Quantum information theory for machine learning

Abstract

The remarkable successes of machine learning and of deep learning in particular during the last decade have caused an explosive growth of interest in the field. Meanwhile, there are still significant gaps in our understanding of the processes involved, making the area a very promising topic for theoretical investigation. A particularly interesting idea that has received a lot of attention recently is the claim that the successive transformations performed by deep neural networks behave similarly to the renormalization group flows of statistical mechanics. In the light of this it is natural also to consider numerical renormalization algorithms as interesting candidates for performing general machine learning. It turns out that both the DMRG and the more recent Entanglement Renormalization algorithm from numerical quantum mechanics are quite well suited for this purpose. Both of these algorithms are most naturally described using the language of tensor networks, which are graph based representations of multilinear tensors, typically used for the description of quantum states. This thesis discusses machine learning with tensor networks from a holistic perspective and makes a review of some of the recent work on the subject. Also of significant interest is the study of expressive power of neural networks. A recent proposal suggests employing quantum entanglement entropy as a measure of a models ability to represent complex correlations between input regions. We study the interpretability and implications of such a measure as well as its relations to the quantum version of the max- ow/min-cut theorem, which relates the entanglement entropy of a tensor network state to the minimal cut in its graph. A generalization of said theorem is found, leading us to alternate, and very simple, proofs of some already known scaling laws of quantum entanglement in Boltzmann machines and convolutional arithmetic circuits, which are derivative of standard convolutional neural networks.