# Trainability barriers and opportunities in quantum generative modeling

arXiv: https://arxiv.org/abs/2305.02881 · HTML: https://mlsanity.com/paper/2305.02881/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 123 references that are in the map.
- First posted on arXiv 4 May 2023.

## Builds on (most cited first, 30 of 47)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Absence of Barren Plateaus in Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2011.02966/) (2020)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [Exploiting symmetry in variational quantum machine learning](https://mlsanity.com/paper/2205.06217/) (2022)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [On the Quantum versus Classical Learnability of Discrete Distributions](https://mlsanity.com/paper/2007.14451/) (2020)
- [Trainability of Dissipative Perceptron-Based Quantum Neural Networks](https://mlsanity.com/paper/2005.12458/) (2020)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Equivalence of quantum barren plateaus to cost concentration and narrow gorges](https://mlsanity.com/paper/2104.05868/) (2021)
- [Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.09974/) (2022)
- [Out-of-distribution generalization for learning quantum dynamics](https://mlsanity.com/paper/2204.10268/) (2022)
- [Enhancing Generative Models via Quantum Correlations](https://mlsanity.com/paper/2101.08354/) (2021)

## Built on by (most cited first, 30 of 36)

- [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)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [Quantum-Noise-Driven Generative Diffusion Models](https://mlsanity.com/paper/2308.12013/) (2023)
- [On the Sample Complexity of Quantum Boltzmann Machine Learning](https://mlsanity.com/paper/2306.14969/) (2023)
- [On fundamental aspects of quantum extreme learning machines](https://mlsanity.com/paper/2312.15124/) (2023)
- [Variational quantum simulation: a case study for understanding warm starts](https://mlsanity.com/paper/2404.10044/) (2024)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Generative quantum advantage for classical and quantum problems](https://mlsanity.com/paper/2509.09033/) (2025)
- [Symmetry-invariant quantum machine learning force fields](https://mlsanity.com/paper/2311.11362/) (2023)
- [Variational measurement-based quantum computation for generative modeling](https://mlsanity.com/paper/2310.13524/) (2023)
- [Expressive equivalence of classical and quantum restricted Boltzmann machines](https://mlsanity.com/paper/2502.17562/) (2025)
- [Hierarchical Learning for Quantum ML: Novel Training Technique for Large-Scale Variational Quantum Circuits](https://mlsanity.com/paper/2311.12929/) (2023)
- [Arbitrary Polynomial Separations in Trainable Quantum Machine Learning](https://mlsanity.com/paper/2402.08606/) (2024)
- [Trainability issues in quantum policy gradients](https://mlsanity.com/paper/2406.09614/) (2024)
- [Quantum-Cognitive Neural Networks: Assessing Confidence and Uncertainty with Human Decision-Making Simulations](https://mlsanity.com/paper/2412.08010/) (2024)
- [Spectral methods: crucial for machine learning, natural for quantum computers?](https://mlsanity.com/paper/2603.24654/) (2026)
- [Overcoming Dimensional Factorization Limits in Discrete Diffusion Models through Quantum Joint Distribution Learning](https://mlsanity.com/paper/2505.05151/) (2025)
- [LLM-Guided Ans"atze Design for Quantum Circuit Born Machines in Financial Generative Modeling](https://mlsanity.com/paper/2509.08385/) (2025)
- [ConQuER: Modular Architectures for Control and Bias Mitigation in IQP Quantum Generative Models](https://mlsanity.com/paper/2509.22551/) (2025)
- [Accelerating Inference for Multilayer Neural Networks with Quantum Computers](https://mlsanity.com/paper/2510.07195/) (2025)
- [Limits of quantum generative models with classical sampling hardness](https://mlsanity.com/paper/2512.24801/) (2025)
- [Characterizing Trainability of Instantaneous Quantum Polynomial Circuit Born Machines](https://mlsanity.com/paper/2602.11042/) (2026)
- [Scaling Quantum Machine Learning without Tricks: Full-Resolution and Diverse Image Generation](https://mlsanity.com/paper/2603.00233/) (2026)
- [Probabilistic modeling over permutations using quantum computers](https://mlsanity.com/paper/2603.22401/) (2026)
- [Hybrid Classical-Quantum architecture for vectorised image classification of hand-written sketches](https://mlsanity.com/paper/2407.06416/) (2024)
- [Variational Inference for Quantum HyperNetworks](https://mlsanity.com/paper/2506.05888/) (2025)
- [Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms](https://mlsanity.com/paper/2506.13865/) (2025)
- [Quantum-Enhanced Generative Models for Rare Event Prediction](https://mlsanity.com/paper/2511.02042/) (2025)

## Abstract (arXiv)

Quantum generative models provide inherently efficient sampling strategies and thus show promise for achieving an advantage using quantum hardware. In this work, we investigate the barriers to the trainability of quantum generative models posed by barren plateaus and exponential loss concentration. We explore the interplay between explicit and implicit models and losses, and show that using quantum generative models with explicit losses such as the KL divergence leads to a new flavour of barren plateaus. In contrast, the implicit Maximum Mean Discrepancy loss can be viewed as the expectation value of an observable that is either low-bodied and provably trainable, or global and untrainable depending on the choice of kernel. In parallel, we find that solely low-bodied implicit losses cannot in general distinguish high-order correlations in the target data, while some quantum loss estimation strategies can. We validate our findings by comparing different loss functions for modelling data from High-Energy-Physics.