Holographic codes from hyperinvariant tensor networks

Creators: Steinberg, Matthew; Feld, Sebastian; Jahn, Alexander

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.

Copyright and License

© The Author(s) 2023. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Acknowledgement

We would like to thank Chris Akers, Charles Cao, Philippe Faist, Jens Eisert, David Elkouss, and John Preskill for helpful discussions and comments. M.S. and S.F. are grateful for financial support from the Intel Corporation. A.J. is supported by the Simons Collaboration on It from Qubit, the US Department of Energy (DE-SC0018407), and the Einstein Research Unit "Perspectives of a quantum digital transformation".

Funding

Open Access funding enabled and organized by Projekt DEAL.

Data Availability

The authors declare that the data supporting the findings of this study are available within the paper.

Code Availability

The Mathematica code used to verify the analytical calculations is available from the corresponding author on request.

Contributions

The paper was prepared jointly by the authors. M.S. developed the tensor network code which is the main result of this paper. S.F. and A.J. provided project guidance, with S.F. contributing to quantum code aspects and AJ developing the connection to holography.

Conflict of Interest

The authors declare no competing interests.

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