# Concentration of Data Encoding in Parameterized Quantum Circuits

arXiv: https://arxiv.org/abs/2206.08273 · HTML: https://mlsanity.com/paper/2206.08273/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 68 references that are in the map.
- First posted on arXiv 16 Jun 2022.

## Builds on (most cited first, 19 of 19)

- [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)
- [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)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Quantum machine learning: a classical perspective](https://mlsanity.com/paper/1707.08561/) (2017)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Robust data encodings for quantum classifiers](https://mlsanity.com/paper/2003.01695/) (2020)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Encoding-dependent generalization bounds for parametrized quantum circuits](https://mlsanity.com/paper/2106.03880/) (2021)
- [Reinforcement learning for optimization of variational quantum circuit architectures](https://mlsanity.com/paper/2103.16089/) (2021)
- [Recent advances for quantum classifiers](https://mlsanity.com/paper/2108.13421/) (2021)
- [Sublinear quantum algorithms for training linear and kernel-based classifiers](https://mlsanity.com/paper/1904.02276/) (2019)
- [VSQL: Variational Shadow Quantum Learning for Classification](https://mlsanity.com/paper/2012.08288/) (2020)
- [Universal discriminative quantum neural networks](https://mlsanity.com/paper/1805.08654/) (2018)

## Built on by (most cited first, 15 of 15)

- [Exponential concentration in quantum kernel methods](https://mlsanity.com/paper/2208.11060/) (2022)
- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [A Survey on Quantum Machine Learning: Current Trends, Challenges, Opportunities, and the Road Ahead](https://mlsanity.com/paper/2310.10315/) (2023)
- [CompressedMediQ: Hybrid Quantum Machine Learning Pipeline for High-Dimensional Neuroimaging Data](https://mlsanity.com/paper/2409.08584/) (2024)
- [Quantum Mixed-State Self-Attention Network](https://mlsanity.com/paper/2403.02871/) (2024)
- [Expressibility-induced Concentration of Quantum Neural Tangent Kernels](https://mlsanity.com/paper/2311.04965/) (2023)
- [ATP: Adaptive Threshold Pruning for Efficient Data Encoding in Quantum Neural Networks](https://mlsanity.com/paper/2503.21815/) (2025)
- [Quantum Natural Language Processing: A Comprehensive Review of Models, Methods, and Applications](https://mlsanity.com/paper/2504.09909/) (2025)
- [Predictive Performance of Deep Quantum Data Re-uploading Models](https://mlsanity.com/paper/2505.20337/) (2025)
- [Security Concerns in Quantum Machine Learning as a Service](https://mlsanity.com/paper/2408.09562/) (2024)
- [Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks](https://mlsanity.com/paper/2309.14980/) (2023)
- [SuperEncoder: Towards Universal Neural Approximate Quantum State Preparation](https://mlsanity.com/paper/2408.05435/) (2024)
- [Quantum Deep Sets and Sequences](https://mlsanity.com/paper/2504.02241/) (2025)
- [Re-uploading quantum data: a universal function approximator for quantum inputs](https://mlsanity.com/paper/2509.18530/) (2025)
- [QTrans: A Quantum Transformer for Sentiment Classification](https://mlsanity.com/paper/2609.12011/) (2026)

## Abstract (arXiv)

Variational quantum algorithms have been acknowledged as a leading strategy to realize near-term quantum advantages in meaningful tasks, including machine learning and combinatorial optimization. When applied to tasks involving classical data, such algorithms generally begin with quantum circuits for data encoding and then train quantum neural networks (QNNs) to minimize target functions. Although QNNs have been widely studied to improve these algorithms' performance on practical tasks, there is a gap in systematically understanding the influence of data encoding on the eventual performance. In this paper, we make progress in filling this gap by considering the common data encoding strategies based on parameterized quantum circuits. We prove that, under reasonable assumptions, the distance between the average encoded state and the maximally mixed state could be explicitly upper-bounded with respect to the width and depth of the encoding circuit. This result in particular implies that the average encoded state will concentrate on the maximally mixed state at an exponential speed on depth. Such concentration seriously limits the capabilities of quantum classifiers, and strictly restricts the distinguishability of encoded states from a quantum information perspective. We further support our findings by numerically verifying these results on both synthetic and public data sets. Our results highlight the significance of quantum data encoding in machine learning tasks and may shed light on future encoding strategies.