  A crucial milestone in the field of quantum simulation and computation is to
demonstrate that a quantum device can compute certain tasks that are impossible
to reproduce by a classical computer with any reasonable resources. Such a
demonstration is referred to as quantum supremacy. One of the most important
questions is to identify setups that exhibit quantum supremacy and can be
implemented with current quantum technology. The two standard candidates are
boson sampling and random quantum circuits. Here, we show that quantum
supremacy can be obtained in generic periodically-driven quantum many-body
systems. Our analysis is based on the eigenstate thermalization hypothesis and
strongly-held conjectures in complexity theory. To illustrate our work, We give
examples of simple disordered Ising chains driven by global magnetic fields and
Bose-Hubbard chains with modulated hoppings. Our proposal opens the way for a
large class of quantum platforms to demonstrate and benchmark quantum
supremacy.
