to be overcome before we can assess their actual performance. The sparse connectivity, resulting from the local interaction between quantum bits in physical hardware implementations, is considered the most severe limitation to the quality of constructing powerful generative unsupervised machine-learning models. Here we use embedding techniques to add redundancy to data sets, allowing us to increase the modeling capacity of quantum annealers. We illustrate our findings by training hardware-embedded graphical models on a binarized data set of handwritten digits and two synthetic data sets in experiments with up to 940 quantum bits. Our model can be trained in quantum hardware without full knowledge of the effective parameters specifying the corresponding quantum Gibbs-like distribution; therefore, this approach avoids the need to infer the effective temperature at each iteration, speeding up learning; it also mitigates the effect of noise in the control parameters, making it robust to deviations from the reference Gibbs distribution. Our approach demonstrates the feasibility of using quantum annealers for implementing generative models, and it provides a suitable framework for benchmarking these quantum technologies on machine-learning-related tasks.