Fig. 1. Schematic energy band structures for (a) quantum Hall effect and quantum anomalous Hall effect, (b) quantum spin Hall effect, (c) a topological superconductor and (d) sche-matic diagram of topological edge/surface states in real space.

Fig. 2. Distribution of normalized topological superconduc-tivity gap in momentum space.

Fig. 3. Schematics of realization of non-Abelian statistics using Majorana bound states in vortex cores of a topological superconductor. Black arrows denote the exchanging paths of two quantum vortices. Superconducting phase takes a 2π jump when a vortex crosses the red cuts.

Fig. 4. (a) Schematics of a heterostructure consisting of a spin-orbital coupling semiconductor nanowire and an s wave superconductor; (b) the band dispersion of the nanowire with finite magnetic field (solid lines) and zero magnetic field (dashed lines).

Fig. 5. (a) Schematic design of a universal quantum gate for Majorana qubit, where the qubit is manipulated by voltage across the Josephson-Majorana junction; (b) the two energy levels of the Majorana qubit depending on the phase difference across the junction; (c)-(e) the LZS oscillation of Majorana qubit under current pulse: (c) a short pulse, (d) a long pulse, (e) a sequence of pulses^[25].

Fig. 6. (a) System with couplings between Majorana bound states and two quantum dots; (b) correlation between the electron occupations on the two quantum dots^[28].

Fig. 7. (a) Schematic of the linear dispersion of surface state of a 3D TI; (b) schematic of a TI/s-SC heterostructure, where the red points denote the Majorana bound states at the center of a quantum vortex^[29].

Fig. 8. (a) Spin-resolved wavefunctions of the low energy quasiparticle states in the vortex core of a topological superconductor; (b) spectrum of the ratio between densities of states for the spin-up and spin-down components^[29].

Fig. 9. (a), (c), (e), (g) Schematic of the device which transports edge Majorana states using gate voltages; (b), (d), (f) corresponding wavefunctions of the edge Majorana states in (a), (c), (e)^[32].

Fig. 10. (a), (b) NOT quantum gate operation of the Majorana qubit; (c), (d) a single-electron pumping based on the edge Majorana states^[34].