# The Expressive Power of Parameterized Quantum Circuits

arXiv: https://arxiv.org/abs/1810.11922 · HTML: https://mlsanity.com/paper/1810.11922/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2018.

## Built on by (most cited first, 18 of 18)

- [Federated Quantum Machine Learning](https://mlsanity.com/paper/2103.12010/) (2021)
- [Variational Quantum Reinforcement Learning via Evolutionary Optimization](https://mlsanity.com/paper/2109.00540/) (2021)
- [Quantum machine learning with differential privacy](https://mlsanity.com/paper/2103.06232/) (2021)
- [Quantum deep recurrent reinforcement learning](https://mlsanity.com/paper/2210.14876/) (2022)
- [Hybrid Quantum-Classical Graph Convolutional Network](https://mlsanity.com/paper/2101.06189/) (2021)
- [Federated Quantum Machine Learning with Differential Privacy](https://mlsanity.com/paper/2310.06973/) (2023)
- [Federated Quantum Long Short-term Memory (FedQLSTM)](https://mlsanity.com/paper/2312.14309/) (2023)
- [Reservoir Computing via Quantum Recurrent Neural Networks](https://mlsanity.com/paper/2211.02612/) (2022)
- [Learning quantum data with the quantum Earth Mover's distance](https://mlsanity.com/paper/2101.03037/) (2021)
- [Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series](https://mlsanity.com/paper/2108.03190/) (2021)
- [eQMARL: Entangled Quantum Multi-Agent Reinforcement Learning for Distributed Cooperation over Quantum Channels](https://mlsanity.com/paper/2405.17486/) (2024)
- [Quantum Embedding Search for Quantum Machine Learning](https://mlsanity.com/paper/2105.11853/) (2021)
- [Generative Invertible Quantum Neural Networks](https://mlsanity.com/paper/2302.12906/) (2023)
- [An efficient combination strategy for hybird quantum ensemble classifier](https://mlsanity.com/paper/2210.06785/) (2022)
- [Hybrid Quantum-Classical Recurrent Neural Networks](https://mlsanity.com/paper/2510.25557/) (2025)
- [Quantum Privacy Aggregation of Teacher Ensembles (QPATE) for Privacy-preserving Quantum Machine Learning](https://mlsanity.com/paper/2401.07464/) (2024)
- [Sculpting Quantum Landscapes: Fubini-Study Metric Conditioning for Geometry Aware Learning in Parameterized Quantum Circuits](https://mlsanity.com/paper/2506.21940/) (2025)
- [Hybrid quantum-classical neural network for sentiment analysis](https://mlsanity.com/paper/2607.01943/) (2026)

## Abstract (arXiv)

Parameterized quantum circuits (PQCs) have been broadly used as a hybrid quantum-classical machine learning scheme to accomplish generative tasks. However, whether PQCs have better expressive power than classical generative neural networks, such as restricted or deep Boltzmann machines, remains an open issue. In this paper, we prove that PQCs with a simple structure already outperform any classical neural network for generative tasks, unless the polynomial hierarchy collapses. Our proof builds on known results from tensor networks and quantum circuits (in particular, instantaneous quantum polynomial circuits). In addition, PQCs equipped with ancillary qubits for post-selection have even stronger expressive power than those without post-selection. We employ them as an application for Bayesian learning, since it is possible to learn prior probabilities rather than assuming they are known. We expect that it will find many more applications in semi-supervised learning where prior distributions are normally assumed to be unknown. Lastly, we conduct several numerical experiments using the Rigetti Forest platform to demonstrate the performance of the proposed Bayesian quantum circuit.