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Register a new userMachine Learning Universal Bosonic Functionals
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Added by
Carlos L. Benavides-Riveros
Publication date
2021
fields
Physics
and research's language is
English
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The one-body reduced density matrix $gamma$ 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 $mathcal{F}[gamma]$ that recovers quantum correlations exactly. Based on a novel decomposition of $gamma$, 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 $mathcal{F}[gamma]$ and its functional derivative. For the Bose-Hubbard model, we present a comparison between our approach and Quantum Monte Carlo.
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In this work we explore the potential of a new data-driven approach to the design of exchange-correlation (XC) functionals. The approach, inspired by convolutional filters in computer vision and surrogate functions from optimization, utilizes convolutions of the electron density to form a feature space to represent local electronic environments and neural networks to map the features to the exchange-correlation energy density. These features are orbital free, and provide a systematic route to including information at various length scales. This work shows that convolutional descriptors are theoretically capable of an exact representation of the electron density, and proposes Maxwell-Cartesian spherical harmonic kernels as a class of rotationally invariant descriptors for the construction of machine-learned functionals. The approach is demonstrated using data from the B3LYP functional on a number of small-molecules containing C, H, O, and N along with a neural network regression model. The machine-learned functionals are compared to standard physical approximations and the accuracy is assessed for the absolute energy of each molecular system as well as formation energies. The results indicate that it is possible to reproduce B3LYP formation energies to within chemical accuracy using orbital-free descriptors with a spatial extent of 0.2 A. The findings provide empirical insight into the spatial range of electron exchange, and suggest that the combination of convolutional descriptors and machine-learning regression models is a promising new framework for XC functional design, although challenges remain in obtaining training data and generating models consistent with pseudopotentials.
Suppose that three kinds of quantum systems are given in some unknown states $ket f^{otimes N}$, $ket{g_1}^{otimes K}$, and $ket{g_2}^{otimes K}$, and we want to decide which textit{template} state $ket{g_1}$ or $ket{g_2}$, each representing the feature of the pattern class ${cal C}_1$ or ${cal C}_2$, respectively, is closest to the input textit{feature} state $ket f$. This is an extension of the pattern matching problem into the quantum domain. Assuming that these states are known a priori to belong to a certain parametric family of pure qubit systems, we derive two kinds of matching strategies. The first is a semiclassical strategy which is obtained by the natural extension of conventional matching strategies and consists of a two-stage procedure: identification (estimation) of the unknown template states to design the classifier (textit{learning} process to train the classifier) and classification of the input system into the appropriate pattern class based on the estimated results. The other is a fully quantum strategy without any intermediate measurement which we might call as the {it universal quantum matching machine}. We present the Bayes optimal solutions for both strategies in the case of K=1, showing that there certainly exists a fully quantum matching procedure which is strictly superior to the straightforward semiclassical extension of the conventional matching strategy based on the learning process.
Machine learning is used to approximate density functionals. For the model problem of the kinetic energy of non-interacting fermions in 1d, mean absolute errors below 1 kcal/mol on test densities similar to the training set are reached with fewer than 100 training densities. A predictor identifies if a test density is within the interpolation region. Via principal component analysis, a projected functional derivative finds highly accurate self-consistent densities. Challenges for application of our method to real electronic structure problems are discussed.
Machine learning is employed to build an energy density functional for self-bound nuclear systems for the first time. By learning the kinetic energy as a functional of the nucleon density alone, a robust and accurate orbital-free density functional for nuclei is established. Self-consistent calculations that bypass the Kohn-Sham equations provide the ground-state densities, total energies, and root-mean-square radii with a high accuracy in comparison with the Kohn-Sham solutions. No existing orbital-free density functional theory comes close to this performance for nuclei. Therefore, it provides a new promising way for future developments of nuclear energy density functionals for the whole nuclear chart.
In recent years, there is a growing interest in using quantum computers for solving combinatorial optimization problems. In this work, we developed a generic, machine learning-based framework for mapping continuous-space inverse design problems into surrogate quadratic unconstrained binary optimization (QUBO) problems by employing a binary variational autoencoder and a factorization machine. The factorization machine is trained as a low-dimensional, binary surrogate model for the continuous design space and sampled using various QUBO samplers. Using the D-Wave Advantage hybrid sampler and simulated annealing, we demonstrate that by repeated resampling and retraining of the factorization machine, our framework finds designs that exhibit figures of merit exceeding those of its training set. We showcase the frameworks performance on two inverse design problems by optimizing (i) thermal emitter topologies for thermophotovoltaic applications and (ii) diffractive meta-gratings for highly efficient beam steering. This technique can be further scaled to leverage future developments in quantum optimization to solve advanced inverse design problems for science and engineering applications.
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