We develop the theory to describe the equilibrium ion positions and phonon modes for a trapped ion quantum simulator in an oblate Paul trap that creates two-dimensional Coulomb crystals in a triangular lattice. By coupling the internal states of the ions to laser beams propagating along the symmetry axis, we study the effective Ising spin-spin interactions that are mediated via the axial phonons and are less sensitive to ion micromotion. We find that the axial mode frequencies permit the programming of Ising interactions with inverse power law spin-spin couplings that can be tuned from uniform to r −3 with DC voltages. Such a trap could allow for interesting new geometrical configurations for quantum simulations on moderately sized systems including frustrated magnetism on triangular lattices or Aharonov-Bohm effects on ion tunneling. The trap also incorporates periodic boundary conditions around loops which could be employed to examine time crystals.

Due to the experimental time constraints of state of the art quantum simulations, the direct preparation of the ground state by adiabatically ramping the field of a transverse-field Ising model becomes more and more difficult as the number of particles increase. We propose a spectroscopy protocol that intentionally creates excitations through diabatic ramping and measures a low-noise observable as a function of time for a constant Hamiltonian to reveal the structure of the coherent dynamics of the resulting many-body states. To simulate experimental data, noise from counting statistics and decoherence error are added. Compressive sensing is then applied to Fourier transform the simulated data into the frequency domain and extract the low-lying energy excitation spectrum. By using compressive sensing, the amount of data in time needed to extract this energy spectrum is sharply reduced, making such experiments feasible with current technology in, for example, ion trap quantum simulators.

We model the bang-bang optimization protocol as a shortcut to adiabaticity in the ground-state preparation of an ion-trap-based quantum simulator. Compared to a locally adiabatic evolution, the bang-bang protocol produces a somewhat lower ground-state probability, but its implementation is so much simpler than the locally adiabatic approach, that it remains an excellent choice to use for maximizing ground-state preparation in systems that cannot be solved with conventional computers. We describe how one can optimize the shortcut and provide specific details for how it can be implemented with current ion-trap-based quantum simulators. Introduction. There has been much recent progress in ion-trap-based quantum simulation. Original experiments focused on adiabatic state preparation [1-3] of the transverse-field Ising model by initially orienting all of the spins along the field axis (in a large initial field) and then ramping the field to zero to create the ground state of the Ising model. But when the system size was increased, and frustrated antiferromagnetic systems were examined, it became clear that these experiments would have a large amount of diabatic excitation [4], which led to the study of excited states [4][5][6][7] and to a protocol that optimizes the field ramp with a locally adiabatic criterion [8]. In addition, other experimental situations were examined, such as Lieb-Robinson bounds [9, 10] and higher-spin cases [11]. Currently, there are two foci for adiabatic state preparation: (i) find shortcuts which will allow the original protocol to be achieved or (ii) use the diabatic excitations as a means to study low-energy excitations. Within the first category, recent work has found an exact shortcut for adiabatic state preparation [12,13] (at least for the nearest-neighbor transverse field Ising model), but the multiple-spin interactions needed to accomplish this goal are too complicated to implement in the current generation of quantum simulators. In the second category, we already mentioned experimental [4,5,7] and theoretical [6] methods to produce or measure specific excitations. It also is possible, in some cases, for the diabatic excitations to resemble an equilibrium thermal state, especially for ferromagnetic systems [14].

We discovered an error in the units of time in the paper. The time units need to be multiplied by N , the number of ions in the crystal. Since all results in the paper are for N = 400, all time values must be multiplied by 400. The easiest fix for this, is to change the time-axis labels in all relevant figures (listed below) to tJ 0 /400. These corrections affect the following figures: (i) Fig. 4 (the horizontal axis becomes tJ 0 /400, the red curve is for τ = 4000/J 0 , and the green curve for τ = 10,000/J 0); (ii) Fig. 6 (τ d J 0 = 4000 in the label); (iii) Fig. 7 [the horizontal axis of Figs. 7(b) and 7(c) and the labels become τ d J 0 = 4000]; (iv) Fig. 9 (labels τ ramp J 0 are equal to 400 for black, 800 for red, and 1600 for green, and also in the caption); and (v) Figs. 10, 11, and 13 (the horizontal axis becomes t meas J 0 /400). In addition, the following changes need to be made in the text: Five lines below Eq. (14), the sentences should read, "We used decoherence times of τ d J 0 = 10 000 and 4000 in Fig. 4(a). The total time during our simulations is approximately 32 ms, this is with J 0 = 2π × 1.6 kHz." In Sec. III, the first paragraph discusses τ ramp J 0 values, which need to be multiplied by 400 to be 1600 and 800. Also, the second sentence of the first paragraph should read, "We used J 0 = 2π × 1.6 kHz." On page 10, the first paragraph of simulated data discusses τ d J 0 values, which must be multiplied by 400 to become 10 000 and 400. On page 11, τ d J 0 = 10 000. These corrections do not change any of our conclusions, but they do make some of the run times a bit long for current state-of-the-art experiments if J 0 is on the order of 2π × 1.6 kHz. For larger J 0 , this issue becomes less important.

Recent work proposed a variant on Ramsey interferometry for coupled spin-1/2 systems that directly measures the retarded spin-spin Green's function. We expand on that work by investigating nonequilibrium retarded spin-spin Green's functions within the transverse-field Ising model. We derive the lowest four spectral moments to understand the short-time behavior and we employ a Lehmann-like representation to determine the spectral behavior. We simulate a Ramsey protocol for a nonequilibrium quantum spin system that consists of a coherent superposition of the ground state and diabatically excited higher-energy states via a temporally ramped transverse magnetic field. We then apply the Ramsey spectroscopy protocol to the final Hamiltonian, which has a constant transverse field. The short-time behavior directly relates to Lieb-Robinson bounds for the transport of many-body correlations, while the long-time behavior relates to the excitation spectra of the Hamiltonian. Compressive sensing is employed in the data analysis to efficiently extract that spectra.arXiv:1512.05316v1 [quant-ph]