Quantum Manipulation of Trapped Ions in Two Dimensional Coulomb Crystals

We show that a large number of ions forming a 2D Coulomb crystal provides an almost ideal system for scalable quantum computation and quantum simulation. In particular, the coupling of the internal states to the motion of the ions transverse to the crystal plane allows one to implement two-qubit quantum gates. We analyze in detail the decoherence induced by anharmonic couplings, and show that very high gate fidelities can be achieved with current experimental setups.

Figure 1

Quantum gate in a 2D Coulomb crystal: standing waves induce a state-dependent dipole force on two nearest neighbors in a triangular lattice.

Figure 2

(a) Spectrum of the vibrational modes with wave vector $\mathbf{q}$ along the $\mathbf{x}$ direction. $N={10}^4$ ions, ${\omega }_z=50{\omega }_{\mathrm{xy}}$, and we consider periodic boundary conditions. (b) Continuous lines: $\mathcal{E}$ (error without phase correction). Dashed lines: ${\mathcal{E}}^{\prime }$ (error with correction of the gate time). Thick lines and thin lines correspond to ${\omega }_{\mathrm{xy}}=20\mathrm{kHz}$ and ${\omega }_{\mathrm{xy}}=200\mathrm{kHz}$, respectively. $N={10}^4$ ions and $\Gamma /{\omega }_{\mathrm{xy}}=0.05$.