This study reports the experimental observation of anomalous transport in a kicked quasicrystal, a system with a rich phase diagram that includes fully localized, fully delocalized, and multifractal states. The research demonstrates that the kicked Aubry–André–Harper (kAAH) model exhibits anomalous transport across a broad range of parameters, with transport properties showing re-entrant delocalization and sub-ballistic transport. The experimental realization of the kAAH model is enabled by new Floquet engineering techniques, which expand the accessible parameter space by five orders of magnitude. The study also presents a theoretical explanation of these phenomena based on eigenstate multifractality. The experiments involve loading a Bose-Einstein condensate of strontium atoms into an optical lattice and applying periodic kicks to realize the kAAH Hamiltonian. The phase diagram is mapped by measuring the long-time evolution of the density distribution of an initial wave packet. The results show that the system exhibits multifractal states in an extended parameter regime, leading to anomalous transport. The study also demonstrates the use of apodized Floquet engineering to suppress interband heating and improve the performance of the quantum simulator. The findings highlight the importance of multifractality in understanding the transport properties of driven quantum systems and open new avenues for exploring exotic states of matter in quantum simulations. The results are consistent with theoretical predictions and provide insights into the behavior of driven quantum systems in the presence of disorder and periodic driving.This study reports the experimental observation of anomalous transport in a kicked quasicrystal, a system with a rich phase diagram that includes fully localized, fully delocalized, and multifractal states. The research demonstrates that the kicked Aubry–André–Harper (kAAH) model exhibits anomalous transport across a broad range of parameters, with transport properties showing re-entrant delocalization and sub-ballistic transport. The experimental realization of the kAAH model is enabled by new Floquet engineering techniques, which expand the accessible parameter space by five orders of magnitude. The study also presents a theoretical explanation of these phenomena based on eigenstate multifractality. The experiments involve loading a Bose-Einstein condensate of strontium atoms into an optical lattice and applying periodic kicks to realize the kAAH Hamiltonian. The phase diagram is mapped by measuring the long-time evolution of the density distribution of an initial wave packet. The results show that the system exhibits multifractal states in an extended parameter regime, leading to anomalous transport. The study also demonstrates the use of apodized Floquet engineering to suppress interband heating and improve the performance of the quantum simulator. The findings highlight the importance of multifractality in understanding the transport properties of driven quantum systems and open new avenues for exploring exotic states of matter in quantum simulations. The results are consistent with theoretical predictions and provide insights into the behavior of driven quantum systems in the presence of disorder and periodic driving.