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Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence (1503.06237v2)

Published 20 Mar 2015 in hep-th and quant-ph

Abstract: We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.

Citations (841)

Summary

  • The paper presents tensor network models using perfect tensors that emulate the AdS/CFT entanglement structure and exactly realize the Ryu-Takayanagi formula.
  • The models support bulk reconstruction by encoding logical qubits in multiple boundary representations, showcasing robust quantum error-correcting capabilities.
  • Numerical analyses reveal erasure thresholds that offer actionable insights into resilient quantum information encoding within holographic frameworks.

Holographic Quantum Error-Correcting Codes: Insights and Implications

In the paper titled "Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence," the authors propose a novel approach for understanding the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence via quantum error-correcting codes structured in tensor networks. This research introduces a theoretical framework which aligns quantum error-correcting codes with the conceptual constructs of holography, particularly focusing on the entanglement aspects of the AdS/CFT duality.

Central to this research is the construction of toy models using holographic tensor networks. These networks are composed of "perfect" tensors, embodying maximal entanglement properties and thereby enabling a transformation from the bulk Hilbert space to the boundary Hilbert space. Through this construction, the paper models features analogous to those in the AdS/CFT correspondence, notably the Ryu-Takayanagi (RT) formula and multipartite entanglement properties such as tripartite information negativity.

Key Insights

  1. Tensor Network Structure: The models utilize perfect tensors laid out on hyperbolic tessellations. Each tensor represents an isometry, ensuring that the networks simulate the entanglement structure anticipated in AdS/CFT. Moreover, the encoding of logical and physical degrees of freedom manifests within these tensor networks, akin to the bulk and boundary in AdS/CFT.
  2. Ryu-Takayanagi Formula Representation: In this setting, the RT formula, which relates the boundary entanglement entropy with a minimal surface in the bulk, is shown to hold exactly across various contexts. This exactness is particularly true for connected boundary regions within these tensor network models, thereby reinforcing their fidelity to the duality's entanglement-factorization aspect.
  3. Quantum Error-Correcting Features and Bulk Reconstruction: A notable focus of the paper is the realization of AdS-Rindler reconstruction, which permits bulk local operators to be represented in multiple distinct ways on the boundary. This is achieved through the network's structure, which supports the code's logical operations and aligns with the holographic dictionary postulated in AdS/CFT.
  4. Numerical Insights and Erasure Thresholds: The research includes numerical analysis and derivations of erasure thresholds for these codes, illustrating the conditions necessary for robust logical qubit encoding and decoding against random erasure errors. Particularly, configurations that alternate pentagons and hexagons demonstrate threshold behavior, providing insights into quantum error correction's resilience in such holographic models.
  5. Implications and Extensions: The implications of this paper are manifold, spanning potential advancements in fault-tolerant quantum computing where such holographic codes could offer new paradigms for dealing with noise through geometric and topological means. Furthermore, they provide theoretical frameworks warranting exploration of emergent spacetime geometries via quantum information perspectives.

Theoretical and Practical Implications

The integration of quantum error-correcting codes within the holographic paradigm offers a compelling synthesis of quantum information theory and quantum gravity concepts. The explicit use of tensor networks to mimic AdS/CFT correspondence suggests broader applications in quantum simulation and quantum causality. Moreover, this research prompts a reevaluation of traditional boundaries between bulk gravitational dynamics and the quantum informational processing on the boundary, potentially influencing future developments in high-energy physics and quantum information science.

Looking forward, further investigation may focus on extending these models to higher-dimensional and dynamic settings, probing the fine details of bulk operator reconstruction, and enhancing the understanding of black-hole microstates and wormhole geometries within this framework. The intersection of these fields could yield substantial contributions to the foundations of quantum gravity and error correction in quantum computing technology.

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