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

Abstract

We study the simulation of the topological phases in three subsequent dimensions with quantum walks. We are mainly focused on the completion of a table for the protocols of the quantum walk that could simulate different family of the topological phases in one, two dimensions and take the first initiatives to build necessary protocols for three-dimensional cases. We also highlight the possible boundary states that can be observed for each protocol in different dimensions and extract the conditions for their emergences or absences. To further enrich the simulation of the topological phenomenas, we include step-dependent coins in the evolution operators of the quantum walks. Consequently, this leads to step-dependency of the simulated topological phenomenas and their properties which in turn introduce dynamicality as a feature to simulated topological phases and boundary states. This dynamicality provides the step-number of the quantum walk as a mean to control and engineer the number of topological phases and boundary states, their populations, types and even occurrences.

Fidelity susceptibility near topological phase transitions in quantum walks

“…For this purpose, we turn to the quantum walks, which are proposed to be universal primitives [23] that can simulate a variety of quantum systems and phenomena [24,25], including topological materials [26]. The flexibility and controllability of the quantum walks help to simulate these topological phases [27], which include all symmetry classes and edge states in one-(1D) and twodimensional (2D) systems [26][27][28][29][30][31], some others in threedimension (3D) [32], as well as to directly probe the topo-logical invariants [33]. The topological phase transitions [34] and the possibility to invoke bulk-boundary correspondence have also been addressed for quantum walks.…”