# Reduced Density Matrices Through Machine Learning

arXiv: https://arxiv.org/abs/2511.07367 · HTML: https://mlsanity.com/paper/2511.07367/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 39 references that are in the map.
- First posted on arXiv 10 Nov 2025.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [Transformers Can Achieve Length Generalization But Not Robustly](https://mlsanity.com/paper/2402.09371/) (2024)
- [A Self-Attention Ansatz for Ab-initio Quantum Chemistry](https://mlsanity.com/paper/2211.13672/) (2022)
- [Scalable-Softmax Is Superior for Attention](https://mlsanity.com/paper/2501.19399/) (2025)
- [Gauge Equivariant Neural Networks for 2+1D U(1) Gauge Theory Simulations in Hamiltonian Formulation](https://mlsanity.com/paper/2211.03198/) (2022)
- [Simulating 2+1D Lattice Quantum Electrodynamics at Finite Density with Neural Flow Wavefunctions](https://mlsanity.com/paper/2212.06835/) (2022)
- [Pairing-based graph neural network for simulating quantum materials](https://mlsanity.com/paper/2311.02143/) (2023)
- [Solving the Hubbard model with Neural Quantum States](https://mlsanity.com/paper/2507.02644/) (2025)
- [Is attention all you need to solve the correlated electron problem?](https://mlsanity.com/paper/2502.05383/) (2025)

## Built on by (most cited first, 1 of 1)

- [Representability-Aware Neural Networks for Reduced Density Matrices: Application to Fractional Chern Insulators](https://mlsanity.com/paper/2605.20326/) (2026)

## Abstract (arXiv)

n-particle reduced density matrices (n-RDMs) play a central role in understanding correlated phases of matter, but their calculation is often computationally inefficient for strongly-correlated states at large system sizes. In this work, we use neural network (NN) architectures to accelerate and even predict n-RDMs for large systems. Our underlying intuition is that, for gapped states, n-RDMs are often smooth functions over the Brillouin zone (BZ) and are therefore interpolable, allowing NNs trained on small-size systems to predict large-size ones. Building on this, we devise two NNs: (i) a self-attention NN that maps random RDMs to physical ones, and (ii) a Sinusoidal Representation Network (SIREN) that directly maps momentum-space coordinates to RDM values. We test the NNs on RDMs in three 2D models: the pair-pair correlation functions of the Richardson model of superconductivity, the translationally-invariant Hartree-Fock (HF) 1-RDM in a four-band repulsive model, and the translation-breaking HF 1-RDM in the half-filled Hubbard model. We find that a SIREN trained on a 6× 6 momentum mesh and a SIREN trained on 4 tilted meshes (each of which has 12 momentum points) can predict the 18× 18 pair-pair correlation function with a relative accuracy of 94.29% and 93.77%, respectively. NNs trained on 6× 6 and 8× 8 meshes provide high-quality initial guesses for 50× 50 translation-invariant HF and 30× 30 fully translation-breaking-allowed HF, reducing the required number of iterations by up to 91.63% and 92.78%, respectively, compared to random initializations. Our results illustrate the potential of NN-based methods for interpolable n-RDMs, which might open a new avenue for future research on strongly correlated phases.