Tensor Network Computations That Capture Strict Variationality, Volume Law Behavior, and the Efficient Representation of Neural Network States

TL;DR

This paper introduces tensor network functions that allow for exact variational estimates and efficient representation of neural network states, overcoming previous computational limitations and enabling new applications in quantum and neural network modeling.

Contribution

It presents a novel class of tensor network states called tensor network functions that do not require approximate contractions, expanding the scope of tensor network applications.

Findings

Enables strict variational energy estimates on loopy graphs.

Captures aspects of volume law time evolution.

Maps feed-forward neural networks onto efficient tensor network functions.

Abstract

We introduce a change of perspective on tensor network states that is defined by the computational graph of the contraction of an amplitude. The resulting class of states, which we refer to as tensor network functions, inherit the conceptual advantages of tensor network states while removing computational restrictions arising from the need to converge approximate contractions. We use tensor network functions to compute strict variational estimates of the energy on loopy graphs, analyze their expressive power for ground-states, show that we can capture aspects of volume law time evolution, and provide a mapping of general feed-forward neural nets onto efficient tensor network functions. Our work expands the realm of computable tensor networks to ones where accurate contraction methods are not available, and opens up new avenues to use tensor networks.