Deep learning of spatial densities in inhomogeneous correlated quantum systems

TL;DR

The paper addresses rapid, scalable prediction of spatial densities in inhomogeneous, correlated quantum lattice systems where the Hilbert space grows exponentially. It introduces a deep convolutional network that takes a potential landscape $V(x)$ with the chemical potential $\mu$ encoded as input via $V(x)-\mu$ and outputs spatial maps of multiple observables (e.g., ${\langle \hat{n}(x) \rangle}$, ${\langle \hat{I}(x) \rangle}$, and relevant correlators) across a family of models with varying couplings $J, U, U', \dots$. Training is performed on random potentials generated in 1D (uniform white noise) and 2D (Gaussian random fields) from which labels are obtained by exact diagonalization or QMC; the network generalizes to larger sizes and captures interference, interactions, and phase transitions. The approach yields dramatic inference speedups (e.g., up to $5\cdot10^{5}$-fold in 40-site 2D systems) and enables inverse design and Hamiltonian learning by differentiable mapping from density to potential, opening routes to rapid exploration of quantum materials and quantum simulators.

Abstract

Machine learning has made important headway in helping to improve the treatment of quantum many-body systems. A domain of particular relevance are correlated inhomogeneous systems. What has been missing so far is a general, scalable deep-learning approach that would enable the rapid prediction of spatial densities for strongly correlated systems in arbitrary potentials. In this work, we present a straightforward scheme, where we learn to predict densities using convolutional neural networks trained on random potentials. While we demonstrate this approach in 1D and 2D lattice models using data from numerical techniques like Quantum Monte Carlo, it is directly applicable as well to training data obtained from experimental quantum simulators. We train networks that can predict the densities of multiple observables simultaneously and that can predict for a whole class of many-body lattice models, for arbitrary system sizes. We show that our approach can handle well the interplay of interference and interactions and the behaviour of models with phase transitions in inhomogeneous situations, and we also illustrate the ability to solve inverse problems, finding a potential for a desired density.

Figures (5)

• Figure 1: Overview of the approach. (a) The convolutional neural network maps a potential landscape of arbitrary size, together with model parameters, to the spatial maps of multiple observables (including correlators, here denoted as 'noise'). (b) Training procedure. Any of a set of numerical methods are used to turn random potentials into spatial maps, providing the training data. The dependence of energy on particle number is recorded as well and gives access to the chemical potential.
• Figure 2: Performance of the network addressing the interplay of interactions and interference. (a) Density for a step potential (never seen during training), displaying Friedel oscillations in a 1D system of interacting fermions. The NN prediction is very close to the almost exact result (DMRG), in contrast to Hartree-Fock. (b) Density evolution vs. potential step height, comparing the three different approaches; again the NN performs very well. (c) Predictions for multiple observables, obtained from the single (multi-head) network, evaluated for a step potential in a smaller system size. We compare the network output with the exact diagonalization (ED) result.
• Figure 3: Results for a 2D model with phase transitions. (a) Phase diagram for the 2D Hubbard model with nearest-neighbor coupling, vs. chemical potential and hopping strength for $4U'/U = 1.0$. The NN (trained on finite-T QMC results) is asked to predict the density for a flat potential, after having been trained on random potentials. We plot the difference between the maximum and the minimum of the density. The indicated phase boundaries mark the ground-state phase transitions extracted from ohgoeGroundStatePhaseDiagram2012. Checkerboard solids $\textrm{CB}_1$ and $\textrm{CB}_2$ with filling factors $\rho=3/2,1/2$, supersolid (SS), superfluid (SF), Mott insulator (MI, $\rho = 1$). (b) Cuts through (a), comparing predictions of the network with those obtained from QMC (with max. and min. densities). QMC densities in the SF phase are not perfectly constant in space, due to sampling noise. (c) Predictions (NN vs. QMC) for a 2D potential well. Right plot shows evolution of density along a 1D cut for varying well depth. (d) Predictions for a harmonic potential, changing the chemical potential. (e) Wall-clock time for QMC calculations vs. system size (for a fixed convergence criterion), and the same for NN evaluation (inference time). (f) Test error (mean squared error) vs. training set size, indicating fluctuations (standard deviation $\sigma$) over several training runs. (g) For given parameters (4J/U=0.25,zU'/U=1.0) the network (NN) is used to "invert" a prescribed density $\rho$ to produce a corresponding potential (Inverted), see main text, which is tested by using both the NN and QMC for recovering the density.
• Figure 4: Architecture of the 2D model. a) The residual blocks take a preactivation structure 10.1007/978-3-319-46493-0_38. b) Overall network architecture with residual blocks (purple) and BiFPN structure.
• Figure 5: Loss curves during the training of the 2D model.