# Capacity and quantum geometry of parametrized quantum circuits

arXiv: https://arxiv.org/abs/2102.01659 · HTML: https://mlsanity.com/paper/2102.01659/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 46 references that are in the map.
- First posted on arXiv 2 Feb 2021.

## Builds on (most cited first, 3 of 3)

- [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)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)

## Built on by (most cited first, 30 of 32)

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Understanding quantum machine learning also requires rethinking generalization](https://mlsanity.com/paper/2306.13461/) (2023)
- [Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits](https://mlsanity.com/paper/2203.09376/) (2022)
- [Recent advances for quantum classifiers](https://mlsanity.com/paper/2108.13421/) (2021)
- [Quantum machine learning of large datasets using randomized measurements](https://mlsanity.com/paper/2108.01039/) (2021)
- [Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification](https://mlsanity.com/paper/2301.01597/) (2022)
- [Explaining Quantum Circuits with Shapley Values: Towards Explainable Quantum Machine Learning](https://mlsanity.com/paper/2301.09138/) (2023)
- [Optimal training of variational quantum algorithms without barren plateaus](https://mlsanity.com/paper/2104.14543/) (2021)
- [Stochastic noise can be helpful for variational quantum algorithms](https://mlsanity.com/paper/2210.06723/) (2022)
- [Quantum Neural Network Classifiers: A Tutorial](https://mlsanity.com/paper/2206.02806/) (2022)
- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Statistical Complexity of Quantum Learning](https://mlsanity.com/paper/2309.11617/) (2023)
- [Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric](https://mlsanity.com/paper/2303.13462/) (2023)
- [Effects of noise on the overparametrization of quantum neural networks](https://mlsanity.com/paper/2302.05059/) (2023)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Double descent in quantum kernel methods](https://mlsanity.com/paper/2501.10077/) (2025)
- [Mitigating barren plateaus of variational quantum eigensolvers](https://mlsanity.com/paper/2205.13539/) (2022)
- [Data re-uploading in Quantum Machine Learning for time series: application to traffic forecasting](https://mlsanity.com/paper/2501.12776/) (2025)
- [sQUlearn -- A Python Library for Quantum Machine Learning](https://mlsanity.com/paper/2311.08990/) (2023)
- [Entanglement Diagnostics for Efficient Quantum Computation](https://mlsanity.com/paper/2102.12534/) (2021)
- [Beyond Ans"atze: Learning Quantum Circuits as Unitary Operators](https://mlsanity.com/paper/2203.00601/) (2022)
- [Data-Dependent Generalization Bounds for Parameterized Quantum Models Under Noise](https://mlsanity.com/paper/2412.11451/) (2024)
- [Quantum-enhanced causal discovery for a small number of samples](https://mlsanity.com/paper/2501.05007/) (2025)
- [Variational Optimization for Quantum Problems using Deep Generative Networks](https://mlsanity.com/paper/2404.18041/) (2024)
- [A Fast Quantum Image Compression Algorithm based on Taylor Expansion](https://mlsanity.com/paper/2502.10684/) (2025)
- [Quantum Complex-Valued Self-Attention Model](https://mlsanity.com/paper/2503.19002/) (2025)
- [Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms](https://mlsanity.com/paper/2506.13865/) (2025)
- [FD4QC: Application of Classical and Quantum-Hybrid Machine Learning for Financial Fraud Detection A Technical Report](https://mlsanity.com/paper/2507.19402/) (2025)
- [From Reachability to Learnability: Geometric Design Principles for Quantum Neural Networks](https://mlsanity.com/paper/2603.03071/) (2026)

## Abstract (arXiv)

To harness the potential of noisy intermediate-scale quantum devices, it is paramount to find the best type of circuits to run hybrid quantum-classical algorithms. Key candidates are parametrized quantum circuits that can be effectively implemented on current devices. Here, we evaluate the capacity and trainability of these circuits using the geometric structure of the parameter space via the effective quantum dimension, which reveals the expressive power of circuits in general as well as of particular initialization strategies. We assess the expressive power of various popular circuit types and find striking differences depending on the type of entangling gates used. Particular circuits are characterized by scaling laws in their expressiveness. We identify a transition in the quantum geometry of the parameter space, which leads to a decay of the quantum natural gradient for deep circuits. For shallow circuits, the quantum natural gradient can be orders of magnitude larger in value compared to the regular gradient; however, both of them can suffer from vanishing gradients. By tuning a fixed set of circuit parameters to randomized ones, we find a region where the circuit is expressive, but does not suffer from barren plateaus, hinting at a good way to initialize circuits. We show an algorithm that prunes redundant parameters of a circuit without affecting its effective dimension. Our results enhance the understanding of parametrized quantum circuits and can be immediately applied to improve variational quantum algorithms.