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

TL;DR

This paper explores simulating topological phases across one, two, and three dimensions using quantum walks, introducing step-dependent coins to control and engineer boundary states and topological properties dynamically.

Contribution

It provides a comprehensive protocol table for simulating topological phases in multiple dimensions and introduces step-dependent coins to dynamically control topological features.

Findings

Protocols for 1D, 2D, and initial 3D topological phase simulation

Step-dependent coins enable dynamic control of topological states

Boundary states depend on protocol and can be engineered

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.