Simulating how many-body quantum systems evolve out of equilibrium, for example after a sudden change (“quench”), is a central challenge in quantum physics, relevant for quantum materials, quantum information, and quantum simulation platforms. Yet, while classical algorithms can sometimes handle such tasks, their resource demands (time, computational power, energy) often scale up drastically as system size grows. In this work, the authors deliver a direct comparison between state-of-the-art classical simulation methods and real quantum hardware based on neutral-atom (Rydberg) quantum processors operated in analog mode. They evaluate both run-time and energy consumption, by combining experimental data from a quantum processing unit with numerical benchmarks on classical hardware and fitting scaling laws for both regimes. Their analysis shows that neutral-atom quantum devices are already competitive compared to classical simulation: for sufficiently large systems and long evolutions, the quantum hardware can compete with classical methods while using significantly less energy. These findings suggest that analog quantum simulators are not only a path to computational advantage for hard quantum-dynamics problems, but also a potentially more sustainable and energy-efficient way to compute. This work thus outlines a concrete roadmap and benchmark for when, where, and why quantum hardware could outperform classical machines in studying non-equilibrium quantum dynamics.