Tensor chain and constraints in tensor networks

Abstract

This paper accompanies with our recent work on quantum error correction (QEC) and entanglement spectrum (ES) in tensor networks (arXiv:1806.05007). We propose a general framework for planar tensor network state with tensor constraints as a model for $Ad{S}_3/CF{T}_2$ correspondence, which could be viewed as a generalization of hyperinvariant tensor networks recently proposed by Evenbly. We elaborate our proposal on tensor chains in a tensor network by tiling $H^2$ space and provide a diagrammatical description for general multi-tensor constraints in terms of tensor chains, which forms a generalized greedy algorithm. The behavior of tensor chains under the action of greedy algorithm is investigated in detail. In particular, for a given set of tensor constraints, a critically protected (CP) tensor chain can be figured out and evaluated by its average reduced interior angle. We classify tensor networks according to their ability of QEC and the flatness of ES. The corresponding geometric description of critical protection over the hyperbolic space is also given.

arXiv comment63 pages, 36 figures

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Cited by in corpus (4)

• Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings
• Surface growth scheme for bulk reconstruction and tensor network
• Thread/State correspondence: from bit threads to qubit threads
• Quantum error correction and entanglement spectrum in tensor networks