Speaker: Thomas Barthel
Affiliation: Duke University.
Date: Tuesday, 6 June 2023 at 12:00
Location: Online seminar
Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states
Vanishing gradients can pose substantial obstacles for high-dimensional
optimization problems. Here we consider energy minimization problems for
quantum many-body systems with extensive Hamiltonians, 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 energy gradient decreases exponentially with increasing system
size. This occurs, for example, for quantum neural networks and for brickwall
quantum circuits when the depth increases polynomially in the system size. Here
we show that the 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 gradient
variance provide an analytical guarantee for the trainability of randomly
initialized tensor network states (TNS) and motivate certain initialization
schemes. In a suitable representation, unitary tensors that parametrize the TNS
are sampled according to the uniform Haar measure. We employ a Riemannian
formulation of the gradient based optimizations which simplifies the analytical
evaluation.