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We study the simulation of the topological phases in three subsequentdimensions with quantum walks. We are mainly focused on the completion of atable for the protocols of the quantum walk that could simulate differentfamily of the topological phases in one, two dimensions and take the firstinitiatives to build necessary protocols for three-dimensional cases. We alsohighlight the possible boundary states that can be observed for each protocolin different dimensions and extract the conditions for their emergences orabsences. To further enrich the simulation of the topological phenomenas, weinclude step-dependent coins in the evolution operators of the quantum walks.Consequently, this leads to step-dependency of the simulated topologicalphenomenas and their properties which in turn introduce dynamicality as afeature to simulated topological phases and boundary states. This dynamicalityprovides the step-number of the quantum walk as a mean to control and engineerthe number of topological phases and boundary states, their populations, typesand even occurrences.

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