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Vanishing gradients can pose substantial obstacles for high-dimensionaloptimization problems. Here we consider energy minimization problems forquantum many-body systems with extensive Hamiltonians, which can be studied onclassical computers or in the form of variational quantum eigensolvers onquantum computers. Barren plateaus correspond to scenarios where the averageamplitude of the energy gradient decreases exponentially with increasing systemsize. This occurs, for example, for quantum neural networks and for brickwallquantum circuits when the depth increases polynomially in the system size. Herewe prove that the variational optimization problems for matrix product states,tree tensor networks, and the multiscale entanglement renormalization ansatzare free of barren plateaus. The derived scaling properties for the gradientvariance provide an analytical guarantee for the trainability of randomlyinitialized tensor network states (TNS) and motivate certain initializationschemes. In a suitable representation, unitary tensors that parametrize the TNSare sampled according to the uniform Haar measure. We employ a Riemannianformulation of the gradient based optimizations which simplifies the analyticalevaluation.

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