Quantum Machine Learning Matrix Product States

Abstract

Matrix product states minimize bipartite correlations to compress the classical data representing quantum states. Matrix product state algorithms and similar tools---called tensor network methods---form the backbone of modern numerical methods used to simulate many-body physics. Matrix product states have a further range of applications in machine learning. Finding matrix product states is in general a computationally challenging task, a computational task which we show quantum computers can accelerate. We present a quantum algorithm which returns a classical description of a $k$-rank matrix product state approximating an eigenvector given black-box access to a unitary matrix. Each iteration of the optimization requires $O(n\cdot k^2)$ quantum gates, yielding sufficient conditions for our quantum variational algorithm to terminate in polynomial-time.