Improving the efficiency of variational tensor network algorithms

Abstract

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 $\kappa$, then the environment of any other tensor from $\mathcal{T}$ can be evaluated with identical cost $\kappa$. Moreover, we describe how the set of all single tensor environments from $\mathcal{T}$ can be simultaneously evaluated with fixed cost $3\kappa$. 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.

12 pages, 8 figures, RevTex 4.1, includes reference implementation. Software updated to v1.02: Resolved two scenarios in which multienv would generate errors for valid inputs

References in corpus (7)

• The density-matrix renormalization group in the age of matrix product states
• Real time evolution using the density matrix renormalization group
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Entanglement entropy for the long range Ising chain
• On Rényi entropies of disjoint intervals in conformal field theory
• Simulation of time evolution with the MERA

Cited by in corpus (14)

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• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
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• Unifying Projected Entangled Pair States contractions
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• Automatic Contraction of Unstructured Tensor Networks
• Efficient Decomposition of High-Rank Tensors
• Algorithms for Tensor Network Contraction Ordering
• Efficient Contraction of Large Tensor Networks for Weighted Model Counting through Graph Decompositions
• Chiral bosonic quantum spin liquid in the integer-spin Heisenberg-Kitaev model
• Scaling of contraction costs for entanglement renormalization algorithms including tensor Trotterization and variational Monte Carlo
• GuiTeNet: A graphical user interface for tensor networks
• On the Optimal Linear Contraction Order of Tree Tensor Networks, and Beyond
• Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method