# An end-to-end trainable hybrid classical-quantum classifier

arXiv: https://arxiv.org/abs/2102.02416 · HTML: https://mlsanity.com/paper/2102.02416/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 26 references that are in the map.
- First posted on arXiv 4 Feb 2021.

## Builds on (most cited first, 13 of 13)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [On the Expressive Power of Deep Learning: A Tensor Analysis](https://mlsanity.com/paper/1509.05009/) (2015)
- [Reinforcement Learning with Quantum Variational Circuits](https://mlsanity.com/paper/2008.07524/) (2020)
- [Supervised Learning with Quantum-Inspired Tensor Networks](https://mlsanity.com/paper/1605.05775/) (2016)
- [From probabilistic graphical models to generalized tensor networks for supervised learning](https://mlsanity.com/paper/1806.05964/) (2018)
- [TensorNetwork for Machine Learning](https://mlsanity.com/paper/1906.06329/) (2019)
- [Hybrid quantum-classical classifier based on tensor network and variational quantum circuit](https://mlsanity.com/paper/2011.14651/) (2020)
- [Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Design](https://mlsanity.com/paper/1704.01552/) (2017)
- [Matrix Product State for Feature Extraction of Higher-Order Tensors](https://mlsanity.com/paper/1503.00516/) (2015)

## Built on by (most cited first, 30 of 51)

- [Federated Quantum Machine Learning](https://mlsanity.com/paper/2103.12010/) (2021)
- [A Survey on Quantum Reinforcement Learning](https://mlsanity.com/paper/2211.03464/) (2022)
- [Variational Quantum Reinforcement Learning via Evolutionary Optimization](https://mlsanity.com/paper/2109.00540/) (2021)
- [Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression](https://mlsanity.com/paper/2206.04804/) (2022)
- [Quantum deep recurrent reinforcement learning](https://mlsanity.com/paper/2210.14876/) (2022)
- [Learning to Program Variational Quantum Circuits with Fast Weights](https://mlsanity.com/paper/2402.17760/) (2024)
- [Quantum Architecture Search via Continual Reinforcement Learning](https://mlsanity.com/paper/2112.05779/) (2021)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [Asynchronous training of quantum reinforcement learning](https://mlsanity.com/paper/2301.05096/) (2023)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Federated Quantum Machine Learning with Differential Privacy](https://mlsanity.com/paper/2310.06973/) (2023)
- [Learning to Measure Quantum Neural Networks](https://mlsanity.com/paper/2501.05663/) (2025)
- [Federated Quantum Natural Gradient Descent for Quantum Federated Learning](https://mlsanity.com/paper/2209.00564/) (2022)
- [Optimizing Quantum Federated Learning Based on Federated Quantum Natural Gradient Descent](https://mlsanity.com/paper/2303.08116/) (2023)
- [Efficient quantum recurrent reinforcement learning via quantum reservoir computing](https://mlsanity.com/paper/2309.07339/) (2023)
- [Differentiable Quantum Architecture Search in Asynchronous Quantum Reinforcement Learning](https://mlsanity.com/paper/2407.18202/) (2024)
- [Quantum-Train-Based Distributed Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2412.08845/) (2024)
- [Variational Quanvolutional Neural Networks with enhanced image encoding](https://mlsanity.com/paper/2106.07327/) (2021)
- [Quantum Machine Learning Architecture Search via Deep Reinforcement Learning](https://mlsanity.com/paper/2407.20147/) (2024)
- [A Hybrid Quantum-Classical Neural Network Architecture for Binary Classification](https://mlsanity.com/paper/2201.01820/) (2022)
- [Quantum Advantage Actor-Critic for Reinforcement Learning](https://mlsanity.com/paper/2401.07043/) (2024)
- [Quantum Gradient Class Activation Map for Model Interpretability](https://mlsanity.com/paper/2408.05899/) (2024)
- [Evolutionary Optimization for Designing Variational Quantum Circuits with High Model Capacity](https://mlsanity.com/paper/2412.12484/) (2024)
- [Quantum Federated Learning With Quantum Networks](https://mlsanity.com/paper/2310.15084/) (2023)
- [An Introduction to Quantum Reinforcement Learning (QRL)](https://mlsanity.com/paper/2409.05846/) (2024)
- [Adaptive Non-local Observable on Quantum Neural Networks](https://mlsanity.com/paper/2504.13414/) (2025)
- [Differentiable Quantum Architecture Search in Quantum-Enhanced Neural Network Parameter Generation](https://mlsanity.com/paper/2505.09653/) (2025)
- [Learning to Program Quantum Measurements for Machine Learning](https://mlsanity.com/paper/2505.13525/) (2025)
- [TensorHyper-VQC: A Tensor-Train-Guided Hypernetwork for Robust and Scalable Variational Quantum Computing](https://mlsanity.com/paper/2508.01116/) (2025)
- [Random Quantum Neural Networks (RQNN) for Noisy Image Recognition](https://mlsanity.com/paper/2203.01764/) (2022)

## Abstract (arXiv)

We introduce a hybrid model combining a quantum-inspired tensor network and a variational quantum circuit to perform supervised learning tasks. This architecture allows for the classical and quantum parts of the model to be trained simultaneously, providing an end-to-end training framework. We show that compared to the principal component analysis, a tensor network based on the matrix product state with low bond dimensions performs better as a feature extractor for the input data of the variational quantum circuit in the binary and ternary classification of MNIST and Fashion-MNIST datasets. The architecture is highly adaptable and the classical-quantum boundary can be adjusted according the availability of the quantum resource by exploiting the correspondence between tensor networks and quantum circuits.