  Quantum supremacy is the ability of quantum processors to outperform
classical computers at certain tasks. In digital random quantum circuit
approaches for supremacy, the output distribution produced is described by the
Porter-Thomas (PT) distribution. In this regime, the system uniformly explores
its entire Hilbert space, which makes simulating such quantum dynamics with
classical computational resources impossible for large systems. However, the
latter has no direct application so far in solving a specific problem. In this
work, we show that the same sampling complexity can be achieved from driven
analog quantum processors, with less stringent requirements for coherence and
control. More importantly, we discuss how to apply this approach to solve
problems in quantum simulations of phases of matter and machine learning.
Specifically, we consider a simple quantum spin chain with nearest-neighbor
interactions driven by a global magnetic field. We show how quantum supremacy
is achieved as a consequence of the thermalization due to the interplay between
the disorder and the driven many-body dynamics. We analyze how the achieved PT
distribution can be used as an accessible reference distribution to probe the
many-body localization (MBL) phase transition. In the second part of our work,
we show how our setup can be used for generative modeling machine learning
tasks. We propose a novel variational hybrid quantum-classical approach,
exploiting the system's inherent tunable MBL dynamics, to train the device to
learn distributions of complex classical data. The performance of our training
protocol depends solely on the phase that the quantum system is in, which makes
fine-tuning of local parameters not necessary. The protocol is implementable in
a range of driven quantum many-body systems, compatible with noisy
intermediate-scale quantum devices.
