# Learning Energy-Based Representations of Quantum Many-Body States

arXiv: https://arxiv.org/abs/2304.04058 · HTML: https://mlsanity.com/paper/2304.04058/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 63 references that are in the map.
- First posted on arXiv 8 Apr 2023.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [Searching for Activation Functions](https://mlsanity.com/paper/1710.05941/) (2017)
- [Maximum Entropy Generators for Energy-Based Models](https://mlsanity.com/paper/1901.08508/) (2019)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models](https://mlsanity.com/paper/1605.07252/) (2016)
- [Deep Directed Generative Models with Energy-Based Probability Estimation](https://mlsanity.com/paper/1606.03439/) (2016)
- [Optimal structure and parameter learning of Ising models](https://mlsanity.com/paper/1612.05024/) (2016)
- [Efficient Learning of Discrete Graphical Models](https://mlsanity.com/paper/1902.00600/) (2019)
- [Stabiliser states are efficiently PAC-learnable](https://mlsanity.com/paper/1705.00345/) (2017)
- [Learning Ising Models with Independent Failures](https://mlsanity.com/paper/1902.04728/) (2019)
- [Optimal learning of quantum Hamiltonians from high-temperature Gibbs states](https://mlsanity.com/paper/2108.04842/) (2021)
- [Learning of Discrete Graphical Models with Neural Networks](https://mlsanity.com/paper/2006.11937/) (2020)
- [Positive-definite parametrization of mixed quantum states with deep neural networks](https://mlsanity.com/paper/2206.13488/) (2022)

## Built on by (most cited first, 2 of 2)

- [Quantum State Generation with Structure-Preserving Diffusion Model](https://mlsanity.com/paper/2404.06336/) (2024)
- [Discrete Diffusion with Sample-Efficient Estimators for Conditionals](https://mlsanity.com/paper/2602.20293/) (2026)

## Abstract (arXiv)

Efficient representation of quantum many-body states on classical computers is a problem of enormous practical interest. An ideal representation of a quantum state combines a succinct characterization informed by the system's structure and symmetries, along with the ability to predict the physical observables of interest. A number of machine learning approaches have been recently used to construct such classical representations [1-6] which enable predictions of observables [7] and account for physical symmetries [8]. However, the structure of a quantum state gets typically lost unless a specialized ansatz is employed based on prior knowledge of the system [9-12]. Moreover, most such approaches give no information about what states are easier to learn in comparison to others. Here, we propose a new generative energy-based representation of quantum many-body states derived from Gibbs distributions used for modeling the thermal states of classical spin systems. Based on the prior information on a family of quantum states, the energy function can be specified by a small number of parameters using an explicit low-degree polynomial or a generic parametric family such as neural nets, and can naturally include the known symmetries of the system. Our results show that such a representation can be efficiently learned from data using exact algorithms in a form that enables the prediction of expectation values of physical observables. Importantly, the structure of the learned energy function provides a natural explanation for the hardness of learning for a given class of quantum states.