Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms

TL;DR

This work introduces two approaches that achieve a constant sample complexity, independent of system size $n$, for learning ground state properties and is the first rigorous sample complexity bound on a neural network model for predicting ground state properties.

Abstract

A fundamental problem in quantum many-body physics is that of finding ground states of local Hamiltonians. A number of recent works gave provably efficient machine learning (ML) algorithms for learning ground states. Specifically, [Huang et al. Science 2022], introduced an approach for learning properties of the ground state of an $n$-qubit gapped local Hamiltonian $H$ from only $n^{\mathcal{O}(1)}$ data points sampled from Hamiltonians in the same phase of matter. This was subsequently improved by [Lewis et al. Nature Communications 2024], to $\mathcal{O}(\log n)$ samples when the geometry of the $n$-qubit system is known. In this work, we introduce two approaches that achieve a constant sample complexity, independent of system size $n$, for learning ground state properties. Our first algorithm consists of a simple modification of the ML model used by Lewis et al. and applies to a property of interest known beforehand. Our second algorithm, which applies even if a description of the property is not known, is a deep neural network model. While empirical results showing the performance of neural networks have been demonstrated, to our knowledge, this is the first rigorous sample complexity bound on a neural network model for predicting ground state properties. We also perform numerical experiments that confirm the improved scaling of our approach compared to earlier results.

Figures (7)

• Figure 1: A deep network model for predicting ground state properties. Given a vector $x \in [-1,1]^m$ that parameterizes a quantum many-body Hamiltonian $H(x)$, the algorithm uses geometric structure to create "local" neural network models $f_{P_i}^{\theta_{P_i}}$. The ML algorithm then combines the outputs of these local models to predict a property $\tr(O\rho(x))$, where $\rho(x)$ is the ground state of $H(x)$. Here, we decompose $O = \sum_{i=1}^M \alpha_{P_i} P_i$ for Pauli operators $P_i$, where the final layer takes a linear combination of the outputs of the local models weighted by some trainable parameters $w_{P_i}$ that intuitively should approximate the Pauli coefficients $\alpha_{P_i}$.
• Figure 2: Numerical experiments.(Left) Comparison with previous methods. Each point indicates the prediction error (RMSE) of our deep learning model or the regression model of lewis2024improved, fixing the training set size $N = 3686$ and the size of the local neighorbood $\delta_1 = 0$ (\ref{['eq:ip-main']}). We train both algorithms on either LDS or uniformly random points, which achieve similar performance. (Center) Scaling with training size. Each point indicates the prediction error of our deep learning model given LDS training data for various $\delta_1$ and training data sizes. (Right) Neural network weights and training error. Blue points correspond to the training error of the neural network model. Red points correspond to the $\ell_1$ norm of parameters in the last layer or the largest absolute value of the parameters of the neural network, fixing $N = 3686$ and $\delta_1=1$. This shows that the assumptions in \ref{['thm:generalization_highlevel']} are achieved in practice. The shaded areas denote the 1-sigma error bars across the assessed ground state properties.
• Figure 3: Transformed low-discrepancy sequences. The blue circles correspond to two-dimensional uniform Sobol points $x$. The orange triangles indicate the corresponding Sobol points with respect to the CDF of the standard normal distribution, denoted by $\Phi$. The latter forms a low-discrepancy-sequence with respect to the Borel measure $\mu=\Phi$.
• Figure 4: Training/Prediction Error vs. System Size. This figure shows the scaling of the training (left) and prediction (right) RMSE with respect to system size for different values of $\delta_1$. All training sets are distributed as Sobol sequences and were trained on $N=3686$ samples. The shaded areas denote the 1-sigma error bars across the assessed ground state properties.
• Figure 5: Training/Prediction Error vs. Local Neighborhood Size. This figure shows the scaling of the training (left) and prediction (right) RMSE with respect to the local neighborhood size $\delta_1$. All training sets are of size $N=3686$ with system size $9\times 5$. The shaded areas denote the 1-sigma error bars across the assessed ground state properties.

Theorems & Definitions (78)

• Theorem 1: Informal
• Theorem 2: Informal
• Theorem 3: Theorem 1 in lewis2024improved
• Theorem 4: Constant sample complexity
• Corollary 1: Learning representations of ground states
• Theorem 5: Neural network sample complexity guarantee
• Corollary 2: Learning representations of ground states with neural networks
• Theorem 6
• Theorem 7: Theorem 5 in lewis2024improved
• Lemma 1: Lemma 15 in lewis2024improved