# Supervised Learning with Quantum-Inspired Tensor Networks

arXiv: https://arxiv.org/abs/1605.05775 · HTML: https://mlsanity.com/paper/1605.05775/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2016.

## Built on by (most cited first, 30 of 36)

- [An end-to-end trainable hybrid classical-quantum classifier](https://mlsanity.com/paper/2102.02416/) (2021)
- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [High-Order Pooling for Graph Neural Networks with Tensor Decomposition](https://mlsanity.com/paper/2205.11691/) (2022)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Differentiable Physics: A Position Piece](https://mlsanity.com/paper/2109.07573/) (2021)
- [Quantum-inspired event reconstruction with Tensor Networks: Matrix Product States](https://mlsanity.com/paper/2106.08334/) (2021)
- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [Classification of the Fashion-MNIST Dataset on a Quantum Computer](https://mlsanity.com/paper/2403.02405/) (2024)
- [More Efficient Sampling for Tensor Decomposition With Worst-Case Guarantees](https://mlsanity.com/paper/2110.07631/) (2021)
- [Classical versus Quantum: comparing Tensor Network-based Quantum Circuits on LHC data](https://mlsanity.com/paper/2202.10471/) (2022)
- [A Matrix Product State Model for Simultaneous Classification and Generation](https://mlsanity.com/paper/2406.17441/) (2024)
- [Optimizing Tensor Network Contraction Using Reinforcement Learning](https://mlsanity.com/paper/2204.09052/) (2022)
- [tn4ml: Tensor Network Training and Customization for Machine Learning](https://mlsanity.com/paper/2502.13090/) (2025)
- [Position: Tensor Networks are a Valuable Asset for Green AI](https://mlsanity.com/paper/2205.12961/) (2022)
- [Positive unlabeled learning with tensor networks](https://mlsanity.com/paper/2211.14085/) (2022)
- [Tensor-Train Networks for Learning Predictive Modeling of Multidimensional Data](https://mlsanity.com/paper/2101.09184/) (2021)
- [Neural Tangent Kernel of Matrix Product States: Convergence and Applications](https://mlsanity.com/paper/2111.14046/) (2021)
- [Deep tensor networks with matrix product operators](https://mlsanity.com/paper/2209.09098/) (2022)
- [Distributive Pre-Training of Generative Modeling Using Matrix-Product States](https://mlsanity.com/paper/2306.14787/) (2023)
- [Using matrix-product states for time-series machine learning](https://mlsanity.com/paper/2412.15826/) (2024)
- [Deep Sequence Modeling with Quantum Dynamics: Language as a Wave Function](https://mlsanity.com/paper/2602.22255/) (2026)
- [Rademacher Random Projections with Tensor Networks](https://mlsanity.com/paper/2110.13970/) (2021)
- [Learning Circular Hidden Quantum Markov Models: A Tensor Network Approach](https://mlsanity.com/paper/2111.01536/) (2021)
- [Group-invariant tensor train networks for supervised learning](https://mlsanity.com/paper/2206.15051/) (2022)
- [Tensorizing flows: a tool for variational inference](https://mlsanity.com/paper/2305.02460/) (2023)
- [Quantum-Inspired Robust and Scalable SAR Object Classification](https://mlsanity.com/paper/2604.25755/) (2026)
- [Quantum Machine Learning for Health State Diagnosis and Prognostics](https://mlsanity.com/paper/2108.12265/) (2021)
- [Entangling Machine Learning with Quantum Tensor Networks](https://mlsanity.com/paper/2403.12969/) (2024)
- [FLAASH: Flexible Accelerator Architecture for Sparse High-Order Tensor Contraction](https://mlsanity.com/paper/2404.16317/) (2024)
- [HMAE: Self-Supervised Few-Shot Learning for Quantum Spin Systems](https://mlsanity.com/paper/2505.03140/) (2025)

## Abstract (arXiv)

Tensor networks are efficient representations of high-dimensional tensors which have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing such networks can be adapted to supervised learning tasks by using matrix product states (tensor trains) to parameterize models for classifying images. For the MNIST data set we obtain less than 1% test set classification error. We discuss how the tensor network form imparts additional structure to the learned model and suggest a possible generative interpretation.