Entanglement and Tensor Networks for Supervised Image Classification

John Michael Martyn, Guifre Vidal, Chase Riley Roberts, Stefan Leichenauer

Abstract:

Tensor networks, originally designed to address computational problems in quantum many-body
physics, have recently been applied to machine learning tasks. However, compared to quantum
physics, where the reasons for the success of tensor network approaches over the last 30 years is well
understood, very little is yet known about why these techniques work for machine learning. The goal
of this paper is to investigate entanglement properties of tensor network models in a current machine
learning application, in order to uncover general principles that may guide future developments. We
revisit the use of tensor networks for supervised image classification using the MNIST data set of
handwritten digits, as pioneered by Stoudenmire and Schwab [Adv. in Neur. Inform. Proc. Sys. 29,
4799 (2016)]. Firstly we hypothesize about which state the tensor network might be learning during
training. For that purpose, we propose a plausible candidate state $\Sigma_\ell\rangle$ (built as a superposition
of product states corresponding to images in the training set) and investigate its entanglement
properties. We conclude that $\Sigma_\ell\rangle$ is so robustly entangled that it cannot be approximated by
the tensor network used in that work, which must therefore be representing a very different state.
Secondly, we use tensor networks with a block product structure, in which entanglement is restricted
within small blocks of n × n pixels/qubits. We find that these states are extremely expressive (e.g.
training accuracy of 99.97% already for n = 2), suggesting that long-range entanglement may not
be essential for image classification. However, in our current implementation, optimization leads to
over-fitting, resulting in test accuracies that are not competitive with other current approaches.