Shortcuts to non-Abelian braiding

Topological quantum information processing relies on adiabatic braiding of non-Abelian quasiparticles. Performing the braiding operations in finite time introduces transitions out of the ground-state manifold and deviations from the non-Abelian Berry phase. We show that these errors can be eliminated by suitably designed counterdiabatic correction terms in the Hamiltonian. We implement the resulting shortcuts to adiabaticity for simple protocols of non-Abelian braiding and show that the error suppression can be substantial even for approximate realizations of the counterdiabatic terms.

Figure 1

(a) $\mathsf{Y}$ junction with central Majorana ${\gamma }_0$ and three outer Majoranas ${\gamma }_j$ ($j=1,2,3$). The outer Majoranas are coupled to the inner Majoranas with strength ${\Delta }_j$. (b) Basic step of the braiding procedure, moving a zero-energy Majorana from the end of wire 1 to the end of wire 3 by varying the ${\Delta }_j$. Dark (light) wires indicate zero (nonzero) couplings ${\Delta }_j$. Dark red circles correspond to zero-energy Majoranas, and light blue circles indicate Majoranas acquiring a finite energy by coupling. In the intermediate step, the zero-energy Majorana is delocalized over the three Majoranas along the light wires. (c) Three steps as described in (b) result in braiding the zero-energy Majoranas ${\gamma }_1$ and ${\gamma }_2$.

Figure 2

(a) Minimal implementation required for braiding with the shortcut protocol. The additional couplings needed for the shortcut protocol are shown in blue. (b) Wire network with many Majoranas allowing for pairwise exchanges of neighboring Majoranas including the shortcut protocol. Implementing the shortcut merely requires the addition of local couplings within the network.

Figure 3

Diabatic errors vs duration of braiding protocol for the transition probability out of the degenerate subspace of the initial state. The inset shows the phase error relative to the non-Abelian Berry phase. For both quantities, curves are shown in the absence of counterdiabatic terms $[\lambda =0$ in Eq. (18)] and with counterdiabatic terms with 10% ($\lambda =0.9$) and 30% ($\lambda =0.7$) relative error. There would be no diabatic error if the counterdiabatic errors were implemented exactly.