Absence of barren plateaus and scaling of energy gradients in the optimization of isometric tensor network states*

Barren plateaus can pose substantial obstacles for high-dimensional optimization problems. Here we consider optimization problems for quantum many-body systems which can be studied on classical computers or in the form of variational quantum eigensolvers on quantum computers. Barren plateaus correspond to scenarios where the average amplitude of the cost function gradient decreases exponentially with increasing system size. This occurs, for example, for quantum neural networks. Here we show that variational optimization problems for matrix product states, tree tensor networks, and the multiscale entanglement renormalization ansatz are free of barren plateaus. The derived scaling properties for the average gradient variance provide an analytical guarantee for the trainability of randomly initialized tensor network states (TNS). In a suitable representation, unitary tensors that parametrize the TNS are sampled according to the uniform Haar measure. In contrast to other works, we employ a Riemannian formulation of the gradient based optimizations which makes the analytical evaluation rather simple and may be useful in other contexts.