Superconductors hosting long-sought excitations called Majorana fermions may be ultimately used as qubits of fault-tolerant topological quantum computers. A crucial challenge toward the topological quantum computer is to implement quantum operation of nearly degenerate quantum states as a dynamical process of Majorana fermions. In this paper, we investigate the braiding dynamics of Majorana fermions on superconducting nanowires. In a finite size system, a nonadiabatic dynamical process dominates the non-Abelian braiding that operates qubits of Majorana fermions. Our simulations clarify how qubits behave in the real-time braiding process, and elucidate the optimum condition of superconducting nanowires for efficient topological quantum operation.

Energy gap and wave function in thin films of topological insulator is studied, based on tight-binding model. It is revealed that thickness dependence of the magnitude of energy gap is composed of damping and oscillation. The damped behavior originates from the presence of gapless surface Dirac cone in the infinite thickness limit. On the other hand, the oscillatory behavior stems from electronic properties in the thin thickness limit.