# Quantum Tensor Networks, Stochastic Processes, and Weighted Automata

arXiv: https://arxiv.org/abs/2010.10653 · HTML: https://mlsanity.com/paper/2010.10653/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2020.

## Built on by (most cited first, 9 of 9)

- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework](https://mlsanity.com/paper/2106.15666/) (2021)
- [Learning finitely correlated states: stability of the spectral reconstruction](https://mlsanity.com/paper/2312.07516/) (2023)
- [Deep tensor networks with matrix product operators](https://mlsanity.com/paper/2209.09098/) (2022)
- [Lower and Upper Bounds on the VC-Dimension of Tensor Network Models](https://mlsanity.com/paper/2106.11827/) (2021)
- [Learning Circular Hidden Quantum Markov Models: A Tensor Network Approach](https://mlsanity.com/paper/2111.01536/) (2021)
- [Tensor Network Estimation of Distribution Algorithms](https://mlsanity.com/paper/2412.19780/) (2024)
- [Identifiability and minimality bounds of quantum and post-quantum models of classical stochastic processes](https://mlsanity.com/paper/2509.03004/) (2025)

## Abstract (arXiv)

Modeling joint probability distributions over sequences has been studied from many perspectives. The physics community developed matrix product states, a tensor-train decomposition for probabilistic modeling, motivated by the need to tractably model many-body systems. But similar models have also been studied in the stochastic processes and weighted automata literature, with little work on how these bodies of work relate to each other. We address this gap by showing how stationary or uniform versions of popular quantum tensor network models have equivalent representations in the stochastic processes and weighted automata literature, in the limit of infinitely long sequences. We demonstrate several equivalence results between models used in these three communities: (i) uniform variants of matrix product states, Born machines and locally purified states from the quantum tensor networks literature, (ii) predictive state representations, hidden Markov models, norm-observable operator models and hidden quantum Markov models from the stochastic process literature,and (iii) stochastic weighted automata, probabilistic automata and quadratic automata from the formal languages literature. Such connections may open the door for results and methods developed in one area to be applied in another.