# On the Sample Complexity of Quantum Boltzmann Machine Learning

arXiv: https://arxiv.org/abs/2306.14969 · HTML: https://mlsanity.com/paper/2306.14969/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 89 references that are in the map.
- First posted on arXiv 26 Jun 2023.

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

- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Quantum machine learning: a classical perspective](https://mlsanity.com/paper/1707.08561/) (2017)
- [Bias and Generalization in Deep Generative Models: An Empirical Study](https://mlsanity.com/paper/1811.03259/) (2018)
- [Quantum Variational Autoencoder](https://mlsanity.com/paper/1802.05779/) (2018)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [The Complexity of Making the Gradient Small in Stochastic Convex Optimization](https://mlsanity.com/paper/1902.04686/) (2019)
- [Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models](https://mlsanity.com/paper/1609.02542/) (2016)
- [Reinforcement Learning Using Quantum Boltzmann Machines](https://mlsanity.com/paper/1612.05695/) (2016)
- [Generative training of quantum Boltzmann machines with hidden units](https://mlsanity.com/paper/1905.09902/) (2019)
- [Fast and robust quantum state tomography from few basis measurements](https://mlsanity.com/paper/2009.08216/) (2020)
- [Searching for collective behavior in a network of real neurons](https://mlsanity.com/paper/1306.3061/) (2013)
- [Quantum machine learning and quantum biomimetics: A perspective](https://mlsanity.com/paper/2004.12076/) (2020)
- [Optimal learning of quantum Hamiltonians from high-temperature Gibbs states](https://mlsanity.com/paper/2108.04842/) (2021)
- [Generalization Metrics for Practical Quantum Advantage in Generative Models](https://mlsanity.com/paper/2201.08770/) (2022)
- [Quantum-assisted associative adversarial network: Applying quantum annealing in deep learning](https://mlsanity.com/paper/1904.10573/) (2019)
- [Power of Quantum Generative Learning](https://mlsanity.com/paper/2205.04730/) (2022)
- [Toward a Generalization Metric for Deep Generative Models](https://mlsanity.com/paper/2011.00754/) (2020)

## Built on by (most cited first, 19 of 19)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [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)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms](https://mlsanity.com/paper/2405.18489/) (2024)
- [Expressive equivalence of classical and quantum restricted Boltzmann machines](https://mlsanity.com/paper/2502.17562/) (2025)
- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)
- [Overcoming Dimensional Factorization Limits in Discrete Diffusion Models through Quantum Joint Distribution Learning](https://mlsanity.com/paper/2505.05151/) (2025)
- [Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems](https://mlsanity.com/paper/2508.09103/) (2025)
- [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)
- [Generative modeling using evolved quantum Boltzmann machines](https://mlsanity.com/paper/2512.02721/) (2025)
- [Trainability Beyond Linearity in Variational Quantum Objectives](https://mlsanity.com/paper/2604.18846/) (2026)
- [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)
- [Stacking the Deck: Tunable Trainability in Stacked LCUs](https://mlsanity.com/paper/2607.24686/) (2026)
- [Verifiable Sampling for Delegated Generative Quantum Machine Learning](https://mlsanity.com/paper/2610.13211/) (2026)

## Abstract (arXiv)

Quantum Boltzmann machines (QBMs) are machine-learning models for both classical and quantum data. We give an operational definition of QBM learning in terms of the difference in expectation values between the model and target, taking into account the polynomial size of the data set. By using the relative entropy as a loss function this problem can be solved without encountering barren plateaus. We prove that a solution can be obtained with stochastic gradient descent using at most a polynomial number of Gibbs states. We also prove that pre-training on a subset of the QBM parameters can only lower the sample complexity bounds. In particular, we give pre-training strategies based on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. We verify these models and our theoretical findings numerically on a quantum and a classical data set. Our results establish that QBMs are promising machine learning models.