# Shadows of quantum machine learning

arXiv: https://arxiv.org/abs/2306.00061 · HTML: https://mlsanity.com/paper/2306.00061/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 58 references that are in the map.
- First posted on arXiv 31 May 2023.

## Builds on (most cited first, 19 of 19)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Parametrized quantum policies for reinforcement learning](https://mlsanity.com/paper/2103.05577/) (2021)
- [Encoding-dependent generalization bounds for parametrized quantum circuits](https://mlsanity.com/paper/2106.03880/) (2021)
- [On the Quantum versus Classical Learnability of Discrete Distributions](https://mlsanity.com/paper/2007.14451/) (2020)
- [Classical surrogates for quantum learning models](https://mlsanity.com/paper/2206.11740/) (2022)
- [Quantum machine learning of large datasets using randomized measurements](https://mlsanity.com/paper/2108.01039/) (2021)
- [Structural risk minimization for quantum linear classifiers](https://mlsanity.com/paper/2105.05566/) (2021)
- [On establishing learning separations between classical and quantum machine learning with classical data](https://mlsanity.com/paper/2208.06339/) (2022)

## Built on by (most cited first, 30 of 31)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [On the relation between trainability and dequantization of variational quantum learning models](https://mlsanity.com/paper/2406.07072/) (2024)
- [Potential and limitations of random Fourier features for dequantizing quantum machine learning](https://mlsanity.com/paper/2309.11647/) (2023)
- [Training robust and generalizable quantum models](https://mlsanity.com/paper/2311.11871/) (2023)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Training-efficient density quantum machine learning](https://mlsanity.com/paper/2405.20237/) (2024)
- [Consensus-based Distributed Quantum Kernel Learning for Speech Recognition](https://mlsanity.com/paper/2409.05770/) (2024)
- [Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows](https://mlsanity.com/paper/2402.17911/) (2024)
- [Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits](https://mlsanity.com/paper/2408.12199/) (2024)
- [Quantum delegated and federated learning via quantum homomorphic encryption](https://mlsanity.com/paper/2409.19359/) (2024)
- [Quantum-Informed Machine Learning for Predicting Spatiotemporal Chaos with Practical Quantum Advantage](https://mlsanity.com/paper/2507.19861/) (2025)
- [Quantum computing and artificial intelligence: status and perspectives](https://mlsanity.com/paper/2505.23860/) (2025)
- [Learning unitaries with quantum statistical queries](https://mlsanity.com/paper/2310.02254/) (2023)
- [Quantum Neural Networks in Practice: A Comparative Study with Classical Models from Standard Data Sets to Industrial Images](https://mlsanity.com/paper/2411.19276/) (2024)
- [Prospects for quantum advantage in machine learning from the representability of functions](https://mlsanity.com/paper/2512.15661/) (2025)
- [Quantum Supervised Learning](https://mlsanity.com/paper/2407.17161/) (2024)
- [Distributed Quantum Neural Networks on Distributed Photonic Quantum Computing](https://mlsanity.com/paper/2505.08474/) (2025)
- [New Insights on Unfolding and Fine-tuning Quantum Federated Learning](https://mlsanity.com/paper/2506.20016/) (2025)
- [Enhancing the Scalability of Classical Surrogates for Real-World Quantum Machine Learning Applications](https://mlsanity.com/paper/2508.06131/) (2025)
- [Limits of quantum generative models with classical sampling hardness](https://mlsanity.com/paper/2512.24801/) (2025)
- [Training quantum machine learning models on cloud without uploading the data](https://mlsanity.com/paper/2409.04602/) (2024)
- [Let the Quantum Creep In: Designing Quantum Neural Network Models by Gradually Swapping Out Classical Components](https://mlsanity.com/paper/2409.17583/) (2024)
- [Classical Shadows with Improved Median-of-Means Estimation](https://mlsanity.com/paper/2412.03381/) (2024)
- [Differential equation quantum solvers: engineering measurements to reduce cost](https://mlsanity.com/paper/2503.22656/) (2025)
- [The interplay of robustness and generalization in quantum machine learning](https://mlsanity.com/paper/2506.08455/) (2025)
- [Quanvolutional Neural Networks for Pneumonia Detection: An Efficient Quantum-Assisted Feature Extraction Paradigm](https://mlsanity.com/paper/2510.23660/) (2025)
- [Shallow-circuit Supervised Learning on a Quantum Processor](https://mlsanity.com/paper/2601.03235/) (2026)
- [Parameterized Quantum Circuits as Feature Maps: Representation Quality and Readout Effects in Multispectral Land-Cover Classification](https://mlsanity.com/paper/2604.26675/) (2026)

## Abstract (arXiv)

Quantum machine learning is often highlighted as one of the most promising practical applications for which quantum computers could provide a computational advantage. However, a major obstacle to the widespread use of quantum machine learning models in practice is that these models, even once trained, still require access to a quantum computer in order to be evaluated on new data. To solve this issue, we introduce a new class of quantum models where quantum resources are only required during training, while the deployment of the trained model is classical. Specifically, the training phase of our models ends with the generation of a 'shadow model' from which the classical deployment becomes possible. We prove that: i) this class of models is universal for classically-deployed quantum machine learning; ii) it does have restricted learning capacities compared to 'fully quantum' models, but nonetheless iii) it achieves a provable learning advantage over fully classical learners, contingent on widely-believed assumptions in complexity theory. These results provide compelling evidence that quantum machine learning can confer learning advantages across a substantially broader range of scenarios, where quantum computers are exclusively employed during the training phase. By enabling classical deployment, our approach facilitates the implementation of quantum machine learning models in various practical contexts.