  Machine learning is a promising application of quantum computing, but
challenges remain as near-term devices will have a limited number of physical
qubits and high error rates. Motivated by the usefulness of tensor networks for
machine learning in the classical context, we propose quantum computing
approaches to both discriminative and generative learning, with circuits based
on tree and matrix product state tensor networks that could have benefits for
near-term devices. The result is a unified framework where classical and
quantum computing can benefit from the same theoretical and algorithmic
developments, and the same model can be trained classically then transferred to
the quantum setting for additional optimization. Tensor network circuits can
also provide qubit-efficient schemes where, depending on the architecture, the
number of physical qubits required scales only logarithmically with, or
independently of the input or output data sizes. We demonstrate our proposals
with numerical experiments, training a discriminative model to perform
handwriting recognition using a optimization procedure that could be carried
out on quantum hardware, and testing the noise resilience of the trained model.
