Anderson localization confines the wave function of a quantum particle in a one-dimensional random potential to a volume of the order of the localization length $\xi$. Nonlinear add-ons to the wave dynamics mimic many-body interactions on a mean-field level, and result in escape from the Anderson cage and in unlimited subdiffusion of the interacting cloud. We address quantum corrections to that subdiffusion by (i) using the ultrafast unitary Floquet dynamics of discrete-time quantum walks, (ii) an interaction strength ramping to speed up the subdiffusion, and (iii) an action discretization of the nonlinear terms. We observe the saturation of the cloud expansion of $N$ particles to a volume $\sim N\xi$. We predict and observe a universal intermediate logarithmic expansion regime which connects the mean-field diffusion with the final saturation regime and is entirely controlled by particle number $N$. The temporal window of that regime grows exponentially with the localization length $\xi$.

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• Anderson localization
• Diffusion
• Quantum transport
• Semiclassical physics