# Generalization Error Bound for Quantum Machine Learning in NISQ Era -- A Survey

arXiv: https://arxiv.org/abs/2409.07626 · HTML: https://mlsanity.com/paper/2409.07626/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 113 references that are in the map.
- First posted on arXiv 11 Sep 2024.

## Builds on (most cited first, 30 of 51)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [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)
- [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)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [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)
- [Adding Gradient Noise Improves Learning for Very Deep Networks](https://mlsanity.com/paper/1511.06807/) (2015)
- [Entanglement Induced Barren Plateaus](https://mlsanity.com/paper/2010.15968/) (2020)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [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)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [Towards Quantum Machine Learning with Tensor Networks](https://mlsanity.com/paper/1803.11537/) (2018)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [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)
- [Generalization Error in Deep Learning](https://mlsanity.com/paper/1808.01174/) (2018)
- [Towards understanding the power of quantum kernels in the NISQ era](https://mlsanity.com/paper/2103.16774/) (2021)
- [Bandwidth Enables Generalization in Quantum Kernel Models](https://mlsanity.com/paper/2206.06686/) (2022)

## Built on by (most cited first, 7 of 7)

- [QuXAI: Explainers for Hybrid Quantum Machine Learning Models](https://mlsanity.com/paper/2505.10167/) (2025)
- [Data-Dependent Generalization Bounds for Parameterized Quantum Models Under Noise](https://mlsanity.com/paper/2412.11451/) (2024)
- [Quantum Circuit-Based Learning Models: Bridging Quantum Computing and Machine Learning](https://mlsanity.com/paper/2602.00048/) (2026)
- [Quantum Machine Learning for Cybersecurity: A Taxonomy and Future Directions](https://mlsanity.com/paper/2512.15286/) (2025)
- [Quantum Error Mitigation with Attention Graph Transformers for Burgers Equation Solvers on NISQ Hardware](https://mlsanity.com/paper/2512.23817/) (2025)
- [Multitask Learning for Earth Observation Data Classification with Hybrid Quantum Network](https://mlsanity.com/paper/2601.22195/) (2026)
- [From Reachability to Learnability: Geometric Design Principles for Quantum Neural Networks](https://mlsanity.com/paper/2603.03071/) (2026)

## Abstract (arXiv)

Despite the mounting anticipation for the quantum revolution, the success of Quantum Machine Learning (QML) in the Noisy Intermediate-Scale Quantum (NISQ) era hinges on a largely unexplored factor: the generalization error bound, a cornerstone of robust and reliable machine learning models. Current QML research, while exploring novel algorithms and applications extensively, is predominantly situated in the context of noise-free, ideal quantum computers. However, Quantum Circuit (QC) operations in NISQ-era devices are susceptible to various noise sources and errors. In this article, we conduct a Systematic Mapping Study (SMS) to explore the state-of-the-art generalization bound for supervised QML in NISQ-era and analyze the latest practices in the field. Our study systematically summarizes the existing computational platforms with quantum hardware, datasets, optimization techniques, and the common properties of the bounds found in the literature. We further present the performance accuracy of various approaches in classical benchmark datasets like the MNIST and IRIS datasets. The SMS also highlights the limitations and challenges in QML in the NISQ era and discusses future research directions to advance the field. Using a detailed Boolean operators query in five reliable indexers, we collected 544 papers and filtered them to a small set of 37 relevant articles. This filtration was done following the best practice of SMS with well-defined research questions and inclusion and exclusion criteria.