# Expressive equivalence of classical and quantum restricted Boltzmann machines

arXiv: https://arxiv.org/abs/2502.17562 · HTML: https://mlsanity.com/paper/2502.17562/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 37 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Builds on (most cited first, 11 of 11)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [On the Convergence Proof of AMSGrad and a New Version](https://mlsanity.com/paper/1904.03590/) (2019)
- [On the Sample Complexity of Quantum Boltzmann Machine Learning](https://mlsanity.com/paper/2306.14969/) (2023)
- [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)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Expressive Power and Approximation Errors of Restricted Boltzmann Machines](https://mlsanity.com/paper/1406.3140/) (2014)
- [On Quantum Circuits for Discrete Graphical Models](https://mlsanity.com/paper/2206.00398/) (2022)

## Built on by (most cited first, 7 of 7)

- [Exponential capacity scaling of classical GANs compared to hybrid latent style-based quantum GANs](https://mlsanity.com/paper/2601.05036/) (2026)
- [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)
- [Quantum Boltzmann Machines for Sample-Efficient Reinforcement Learning](https://mlsanity.com/paper/2511.04856/) (2025)
- [Breaking QAOA's Fixed Target Hamiltonian Barrier: A Fully Connected Quantum Boltzmann Machine via Bilevel Optimization](https://mlsanity.com/paper/2605.07473/) (2026)
- [Variational Quantum Conditional Boltzmann Machines for Time-Series Forecasting: Architectures, Symmetric Hyperparameter Evaluation, and a Nonlinear Benchmark](https://mlsanity.com/paper/2607.24065/) (2026)
- [Verifiable Sampling for Delegated Generative Quantum Machine Learning](https://mlsanity.com/paper/2610.13211/) (2026)

## Abstract (arXiv)

Quantum computers offer the potential for efficiently sampling from complex probability distributions, attracting increasing interest in generative modeling within quantum machine learning. This surge in interest has driven the development of numerous generative quantum models, yet their trainability and scalability remain significant challenges. A notable example is a quantum restricted Boltzmann machine (QRBM), which is based on the Gibbs state of a parameterized non-commuting Hamiltonian. While QRBMs are expressive, their non-commuting Hamiltonians make gradient evaluation computationally demanding, even on fault-tolerant quantum computers. In this work, we propose a semi-quantum restricted Boltzmann machine (sqRBM), a model designed for classical data that mitigates the challenges associated with previous QRBM proposals. The sqRBM Hamiltonian is commuting in the visible subspace while remaining non-commuting in the hidden subspace. This structure allows us to derive closed-form expressions for both output probabilities and gradients. Leveraging these analytical results, we demonstrate that sqRBMs share a close relationship with classical restricted Boltzmann machines (RBM). Our theoretical analysis predicts that, to learn a given probability distribution, an RBM requires three times as many hidden units as an sqRBM, while both models have the same total number of parameters. We validate these findings through numerical simulations involving up to 100 units. Our results suggest that sqRBMs could enable practical quantum machine learning applications in the near future by significantly reducing quantum resource requirements.