# VSQL: Variational Shadow Quantum Learning for Classification

arXiv: https://arxiv.org/abs/2012.08288 · HTML: https://mlsanity.com/paper/2012.08288/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Dec 2020.

## Built on by (most cited first, 17 of 17)

- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [Power and limitations of single-qubit native quantum neural networks](https://mlsanity.com/paper/2205.07848/) (2022)
- [Concentration of Data Encoding in Parameterized Quantum Circuits](https://mlsanity.com/paper/2206.08273/) (2022)
- [Quantum Neural Network Classifiers: A Tutorial](https://mlsanity.com/paper/2206.02806/) (2022)
- [QUILT: Effective Multi-Class Classification on Quantum Computers Using an Ensemble of Diverse Quantum Classifiers](https://mlsanity.com/paper/2309.15056/) (2023)
- [Investigating and Mitigating Barren Plateaus in Variational Quantum Circuits: A Survey](https://mlsanity.com/paper/2407.17706/) (2024)
- [Toward Physically Realizable Quantum Neural Networks](https://mlsanity.com/paper/2203.12092/) (2022)
- [Mitigating barren plateaus of variational quantum eigensolvers](https://mlsanity.com/paper/2205.13539/) (2022)
- [Post-variational quantum neural networks](https://mlsanity.com/paper/2307.10560/) (2023)
- [SLIQ: Quantum Image Similarity Networks on Noisy Quantum Computers](https://mlsanity.com/paper/2309.15259/) (2023)
- [MORE: Measurement and Correlation Based Variational Quantum Circuit for Multi-classification](https://mlsanity.com/paper/2307.11875/) (2023)
- [Enhancing the Trainability of Variational Quantum Circuits with Regularization Strategies](https://mlsanity.com/paper/2405.01606/) (2024)
- [Large Language Models Can Help Mitigate Barren Plateaus in Quantum Neural Networks](https://mlsanity.com/paper/2502.13166/) (2025)
- [Optimisation-free Classification and Density Estimation with Quantum Circuits](https://mlsanity.com/paper/2203.14452/) (2022)
- [QUPID: A Partitioned Quantum Neural Network for Anomaly Detection in Smart Grid](https://mlsanity.com/paper/2601.11500/) (2026)
- [Transformer fault diagnosis using an efficient simulation-driven variational quantum classifier with domain-aware feature encoding](https://mlsanity.com/paper/2609.20214/) (2026)
- [QuPID: Quantum Parameter-Efficient Input-Dependent Retrieval Adaptation for Medical RAG](https://mlsanity.com/paper/2609.33351/) (2026)

## Abstract (arXiv)

Classification of quantum data is essential for quantum machine learning and near-term quantum technologies. In this paper, we propose a new hybrid quantum-classical framework for supervised quantum learning, which we call Variational Shadow Quantum Learning (VSQL). Our method in particular utilizes the classical shadows of quantum data, which fundamentally represent the side information of quantum data with respect to certain physical observables. Specifically, we first use variational shadow quantum circuits to extract classical features in a convolution way and then utilize a fully-connected neural network to complete the classification task. We show that this method could sharply reduce the number of parameters and thus better facilitate quantum circuit training. Simultaneously, less noise will be introduced since fewer quantum gates are employed in such shadow circuits. Moreover, we show that the Barren Plateau issue, a significant gradient vanishing problem in quantum machine learning, could be avoided in VSQL. Finally, we demonstrate the efficiency of VSQL in quantum classification via numerical experiments on the classification of quantum states and the recognition of multi-labeled handwritten digits. In particular, our VSQL approach outperforms existing variational quantum classifiers in the test accuracy in the binary case of handwritten digit recognition and notably requires much fewer parameters.