  The search for an application of near-term quantum devices is widespread.
Quantum Machine Learning is touted as a potential utilisation of such devices,
particularly those which are out of the reach of the simulation capabilities of
classical computers. In this work, we propose a generative Quantum Machine
Learning Model, called the Ising Born Machine (IBM), which we show cannot, in
the worst case, and up to suitable notions of error, be simulated efficiently
by a classical device. We also show this holds for all the circuit families
encountered during training. In particular, we explore quantum circuit learning
using non-universal circuits derived from Ising Model Hamiltonians, which are
implementable on near term quantum devices.
  We propose two novel training methods for the IBM by utilising the Stein
Discrepancy and the Sinkhorn Divergence cost functions. We show numerically,
both using a simulator within Rigetti's Forest platform and on the Aspen-1 16Q
chip, that the cost functions we suggest outperform the more commonly used
Maximum Mean Discrepancy (MMD) for differentiable training. We also propose an
improvement to the MMD by proposing a novel utilisation of quantum kernels
which we demonstrate provides improvements over its classical counterpart. We
discuss the potential of these methods to learn `hard' quantum distributions, a
feat which would demonstrate the advantage of quantum over classical computers,
and provide the first formal definitions for what we call `Quantum Learning
Supremacy'. Finally, we propose a novel view on the area of quantum circuit
compilation by using the IBM to `mimic' target quantum circuits using classical
output data only.
