Braiding of Atomic Majorana Fermions in Wire Networks and Implementation of the Deutsch-Jozsa Algorithm

We propose an efficient protocol for braiding Majorana fermions realized as edge states in atomic wire networks, and demonstrate its robustness against experimentally relevant errors. The braiding of two Majorana fermions located on one side of two adjacent wires requires only a few local operations on this side which can be implemented using local site addressing available in current experiments with cold atoms and molecules. Based on this protocol we provide an experimentally feasible implementation of the Deutsch-Jozsa algorithm for two qubits in a topologically protected way.

Figure 1

Realization of an array of one-dimensional Kitaev wires in an optical lattice setup: atoms (red circles) can hop between neighboring sites (blue circles) with strength $J$ along the individual wires. The pairing term of strength $\Delta$ can be realized by a Raman induced dissociation of Cooper pairs (or Feshbach molecules) forming an atomic BCS reservoir.

Figure 2

Braiding protocol for two perfect quantum wires. The zero-energy Majorana modes that are initially on the upper (lower) wire are shown as yellow (black) spheres, while the blue ones correspond to the Majorana operators, which are coupled into finite-energy fermionic modes. Coupling of Majorana operators via hopping and pairing (Kitaev coupling) is indicated by gray solid links, while the coupling via hopping only is shown as a dashed link.

Figure 3

Evolution of the Majorana correlation functions $⟨i{\gamma }_L^{(1)}{\gamma }_R^{(1)}⟩$ (red circle), $⟨i{\gamma }_L^{(2)}{\gamma }_R^{(2)}⟩$ (blue downward triangle), $⟨i{\gamma }_L^{(2)}{\gamma }_R^{(1)}⟩$ (red square), and $⟨i{\gamma }_L^{(1)}{\gamma }_R^{(2)}⟩$ (blue upward triangle) during the braiding protocol with errors (a) $\alpha =0.05$ and (b) $\alpha =0.01$ in the local operations for two nonideal quantum wires with $|\Delta |=1.5J$ and $\mu =0$. Markers are only drawn in regions where the correlation functions are nonzero.

Figure 4

Braid group in a setup of three wires. We present the real-time evolution of the correlations functions $i⟨{\gamma }_L^{(n)}{\gamma }_R^{(m)}⟩$ under the action of (a) $U_2{U}_1$, (b) $U_1{U}_2$, and (c) $U_1{U}_2{U}_1$ for a chain of the length $L=40$ with $|\Delta |=1.5J$ and $\mu =0$. Markers are only drawn in regions where the correlation functions are nonzero.

Figure 5

(a) Setup and implementation of the oracle Deutsch-Jozsa algorithm for two qubits via braiding. (b) Implementation of the oracle unitary $U_g$ via braiding (see main text).