  Controllable, coherent many-body systems can provide insights into the
fundamental properties of quantum matter, enable the realization of new quantum
phases and could ultimately lead to computational systems that outperform
existing computers based on classical approaches. Here we demonstrate a method
for creating controlled many-body quantum matter that combines
deterministically prepared, reconfigurable arrays of individually trapped cold
atoms with strong, coherent interactions enabled by excitation to Rydberg
states. We realize a programmable Ising-type quantum spin model with tunable
interactions and system sizes of up to 51 qubits. Within this model, we observe
phase transitions into spatially ordered states that break various discrete
symmetries, verify the high-fidelity preparation of these states and
investigate the dynamics across the phase transition in large arrays of atoms.
In particular, we observe robust manybody dynamics corresponding to persistent
oscillations of the order after a rapid quantum quench that results from a
sudden transition across the phase boundary. Our method provides a way of
exploring many-body phenomena on a programmable quantum simulator and could
enable realizations of new quantum algorithms.
