  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. As a new
approach, we consider here the quantum earth mover's (EM) or Wasserstein-1
distance, recently proposed in [De Palma et al., arXiv:2009.04469] 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. Our qWGAN requires resources polynomial in
the number of qubits, and our numerical experiments demonstrate that it is
capable of learning a diverse set of quantum data.
