Improving the efficiency of variational tensor network algorithms

We present several results relating to the contraction of generic tensor networks and discuss their application to the simulation of quantum many-body systems using variational approaches based upon tensor network states. Given a closed tensor network mathcal{T}, we prove that if the environment of a single tensor from the network can be evaluated with computational cost κ, then the environment of any other tensor from mathcal{T} can be evaluated with identical cost κ. Moreover, we describe how the set of all single tensor environments from mathcal{T} can be simultaneously evaluated with fixed cost 3κ. The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a Multi-scale Entanglement Renormalization Ansatz (MERA) for the ground state of a 1D quantum system, where they are shown to substantially reduce the computation time.