Quantum information theory for machine learning
Typ
Examensarbete för masterexamen
Program
Publicerad
2019
Författare
Wadman, Rikard
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
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.
Beskrivning
Ämne/nyckelord
tensor networks, machine learning, quantum information, multilinear algebra, network theory, Convolutional arithmetic circuits, Boltzmann machines, entanglement, DMRG, MERA