  Tensor network algorithms seek to minimize correlations to compress the
classical data representing quantum states. Tensor network algorithms and
similar tools---called tensor network methods---form the backbone of modern
numerical methods used to simulate many-body physics and have a further range
of applications in machine learning. Finding and contracting tensor network
states is a computational task which quantum computers might be used to
accelerate. We present a quantum algorithm which returns a classical
description of a rank-$r$ tensor network state satisfying an area law and
approximating an eigenvector given black-box access to a unitary matrix. Our
work creates a bridge between several contemporary approaches, including tensor
networks, the variational quantum eigensolver (VQE), quantum approximate
optimization (QAOA), and quantum computation.
