  The nonequilibrium quantum dynamics of closed many-body systems is a rich yet
challenging field. While recent progress for periodically driven (Floquet)
systems has yielded a number of rigorous results, our understanding on quantum
many-body systems driven by rapidly varying but a- and quasi-periodic driving
is still limited. Here, we derive rigorous, non-perturbative, bounds on the
heating rate in quantum many-body systems under Thue-Morse quasi-periodic
driving and under random multipolar driving, the latter being a tunably
randomized variant of the former. In the process, we derive a static effective
Hamiltonian that describes the transient prethermal state, including the
dynamics of local observables. Our bound for Thue-Morse quasi-periodic driving
suggests that the heating time scales like $(\omega/g)^{-C\ln(\omega/g)}$ with
a positive constant $C$ and a typical energy scale $g$ of the Hamiltonian, in
agreement with our numerical simulations.
