# Reduction of finite sampling noise in quantum neural networks

arXiv: https://arxiv.org/abs/2306.01639 · HTML: https://mlsanity.com/paper/2306.01639/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 63 references that are in the map.
- First posted on arXiv 2 Jun 2023.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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 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)
- [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)
- [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)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [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)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Effect of barren plateaus on gradient-free optimization](https://mlsanity.com/paper/2011.12245/) (2020)
- [Training Quantum Embedding Kernels on Near-Term Quantum Computers](https://mlsanity.com/paper/2105.02276/) (2021)
- [Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces](https://mlsanity.com/paper/2009.00298/) (2020)
- [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)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?](https://mlsanity.com/paper/2109.01051/) (2021)
- [Quantum Gaussian Process Regression for Bayesian Optimization](https://mlsanity.com/paper/2304.12923/) (2023)
- [Quantum neural networks force fields generation](https://mlsanity.com/paper/2203.04666/) (2022)

## Built on by (most cited first, 13 of 13)

- [Quantum Kernel Methods under Scrutiny: A Benchmarking Study](https://mlsanity.com/paper/2409.04406/) (2024)
- [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)
- [Reinforcement learning-based architecture search for quantum machine learning](https://mlsanity.com/paper/2406.02717/) (2024)
- [Optimal training of finitely-sampled quantum reservoir computers for forecasting of chaotic dynamics](https://mlsanity.com/paper/2409.01394/) (2024)
- [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)
- [sQUlearn -- A Python Library for Quantum Machine Learning](https://mlsanity.com/paper/2311.08990/) (2023)
- [A Numerical Gradient Inversion Attack in Variational Quantum Neural-Networks](https://mlsanity.com/paper/2504.12806/) (2025)
- [QuXAI: Explainers for Hybrid Quantum Machine Learning Models](https://mlsanity.com/paper/2505.10167/) (2025)
- [Data Efficient Prediction of excited-state properties using Quantum Neural Networks](https://mlsanity.com/paper/2412.09423/) (2024)
- [Quantum Reservoir Computing for Short-Term Power Load Forecasting in Resource-Constrained Energy Systems](https://mlsanity.com/paper/2606.12806/) (2026)
- [AutoQML: A Framework for Automated Quantum Machine Learning](https://mlsanity.com/paper/2502.21025/) (2025)
- [Measurement-Budget Allocation in Quantum Learning with Finite-Shot Generalization Guarantees](https://mlsanity.com/paper/2608.24891/) (2026)
- [Adaptive-Shot Hybrid Quantum Anomaly Detection for Tactile Internet Security: Reliability-Aware Measurement Allocation Under Resource Constraints](https://mlsanity.com/paper/2610.05835/) (2026)

## Abstract (arXiv)

Quantum neural networks (QNNs) use parameterized quantum circuits with data-dependent inputs and generate outputs through the evaluation of expectation values. Calculating these expectation values necessitates repeated circuit evaluations, thus introducing fundamental finite-sampling noise even on error-free quantum computers. We reduce this noise by introducing the variance regularization, a technique for reducing the variance of the expectation value during the quantum model training. This technique requires no additional circuit evaluations if the QNN is properly constructed. Our empirical findings demonstrate the reduced variance speeds up the training and lowers the output noise as well as decreases the number of necessary evaluations of gradient circuits. This regularization method is benchmarked on the regression of multiple functions and the potential energy surface of water. We show that in our examples, it lowers the variance by an order of magnitude on average and leads to a significantly reduced noise level of the QNN. We finally demonstrate QNN training on a real quantum device and evaluate the impact of error mitigation. Here, the optimization is feasible only due to the reduced number of necessary shots in the gradient evaluation resulting from the reduced variance.