# Ab-initio study of interacting fermions at finite temperature with neural canonical transformation

arXiv: https://arxiv.org/abs/2105.08644 · HTML: https://mlsanity.com/paper/2105.08644/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 115 references that are in the map.
- First posted on arXiv 18 May 2021.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [Neural Network Renormalization Group](https://mlsanity.com/paper/1802.02840/) (2018)
- [The Expressive Power of a Class of Normalizing Flow Models](https://mlsanity.com/paper/2006.00392/) (2020)
- [Monge-Amp`ere Flow for Generative Modeling](https://mlsanity.com/paper/1809.10188/) (2018)
- [Better, Faster Fermionic Neural Networks](https://mlsanity.com/paper/2011.07125/) (2020)
- [Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm](https://mlsanity.com/paper/1910.02071/) (2019)
- [Exchangeable Neural ODE for Set Modeling](https://mlsanity.com/paper/2008.02676/) (2020)
- [Neural Canonical Transformation with Symplectic Flows](https://mlsanity.com/paper/1910.00024/) (2019)
- [Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states](https://mlsanity.com/paper/2008.00118/) (2020)
- [Inferring the quantum density matrix with machine learning](https://mlsanity.com/paper/1904.05903/) (2019)

## Built on by (most cited first, 16 of 16)

- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (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)
- [m^ast of two-dimensional electron gas: a neural canonical transformation study](https://mlsanity.com/paper/2201.03156/) (2022)
- [Deep Variational Free Energy Approach to Dense Hydrogen](https://mlsanity.com/paper/2209.06095/) (2022)
- [Ab-initio variational wave functions for the time-dependent many-electron Schr"odinger equation](https://mlsanity.com/paper/2403.07447/) (2024)
- [Deep generative modelling of canonical ensemble with differentiable thermal properties](https://mlsanity.com/paper/2404.18404/) (2024)
- [Numerical and geometrical aspects of flow-based variational quantum Monte Carlo](https://mlsanity.com/paper/2203.14824/) (2022)
- [Waveflow: boundary-conditioned normalizing flows applied to fermionic wavefunctions](https://mlsanity.com/paper/2211.14839/) (2022)
- [Artificial intelligence for artificial materials: moir'e atom](https://mlsanity.com/paper/2303.08162/) (2023)
- [Sorting Out Quantum Monte Carlo](https://mlsanity.com/paper/2311.05598/) (2023)
- [A Theoretical Framework for an Efficient Normalizing Flow-Based Solution to the Electronic Schrodinger Equation](https://mlsanity.com/paper/2406.00047/) (2024)
- [Inducing Riesz and orthonormal bases in L2 via composition operators](https://mlsanity.com/paper/2406.18613/) (2024)
- [Accurate and thermodynamically consistent hydrogen equation of state for planetary modeling with flow matching](https://mlsanity.com/paper/2501.10594/) (2025)
- [Convergence theory for Hermite approximations under adaptive coordinate transformations](https://mlsanity.com/paper/2604.16975/) (2026)
- [Quantum Dynamics via Score Matching on Bohmian Trajectories](https://mlsanity.com/paper/2604.25137/) (2026)

## Abstract (arXiv)

We present a variational density matrix approach to the thermal properties of interacting fermions in the continuum. The variational density matrix is parametrized by a permutation equivariant many-body unitary transformation together with a discrete probabilistic model. The unitary transformation is implemented as a quantum counterpart of neural canonical transformation, which incorporates correlation effects via a flow of fermion coordinates. As the first application, we study electrons in a two-dimensional quantum dot with an interaction-induced crossover from Fermi liquid to Wigner molecule. The present approach provides accurate results in the low-temperature regime, where conventional quantum Monte Carlo methods face severe difficulties due to the fermion sign problem. The approach is general and flexible for further extensions, thus holds the promise to deliver new physical results on strongly correlated fermions in the context of ultracold quantum gases, condensed matter, and warm dense matter physics.