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Convolutional neural networks (CNNs) have been employed along withVariational Monte Carlo methods for finding the ground state of quantummany-body spin systems with great success. In order to do so, however, a CNNwith only linearly many variational parameters has to circumvent the ``curse ofdimensionality'' and successfully approximate a wavefunction on anexponentially large Hilbert space. In our work, we provide a theoretical andexperimental analysis of how the CNN optimizes learning for spin systems, andinvestigate the CNN's low dimensional approximation. We first quantify the roleplayed by physical symmetries of the underlying spin system during training. Weincorporate our insights into a new training algorithm and demonstrate itsimproved efficiency, accuracy and robustness. We then further investigate theCNN's ability to approximate wavefunctions by looking at the entanglementspectrum captured by the size of the convolutional filter. Our insights revealthe CNN to be an ansatz fundamentally centered around the occurrence statisticsof $K$-motifs of the input strings. We use this motivation to provide theshallow CNN ansatz with a unifying theoretical interpretation in terms of otherwell-known statistical and physical ansatzes such as the maximum entropy(MaxEnt) and entangled plaquette correlator product states (EP-CPS). Usingregression analysis, we find further relationships between the CNN'sapproximations of the different motifs' expectation values. Our results allowus to gain a comprehensive, improved understanding of how CNNs successfullyapproximate quantum spin Hamiltonians and to use that understanding to improveCNN performance.

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