  Although the laws of classical physics are deterministic, thermodynamics
gives rise to an arrow of time through irreversible processes. In quantum
mechanics the unitary nature of the time evolution makes it intrinsically
reversible, however the question of how to revert an unknown time evolution
nevertheless remains. Remarkably, there have been several recent demonstrations
of protocols for reverting unknown unitaries in scenarios where even the
interactions with the target system are unknown. The practical use of these
universal rewinding protocols is limited by their probabilistic nature, raising
the fundamental question of whether time-reversal could be performed
deterministically. Here we show that quantum physics indeed allows for
deterministic universal time-reversal by exploiting the non-commuting nature of
quantum operators, and demonstrate a recursive protocol for two-level quantum
systems with an arbitrarily high probability of success. Using a photonic
platform we demonstrate our protocol, reverting the discrete time evolution of
a polarization state with an average state fidelity of over 95%. Our protocol,
requiring no knowledge of the quantum process to be rewound, is optimal in its
running time, and brings quantum rewinding into a regime of practical
relevance.
