# Introducing Non-Linear Activations into Quantum Generative Models

arXiv: https://arxiv.org/abs/2205.14506 · HTML: https://mlsanity.com/paper/2205.14506/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 37 references that are in the map.
- First posted on arXiv 28 May 2022.

## Builds on (most cited first, 14 of 14)

- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [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)
- [Stochastic Variational Video Prediction](https://mlsanity.com/paper/1710.11252/) (2017)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Experimental Quantum Generative Adversarial Networks for Image Generation](https://mlsanity.com/paper/2010.06201/) (2020)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [Quantum Generative Models for Small Molecule Drug Discovery](https://mlsanity.com/paper/2101.03438/) (2021)
- [Quantum Neuron: an elementary building block for machine learning on quantum computers](https://mlsanity.com/paper/1711.11240/) (2017)
- [An Introduction to Image Synthesis with Generative Adversarial Nets](https://mlsanity.com/paper/1803.04469/) (2018)
- [Maximum Mean Discrepancy for Generalization in the Presence of Distribution and Missingness Shift](https://mlsanity.com/paper/2111.10344/) (2021)
- [Quantum Unsupervised and Supervised Learning on Superconducting Processors](https://mlsanity.com/paper/1909.04226/) (2019)

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

- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [Quantum Natural Language Processing](https://mlsanity.com/paper/2403.19758/) (2024)
- [Learning with SASQuaTCh: a Novel Variational Quantum Transformer Architecture with Kernel-Based Self-Attention](https://mlsanity.com/paper/2403.14753/) (2024)
- [A supplemental investigation of non-linearity in quantum generative models with respect to simulatability and optimization](https://mlsanity.com/paper/2302.00788/) (2023)
- [Quantum Scrambling Born Machine](https://mlsanity.com/paper/2602.17281/) (2026)

## Abstract (arXiv)

Due to the linearity of quantum mechanics, it remains a challenge to design quantum generative machine learning models that embed non-linear activations into the evolution of the statevector. However, some of the most successful classical generative models, such as those based on neural networks, involve highly non-linear dynamics for quality training. In this paper, we explore the effect of these dynamics in quantum generative modeling by introducing a model that adds non-linear activations via a neural network structure onto the standard Born Machine framework - the Quantum Neuron Born Machine (QNBM). To achieve this, we utilize a previously introduced Quantum Neuron subroutine, which is a repeat-until-success circuit with mid-circuit measurements and classical control. After introducing the QNBM, we investigate how its performance depends on network size, by training a 3-layer QNBM with 4 output neurons and various input and hidden layer sizes. We then compare our non-linear QNBM to the linear Quantum Circuit Born Machine (QCBM). We allocate similar time and memory resources to each model, such that the only major difference is the qubit overhead required by the QNBM. With gradient-based training, we show that while both models can easily learn a trivial uniform probability distribution, on a more challenging class of distributions, the QNBM achieves an almost 3x smaller error rate than a QCBM with a similar number of tunable parameters. We therefore provide evidence that suggests that non-linearity is a useful resource in quantum generative models, and we put forth the QNBM as a new model with good generative performance and potential for quantum advantage.