  Quantifying how far the output of a learning algorithm is from its target is
an essential task in machine learning. However, in quantum settings, the loss
landscapes of commonly used distance metrics often produce undesirable outcomes
such as poor local minima and exponentially decaying gradients. To overcome
these obstacles, we consider here the recently proposed quantum earth mover's
(EM) or Wasserstein-1 distance as a quantum analog to the classical EM
distance. We show that the quantum EM distance possesses unique properties, not
found in other commonly used quantum distance metrics, that make quantum
learning more stable and efficient. We propose a quantum Wasserstein generative
adversarial network (qWGAN) which takes advantage of the quantum EM distance
and provides an efficient means of performing learning on quantum data. We
provide examples where our qWGAN is capable of learning a diverse set of
quantum data with only resources polynomial in the number of qubits.
