# From Tensor Network Quantum States to Tensorial Recurrent Neural Networks

arXiv: https://arxiv.org/abs/2206.12363 · HTML: https://mlsanity.com/paper/2206.12363/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 20 references that are in the map.
- First posted on arXiv 24 Jun 2022.

## Builds on (most cited first, 1 of 1)

- [Variational Neural Annealing](https://mlsanity.com/paper/2101.10154/) (2021)

## Built on by (most cited first, 14 of 14)

- [Tensor Networks Meet Neural Networks: A Survey and Future Perspectives](https://mlsanity.com/paper/2302.09019/) (2023)
- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Tensor networks for interpretable and efficient quantum-inspired machine learning](https://mlsanity.com/paper/2311.11258/) (2023)
- [L2M: Mutual Information Scaling Law for Long-Context Language Modeling](https://mlsanity.com/paper/2503.04725/) (2025)
- [ANTN: Bridging Autoregressive Neural Networks and Tensor Networks for Quantum Many-Body Simulation](https://mlsanity.com/paper/2304.01996/) (2023)
- [Certifying almost all quantum states with few single-qubit measurements](https://mlsanity.com/paper/2404.07281/) (2024)
- [Investigating Topological Order using Recurrent Neural Networks](https://mlsanity.com/paper/2303.11207/) (2023)
- [When can classical neural networks represent quantum states?](https://mlsanity.com/paper/2410.23152/) (2024)
- [Parallel Scan Recurrent Neural Quantum States for Scalable Variational Monte Carlo](https://mlsanity.com/paper/2605.13807/) (2026)
- [Adaptive Neural Quantum States: A Recurrent Neural Network Perspective](https://mlsanity.com/paper/2507.18700/) (2025)
- [Simulating Non-Markovian Open Quantum Dynamics with Neural Quantum States](https://mlsanity.com/paper/2404.11093/) (2024)
- [Seeding neural network quantum states with tensor network states](https://mlsanity.com/paper/2506.23550/) (2025)
- [Gaussian Universality and Its Breakdown in Tensor-Network Machine Learning](https://mlsanity.com/paper/2610.06080/) (2026)

## Abstract (arXiv)

We show that any matrix product state (MPS) can be exactly represented by a recurrent neural network (RNN) with a linear memory update. We generalize this RNN architecture to 2D lattices using a multilinear memory update. It supports perfect sampling and wave function evaluation in polynomial time, and can represent an area law of entanglement entropy. Numerical evidence shows that it can encode the wave function using a bond dimension lower by orders of magnitude when compared to MPS, with an accuracy that can be systematically improved by increasing the bond dimension.