# Learning Nonlinear Input-Output Maps with Dissipative Quantum Systems

## TL;DR

This paper develops a theory for using dissipative quantum systems to learn nonlinear input-output maps with fading memory, showing they can be universal approximators and potentially outperform classical methods with fewer resources.

## Contribution

It introduces a theoretical framework for dissipative quantum systems as universal learners of fading memory maps and provides an example class with provable universality.

## Key findings

- Small quantum systems can match classical performance
- Quantum systems have potential to surpass classical methods
- Exponential Hilbert space offers resource advantages

## Abstract

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.