Quantum computing is an emerging, interdisciplinary field of science in the intersection of computer science, mathematics and physics. Recent experimental advances in building a physical quantum computer show the urgency of finding possible applications. On the other hand to date we only have very small quantum computers, which are mostly useful for proof of concept demonstrations, thus for the time being one needs to focus on building and understanding the underlying mathematical theory. A particularly interesting aspect of quantum computing is quantum machine learning, which also needs a more firm theoretical understanding, because many of the recent developments are based on heuristic approaches which cannot be properly tested yet, due to the limitations of the available hardware. This proposal outlines new approaches and ideas for quantum algorithm development, and attempts to improve some aspects of the theory of quantum machine learning, while also encompasses some fundamental theoretical questions. The described ideas are all related to the problem of finding large-scale structures in various objects. Since quantum computers tend to be quite efficient at recognizing patterns, it is a promising angle of approach. The relevant ideas are inspired by multiple related disciplines, and several of the proposed tools were recently co-developed by the applicant. The supervisor has an outstanding track record in developing the mathematical theory of large-scale structures emerging in graphs, groups and networks, while the applicant has demonstrated strong problem solving skills and the ability of developing novel quantum algorithms, which promises a fruitful collaboration in the implementation of the proposed action.

Graph limit theory is a rapidly developing field of mathematics. Mszros proposes to use ideas from graph limit theory to answer questions about random simplicial complexes. The motivation for investigating the cokernels of random integral matrices comes from the number theoretic conjectures of Cohen and Lenstra. Mszros plans to make use of the fact that the questions arising studying the homology of random simplicial complexes are very similar to questions about the cokernels of random integral matrices. Thus, ideas can be transferred from one area to the other. Mszros also proposes to investigate the expansion properties of random simplicial complexes. Many researchers at the Rnyi Institute (the host institute) played a key role in developing graph limit theory and finding applications of this theory in several areas including ergodic theory, group theory, probability theory, statistical physics and computer science. Large part of Mszross past and proposed projects relies on these techniques. Building on the research that he started during his three years at the University of Toronto as a postdoctoral fellow, Mszros will bring new questions and areas where these techniques can be applied like stochastic topology. Mszross work will not only provide new areas of applications, but hopefully will open new directions in graph limit theory. The arising new topics in graph limit theory can be of great interest to many members of the host institute. The fellowship would provide Mszros an excellent opportunity for many fruitful collaborations to explore these topics further. He is also looking forward to improving his organization, management and communication skills to become a more well rounded researcher.