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arXiv:2005.10672 · Faraday Discussions

Insights into one-body density matrices using deep learning

Jack Wetherell, Andrea Costamagna, Matteo Gatti, Lucia Reining

Published Jan 1, 2020• 6 citations
arXivHTMLPDF

Abstract

The one-body reduced density matrix (1-RDM) of a many-body system at zero temperature gives direct access to many observables, such as the charge density, kinetic energy and occupation numbers. It would be desirable to express it as a simple functional of the density or of other local observables, but to date satisfactory approximations have not yet been found. Deep learning is the state of the art approach to performing high dimensional regressions and classification tasks, and is becoming widely used in the condensed matter community to develop increasingly accurate density functionals. Autoencoders are deep learning models that perform efficient dimensionality reduction, allowing the distillation of data to the fundamental features needed to represent it. By training autoencoders on a large data-set of 1-RDMs from exactly solvable real-space model systems, and performing principal component analysis, the machine learns to what extent the data can be compressed and hence how it is constrained. We gain insight into these machine learned constraints and employ them to inform approximations to the 1-RDM as a functional of the charge density. We exploit known physical properties of the 1-RDM in the simplest possible cases to perform feature engineering, where we inform the structure of the models from known mathematical relations, allowing us to integrate existing understanding into the machine learning methods. By comparing various deep learning approaches we gain insight into what physical features of the density matrix are most amenable to machine learning, utilising both known and learned characteristics.

DOI
10.1039/d0fd00061b

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The one-body reduced density matrix (1-RDM) of a many-body system at zero temperature gives direct access to many observables, such as the charge density, kinetic energy and occupation numbers. It would be desirable to express it as a simple functional of the density or of other local observables, but to date satisfactory approximations have not yet been found. Deep learning is the state of the art approach to performing high dimensional regressions and classification tasks, and is becoming widely used in the condensed matter community to develop increasingly accurate density functionals. Autoencoders are deep learning models that perform efficient dimensionality reduction, allowing the distillation of data to the fundamental features needed to represent it. By training autoencoders on a large data-set of 1-RDMs from exactly solvable real-space model systems, and performing principal component analysis, the machine learns to what extent the data can be compressed and hence how it is constrained. We gain insight into these machine learned constraints and employ them to inform approximations to the 1-RDM as a functional of the charge density. We exploit known physical properties of the 1-RDM in the simplest possible cases to perform feature engineering, where we inform the structure of the models from known mathematical relations, allowing us to integrate existing understanding into the machine learning methods. By comparing various deep learning approaches we gain insight into what physical features of the density matrix are most amenable to machine learning, utilising both known and learned characteristics.
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The one-body reduced density matrix γγγ plays a fundamental role in describing and predicting quantum features of bosonic systems, such as Bose-Einstein condensation. The recently proposed reduced density matrix functional theory for bosonic ground states establishes the existence of a universal functional F[γ]\mathcal{F}[γ]F[γ] that recovers quantum correlations exactly. Based on a novel decomposition of γγγ, we have developed a method to design reliable approximations for such universal functionals: our results suggest that for translational invariant systems the constrained search approach of functional theories can be transformed into an unconstrained problem through a parametrization of an Euclidian space. This simplification of the search approach allows us to use standard machine-learning methods to perform a quite efficient computation of both F[γ]\mathcal{F}[γ]F[γ] and its functional derivative. For the Bose-Hubbard model, we present a comparison between our approach and Quantum Monte Carlo.
From drug design to material science, computational chemistry has become a valuable tool for many fields of applications in the past decades. Quantum chemistry in particular, can provide accurate descriptions of molecules and solids by relying on principles of quantum mechanics with the Schrödinger equation at its core. Unfortunately this equation becomes prohibitively expensive to solve for systems made of more than a few electrons and one has to rely on other approaches. Among the most popular is Density Functional Theory, relatively inexpensive and exact in principle. One has, however, to introduce approximations in practice, deteriorating its accuracy for challenging, so-called strongly correlated, systems. On the contrary, other methods like CASSCF remain reliable for these systems, but although cheaper than the solution of the Schrödinger equation, remain too expensive for extended systems. There is therefore room for methods in between, that could handle large correlated systems with reasonable accuracy. One such method is the One Body Reduced Density Matrix (1-RDM) Functional Theory (RDMFT), based on the fact that an electronic system can, in principle exactly, be described by its 1-RDM only. Although its theory dates back to the 1970s, computational hindrances have prevented this method from shining so far. Indeed, computations in RDMFT rely on a constrained optimization of the eigenvalues and eigenvectors of the 1-RDM, that can be reformulated as an unconstrained optimization by parameterizing the latter but in both cases suffers from convergence problems. In this thesis, the role of this parametrization is analyzed and motivates a now second order optimization algorithm, which improves the convergence speed by two orders of magnitude w.r.t. the state of the art algorithm. This understanding of the problem also inspires an alternative approach with the 1-RDM as the optimization variable. This however requires the development of new approximations and a first investigation in that direction, based on Machine Learning is performed. Lastly, a generalization of the method to excited states is attempted, highlighting some key computational challenges.
A striking consequence of the Hohenberg-Kohn theorem of density functional theory is the existence of a bijection between the local density and the ground-state many-body wave function. Here we study the problem of constructing approximations to the Hohenberg-Kohn map using a statistical learning approach. Using supervised deep learning with synthetic data, we show that this map can be accurately constructed for a chain of one-dimensional interacting spinless fermions in different phases of this model including the charge ordered Mott insulator and metallic phases and the critical point separating them. However, we also find that the learning is less effective across quantum phase transitions, suggesting an intrinsic difficulty in efficiently learning nonsmooth functional relations. We further study the problem of directly reconstructing complex observables from simple local density measurements, proposing a scheme amenable to statistical learning from experimental data.
Density estimation is a fundamental task in statistics and machine learning that aims to estimate, from a set of samples, the probability density function of the distribution that generated them. There are different methods for addressing this problem but recently deep-neural density estimation methods have emerged as a powerful alternative. This paper presents a novel method for neural density estimation based on density matrices and adaptive Fourier features. Density matrices are commonly used in quantum mechanics to represent the quantum state of a physical system. In this work, they are used to estimate probability densities using an operation called quantum measurement. The proposed method can be trained without optimization using an averaging operation over the samples of the training dataset. It can also be integrated with deep learning architectures and trained using gradient descent. The performance of the proposed method was evaluated on a range of synthetic and real datasets and compared with fast kernel density estimation and state-of-the-art neural density estimation methods. The results demonstrate that the proposed method achieves competitive performance while being faster and more efficient than existing methods.
Abstract A density matrix describes the statistical state of a quantum system. It is a powerful formalism to represent both the quantum and classical uncertainty of quantum systems and to express different statistical operations such as measurement, system combination and expectations as linear algebra operations. This paper explores how density matrices can be used as a building block for machine learning models exploiting their ability to straightforwardly combine linear algebra and probability. One of the main results of the paper is to show that density matrices coupled with random Fourier features could approximate arbitrary probability distributions over Rn\mathbb {R}^{n}Rn R n . Based on this finding the paper builds different models for density estimation, classification and regression. These models are differentiable, so it is possible to integrate them with other differentiable components, such as deep learning architectures and to learn their parameters using gradient-based optimization. In addition, the paper presents optimization-less training strategies based on estimation and model averaging. The models are evaluated in benchmark tasks and the results are reported and discussed.