# Learning quantum data with the quantum Earth Mover's distance

arXiv: https://arxiv.org/abs/2101.03037 · HTML: https://mlsanity.com/paper/2101.03037/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 52 of its 152 references that are in the map.
- First posted on arXiv 8 Jan 2021.

## Builds on (most cited first, 30 of 52)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Self-Attention Generative Adversarial Networks](https://mlsanity.com/paper/1805.08318/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Recent Advances in Convolutional Neural Networks](https://mlsanity.com/paper/1512.07108/) (2015)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [DualGAN: Unsupervised Dual Learning for Image-to-Image Translation](https://mlsanity.com/paper/1704.02510/) (2017)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Absence of Barren Plateaus in Quantum Convolutional Neural Networks](https://mlsanity.com/paper/2011.02966/) (2020)
- [Stabilizing Training of Generative Adversarial Networks through Regularization](https://mlsanity.com/paper/1705.09367/) (2017)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [CAN: Creative Adversarial Networks, Generating "Art" by Learning About Styles and Deviating from Style Norms](https://mlsanity.com/paper/1706.07068/) (2017)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations](https://mlsanity.com/paper/1903.05895/) (2019)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [Continuous-variable quantum neural networks](https://mlsanity.com/paper/1806.06871/) (2018)
- [Quantum Adversarial Machine Learning](https://mlsanity.com/paper/2001.00030/) (2019)
- [Tunable Efficient Unitary Neural Networks (EUNN) and their application to RNNs](https://mlsanity.com/paper/1612.05231/) (2016)

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

- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Generative quantum machine learning via denoising diffusion probabilistic models](https://mlsanity.com/paper/2310.05866/) (2023)
- [Hybrid Quantum-Classical Generative Adversarial Network for High Resolution Image Generation](https://mlsanity.com/paper/2212.11614/) (2022)
- [Explaining Quantum Circuits with Shapley Values: Towards Explainable Quantum Machine Learning](https://mlsanity.com/paper/2301.09138/) (2023)
- [Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series](https://mlsanity.com/paper/2108.03190/) (2021)
- [Quantum Generative Adversarial Networks: Bridging Classical and Quantum Realms](https://mlsanity.com/paper/2312.09939/) (2023)
- [Trained quantum neural networks are Gaussian processes](https://mlsanity.com/paper/2402.08726/) (2024)
- [Quantum Capsule Networks](https://mlsanity.com/paper/2201.01778/) (2022)
- [Energy-dependent barren plateau in bosonic variational quantum circuits](https://mlsanity.com/paper/2305.01799/) (2023)
- [Scaling Quantum Machine Learning without Tricks: Full-Resolution and Diverse Image Generation](https://mlsanity.com/paper/2603.00233/) (2026)
- [Hybrid Quantum Generative Adversarial Networks for Molecular Simulation and Drug Discovery](https://mlsanity.com/paper/2212.07826/) (2022)
- [Quantum Wasserstein GANs for State Preparation at Unseen Points of a Phase Diagram](https://mlsanity.com/paper/2309.09543/) (2023)
- [Mean-field limit from general mixtures of experts to quantum neural networks](https://mlsanity.com/paper/2501.14660/) (2025)

## Abstract (arXiv)

Quantifying how far the output of a learning algorithm is from its target is an essential task in machine learning. However, in quantum settings, the loss landscapes of commonly used distance metrics often produce undesirable outcomes such as poor local minima and exponentially decaying gradients. To overcome these obstacles, we consider here the recently proposed quantum earth mover's (EM) or Wasserstein-1 distance as a quantum analog to the classical EM distance. We show that the quantum EM distance possesses unique properties, not found in other commonly used quantum distance metrics, that make quantum learning more stable and efficient. We propose a quantum Wasserstein generative adversarial network (qWGAN) which takes advantage of the quantum EM distance and provides an efficient means of performing learning on quantum data. We provide examples where our qWGAN is capable of learning a diverse set of quantum data with only resources polynomial in the number of qubits.