Toward simulation of topological phenomena with one-, two-, and three-dimensional quantum walks

We study the simulation of the topological phases in three subsequent dimensions with quantum walks. We focus mainly on the completion of a table for the protocols of the quantum walk that could simulate different families of the topological phases in one, two, and three dimensions. We also highlight the possible boundary states that can be observed for each protocol in different dimensions and extract the conditions for their emergences. To further enrich the simulation of the topological phenomena, we include step-dependent coins in the evolution operators of the quantum walks. This leads to step dependence of the simulated topological phenomena and their properties which introduces dynamicity as a feature of simulated topological phases and boundary states. This dynamicity provides the step number of the quantum walk as a means to control and engineer the numbers of topological phases and boundary states, their numbers, types, and even occurrences.