# Quantum activation functions for quantum neural networks

arXiv: https://arxiv.org/abs/2201.03700 · HTML: https://mlsanity.com/paper/2201.03700/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 10 Jan 2022.

## Builds on (most cited first, 11 of 11)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Quantum Neuron: an elementary building block for machine learning on quantum computers](https://mlsanity.com/paper/1711.11240/) (2017)
- [Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits](https://mlsanity.com/paper/2001.00550/) (2020)
- [Satellite Image Classification with Deep Learning](https://mlsanity.com/paper/2010.06497/) (2020)
- [Quantum perceptron over a field and neural network architecture selection in a quantum computer](https://mlsanity.com/paper/1602.00709/) (2016)
- [A Simple Quantum Neural Net with a Periodic Activation Function](https://mlsanity.com/paper/1804.07633/) (2018)
- [Neural Networks Architecture Evaluation in a Quantum Computer](https://mlsanity.com/paper/1711.04759/) (2017)
- [Weightless neural network parameters and architecture selection in a quantum computer](https://mlsanity.com/paper/1601.03277/) (2016)

## Built on by (most cited first, 10 of 10)

- [Learning with SASQuaTCh: a Novel Variational Quantum Transformer Architecture with Kernel-Based Self-Attention](https://mlsanity.com/paper/2403.14753/) (2024)
- [Quantum-Cognitive Neural Networks: Assessing Confidence and Uncertainty with Human Decision-Making Simulations](https://mlsanity.com/paper/2412.08010/) (2024)
- [Enabling Non-Linear Quantum Operations through Variational Quantum Splines](https://mlsanity.com/paper/2303.04788/) (2023)
- [Assessing the Impact of Noise on Quantum Neural Networks: An Experimental Analysis](https://mlsanity.com/paper/2311.14057/) (2023)
- [Quantum Machine Learning in Drug Discovery: Applications in Academia and Pharmaceutical Industries](https://mlsanity.com/paper/2409.15645/) (2024)
- [Efficient Quantum Circuits for Machine Learning Activation Functions including Constant T-depth ReLU](https://mlsanity.com/paper/2404.06059/) (2024)
- [A Tutorial on the Use of Physics-Informed Neural Networks to Compute the Spectrum of Quantum Systems](https://mlsanity.com/paper/2407.20669/) (2024)
- [Hybrid Quantum Neural Networks for Efficient Protein-Ligand Binding Affinity Prediction](https://mlsanity.com/paper/2509.11046/) (2025)
- [Training quantum machine learning models on cloud without uploading the data](https://mlsanity.com/paper/2409.04602/) (2024)
- [Solving the encoding bottleneck: of the HHL algorithm, by the HHL algorithm](https://mlsanity.com/paper/2502.13534/) (2025)

## Abstract (arXiv)

The field of artificial neural networks is expected to strongly benefit from recent developments of quantum computers. In particular, quantum machine learning, a class of quantum algorithms which exploit qubits for creating trainable neural networks, will provide more power to solve problems such as pattern recognition, clustering and machine learning in general. The building block of feed-forward neural networks consists of one layer of neurons connected to an output neuron that is activated according to an arbitrary activation function. The corresponding learning algorithm goes under the name of Rosenblatt perceptron. Quantum perceptrons with specific activation functions are known, but a general method to realize arbitrary activation functions on a quantum computer is still lacking. Here we fill this gap with a quantum algorithm which is capable to approximate any analytic activation functions to any given order of its power series. Unlike previous proposals providing irreversible measurement--based and simplified activation functions, here we show how to approximate any analytic function to any required accuracy without the need to measure the states encoding the information. Thanks to the generality of this construction, any feed-forward neural network may acquire the universal approximation properties according to Hornik's theorem. Our results recast the science of artificial neural networks in the architecture of gate-model quantum computers.