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 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 T can be
evaluated with identical cost κ. Moreover, we describe how the set of
all single tensor environments from 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.Comment: 12 pages, 8 figures, RevTex 4.1, includes reference implementation.
Software updated to v1.02: Resolved two scenarios in which multienv would
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