# Positive-definite parametrization of mixed quantum states with deep neural networks

arXiv: https://arxiv.org/abs/2206.13488 · HTML: https://mlsanity.com/paper/2206.13488/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 24 references that are in the map.
- First posted on arXiv 27 Jun 2022.

## Builds on (most cited first, 3 of 3)

- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Better, Faster Fermionic Neural Networks](https://mlsanity.com/paper/2011.07125/) (2020)
- [Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo](https://mlsanity.com/paper/2106.10558/) (2021)

## Built on by (most cited first, 4 of 4)

- [Quantum State Generation with Structure-Preserving Diffusion Model](https://mlsanity.com/paper/2404.06336/) (2024)
- [Empirical Sample Complexity of Neural Network Mixed State Reconstruction](https://mlsanity.com/paper/2307.01840/) (2023)
- [Learning Energy-Based Representations of Quantum Many-Body States](https://mlsanity.com/paper/2304.04058/) (2023)
- [Fermionic neural Gibbs states](https://mlsanity.com/paper/2512.04663/) (2025)

## Abstract (arXiv)

We introduce the Gram-Hadamard Density Operator (GHDO), a new deep neural-network architecture that can encode positive semi-definite density operators of exponential rank with polynomial resources. We then show how to embed an autoregressive structure in the GHDO to allow direct sampling of the probability distribution. These properties are especially important when representing and variationally optimizing the mixed quantum state of a system interacting with an environment. Finally, we benchmark this architecture by simulating the steady state of the dissipative transverse-field Ising model. Estimating local observables and the R'enyi entropy, we show significant improvements over previous state-of-the-art variational approaches.