This question is similar to the Phys.SE post Retrodiction in Quantum Mechanics, however, it addresses a different issue: how would you design a machine that can measure a simple quantum system and "rewind it"? The machine is allowed to generate as much entropy as it needs to to do so.
Lets say two molecules (~10 atoms each) smash into each-other (~30 km/s relative velocity) and spew atoms all over the place. You try to reconstruct the initial state by precisely measuring the position and velocity of each atom in the cloud and rewinding the dynamics. In classical, deterministic physics, this is doable. In quantum mechanics you can't precisely measure the position and momentum/velocity.
However, it may be possible to design the detector to measure position precisely, and put the detector walls far enough away so that the prior momentum can be inferred based on time-of flight. If this will work, will the walls have to get exponentially further as the number of atoms increases?
The ultimate "quantum scrambler" is a black hole. Even a plank mass hole is 10^19 heavier than a hydrogen atom. The walls may have to be placed farther away than the de-sitter horizon, making it impossible to extract any information from even the tiniest hole.