Quantum Machine Learning Tensor Network States

Tensor network states 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 tensor states is in general a computationally challenging task, a computational task which quantum computers might be used to accelerate. We present a quantum algorithm which returns a classical description of a $k$-rank tensor network state satisfying an area law and approximating an eigenvector given black-box access to a unitary matrix. Each iteration of the optimization requires $O(n\cdot k^2)$ quantum gates.