Optimization at the boundary of the tensor network variety

Abstract

Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g. as in projected entangled pair states (PEPS), then the set of tensor network states of given bond dimension is not closed. Its closure is the tensor network variety. Recent work has shown that states on the boundary of this variety can yield more efficient representations for states of physical interest, but it remained unclear how to systematically find and optimize over such representations. We address this issue by defining a new ansatz class of states that includes states at the boundary of the tensor network variety of given bond dimension. We show how to optimize over this class in order to find ground states of local Hamiltonians by only slightly modifying standard algorithms and code for tensor networks. We apply this new method to a different of models and observe favorable energies and runtimes when compared with standard tensor network methods.

20 pages, 6 figures. Close to published version

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• Criticality, the area law, and the computational power of PEPS
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
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Cited by in corpus (11)

• On the closedness and geometry of tensor network state sets
• The minimal canonical form of a tensor network
• Dimension of Tensor Network varieties
• The resource theory of tensor networks
• The Linear Span of Uniform Matrix Product States
• Classifying Entanglement by Algebraic Geometry
• Persistent Tensors and Multiqudit Entanglement Transformation
• Partial Degeneration of Tensors
• Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
• Quantum max-flow in the bridge graph
• The product structure of MPS-under-permutations