Resetting uncontrolled quantum systems

We consider a scenario where we wish to bring a closed system of known Hilbert space dimension dS (the target), subject to an unknown Hamiltonian evolution, back to its quantum state at a past time t0. The target is out of our control: this means that we ignore both its free Hamiltonian and how the system interacts with other quantum systems we may use to influence it. Under these conditions, we prove that there exist protocols within the framework of non-relativistic quantum physics which reset the target system to its exact quantum state at t0. Each "resetting protocol" is successful with non-zero probability for all possible free Hamiltonians and interaction unitaries, save a subset of zero measure. When the target is a qubit and the interaction is sampled from the Haar measure, the simplest resetting circuits have a significant average probability of success and their implementation is within reach of current quantum technologies. Finally, we find that, in case the resetting protocol fails, it is possible to run a further protocol that, if successful, undoes both the natural evolution of the target and the effects of the failed protocol over the latter. By chaining in this fashion several such protocols, one can substantially increase the overall probability of a successful resetting.