Implementing two-qubit phase gates by exchanging non-Abelian quasiparticles

We study how to implement two-qubit phase gates by exchanging non-Abelian quasiparticles. We firstly investigate quantum dynamics of a single trapped ion with two stable electronic ground states and with a larger energy gap from the rest of the spectrum, which is held in the Lamb–Dicke regime of a driven optical lattice. A set of degenerate Schrödinger's cat states with the same expected energy is found, and wavepackets of the probability densities occupying different spin states are identical to the quasiparticles obeying the proposed non-Abelian interchange. The controlled transitions between different instantaneous degenerate ground states are illustrated for an array of $\delta$-shaped laser pulses. Making use of the mathematical equivalence between the single-ion system and the center-of-mass system of two trapped ions, the two-qubit phase gates are implemented by exchanging the non-Abelian quasiparticles of the center-of-mass motion via the periodic state-dependent forces. Such phase gates depend on geometric and topological properties of the system, which makes them resistant to certain errors. The results can be justified with the current experimental capability and may be extended to an array of weakly coupled trapped-ion pairs for demonstrating the non-Abelian statistics of the quasiparticles and for encoding the topological qubits.