# Encoding-dependent generalization bounds for parametrized quantum circuits

arXiv: https://arxiv.org/abs/2106.03880 · HTML: https://mlsanity.com/paper/2106.03880/

## Key facts

- Cited by 82 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 17 of 17)

- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [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)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Noisy intermediate-scale quantum (NISQ) algorithms](https://mlsanity.com/paper/2101.08448/) (2021)
- [On the Quantum versus Classical Learnability of Discrete Distributions](https://mlsanity.com/paper/2007.14451/) (2020)
- [Input Redundancy for Parameterized Quantum Circuits](https://mlsanity.com/paper/1901.11434/) (2019)
- [Pseudo-dimension of quantum circuits](https://mlsanity.com/paper/2002.01490/) (2020)
- [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)
- [Generalization in Quantum Machine Learning: a Quantum Information Perspective](https://mlsanity.com/paper/2102.08991/) (2021)

## Built on by (most cited first, 30 of 82)

- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (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)
- [Experimental quantum adversarial learning with programmable superconducting qubits](https://mlsanity.com/paper/2204.01738/) (2022)
- [Classical surrogates for quantum learning models](https://mlsanity.com/paper/2206.11740/) (2022)
- [Recent advances for quantum classifiers](https://mlsanity.com/paper/2108.13421/) (2021)
- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Shadows of quantum machine learning](https://mlsanity.com/paper/2306.00061/) (2023)
- [Towards quantum enhanced adversarial robustness in machine learning](https://mlsanity.com/paper/2306.12688/) (2023)
- [Exponential separations between classical and quantum learners](https://mlsanity.com/paper/2306.16028/) (2023)
- [QSAN: A Near-term Achievable Quantum Self-Attention Network](https://mlsanity.com/paper/2207.07563/) (2022)
- [On the relation between trainability and dequantization of variational quantum learning models](https://mlsanity.com/paper/2406.07072/) (2024)
- [Learning Quantum Processes and Hamiltonians via the Pauli Transfer Matrix](https://mlsanity.com/paper/2212.04471/) (2022)
- [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)
- [The dilemma of quantum neural networks](https://mlsanity.com/paper/2106.04975/) (2021)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [High Dimensional Quantum Machine Learning With Small Quantum Computers](https://mlsanity.com/paper/2203.13739/) (2022)
- [On fundamental aspects of quantum extreme learning machines](https://mlsanity.com/paper/2312.15124/) (2023)
- [On the expressivity of embedding quantum kernels](https://mlsanity.com/paper/2309.14419/) (2023)
- [Sample Complexity of Learning Parametric Quantum Circuits](https://mlsanity.com/paper/2107.09078/) (2021)
- [Dynamical simulation via quantum machine learning with provable generalization](https://mlsanity.com/paper/2204.10269/) (2022)
- [Concentration of Data Encoding in Parameterized Quantum Circuits](https://mlsanity.com/paper/2206.08273/) (2022)
- [On establishing learning separations between classical and quantum machine learning with classical data](https://mlsanity.com/paper/2208.06339/) (2022)
- [Learning quantum states and unitaries of bounded gate complexity](https://mlsanity.com/paper/2310.19882/) (2023)
- [Quantum Machine Learning on Near-Term Quantum Devices: Current State of Supervised and Unsupervised Techniques for Real-World Applications](https://mlsanity.com/paper/2307.00908/) (2023)
- [Incremental Data-Uploading for Full-Quantum Classification](https://mlsanity.com/paper/2205.03057/) (2022)

## Abstract (arXiv)

A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.