mlsanityData to 14 Oct 2026

On the Sample Complexity of Quantum Boltzmann Machine Learning

arXiv 2306.14969 ↗First posted 26 Jun 2023quant-ph cs.LGcited by 19

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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.

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

27 of 89 references.

References · 27

  1. 2021On the Opportunities and Risks of Foundation Modelscited by 4,033
  2. 2015Automatic differentiation in machine learning: a surveycited by 855
  3. 2018Barren plateaus in quantum neural network training landscapescited by 630
  4. 2019Parameterized quantum circuits as machine learning modelscited by 332
  5. 2020Predicting Many Properties of a Quantum System from Very Few Measurementscited by 233
  6. 2023Challenges and Opportunities in Quantum Machine Learningcited by 203
  7. 2023Handbook of Convergence Theorems for (Stochastic) Gradient Methodscited by 163
  8. 2020Better Theory for SGD in the Nonconvex Worldcited by 158
  9. 2020Entanglement Induced Barren Plateauscited by 101
  10. 2021Information-theoretic bounds on quantum advantage in machine learningcited by 96
  11. 2017Quantum machine learning: a classical perspectivecited by 95
  12. 2018Bias and Generalization in Deep Generative Models: An Empirical Studycited by 83
15 more
  1. 2018Quantum Variational Autoencodercited by 41
  2. 2020Sample-efficient learning of quantum many-body systemscited by 37
  3. 2023Trainability barriers and opportunities in quantum generative modelingcited by 36
  4. 2019The Complexity of Making the Gradient Small in Stochastic Convex Optimizationcited by 34
  5. 2016Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Modelscited by 26
  6. 2016Reinforcement Learning Using Quantum Boltzmann Machinescited by 20
  7. 2019Generative training of quantum Boltzmann machines with hidden unitscited by 14
  8. 2020Fast and robust quantum state tomography from few basis measurementscited by 10
  9. 2013Searching for collective behavior in a network of real neuronscited by 9
  10. 2020Quantum machine learning and quantum biomimetics: A perspectivecited by 8
  11. 2021Optimal learning of quantum Hamiltonians from high-temperature Gibbs statescited by 8
  12. 2022Generalization Metrics for Practical Quantum Advantage in Generative Modelscited by 8
  13. 2019Quantum-assisted associative adversarial network: Applying quantum annealing in deep learningcited by 7
  14. 2022Power of Quantum Generative Learningcited by 6
  15. 2020Toward a Generalization Metric for Deep Generative Modelscited by 5

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Citing papers per year: 2023: 1; 2024: 4; 2025: 9; 2026: 5.

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  1. 2024Barren Plateaus in Variational Quantum Computingcited by 133
  2. 2023Does provable absence of barren plateaus imply classical simulability?cited by 80
  3. 2024Quantum Boltzmann machine learning of ground-state energiescited by 12
  4. 2025A Primer on Quantum Machine Learningcited by 12
  5. 2024Natural gradient and parameter estimation for quantum Boltzmann machinescited by 9
  6. 2024Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithmscited by 7
  7. 2025Expressive equivalence of classical and quantum restricted Boltzmann machinescited by 7
  8. 2025Quantum thermodynamics and semi-definite optimizationcited by 5
  9. 2025Overcoming Dimensional Factorization Limits in Discrete Diffusion Models through Quantum Joint Distribution Learningcited by 2
  10. 2025Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systemscited by 2
  11. 2025Structured quantum learning via em algorithm for Boltzmann machinescited by 1
  12. 2025Density Operator Expectation Maximizationcited by 1
7 more
  1. 2025Quantum Boltzmann Machines for Sample-Efficient Reinforcement Learningcited by 0
  2. 2025Generative modeling using evolved quantum Boltzmann machinescited by 0
  3. 2026Trainability Beyond Linearity in Variational Quantum Objectivescited by 0
  4. 2026Breaking QAOA's Fixed Target Hamiltonian Barrier: A Fully Connected Quantum Boltzmann Machine via Bilevel Optimizationcited by 0
  5. 2026Variational Quantum Conditional Boltzmann Machines for Time-Series Forecasting: Architectures, Symmetric Hyperparameter Evaluation, and a Nonlinear Benchmarkcited by 0
  6. 2026Stacking the Deck: Tunable Trainability in Stacked LCUscited by 0
  7. 2026Verifiable Sampling for Delegated Generative Quantum Machine Learningcited by 0