# Learning Nonlinear Input-Output Maps with Dissipative Quantum Systems

arXiv: https://arxiv.org/abs/1901.01653 · HTML: https://mlsanity.com/paper/1901.01653/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2019.

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

- [Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs](https://mlsanity.com/paper/2307.12904/) (2023)
- [Learning Temporal Quantum Tomography](https://mlsanity.com/paper/2103.13973/) (2021)
- [Classical and Quantum Physical Reservoir Computing for Onboard Artificial Intelligence Systems: A Perspective](https://mlsanity.com/paper/2407.04717/) (2024)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [Feedback-driven recurrent quantum neural network universality](https://mlsanity.com/paper/2506.16332/) (2025)
- [Stochastic Reservoir Computers](https://mlsanity.com/paper/2405.12382/) (2024)
- [Nonlinear Autoregression with Convergent Dynamics on Novel Computational Platforms](https://mlsanity.com/paper/2108.08001/) (2021)
- [Quantum Noise-Induced Reservoir Computing](https://mlsanity.com/paper/2207.07924/) (2022)
- [Quantum Reservoir Computing and Risk Bounds](https://mlsanity.com/paper/2501.08640/) (2025)
- [QuaRK: A Quantum Reservoir Kernel for Time Series Learning](https://mlsanity.com/paper/2602.13531/) (2026)
- [Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach](https://mlsanity.com/paper/2602.18377/) (2026)
- [A Solvable Molecular Switch Model for Stable Temporal Information Processing](https://mlsanity.com/paper/2508.15451/) (2025)
- [Quantum Reservoir Computing: Recent Advances and Future Directions](https://mlsanity.com/paper/2607.18552/) (2026)

## Abstract (arXiv)

In this paper, we develop a theory of learning nonlinear input-output maps with fading memory by dissipative quantum systems, as a quantum counterpart of the theory of approximating such maps using classical dynamical systems. The theory identifies the properties required for a class of dissipative quantum systems to be em universal, in that any input-output map with fading memory can be approximated arbitrarily closely by an element of this class. We then introduce an example class of dissipative quantum systems that is provably universal. Numerical experiments illustrate that with a small number of qubits, this class can achieve comparable performance to classical learning schemes with a large number of tunable parameters. Further numerical analysis suggests that the exponentially increasing Hilbert space presents a potential resource for dissipative quantum systems to surpass classical learning schemes for input-output maps.