# Generalization in quantum machine learning from few training data

arXiv: https://arxiv.org/abs/2111.05292 · HTML: https://mlsanity.com/paper/2111.05292/

## Key facts

- Cited by 216 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 124 references that are in the map.
- First posted on arXiv 9 Nov 2021.

## Builds on (most cited first, 30 of 37)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [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)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Absence of Barren Plateaus in Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2011.02966/) (2020)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Encoding-dependent generalization bounds for parametrized quantum circuits](https://mlsanity.com/paper/2106.03880/) (2021)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [Trainability of Dissipative Perceptron-Based Quantum Neural Networks](https://mlsanity.com/paper/2005.12458/) (2020)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Towards understanding the power of quantum kernels in the NISQ era](https://mlsanity.com/paper/2103.16774/) (2021)
- [Representation Learning via Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2111.04225/) (2021)
- [On barren plateaus and cost function locality in variational quantum algorithms](https://mlsanity.com/paper/2011.10530/) (2020)
- [Input Redundancy for Parameterized Quantum Circuits](https://mlsanity.com/paper/1901.11434/) (2019)
- [Subtleties in the trainability of quantum machine learning models](https://mlsanity.com/paper/2110.14753/) (2021)
- [Pseudo-dimension of quantum circuits](https://mlsanity.com/paper/2002.01490/) (2020)
- [Reformulation of the No-Free-Lunch Theorem for Entangled Data Sets](https://mlsanity.com/paper/2007.04900/) (2020)
- [Entangled Datasets for Quantum Machine Learning](https://mlsanity.com/paper/2109.03400/) (2021)
- [On the statistical complexity of quantum circuits](https://mlsanity.com/paper/2101.06154/) (2021)
- [Structural risk minimization for quantum linear classifiers](https://mlsanity.com/paper/2105.05566/) (2021)

## Built on by (most cited first, 30 of 216)

- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Does provable absence of barren plateaus imply classical simulability?](https://mlsanity.com/paper/2312.09121/) (2023)
- [Exploiting symmetry in variational quantum machine learning](https://mlsanity.com/paper/2205.06217/) (2022)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.09974/) (2022)
- [Out-of-distribution generalization for learning quantum dynamics](https://mlsanity.com/paper/2204.10268/) (2022)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits](https://mlsanity.com/paper/2203.09376/) (2022)
- [Quantum Self-Attention Neural Networks for Text Classification](https://mlsanity.com/paper/2205.05625/) (2022)
- [Equivariant quantum circuits for learning on weighted graphs](https://mlsanity.com/paper/2205.06109/) (2022)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)
- [Recent Advances for Quantum Neural Networks in Generative Learning](https://mlsanity.com/paper/2206.03066/) (2022)
- [Quantum Convolutional Neural Networks are Effectively Classically Simulable](https://mlsanity.com/paper/2408.12739/) (2024)
- [Learning to predict arbitrary quantum processes](https://mlsanity.com/paper/2210.14894/) (2022)
- [On quantum backpropagation, information reuse, and cheating measurement collapse](https://mlsanity.com/paper/2305.13362/) (2023)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Machine Learning for Anomaly Detection in Particle Physics](https://mlsanity.com/paper/2312.14190/) (2023)
- [Benchmarking Adversarially Robust Quantum Machine Learning at Scale](https://mlsanity.com/paper/2211.12681/) (2022)
- [Towards quantum enhanced adversarial robustness in machine learning](https://mlsanity.com/paper/2306.12688/) (2023)
- [On the relation between trainability and dequantization of variational quantum learning models](https://mlsanity.com/paper/2406.07072/) (2024)
- [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)
- [Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix](https://mlsanity.com/paper/2212.04471/) (2022)
- [Applying QNLP to sentiment analysis in finance](https://mlsanity.com/paper/2307.11788/) (2023)
- [Analytic theory for the dynamics of wide quantum neural networks](https://mlsanity.com/paper/2203.16711/) (2022)
- [Symmetric Pruning in Quantum Neural Networks](https://mlsanity.com/paper/2208.14057/) (2022)
- [Quantum anomaly detection in the latent space of proton collision events at the LHC](https://mlsanity.com/paper/2301.10780/) (2023)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)

## Abstract (arXiv)

Modern quantum machine learning (QML) methods involve variationally optimizing a parameterized quantum circuit on a training data set, and subsequently making predictions on a testing data set (i.e., generalizing). In this work, we provide a comprehensive study of generalization performance in QML after training on a limited number N of training data points. We show that the generalization error of a quantum machine learning model with T trainable gates scales at worst as √T/N. When only K ll T gates have undergone substantial change in the optimization process, we prove that the generalization error improves to √K / N. Our results imply that the compiling of unitaries into a polynomial number of native gates, a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. We also show that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set. Other potential applications include learning quantum error correcting codes or quantum dynamical simulation. Our work injects new hope into the field of QML, as good generalization is guaranteed from few training data.