Putting machine learning to the test in a quantum many-body system

TL;DR

This work rigorously benchmarks HubbardNet on the Bose-Hubbard model in 1D and 2D, achieving sub-percent errors in ground-state energies and wave-function fidelities above $99\%$ across $U$ spanning superfluid, Mott-insulator, and chaotic regimes. It introduces physics-informed output activations and a fractal-dimension ($D_q$) based training strategy to predict excited-state structure, enabling spectrum-wide insights without full Gram-Schmidt towers. The results demonstrate that ML can qualitatively capture localization, multifractality, and chaotic transitions in many-body eigenstates, offering a rapid exploratory tool that complements traditional numerical methods. These findings highlight potential pathways for ML-assisted quantum many-body studies, while acknowledging scaling limitations and the need for further refinement for deep-spectrum accuracy.

Abstract

Quantum many-body systems pose a formidable computational challenge due to the exponential growth of their Hilbert space. While machine learning (ML) has shown promise as an alternative paradigm, most applications remain at the proof-of-concept stage, focusing narrowly on energy estimation at the lower end of the spectrum. Here, we push ML beyond this frontier by extensively testing HubbardNet, a deep neural network architecture for the Bose-Hubbard model. Pushing improvements in the optimizer and learning rates, and introducing physics-informed output activations that can resolve extremely small wave-function amplitudes, we achieve ground-state energy errors reduced by orders of magnitude and wave-function fidelities exceeding 99%. We further assess physical relevance by analysing generalized inverse participation ratios and multifractal dimensions for ground and excited states in one and two dimensions, demonstrating that optimized ML models reproduce localization, delocalization, and multifractality trends across the spectrum. Crucially, these qualitative predictions remain robust across four decades of the interaction strength, e.g. spanning across superfluid, Mott-insulating, as well as quantum chaotic regimes. Together, these results suggest ML as a viable qualitative predictor of many-body structure, complementing the quantitative strengths of exact diagonalization and tensor-network methods.

Figures (11)

• Figure 1: Variation of the fractal dimension $D_1$ (colour scale) as a function of interaction strength $U$ and scaled energy $\varepsilon$ for (top panel) all $1716$ states of a 1D BHH with system size $M=7$ and particle number $N=7$, and (bottom panel) a 2D BHH on a $4\times4$ square lattice with particle number $N=3$ for all $816$ states. The grey lines in both panels indicate the upper and lower bounds for the $50$ states closest to $\varepsilon=0.5$. These trends serve as reference data for the ML predictions analysed in Sections \ref{['sec:ground-state']} and \ref{['sec:excited-states']}.
• Figure 2: The MLP structure of HubbardNet with $M=4$. The green circles represent the input layer with the $n_i$'s, $U$, and $N$ as the input parameters, while the purple circles denote the $4$ hidden layers. The output neuron is indicated by the red circle.
• Figure 3: Example of a typical learning curve, indicating the loss function minus its minimum value (reached at convergence), $\Delta \mathcal{L}$, as a function of training steps for energy-based training of the 1D ground state for all $U$ values with $M=N=7$.
• Figure 4: Accuracy of the energy-based ground state training in 1D: The upper panel shows $E_0$ for $M=N=7$ as a function of interaction $U$ in terms of the NN (blue $\times$) and the ED (solid black line). The $\exp (u)$ label indicates that the exponential $\sigma$ has been used (see text). The middle panel gives $\delta E_0$ [Eq. \ref{['eq:relativeEkerror']}] with the horizontal dashed line indicating a $10^{-2} \equiv 1\%$ value. The infidelity $1-\mathcal{F}$ is shown in the lower panel. The nine vertical black dashed lines indicate the training values of $U$ [Eq. \ref{['eq:Utrain']}].
• Figure 5: Wave function properties from energy-based ground state training in 1D (cp. Fig. \ref{['fig-1d-ground-state-energy-overlap']}): (a) Coefficients $\psi_0(f)$ and (b) distribution $P(\alpha)$ obtained via the NN ($\times$, red) and the ED (black lines) at $U=0.21$ (top), $U=2.05$ (middle) and $U=20.54$ (bottom) for $M=N=7$. The horizontal Fock-space index in (a) goes from $1$ to $\text{dim}\mathcal{H}=1716$ in integer steps. Lower and upper horizontal axes in each (b) panel show the equivalent values of $\alpha$ and $|\psi|^2$, respectively.