  Classical simulation of quantum systems plays an important role in the study
of many-body phenomena and in the benchmarking and verification of quantum
technologies. Exact simulation is often limited to small systems because the
dimension of the Hilbert space increases exponentially with the size of the
system. For systems that possess a high degree of symmetry, however, classical
simulation can reach much larger sizes. Here, we consider an ensemble of
strongly interacting atoms with permutation symmetry, enabling the simulation
of dynamics of hundreds of atoms at arbitrarily long evolution times. The
system is realized by an ensemble of three-level atoms, where one of the levels
corresponds to a highly excited Rydberg state. In the limit of all-to-all
Rydberg blockade, the Hamiltonian is invariant under permutation of the atoms.
Using techniques from representation theory, we construct a block-diagonal form
of the Hamiltonian, where the size of the largest block increases only linearly
with the system size. We apply this formalism to derive efficient pulse
sequences to prepare arbitrary permutation-invariant quantum states. Moreover,
we study the quantum dynamics following a quench, uncovering a parameter regime
in which the system thermalizes slowly and exhibits pronounced revivals. Our
results create new opportunities for the experimental and theoretical study of
large interacting and nonintegrable quantum systems.
