Fig. 2: Simulation of a \(\sqrt{{\sigma }_{z}}\)-gate in an MSH T-structure.

a–d Spatial plot of the zero-energy N_neq during consecutive times in the gate operation. White arrows (dots) indicate antiferromagnetic in-plane (ferromagnetic out-of-plane) alignment of the magnetic moments. Blue area indicates the superconducting substrate. e Majorana world lines obtained from a projection of the zero-energy N_neq onto the real space x-axis. f Time dependence of the fidelity F_e,o(t) for the even and odd parity states and of the geometric phase difference Δϕ. g Time- and energy-dependent N_neq at a site of the MSH network (see yellow arrow in (a)). h Fidelity F as a function of T_R and ΔT_R. Parameters are (μ, α, Δ, JS) = (−3.993, 0.9, 2.4, 5.2)t_e with a difference of ΔJ = 0.26t_e in the magnetic coupling between alternating sites on the T-structure, (T_R, ΔT_R) = (500, 50)τ_e and Γ = 0.01t_e for the inverse quasi-particle lifetime. These parameters, resulting in a topological superconducting gap of Δ_t ≈ 0.4t_e, were chosen in order to minimize (i) the localization length of the MZMs along the network, and thus (ii) the hybridization between the MZMs. Total system size for all results is 600 sites with dimension (20a₀ × 30a₀).