# Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits

arXiv: https://arxiv.org/abs/2310.07528 · HTML: https://mlsanity.com/paper/2310.07528/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 76 references that are in the map.
- First posted on arXiv 11 Oct 2023.

## Builds on (most cited first, 30 of 34)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks](https://mlsanity.com/paper/1506.01497/) (2015)
- [XLNet: Generalized Autoregressive Pretraining for Language Understanding](https://mlsanity.com/paper/1906.08237/) (2019)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [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)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Optimal Approximation Rate of ReLU Networks in terms of Width and Depth](https://mlsanity.com/paper/2103.00502/) (2021)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [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)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [Experimental Quantum Generative Adversarial Networks for Image Generation](https://mlsanity.com/paper/2010.06201/) (2020)
- [Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces](https://mlsanity.com/paper/2009.00298/) (2020)
- [Experimental quantum adversarial learning with programmable superconducting qubits](https://mlsanity.com/paper/2204.01738/) (2022)
- [Input Redundancy for Parameterized Quantum Circuits](https://mlsanity.com/paper/1901.11434/) (2019)
- [Deep Network Approximation: Beyond ReLU to Diverse Activation Functions](https://mlsanity.com/paper/2307.06555/) (2023)
- [Pseudo-dimension of quantum circuits](https://mlsanity.com/paper/2002.01490/) (2020)
- [Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines](https://mlsanity.com/paper/2207.05865/) (2022)
- [Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression](https://mlsanity.com/paper/2206.04804/) (2022)

## Built on by (most cited first, 10 of 10)

- [Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2509.14026/) (2025)
- [Feedback-driven recurrent quantum neural network universality](https://mlsanity.com/paper/2506.16332/) (2025)
- [Mitigating Exponential Mixed Frequency Growth through Frequency Selection](https://mlsanity.com/paper/2508.10533/) (2025)
- [Hybrid Quantum-Classical Recurrent Neural Networks](https://mlsanity.com/paper/2510.25557/) (2025)
- [Approximation rates of quantum neural networks for periodic functions via Jackson's inequality](https://mlsanity.com/paper/2511.16149/) (2025)
- [SAQNN: Spectral Adaptive Quantum Neural Network as a Universal Approximator](https://mlsanity.com/paper/2602.09718/) (2026)
- [Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs](https://mlsanity.com/paper/2606.20326/) (2026)
- [Learning AC Power Flow Solutions using a Data-Dependent Variational Quantum Circuit](https://mlsanity.com/paper/2509.03495/) (2025)
- [SQUARE: Structured Quantum Representation Adapters as Compact Quadratic Feature Maps for Frozen Language Models](https://mlsanity.com/paper/2609.37134/) (2026)
- [Harmonic analysis and circuit approximation of Sobolev functions on quantum input spaces](https://mlsanity.com/paper/2610.13477/) (2026)

## Abstract (arXiv)

Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.