Recent research has revealed exciting relations between topology, quantum mechanics, condensed-matter physics and computer science: There are strong indications that topological properties associated with exotic quantum systems can provide noise resilience and fault tolerance in quantum-computing architectures. Intense experimental efforts, supported by both private and public investors, are aiming to realize such architectures, the so-called Majorana qubits and the surface code being the two prime examples. The task of the PhD student is to familiarize with the basic concepts of the field, and to develop new theories, both on a fundamental and a practical level, that will faciliate the experimental realization of topology-based ideas in quantum systems.