# Inferring the quantum density matrix with machine learning

arXiv: https://arxiv.org/abs/1904.05903 · HTML: https://mlsanity.com/paper/1904.05903/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Apr 2019.

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

- [Ab-initio study of interacting fermions at finite temperature with neural canonical transformation](https://mlsanity.com/paper/2105.08644/) (2021)
- [m^ast of two-dimensional electron gas: a neural canonical transformation study](https://mlsanity.com/paper/2201.03156/) (2022)
- [Exponentially improved efficient machine learning for quantum many-body states with provable guarantees](https://mlsanity.com/paper/2304.04353/) (2023)
- [Computing excited states of molecules using normalizing flows](https://mlsanity.com/paper/2308.16468/) (2023)
- [Estimating the Euclidean quantum propagator with deep generative modeling of Feynman paths](https://mlsanity.com/paper/2202.02750/) (2022)
- [Numerical and geometrical aspects of flow-based variational quantum Monte Carlo](https://mlsanity.com/paper/2203.14824/) (2022)
- [Inducing Riesz and orthonormal bases in L2 via composition operators](https://mlsanity.com/paper/2406.18613/) (2024)
- [Quantum Maximum Entropy Inference and Hamiltonian Learning](https://mlsanity.com/paper/2407.11473/) (2024)

## Abstract (arXiv)

We introduce two methods for estimating the density matrix for a quantum system: Quantum Maximum Likelihood and Quantum Variational Inference. In these methods, we construct a variational family to model the density matrix of a mixed quantum state. We also introduce quantum flows, the quantum analog of normalizing flows, which can be used to increase the expressivity of this variational family. The eigenstates and eigenvalues of interest are then derived by optimizing an appropriate loss function. The approach is qualitatively different than traditional lattice techniques that rely on the time dependence of correlation functions that summarize the lattice configurations. The resulting estimate of the density matrix can then be used to evaluate the expectation of an arbitrary operator, which opens the door to new possibilities.