# Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure

arXiv: https://arxiv.org/abs/1710.04833 · HTML: https://mlsanity.com/paper/1710.04833/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2017.

## Built on by (most cited first, 30 of 40)

- [Variational Quantum Reinforcement Learning via Evolutionary Optimization](https://mlsanity.com/paper/2109.00540/) (2021)
- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [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)
- [Machine learning with tree tensor networks, CP rank constraints, and tensor dropout](https://mlsanity.com/paper/2305.19440/) (2023)
- [Privacy-preserving machine learning with tensor networks](https://mlsanity.com/paper/2202.12319/) (2022)
- [Quantum-Inspired Tensor Neural Networks for Partial Differential Equations](https://mlsanity.com/paper/2208.02235/) (2022)
- [More Efficient Sampling for Tensor Decomposition With Worst-Case Guarantees](https://mlsanity.com/paper/2110.07631/) (2021)
- [Explainable Natural Language Processing with Matrix Product States](https://mlsanity.com/paper/2112.08628/) (2021)
- [Classical versus Quantum: comparing Tensor Network-based Quantum Circuits on LHC data](https://mlsanity.com/paper/2202.10471/) (2022)
- [Unsupervised Recognition of Informative Features via Tensor Network Machine Learning and Quantum Entanglement Variations](https://mlsanity.com/paper/2207.06031/) (2022)
- [TensorKrowch: Smooth integration of tensor networks in machine learning](https://mlsanity.com/paper/2306.08595/) (2023)
- [Deep Tree Tensor Networks](https://mlsanity.com/paper/2502.09928/) (2025)
- [Entangled q-Convolutional Neural Nets](https://mlsanity.com/paper/2103.11785/) (2021)
- [Automatically Differentiable Quantum Circuit for Many-qubit State Preparation](https://mlsanity.com/paper/2104.14949/) (2021)
- [Interaction Decompositions for Tensor Network Regression](https://mlsanity.com/paper/2208.06029/) (2022)
- [Tensorization is a powerful but underexplored tool for compression and interpretability of neural networks](https://mlsanity.com/paper/2505.20132/) (2025)
- [Tensor-Networks-based Learning of Probabilistic Cellular Automata Dynamics](https://mlsanity.com/paper/2404.11768/) (2024)
- [Tensorization of neural networks for improved privacy and interpretability](https://mlsanity.com/paper/2501.06300/) (2025)
- [Riemannian Optimization on Tree Tensor Networks with Application in Machine Learning](https://mlsanity.com/paper/2507.21726/) (2025)
- [Non-parametric Semi-Supervised Learning in Many-body Hilbert Space with Rescaled Logarithmic Fidelity](https://mlsanity.com/paper/2107.00195/) (2021)
- [Deep tensor networks with matrix product operators](https://mlsanity.com/paper/2209.09098/) (2022)
- [Quantum-Inspired Tensor Neural Networks for Option Pricing](https://mlsanity.com/paper/2212.14076/) (2022)
- [T-ADAF: Adaptive Data Augmentation Framework for Image Classification Network based on Tensor T-product Operator](https://mlsanity.com/paper/2306.04240/) (2023)
- [Cons-training Tensor Networks: Embedding and Optimization Over Discrete Linear Constraints](https://mlsanity.com/paper/2405.09005/) (2024)
- [Universal scaling laws in quantum-probabilistic machine learning by tensor network towards interpreting representation and generalization powers](https://mlsanity.com/paper/2410.09703/) (2024)
- [Stack operation of tensor networks](https://mlsanity.com/paper/2203.16338/) (2022)
- [Intelligent diagnostic scheme for lung cancer screening with Raman spectra data by tensor network machine learning](https://mlsanity.com/paper/2303.06340/) (2023)
- [Application of Tensor Neural Networks to Pricing Bermudan Swaptions](https://mlsanity.com/paper/2304.09750/) (2023)

## Abstract (arXiv)

The resemblance between the methods used in quantum-many body physics and in machine learning has drawn considerable attention. In particular, tensor networks (TNs) and deep learning architectures bear striking similarities to the extent that TNs can be used for machine learning. Previous results used one-dimensional TNs in image recognition, showing limited scalability and flexibilities. In this work, we train two-dimensional hierarchical TNs to solve image recognition problems, using a training algorithm derived from the multi-scale entanglement renormalization ansatz. This approach introduces mathematical connections among quantum many-body physics, quantum information theory, and machine learning. While keeping the TN unitary in the training phase, TN states are defined, which encode classes of images into quantum many-body states. We study the quantum features of the TN states, including quantum entanglement and fidelity. We find these quantities could be properties that characterize the image classes, as well as the machine learning tasks.