  We simulate various topological phenomena in condense matter, such as
formation of different topological phases, boundary and edge states, through
two types of quantum walk with step-dependent coins. Particularly, we show that
one-dimensional quantum walk with step-dependent coin simulates all types of
topological phases in BDI family, as well as all types of boundary and edge
states. In addition, we show that step-dependent coins provide the number of
steps as a controlling factor over the simulations. In fact, with tuning number
of steps, we can determine the occurrences of boundary, edge states and
topological phases, their types and where they should be located. These two
features make quantum walks versatile and highly controllable simulators of
topological phases, boundary, edge states, and topological phase transitions.
We also report on emergences of cell-like structures for simulated topological
phenomena. Each cell contains all types of boundary (edge) states and
topological phases of BDI family.
