Holographic Codes from Hyperinvariant Tensor Networks

Abstract

Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code's logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.

10 pages, 7 figures

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• Far from Perfect: Quantum Error Correction with (Hyperinvariant) Evenbly Codes
• Vertex Representation of Hyperbolic Tensor Networks
• On Infinite Tensor Networks, Complementary Recovery and Type II Factors
• Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
• Absolutely maximally entangled pure states of multipartite quantum systems
• Biased-Noise Thresholds of Zero-Rate Holographic Codes with Tensor-Network Decoding
• Scalable and fault-tolerant preparation of encoded k-uniform states