  A promising paradigm of quantum computing for achieving practical quantum
advantages is quantum annealing or quantum approximate optimization algorithm,
where the classical problems are encoded in Ising interactions. However, it is
challenging to build a quantum system that can efficiently map any structured
problems. Here, we present a programmable trapped-ion quantum simulator of an
Ising model with all-to-all connectivity with up to four spins. We implement
the spin-spin interactions by using the coupling of trapped ions to multiple
collective motional modes and realize the programmability through phase
modulation of the Raman laser beams that are individually addressed on ions. As
an example, we realize several Ising lattices with different interaction
connectivities, where the interactions can be ferromagnetic or
anti-ferromagnetic. We confirm the programmed interaction geometry by observing
the ground states of the corresponding models through quantum state tomography.
Our experimental demonstrations serve as an important basis for realizing
practical quantum advantages with trapped ions.
