Fig. 5: Effect of spatial symmetry on the gate process.

a Spatial plot of the zero-energy N_neq at t = 0 in the MSH T-structure, showing the two topological regions, left and right of a trivial T-segment, with d denoting the spatial distance between the Majorana zero modes within one pair. Blue and gray areas indicate the presence and absence, respectively, of a superconducting substrate. b Majorana world lines, as obtained from a projection of the zero-energy N_neq onto the x-axis. c Oscillation period of the success and failure probabilities within the even parity sector as a function of distance d between the two MZMs within each pair (see panel (a)). d Success and failure probabilities within the even parity sector as a function of T_R with ΔT_R = 0.2T_R for d = 36a₀.For (a–d), parameters are (μ, α, Δ, JS) = (−4, 0.45, 1.2, 2.6)t_e, and for (b) (T_R, ΔT_R) = (100, 20)τ_e. e Spatial plot of the zero-energy N_neq at t = 0 in the MSH box-and-whisker structure, showing the two topological regions, left and right of the box. f Majorana world lines, as obtained from a projection of the zero-energy N_neq onto the x-axis. Success and failure probabilities in the (g) odd and (h) even parity sectors as a function of T_R with ΔT_R = 0.2T_R. For (e–h), parameters are (μ, α, Δ, JS) = (-3.97, 0.9, 2.4, 5.2)t_e, and for (f) (T_R, ΔT_R) = (120, 24)τ_e and Γ = 0.01t_e.