# Unsupervised Generative Modeling Using Matrix Product States

arXiv: https://arxiv.org/abs/1709.01662 · HTML: https://mlsanity.com/paper/1709.01662/

## Key facts

- Cited by 88 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Sep 2017.

## Built on by (most cited first, 30 of 88)

- [Variational Quantum Reinforcement Learning via Evolutionary Optimization](https://mlsanity.com/paper/2109.00540/) (2021)
- [Enhancing Generative Models via Quantum Correlations](https://mlsanity.com/paper/2101.08354/) (2021)
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- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [Tensor Networks Meet Neural Networks: A Survey and Future Perspectives](https://mlsanity.com/paper/2302.09019/) (2023)
- [Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus](https://mlsanity.com/paper/2102.01828/) (2021)
- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Sum of Squares Circuits](https://mlsanity.com/paper/2408.11778/) (2024)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Ab-initio study of interacting fermions at finite temperature with neural canonical transformation](https://mlsanity.com/paper/2105.08644/) (2021)
- [Generative modeling with projected entangled-pair states](https://mlsanity.com/paper/2202.08177/) (2022)
- [A Framework for Demonstrating Practical Quantum Advantage: Racing Quantum against Classical Generative Models](https://mlsanity.com/paper/2303.15626/) (2023)
- [Machine learning with tree tensor networks, CP rank constraints, and tensor dropout](https://mlsanity.com/paper/2305.19440/) (2023)
- [Tensor Networks for Explainable Machine Learning in Cybersecurity](https://mlsanity.com/paper/2401.00867/) (2023)
- [On the Relationship Between Monotone and Squared Probabilistic Circuits](https://mlsanity.com/paper/2408.00876/) (2024)
- [Tensor networks and efficient descriptions of classical data](https://mlsanity.com/paper/2103.06872/) (2021)
- [Generative modeling via tensor train sketching](https://mlsanity.com/paper/2202.11788/) (2022)
- [Quantum-inspired event reconstruction with Tensor Networks: Matrix Product States](https://mlsanity.com/paper/2106.08334/) (2021)
- [Quantum-Inspired Tensor Neural Networks for Partial Differential Equations](https://mlsanity.com/paper/2208.02235/) (2022)
- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [Generalization Metrics for Practical Quantum Advantage in Generative Models](https://mlsanity.com/paper/2201.08770/) (2022)
- [Scalably learning quantum many-body Hamiltonians from dynamical data](https://mlsanity.com/paper/2209.14328/) (2022)
- [Generative Modeling via Hierarchical Tensor Sketching](https://mlsanity.com/paper/2304.05305/) (2023)
- [Unsupervised Recognition of Informative Features via Tensor Network Machine Learning and Quantum Entanglement Variations](https://mlsanity.com/paper/2207.06031/) (2022)
- [A Matrix Product State Model for Simultaneous Classification and Generation](https://mlsanity.com/paper/2406.17441/) (2024)
- [Discrete generative diffusion models without stochastic differential equations: a tensor network approach](https://mlsanity.com/paper/2407.11133/) (2024)
- [Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework](https://mlsanity.com/paper/2106.15666/) (2021)
- [GANmapper: geographical data translation](https://mlsanity.com/paper/2108.04232/) (2021)

## Abstract (arXiv)

Generative modeling, which learns joint probability distribution from data and generates samples according to it, is an important task in machine learning and artificial intelligence. Inspired by probabilistic interpretation of quantum physics, we propose a generative model using matrix product states, which is a tensor network originally proposed for describing (particularly one-dimensional) entangled quantum states. Our model enjoys efficient learning analogous to the density matrix renormalization group method, which allows dynamically adjusting dimensions of the tensors and offers an efficient direct sampling approach for generative tasks. We apply our method to generative modeling of several standard datasets including the Bars and Stripes, random binary patterns and the MNIST handwritten digits to illustrate the abilities, features and drawbacks of our model over popular generative models such as Hopfield model, Boltzmann machines and generative adversarial networks. Our work sheds light on many interesting directions of future exploration on the development of quantum-inspired algorithms for unsupervised machine learning, which are promisingly possible to be realized on quantum devices.