Discrete-time quantum walks in two-dimensional amorphous topological matter

Abstract

Discrete-time quantum walks have been steadily growing in relevance as a research topic due to their potential applications as building blocks for universal quantum computation and tools for designing fast quantum search algorithms. Recently, it has been discovered that they can be used as a platform for the study of topological phases of matter. In this thesis, we construct a model of amorphous topological matter through a two-dimensional discrete-time quantum walk protocol propagating on a diluted lattice. We perform a computational study of transport properties of this model and discover that the ballistic spread of the quantum walk dissipates into the diffusive regime immediately when any amount of site disorder is introduced into the lattice. We report that the non-trivial topological phase reduces the degree of Anderson localization in the quantum walk and decreases its localization threshold to p ≈ 0.67, which is 17 to 32 percent lower compared to quantum walks in the trivial topological phase. We also observe the onset of anomalous subdiffusion in the vicinity of the localization threshold with diffusion exponent α ≈ 0.9. Our results are reinforced by simulations of the two-dimensional discrete-time quantum walk protocol up to 10000 time steps, surpassing formerly conducted numerical studies by an order of magnitude.