Abstract: Topological quantum materials have recently been a worldwide frontier research topic in condensed matter physics. Over the past decades, the wide investigation of the quantum Hall effect has led to a new paradigm based on the notion of topological order. Through combining the mathematical concept of topology and the energy band theory, scholars have incorporated them into
the theory, calculation, and experiment of solid state electronic materials. Topological materials may have the properties of peculiar surface states and low energy consumption for electron transport. The origin of such effects is that topological quantum states can be protected strictly by symmetry, and they are highly robust to common impurities, defects or disorder, unless the topological property of the system is changed through a quantum control or phase transition. This emerging research field provides numerous possibilities for the future exploration of electronic materials and devices, as well as information technology based on quantum topological systems and computing. For the development of materials science, the introduction of topological concept enables material scientists to study materials with deeper insight and to describe materials using more advanced mathematical tool. This paper will cover the research progress of topological quantum materials from the perspective of computational science, including topological insulators and topological semi-metals.