  We analyze the barren plateau phenomenon in the variational optimization of
quantum circuits inspired by matrix product states (qMPS), tree tensor networks
(qTTN), and the multiscale entanglement renormalization ansatz (qMERA). We
consider as the cost function the expectation value of a Hamiltonian that is a
sum of local terms. For randomly chosen variational parameters we show that the
variance of the cost function gradient decreases exponentially with the
distance of a Hamiltonian term from the canonical centre in the quantum tensor
network. Therefore, as a function of qubit count, for qMPS most gradient
variances decrease exponentially and for qTTN as well as qMERA they decrease
polynomially. We also show that the calculation of these gradients is
exponentially more efficient on a classical computer than on a quantum
computer.
