# Generative training of quantum Boltzmann machines with hidden units

arXiv: https://arxiv.org/abs/1905.09902 · HTML: https://mlsanity.com/paper/1905.09902/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2019.

## Built on by (most cited first, 14 of 14)

- [Recent Advances for Quantum Neural Networks in Generative Learning](https://mlsanity.com/paper/2206.03066/) (2022)
- [On the Sample Complexity of Quantum Boltzmann Machine Learning](https://mlsanity.com/paper/2306.14969/) (2023)
- [Near-Optimal Quantum Algorithms for Multivariate Mean Estimation](https://mlsanity.com/paper/2111.09787/) (2021)
- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [Expressive equivalence of classical and quantum restricted Boltzmann machines](https://mlsanity.com/paper/2502.17562/) (2025)
- [On Quantum Circuits for Discrete Graphical Models](https://mlsanity.com/paper/2206.00398/) (2022)
- [Towards Transfer Learning for Large-Scale Image Classification Using Annealing-based Quantum Boltzmann Machines](https://mlsanity.com/paper/2311.15966/) (2023)
- [Meta-learning of Gibbs states for many-body Hamiltonians with applications to Quantum Boltzmann Machines](https://mlsanity.com/paper/2507.16373/) (2025)
- [Quantum Machine Learning For Classical Data](https://mlsanity.com/paper/2105.03684/) (2021)
- [Structured quantum learning via em algorithm for Boltzmann machines](https://mlsanity.com/paper/2507.21569/) (2025)
- [Density Operator Expectation Maximization](https://mlsanity.com/paper/2507.22786/) (2025)
- [Generative modeling using evolved quantum Boltzmann machines](https://mlsanity.com/paper/2512.02721/) (2025)
- [Breaking QAOA's Fixed Target Hamiltonian Barrier: A Fully Connected Quantum Boltzmann Machine via Bilevel Optimization](https://mlsanity.com/paper/2605.07473/) (2026)
- [Quantum score matching with applications to learning thermal states](https://mlsanity.com/paper/2609.28391/) (2026)

## Abstract (arXiv)

In this article we provide a method for fully quantum generative training of quantum Boltzmann machines with both visible and hidden units while using quantum relative entropy as an objective. This is significant because prior methods were not able to do so due to mathematical challenges posed by the gradient evaluation. We present two novel methods for solving this problem. The first proposal addresses it, for a class of restricted quantum Boltzmann machines with mutually commuting Hamiltonians on the hidden units, by using a variational upper bound on the quantum relative entropy. The second one uses high-order divided difference methods and linear-combinations of unitaries to approximate the exact gradient of the relative entropy for a generic quantum Boltzmann machine. Both methods are efficient under the assumption that Gibbs state preparation is efficient and that the Hamiltonian are given by a sparse row-computable matrix.