# No-Free-Lunch Theories for Tensor-Network Machine Learning Models

arXiv: https://arxiv.org/abs/2412.05674 · HTML: https://mlsanity.com/paper/2412.05674/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 106 references that are in the map.
- First posted on arXiv 7 Dec 2024.

## Builds on (most cited first, 21 of 21)

- [Quantum Natural Language Processing on Near-Term Quantum Computers](https://mlsanity.com/paper/2005.04147/) (2020)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [Tensor Networks Meet Neural Networks: A Survey and Future Perspectives](https://mlsanity.com/paper/2302.09019/) (2023)
- [CompactifAI: Extreme Compression of Large Language Models using Quantum-Inspired Tensor Networks](https://mlsanity.com/paper/2401.14109/) (2024)
- [TensorNetwork for Machine Learning](https://mlsanity.com/paper/1906.06329/) (2019)
- [Anomaly Detection with Tensor Networks](https://mlsanity.com/paper/2006.02516/) (2020)
- [Hybrid quantum-classical classifier based on tensor network and variational quantum circuit](https://mlsanity.com/paper/2011.14651/) (2020)
- [TensorNetwork: A Library for Physics and Machine Learning](https://mlsanity.com/paper/1905.01330/) (2019)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Generative modeling with projected entangled-pair states](https://mlsanity.com/paper/2202.08177/) (2022)
- [Entanglement-guided architectures of machine learning by quantum tensor network](https://mlsanity.com/paper/1803.09111/) (2018)
- [Learning quantum states and unitaries of bounded gate complexity](https://mlsanity.com/paper/2310.19882/) (2023)
- [Federated Hierarchical Tensor Networks: a Collaborative Learning Quantum AI-Driven Framework for Healthcare](https://mlsanity.com/paper/2405.07735/) (2024)
- [Language Modeling Using Tensor Trains](https://mlsanity.com/paper/2405.04590/) (2024)
- [Generalization and Overfitting in Matrix Product State Machine Learning Architectures](https://mlsanity.com/paper/2208.04372/) (2022)
- [Tensor Network-Constrained Kernel Machines as Gaussian Processes](https://mlsanity.com/paper/2403.19500/) (2024)
- [Tensor-Networks-based Learning of Probabilistic Cellular Automata Dynamics](https://mlsanity.com/paper/2404.11768/) (2024)
- [Learning topological states from randomized measurements using variational tensor network tomography](https://mlsanity.com/paper/2406.00193/) (2024)
- [Exponential Machines](https://mlsanity.com/paper/1605.03795/) (2016)
- [Tensor-Train Networks for Learning Predictive Modeling of Multidimensional Data](https://mlsanity.com/paper/2101.09184/) (2021)
- [Separable Power of Classical and Quantum Learning Protocols Through the Lens of No-Free-Lunch Theorem](https://mlsanity.com/paper/2405.07226/) (2024)

## Built on by (most cited first, 1 of 1)

- [When AI meets quantum information: A comprehensive review](https://mlsanity.com/paper/2607.00365/) (2026)

## Abstract (arXiv)

Tensor network machine learning models have shown remarkable versatility in tackling complex data-driven tasks, ranging from quantum many-body problems to classical pattern recognitions. Despite their promising performance, a comprehensive understanding of the underlying assumptions and limitations of these models is still lacking. In this work, we focus on the rigorous formulation of their no-free-lunch theorem -- essential yet notoriously challenging to formalize for specific tensor network machine learning models. In particular, we rigorously analyze the generalization risks of learning target output functions from input data encoded in tensor network states. We first prove a no-free-lunch theorem for machine learning models based on matrix product states, i.e., the one-dimensional tensor network states. Furthermore, we circumvent the challenging issue of calculating the partition function for two-dimensional Ising model, and prove the no-free-lunch theorem for the case of two-dimensional projected entangled-pair state, by introducing the combinatorial method associated to the "puzzle of polyominoes". Our findings reveal the intrinsic limitations of tensor network-based learning models in a rigorous fashion, and open up an avenue for future analytical exploration of both the strengths and limitations of quantum-inspired machine learning frameworks.