  Controlling the interaction graph between spins or qubits in a quantum
simulator allows user-controlled tailoring of native interactions to achieve a
target Hamiltonian. The flexibility of engineering long-ranged phonon-mediated
spin-spin interactions in a trapped ion quantum simulator offers such a
possibility. Trapped ions, a leading candidate for simulating computationally
hard quantum many-body dynamics, are most readily trapped in a linear 1D chain,
limiting their utility for readily simulating higher dimensional spin models.
In this work, we introduce a hybrid method of analog-digital simulation for
simulating 2D spin models and dynamically changing interactions to achieve a
new graph using a linear 1D chain. The method relies on time domain Hamiltonian
engineering through a successive application of Stark shift gradient pulses,
and wherein the pulse sequence can simply be obtained from a Fourier series
decomposition of the target Hamiltonian over the space of lattice couplings. We
focus on engineering 2D rectangular nearest-neighbor spin lattices,
demonstrating that the required control parameters scale linearly with ion
number. This hybrid approach offers compelling possibilities for the use of 1D
chains in the study of Hamiltonian quenches, dynamical phase transitions, and
quantum transport in 2D and 3D. We discuss a possible experimental
implementation of this approach using real experimental parameters.
