Topological qubits are attractive because of the potential to store quantum information in a topologically protected manner; however, they are challenging to realize. This Review surveys the recent attempts to realize topological qubits out of materials systems that combine superconductivity, spin-orbit coupling and a magnetic field, and surveys both theoretical ideas and experimental results.

Among the major avenues that are being pursued for realizing quantum bits, the Majorana-based approach has been the most recent to be launched. It attempts to realize qubits that store quantum information in a topologically protected manner. The quantum information is protected by non-local storage in localized and well-separated Majorana zero modes, and manipulated by exploiting their non-abelian quantum statistics. Realizing these topological qubits is experimentally challenging, requiring superconductivity, helical electrons (created by spin-orbit coupling) and breaking of time-reversal symmetry to all cooperate in an uncomfortable alliance. Over the past decade, several candidate materials systems for realizing Majorana-based topological qubits have been explored, and there is accumulating, though still debated, evidence that zero modes are indeed being realized. This Review surveys the basic physical principles on which these approaches are based, the materials systems that are being developed and the current state of the field. We highlight both the progress that has been made and the challenges that still need to be overcome.