Topologically protected braiding in a single wire using Floquet Majorana modes

Majorana zero modes are a promising platform for topologically protected quantum information processing. Their non-Abelian nature, which is key for performing quantum gates, is most prominently exhibited through braiding. While originally formulated for two-dimensional systems, it has been shown that braiding can also be realized using one-dimensional wires by forming an essentially two-dimensional network. Here, we show that in driven systems far from equilibrium, one can do away with the second spatial dimension altogether by instead using quasienergy as the second dimension. To realize this, we use a Floquet topological superconductor which can exhibit Majorana modes at two special eigenvalues of the evolution operator, 0 and $\pi$, and thus can realize four Majorana modes in a single, driven quantum wire. We describe and numerically evaluate a protocol that realizes a topologically protected exchange of two Majorana zero modes in a single wire by adiabatically modulating the Floquet drive and using the $\pi$ modes as auxiliary degrees of freedom.

Figure 1

Left: Phase diagram of the Floquet system in terms of the strength of the topological (trivial) Hamiltonian $H_0\left(H_1\right)$ in the two-part drive [see Eqs. (3) and (4)]. It is possible to realize four phases characterized by the presence of zero or $\pi$ modes: 1, trivial; 2, MZMs; 3, MPMs; and 4, MZMs and MPMs. The gray crosses mark the sweet spots of the corresponding phases with vanishing correlation lengths. Right: Parametrization used in Eq. (2) to obtain the phase diagram. The parameter $\delta$ quantifies the distance to the critical point and the direction of increasing $\delta$ is indicated in the left panel.

Figure 2

Left: Full braid protocol for a system of $L=20$ sites. The colors correspond to different phases; green (red) crosses indicate the locations of MZMs (MPMs). Right: Schematic representation of the braiding process of two MZMs. In the center region it is possible to convert between MZMs (denoted by 0) and MPMs (denoted by $\pi$). After the right MZM has been converted into an MPM it can be safely moved past the left MZM in the region where time-translational symmetry is preserved.

Figure 3

Errors in the braid protocol, measured by the deviation from unitarity of the evolution in the low-energy subspace (left panel) and deviation in the applied phase (right panel; see main text for definitions of ${\mathrm{\Delta }}_{\mathrm{diab}}$ and ${\mathrm{\Delta }}_{\mathrm{phase}}$) as a function of the number of interpolation steps between stages of the protocol. In the limit of $N_s\to \infty$, adiabaticity is recovered. The errors generally vanish with a lower law; however, for fast protocols ($N_s<200$) an exponential transient behavior is observed. In the phase error, the dependence on $N_s$ is nonmonotonic: for sufficiently slow protocols, the evolution becomes adiabatic with respect to the residual finite-size splitting of Majorana modes. Parameters used are ${\lambda }_{\mathrm{pert}}=0.2$.