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

TL;DR

This paper analyzes how convolutional neural networks (CNNs) effectively approximate quantum spin systems, focusing on their low-dimensional representations, symmetries, and entanglement properties, leading to improved training algorithms and theoretical insights.

Contribution

The work provides a theoretical and experimental analysis of CNNs in quantum spin systems, introduces an improved training algorithm, and offers a unifying interpretation of CNNs as statistical and physical ansatzes.

Findings

CNNs capture the occurrence statistics of K-motifs in input strings.

Incorporating physical symmetries improves CNN training efficiency and accuracy.

CNNs can be interpreted as maximum entropy and entangled plaquette states.

Abstract

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. In order to do so, however, a CNN with only linearly many variational parameters has to circumvent the ``curse of dimensionality'' and successfully approximate a wavefunction on an exponentially large Hilbert space. In our work, we provide a theoretical and experimental analysis of how the CNN optimizes learning for spin systems, and investigate the CNN's low dimensional approximation. We first quantify the role played by physical symmetries of the underlying spin system during training. We incorporate our insights into a new training algorithm and demonstrate its improved efficiency, accuracy and robustness. We then further investigate the CNN's ability to approximate wavefunctions by looking at the entanglement spectrum captured by the size of the convolutional filter. Our insights reveal the CNN to be an ansatz fundamentally centered around the occurrence statistics of $K$-motifs of the input strings. We use this motivation to provide the shallow CNN ansatz with a unifying theoretical interpretation in terms of other well-known statistical and physical ansatzes such as the maximum entropy (MaxEnt) and entangled plaquette correlator product states (EP-CPS). Using regression analysis, we find further relationships between the CNN's approximations of the different motifs' expectation values. Our results allow us to gain a comprehensive, improved understanding of how CNNs successfully approximate quantum spin Hamiltonians and to use that understanding to improve CNN performance.