# Supervised Learning with Projected Entangled Pair States

arXiv: https://arxiv.org/abs/2009.09932 · HTML: https://mlsanity.com/paper/2009.09932/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Sep 2020.

## Built on by (most cited first, 30 of 31)

- [Tensor Networks Meet Neural Networks: A Survey and Future Perspectives](https://mlsanity.com/paper/2302.09019/) (2023)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Tensor networks for interpretable and efficient quantum-inspired machine learning](https://mlsanity.com/paper/2311.11258/) (2023)
- [Generative modeling with projected entangled-pair states](https://mlsanity.com/paper/2202.08177/) (2022)
- [Machine learning with tree tensor networks, CP rank constraints, and tensor dropout](https://mlsanity.com/paper/2305.19440/) (2023)
- [Quantum-inspired event reconstruction with Tensor Networks: Matrix Product States](https://mlsanity.com/paper/2106.08334/) (2021)
- [Privacy-preserving machine learning with tensor networks](https://mlsanity.com/paper/2202.12319/) (2022)
- [Improvements to Gradient Descent Methods for Quantum Tensor Network Machine Learning](https://mlsanity.com/paper/2203.03366/) (2022)
- [Unsupervised Recognition of Informative Features via Tensor Network Machine Learning and Quantum Entanglement Variations](https://mlsanity.com/paper/2207.06031/) (2022)
- [Quantized Fourier and Polynomial Features for more Expressive Tensor Network Models](https://mlsanity.com/paper/2309.05436/) (2023)
- [A Matrix Product State Model for Simultaneous Classification and Generation](https://mlsanity.com/paper/2406.17441/) (2024)
- [Deep Tree Tensor Networks](https://mlsanity.com/paper/2502.09928/) (2025)
- [Classical Simulation of Variational Quantum Classifiers using Tensor Rings](https://mlsanity.com/paper/2201.08878/) (2022)
- [Many-body Approximation for Non-negative Tensors](https://mlsanity.com/paper/2209.15338/) (2022)
- [Tensor-Networks-based Learning of Probabilistic Cellular Automata Dynamics](https://mlsanity.com/paper/2404.11768/) (2024)
- [Tensor Network Based Feature Learning Model](https://mlsanity.com/paper/2512.02547/) (2025)
- [Infinitely Wide Tensor Networks as Gaussian Process](https://mlsanity.com/paper/2101.02333/) (2021)
- [Non-parametric Semi-Supervised Learning in Many-body Hilbert Space with Rescaled Logarithmic Fidelity](https://mlsanity.com/paper/2107.00195/) (2021)
- [Differentiable Programming of Isometric Tensor Networks](https://mlsanity.com/paper/2110.03898/) (2021)
- [Deep tensor networks with matrix product operators](https://mlsanity.com/paper/2209.09098/) (2022)
- [How Informative is the Approximation Error from Tensor Decomposition for Neural Network Compression?](https://mlsanity.com/paper/2305.05318/) (2023)
- [Quantum-inspired Techniques in Tensor Networks for Industrial Contexts](https://mlsanity.com/paper/2404.11277/) (2024)
- [Lower and Upper Bounds on the VC-Dimension of Tensor Network Models](https://mlsanity.com/paper/2106.11827/) (2021)
- [Cons-training Tensor Networks: Embedding and Optimization Over Discrete Linear Constraints](https://mlsanity.com/paper/2405.09005/) (2024)
- [Regularized second-order optimization of tensor-network Born machines](https://mlsanity.com/paper/2501.18691/) (2025)
- [Stack operation of tensor networks](https://mlsanity.com/paper/2203.16338/) (2022)
- [Noisy Tensor Ring approximation for computing gradients of Variational Quantum Eigensolver for Combinatorial Optimization](https://mlsanity.com/paper/2307.03884/) (2023)
- [T'ecnicas Quantum-Inspired en Tensor Networks para Contextos Industriales](https://mlsanity.com/paper/2404.17645/) (2024)
- [Tensor-Augmented Convolutional Neural Networks: Enhancing Expressivity with Generic Tensor Kernels](https://mlsanity.com/paper/2604.08072/) (2026)
- [Quantum-inspired tensor networks in machine learning models](https://mlsanity.com/paper/2604.14287/) (2026)

## Abstract (arXiv)

Tensor networks, a model that originated from quantum physics, has been gradually generalized as efficient models in machine learning in recent years. However, in order to achieve exact contraction, only tree-like tensor networks such as the matrix product states and tree tensor networks have been considered, even for modeling two-dimensional data such as images. In this work, we construct supervised learning models for images using the projected entangled pair states (PEPS), a two-dimensional tensor network having a similar structure prior to natural images. Our approach first performs a feature map, which transforms the image data to a product state on a grid, then contracts the product state to a PEPS with trainable parameters to predict image labels. The tensor elements of PEPS are trained by minimizing differences between training labels and predicted labels. The proposed model is evaluated on image classifications using the MNIST and the Fashion-MNIST datasets. We show that our model is significantly superior to existing models using tree-like tensor networks. Moreover, using the same input features, our method performs as well as the multilayer perceptron classifier, but with much fewer parameters and is more stable. Our results shed light on potential applications of two-dimensional tensor network models in machine learning.