Topologically Decoherence-Protected Qubits with Trapped Ions

We show that trapped ions can be used to simulate a highly symmetrical Hamiltonian with eigenstates naturally protected against local sources of decoherence. This Hamiltonian involves long-range coupling between particles and provides a more efficient protection than nearest neighbor models discussed in previous works. Our results open the perspective of experimentally realizing, in controlled atomic systems, complex entangled states with decoherence times up to 9 orders of magnitude longer than isolated quantum systems.

Figure 1

Two laser beams of slightly different frequencies [blue (gray) and red (dark gray)] produce virtual processes that change the state of one ion and create (annihilate) a phonon mode of $N\times N$ lattice. Each of these laser fields alone does not produce a real ion transition (with or without a phonon mode emission or absorption). The sum of their frequencies is exactly equal to the energy needed to change the state of two ions simultaneously, which is described by the ${\sigma }_{i,j}^{x,y}{\sigma }_{i,k}^{x,y}$ in the effective Hamiltonian. If the phonon mode involved in this process corresponds to global translational motion of all ions (center-of-mass mode), the effective pairwise interaction does not depend on the distance between two ions. It can be restricted to the desired row (column) form if only a subset of ions is illuminated at a given time. The effect of the sequential application of the beams to rows and columns is described by the effective Hamiltonian (1).

Figure 2

An array of $3\times 3$ implemented by a linear trap with 9 ions. To generate the interaction in row 1 (inset a) one applies laser light to ions 1, 2, 3 (inset b), while the column 2 interaction requires light on ions 2, 5, and 8. (inset c). The advantage of this scheme is that it does not require microtrap fabrication. However, it becomes difficult to implement for large arrays ($N>3$) because the distance between the ions in the center of the array gets smaller, making their individual addressing difficult.