Quantum Information on Graphs and Fractals

The primary goal of my thesis is to study the interplay between properties of physical systems (mostly for quantum information processing) and the geometry of these systems. The ambient spaces I have been working with are fractal-type graphs. In many cases analytic computations can be done on these graphs due to their self-similarity. Different scenarios are investigated. Perfect Quantum State Transfer on Graphs and Fractals: We are concerned with identifying graphs and Hamiltonian operators properties that guarantee a perfect quantum state transfer. Toda lattices on weighted Z-graded graphs: We study discrete one dimensional nonlinear equations and their lifts to Z-graded graphs. We prove the existence of radial solitons on Z-graded graphs. Snowflake Domain with Boundary and Interior Energies: We investigate the impact of the fractal boundary on the eigenfunctions of a discrete Laplacian on the Koch Snowflake Domain that takes into account both the interior and the fractal boundary. Harmonic Gradients on Higher Dimensional Sierpinski Gaskets: This project studies the connection between the regularity of a Laplacian of a function and the pointwise existence of its harmonic gradient