  Identifying universal properties of non-equilibrium quantum states is a major
challenge in modern physics. A fascinating prediction is that classical
hydrodynamics emerges universally in the evolution of any interacting quantum
system. Here, we experimentally probe the quantum dynamics of 51 individually
controlled ions, realizing a long-range interacting spin chain. By measuring
space-time resolved correlation functions in an infinite temperature state, we
observe a whole family of hydrodynamic universality classes, ranging from
normal diffusion to anomalous superdiffusion, that are described by L\'evy
flights. We extract the transport coefficients of the hydrodynamic theory,
reflecting the microscopic properties of the system. Our observations
demonstrate the potential for engineered quantum systems to provide key
insights into universal properties of non-equilibrium states of quantum matter.
