Interpreting convolutional neural networks' low-dimensional approximation to quantum spin systems

Convolutional neural networks (CNNs) have been employed along with variational Monte Carlo methods for finding the ground state of quantum many-body spin systems with great success. However, it remains uncertain how CNNs, with a model complexity that scales at most linearly with the number of particles, solve the "curse of dimensionality"and efficiently represent wavefunctions in exponentially large Hilbert spaces. In this work, we use methodologies from information theory, group theory and machine learning, to elucidate how CNN captures relevant physics of quantum systems. We connect CNNs to a class of restricted maximum entropy (MaxEnt) and entangled plaquette correlator product state (EP-CPS) models that approximate symmetry constrained classical correlations between subsystems. For the final part of the puzzle, inspired by similar analyses for matrix product states and tensor networks, we show that the CNNs rely on the spectrum of each subsystem's entanglement Hamiltonians as captured by the size of the convolutional filter. All put together, these allow CNNs to simulate exponential quantum wave functions using a model that scales at most linear in system size as well as provide clues into when CNNs might fail to simulate Hamiltonians. We incorporate our insights into a new training algorithm and demonstrate its improved efficiency, accuracy, and robustness. Finally, we use regression analysis to show how the CNNs solutions can be used to identify salient physical features of the system that are the most relevant to an efficient approximation. Our integrated approach can be extended to similarly analyzing other neural network architectures and quantum spin systems.