# Learning hard quantum distributions with variational autoencoders

arXiv: https://arxiv.org/abs/1710.00725 · HTML: https://mlsanity.com/paper/1710.00725/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Oct 2017.

## Built on by (most cited first, 17 of 17)

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- [Explainable Representation Learning of Small Quantum States](https://mlsanity.com/paper/2306.05694/) (2023)
- [Predicting Properties of Quantum Systems with Conditional Generative Models](https://mlsanity.com/paper/2211.16943/) (2022)
- [Power of Quantum Generative Learning](https://mlsanity.com/paper/2205.04730/) (2022)
- [Quantum computing and artificial intelligence: status and perspectives](https://mlsanity.com/paper/2505.23860/) (2025)
- [Deep Quantum Graph Dreaming: Deciphering Neural Network Insights into Quantum Experiments](https://mlsanity.com/paper/2309.07056/) (2023)
- [Convergence Analysis of Stochastic Gradient Descent with MCMC Estimators](https://mlsanity.com/paper/2303.10599/) (2023)
- [Reinforcement learning to learn quantum states for Heisenberg scaling accuracy](https://mlsanity.com/paper/2412.02334/) (2024)
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- [Regression of high dimensional angular momentum states of light](https://mlsanity.com/paper/2206.09873/) (2022)
- [Neural network quantum state with proximal optimization: a ground-state searching scheme based on variational Monte Carlo](https://mlsanity.com/paper/2210.16493/) (2022)
- [Variational Optimization for Quantum Problems using Deep Generative Networks](https://mlsanity.com/paper/2404.18041/) (2024)
- [Quantum Down Sampling Filter for Variational Auto-encoder](https://mlsanity.com/paper/2501.06259/) (2025)
- [Quantum Walks-Based Adaptive Distribution Generation with Efficient CUDA-Q Acceleration](https://mlsanity.com/paper/2504.13532/) (2025)
- [Learning Minimal Representations of Fermionic Ground States](https://mlsanity.com/paper/2512.11767/) (2025)
- [From Classical to Quantum: Extending Prometheus for Unsupervised Discovery of Phase Transitions in Three Dimensions and Quantum Systems](https://mlsanity.com/paper/2602.14928/) (2026)
- [Generalization of Transformer-Based Neural Quantum States via In-Context Learning](https://mlsanity.com/paper/2610.03463/) (2026)

## Abstract (arXiv)

Studying general quantum many-body systems is one of the major challenges in modern physics because it requires an amount of computational resources that scales exponentially with the size of the system.Simulating the evolution of a state, or even storing its description, rapidly becomes intractable for exact classical algorithms. Recently, machine learning techniques, in the form of restricted Boltzmann machines, have been proposed as a way to efficiently represent certain quantum states with applications in state tomography and ground state estimation. Here, we introduce a new representation of states based on variational autoencoders. Variational autoencoders are a type of generative model in the form of a neural network. We probe the power of this representation by encoding probability distributions associated with states from different classes. Our simulations show that deep networks give a better representation for states that are hard to sample from, while providing no benefit for random states. This suggests that the probability distributions associated to hard quantum states might have a compositional structure that can be exploited by layered neural networks. Specifically, we consider the learnability of a class of quantum states introduced by Fefferman and Umans. Such states are provably hard to sample for classical computers, but not for quantum ones, under plausible computational complexity assumptions. The good level of compression achieved for hard states suggests these methods can be suitable for characterising states of the size expected in first generation quantum hardware.