# Quantum State Generation with Structure-Preserving Diffusion Model

arXiv: https://arxiv.org/abs/2404.06336 · HTML: https://mlsanity.com/paper/2404.06336/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 36 of its 64 references that are in the map.
- First posted on arXiv 9 Apr 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 36)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2304.08818/) (2023)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [Preserve Your Own Correlation: A Noise Prior for Video Diffusion Models](https://mlsanity.com/paper/2305.10474/) (2023)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Residual Corrective Diffusion Modeling for Km-scale Atmospheric Downscaling](https://mlsanity.com/paper/2309.15214/) (2023)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [Quantum Variational Autoencoder](https://mlsanity.com/paper/1802.05779/) (2018)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [Quantum Wasserstein Generative Adversarial Networks](https://mlsanity.com/paper/1911.00111/) (2019)
- [Accurate transition state generation with an object-aware equivariant elementary reaction diffusion model](https://mlsanity.com/paper/2304.06174/) (2023)
- [Generative quantum machine learning via denoising diffusion probabilistic models](https://mlsanity.com/paper/2310.05866/) (2023)

## Built on by (most cited first, 8 of 8)

- [How Discrete and Continuous Diffusion Meet: Comprehensive Analysis of Discrete Diffusion Models via a Stochastic Integral Framework](https://mlsanity.com/paper/2410.03601/) (2024)
- [Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexity](https://mlsanity.com/paper/2405.15986/) (2024)
- [Diffusion Generative Modeling for Spatially Resolved Gene Expression Inference from Histology Images](https://mlsanity.com/paper/2501.15598/) (2025)
- [Evaluating the design space of diffusion-based generative models](https://mlsanity.com/paper/2406.12839/) (2024)
- [DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Models](https://mlsanity.com/paper/2509.21655/) (2025)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Mixed-State Quantum Denoising Diffusion Probabilistic Model](https://mlsanity.com/paper/2411.17608/) (2024)
- [Generative modeling assisted simulation of measurement-altered quantum criticality](https://mlsanity.com/paper/2412.01513/) (2024)

## Abstract (arXiv)

This article considers the generative modeling of the (mixed) states of quantum systems, and an approach based on denoising diffusion model is proposed. The key contribution is an algorithmic innovation that respects the physical nature of quantum states. More precisely, the commonly used density matrix representation of mixed-state has to be complex-valued Hermitian, positive semi-definite, and trace one. Generic diffusion models, or other generative methods, may not be able to generate data that strictly satisfy these structural constraints, even if all training data do. To develop a machine learning algorithm that has physics hard-wired in, we leverage mirror diffusion and borrow the physical notion of von Neumann entropy to design a new map, for enabling strict structure-preserving generation. Both unconditional generation and conditional generation via classifier-free guidance are experimentally demonstrated efficacious, the latter enabling the design of new quantum states when generated on unseen labels.