# Provable learning of quantum states with graphical models

arXiv: https://arxiv.org/abs/2309.09235 · HTML: https://mlsanity.com/paper/2309.09235/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 47 references that are in the map.
- First posted on arXiv 17 Sep 2023.

## Builds on (most cited first, 12 of 12)

- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Efficient quantum tomography II](https://mlsanity.com/paper/1612.00034/) (2016)
- [Efficiently learning Ising models on arbitrary graphs](https://mlsanity.com/paper/1411.6156/) (2014)
- [Quantum Entanglement in Deep Learning Architectures](https://mlsanity.com/paper/1803.09780/) (2018)
- [Learning Graphical Models Using Multiplicative Weights](https://mlsanity.com/paper/1706.06274/) (2017)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [Optimal Quantum Sample Complexity of Learning Algorithms](https://mlsanity.com/paper/1607.00932/) (2016)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Foundations for learning from noisy quantum experiments](https://mlsanity.com/paper/2204.13691/) (2022)
- [Learning Restricted Boltzmann Machines via Influence Maximization](https://mlsanity.com/paper/1805.10262/) (2018)
- [Learning Restricted Boltzmann Machines with Arbitrary External Fields](https://mlsanity.com/paper/1906.06595/) (2019)

## Built on by (most cited first, 3 of 3)

- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [Concept learning of parameterized quantum models from limited measurements](https://mlsanity.com/paper/2408.05116/) (2024)
- [Online Learning of Pure States is as Hard as Mixed States](https://mlsanity.com/paper/2502.00823/) (2025)

## Abstract (arXiv)

The complete learning of an n-qubit quantum state requires samples exponentially in n. Several works consider subclasses of quantum states that can be learned in polynomial sample complexity such as stabilizer states or high-temperature Gibbs states. Other works consider a weaker sense of learning, such as PAC learning and shadow tomography. In this work, we consider learning states that are close to neural network quantum states, which can efficiently be represented by a graphical model called restricted Boltzmann machines (RBMs). To this end, we exhibit robustness results for efficient provable two-hop neighborhood learning algorithms for ferromagnetic and locally consistent RBMs. We consider the Lp-norm as a measure of closeness, including both total variation distance and max-norm distance in the limit. Our results allow certain quantum states to be learned with a sample complexity textitexponentially better than naive tomography. We hence provide new classes of efficiently learnable quantum states and apply new strategies to learn them.