On the descriptive power of Neural-Networks as constrained Tensor Networks with exponentially large bond dimension

Mario Collura, Luca Dell'Anna, Timo Felser, Simone Montangero

SciPost Phys. Core 4, 001 (2021) · published 2 February 2021


In many cases, Neural networks can be mapped into tensor networks with an exponentially large bond dimension. Here, we compare different sub-classes of neural network states, with their mapped tensor network counterpart for studying the ground state of short-range Hamiltonians. We show that when mapping a neural network, the resulting tensor network is highly constrained and thus the neural network states do in general not deliver the naive expected drastic improvement against the state-of-the-art tensor network methods. We explicitly show this result in two paradigmatic examples, the 1D ferromagnetic Ising model and the 2D antiferromagnetic Heisenberg model, addressing the lack of a detailed comparison of the expressiveness of these increasingly popular, variational ans\"atze.

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