  Quantum systems have entered a competitive regime where classical computers
must make approximations to represent highly entangled quantum states. However,
in this beyond-classically-exact regime, fidelity comparisons between quantum
and classical systems have so far been limited to digital quantum devices, and
it remains unsolved how to estimate the actual entanglement content of
experiments. Here we perform fidelity benchmarking and mixed-state entanglement
estimation with a 60-atom analog Rydberg quantum simulator, reaching a high
entanglement entropy regime where exact classical simulation becomes
impractical. Our benchmarking protocol involves extrapolation from comparisons
against an approximate classical algorithm, introduced here, with varying
entanglement limits. We then develop and demonstrate an estimator of the
experimental mixed-state entanglement, finding our experiment is competitive
with state-of-the-art digital quantum devices performing random circuit
evolution. Finally, we compare the experimental fidelity against that achieved
by various approximate classical algorithms, and find that only the algorithm
we introduce is able to keep pace with the experiment on the classical hardware
we employ. Our results enable a new paradigm for evaluating the ability of both
analog and digital quantum devices to generate entanglement in the
beyond-classically-exact regime, and highlight the evolving divide between
quantum and classical systems.
