  Randomness is an essential tool in many disciplines of modern sciences, such
as cryptography, black hole physics, random matrix theory and Monte Carlo
sampling. In quantum systems, random operations can be obtained via random
circuits thanks to so-called q-designs, and play a central role in the fast
scrambling conjecture for black holes. Here we consider a more physically
motivated way of generating random evolutions by exploiting the many-body
dynamics of a quantum system driven with stochastic external pulses. We combine
techniques from quantum control, open quantum systems and exactly solvable
models (via the Bethe-Ansatz) to generate Haar-uniform random operations in
driven many-body systems. We show that any fully controllable system converges
to a unitary q-design in the long-time limit. Moreover, we study the
convergence time of a driven spin chain by mapping its random evolution into a
semigroup with an integrable Liouvillean and finding its gap. Remarkably, we
find via Bethe-Ansatz techniques that the gap is independent of q. We use
mean-field techniques to argue that this property may be typical for other
controllable systems, although we explicitly construct counter-examples via
symmetry breaking arguments to show that this is not always the case. Our
findings open up new physical methods to transform classical randomness into
quantum randomness, via a combination of quantum many-body dynamics and random
driving.
