# Feedback-driven recurrent quantum neural network universality

arXiv: https://arxiv.org/abs/2506.16332 · HTML: https://mlsanity.com/paper/2506.16332/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 78 references that are in the map.
- First posted on arXiv 19 Jun 2025.

## Builds on (most cited first, 27 of 27)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Recurrent Quantum Neural Networks](https://mlsanity.com/paper/2006.14619/) (2020)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [Reservoir Computing Universality With Stochastic Inputs](https://mlsanity.com/paper/1807.02621/) (2018)
- [Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems](https://mlsanity.com/paper/1712.00754/) (2017)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Fading memory echo state networks are universal](https://mlsanity.com/paper/2010.12047/) (2020)
- [Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression](https://mlsanity.com/paper/2206.04804/) (2022)
- [Quantum Recurrent Neural Networks for Sequential Learning](https://mlsanity.com/paper/2302.03244/) (2023)
- [Temporal Information Processing on Noisy Quantum Computers](https://mlsanity.com/paper/2001.09498/) (2020)
- [Approximation and Estimation for High-Dimensional Deep Learning Networks](https://mlsanity.com/paper/1809.03090/) (2018)
- [Risk bounds for reservoir computing](https://mlsanity.com/paper/1910.13886/) (2019)
- [Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs](https://mlsanity.com/paper/2307.12904/) (2023)
- [Learning Nonlinear Input-Output Maps with Dissipative Quantum Systems](https://mlsanity.com/paper/1901.01653/) (2019)
- [Rapid training of quantum recurrent neural networks](https://mlsanity.com/paper/2207.00378/) (2022)
- [Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits](https://mlsanity.com/paper/2310.07528/) (2023)
- [Learning Temporal Quantum Tomography](https://mlsanity.com/paper/2103.13973/) (2021)
- [Deep neural network expressivity for optimal stopping problems](https://mlsanity.com/paper/2210.10443/) (2022)
- [Optimal training of finitely-sampled quantum reservoir computers for forecasting of chaotic dynamics](https://mlsanity.com/paper/2409.01394/) (2024)
- [User Trajectory Prediction in Mobile Wireless Networks Using Quantum Reservoir Computing](https://mlsanity.com/paper/2301.08796/) (2023)
- [State-space systems as dynamic generative models](https://mlsanity.com/paper/2404.08717/) (2024)
- [Role of scrambling and noise in temporal information processing with quantum systems](https://mlsanity.com/paper/2505.10080/) (2025)
- [Quantum Reservoir Computing and Risk Bounds](https://mlsanity.com/paper/2501.08640/) (2025)

## Built on by (most cited first, 5 of 5)

- [A Primer on Quantum Machine Learning](https://mlsanity.com/paper/2511.15969/) (2025)
- [Echoes of the Past: A Unified Perspective on Fading memory and Echo States](https://mlsanity.com/paper/2508.19145/) (2025)
- [Approximation rates of quantum neural networks for periodic functions via Jackson's inequality](https://mlsanity.com/paper/2511.16149/) (2025)
- [Recurrent Quantum Feature Maps for Reservoir Computing](https://mlsanity.com/paper/2604.03469/) (2026)
- [Controllable Quantum Memory Capacity in Quantum Reservoir Networks with Tunable partial-SWAPs](https://mlsanity.com/paper/2605.12713/) (2026)

## Abstract (arXiv)

Quantum reservoir computing uses the dynamics of quantum systems to process temporal data, making it particularly well-suited for machine learning with noisy intermediate-scale quantum devices. Recent developments have introduced feedback-based quantum reservoir systems, which process temporal information with comparatively fewer components and enable real-time computation while preserving the input history. Motivated by their promising empirical performance, in this work, we study the approximation capabilities of feedback-based quantum reservoir computing. More specifically, we are concerned with recurrent quantum neural networks, which are quantum analogues of classical recurrent neural networks. Our results show that regular state-space systems can be approximated using quantum recurrent neural networks without the curse of dimensionality and with the number of qubits only growing logarithmically in the reciprocal of the prescribed approximation accuracy. Notably, our analysis demonstrates that quantum recurrent neural networks are universal with linear readouts, making them both powerful and experimentally accessible. These results pave the way for practical and theoretically grounded quantum reservoir computing with real-time processing capabilities.