# Generative modeling using evolved quantum Boltzmann machines

arXiv: https://arxiv.org/abs/2512.02721 · HTML: https://mlsanity.com/paper/2512.02721/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 53 references that are in the map.
- First posted on arXiv 2 Dec 2025.

## Builds on (most cited first, 27 of 27)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [On Gradient Descent Ascent for Nonconvex-Concave Minimax Problems](https://mlsanity.com/paper/1906.00331/) (2019)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [On the Quantum versus Classical Learnability of Discrete Distributions](https://mlsanity.com/paper/2007.14451/) (2020)
- [Enhancing Generative Models via Quantum Correlations](https://mlsanity.com/paper/2101.08354/) (2021)
- [Extragradient Method: O(1/K) Last-Iterate Convergence for Monotone Variational Inequalities and Connections With Cocoercivity](https://mlsanity.com/paper/2110.04261/) (2021)
- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [On Solving Minimax Optimization Locally: A Follow-the-Ridge Approach](https://mlsanity.com/paper/1910.07512/) (2019)
- [Towards provably efficient quantum algorithms for large-scale machine-learning models](https://mlsanity.com/paper/2303.03428/) (2023)
- [A comprehensive review of Quantum Machine Learning: from NISQ to Fault Tolerance](https://mlsanity.com/paper/2401.11351/) (2024)
- [Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models](https://mlsanity.com/paper/1609.02542/) (2016)
- [Neural Estimation of Statistical Divergences](https://mlsanity.com/paper/2110.03652/) (2021)
- [On the Sample Complexity of Quantum Boltzmann Machine Learning](https://mlsanity.com/paper/2306.14969/) (2023)
- [Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization](https://mlsanity.com/paper/2408.11974/) (2024)
- [Generative training of quantum Boltzmann machines with hidden units](https://mlsanity.com/paper/1905.09902/) (2019)
- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Information-theoretic generalization bounds for learning from quantum data](https://mlsanity.com/paper/2311.05529/) (2023)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Estimating distinguishability measures on quantum computers](https://mlsanity.com/paper/2108.08406/) (2021)
- [Performance Guarantees for Quantum Neural Estimation of Entropies](https://mlsanity.com/paper/2511.19289/) (2025)
- [Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems](https://mlsanity.com/paper/2508.09103/) (2025)
- [HessianFR: An Efficient Hessian-based Follow-the-Ridge Algorithm for Minimax Optimization](https://mlsanity.com/paper/2205.11030/) (2022)

## Abstract (arXiv)

Born-rule generative modeling, a central task in quantum machine learning, seeks to learn probability distributions that can be efficiently sampled by measuring complex quantum states. One hope is for quantum models to efficiently capture probability distributions that are difficult to learn and simulate by classical means alone. Quantum Boltzmann machines were proposed about one decade ago for this purpose, yet efficient training methods have remained elusive. In this paper, I overcome this obstacle by proposing a practical solution that trains quantum Boltzmann machines for Born-rule generative modeling. Two key ingredients in the proposal are the Donsker-Varadhan variational representation of the classical relative entropy and the quantum Boltzmann gradient estimator of [Patel et al., arXiv:2410.12935]. I present the main result for a more general ansatz known as an evolved quantum Boltzmann machine [Minervini et al., arXiv:2501.03367], which combines parameterized real- and imaginary-time evolution. I also show how to extend the findings to other distinguishability measures beyond relative entropy. Finally, I present four different hybrid quantum-classical algorithms for the minimax optimization underlying training, and I discuss their theoretical convergence guarantees.