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A promising paradigm of quantum computing for achieving practical quantumadvantages is quantum annealing or quantum approximate optimization algorithm,where the classical problems are encoded in Ising interactions. However, it ischallenging to build a quantum system that can efficiently map any structuredproblems. Here, we present a programmable trapped-ion quantum simulator of anIsing model with all-to-all connectivity with up to four spins. We implementthe spin-spin interactions by using the coupling of trapped ions to multiplecollective motional modes and realize the programmability through phasemodulation of the Raman laser beams that are individually addressed on ions. Asan example, we realize several Ising lattices with different interactionconnectivities, where the interactions can be ferromagnetic oranti-ferromagnetic. We confirm the programmed interaction geometry by observingthe ground states of the corresponding models through quantum state tomography.Our experimental demonstrations serve as an important basis for realizingpractical quantum advantages with trapped ions.

Realization of programmable Ising models in a trapped-ion quantum simulator