# Probabilistic Modeling with Matrix Product States

arXiv: https://arxiv.org/abs/1902.06888 · HTML: https://mlsanity.com/paper/1902.06888/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Feb 2019.

## Built on by (most cited first, 11 of 11)

- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [Generative modeling with projected entangled-pair states](https://mlsanity.com/paper/2202.08177/) (2022)
- [Generalization Metrics for Practical Quantum Advantage in Generative Models](https://mlsanity.com/paper/2201.08770/) (2022)
- [Explainable Natural Language Processing with Matrix Product States](https://mlsanity.com/paper/2112.08628/) (2021)
- [Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework](https://mlsanity.com/paper/2106.15666/) (2021)
- [Differentiable Programming of Isometric Tensor Networks](https://mlsanity.com/paper/2110.03898/) (2021)
- [Tensor Train for Global Optimization Problems in Robotics](https://mlsanity.com/paper/2206.05077/) (2022)
- [Deep tensor networks with matrix product operators](https://mlsanity.com/paper/2209.09098/) (2022)
- [Entangling Machine Learning with Quantum Tensor Networks](https://mlsanity.com/paper/2403.12969/) (2024)
- [Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization](https://mlsanity.com/paper/2603.12026/) (2026)
- [On the Approximation Complexity of Matrix Product Operator (MPO) Born Machines](https://mlsanity.com/paper/2605.11471/) (2026)

## Abstract (arXiv)

Inspired by the possibility that generative models based on quantum circuits can provide a useful inductive bias for sequence modeling tasks, we propose an efficient training algorithm for a subset of classically simulable quantum circuit models. The gradient-free algorithm, presented as a sequence of exactly solvable effective models, is a modification of the density matrix renormalization group procedure adapted for learning a probability distribution. The conclusion that circuit-based models offer a useful inductive bias for classical datasets is supported by experimental results on the parity learning problem.