# Holographic code[1]

## Description

Block quantum code whose features serve to model aspects of the AdS/CFT holographic duality and, more generally, quantum gravity.

## Notes

## Parent

## Children

- Renormalization group (RG) cat code — The RG cat code encoder has coarse-graining features reminiscent of holography [4].
- Matrix-model code — Matrix-model codes are motivated by the Ads/CFT correspondence because it is manifest in continuous non-Abelian gauge theories with large gauge groups [5].
- Holographic tensor-network code — Holographic codes whose encoders are holographic tensor networks are holographic tensor-network codes.
- Conformal-field theory (CFT) code — CFT codewords lie in the low-energy subspace of a conformal field theory (CFT), e.g., the quantum Ising model at its critical point.
- Kim-Preskill-Tang (KPT) code — The robustness of KPT codes does not rely on arguments from holographic duality, but such codes do aim to describe interiors of black holes.
- SYK code — In a holographic model [6], the large distance of these codes can be interpreted as being due to the emergence of a wormhole.

## Cousins

- Approximate quantum error-correcting code (AQECC) — Universal subspace approximate error correction is used to model black holes [7].
- Approximate operator-algebra QECC — Properties of holographic codes are often quantified in the Heisenberg picture, i.e., in terms of operator algebras [4,8–10].
- 2D lattice stabilizer code — 2D lattice stabilizer codes admit a bulk-boundary correspondence similar to that of holographic codes, namely, the boundary Hilbert space of the former cannot be realized via local degrees of freedom [11].

## References

- [1]
- F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
- [2]
- A. Jahn and J. Eisert, “Holographic tensor network models and quantum error correction: a topical review”, Quantum Science and Technology 6, 033002 (2021) arXiv:2102.02619 DOI
- [3]
- T. Kibe, P. Mandayam, and A. Mukhopadhyay, “Holographic spacetime, black holes and quantum error correcting codes: a review”, The European Physical Journal C 82, (2022) arXiv:2110.14669 DOI
- [4]
- K. Furuya, N. Lashkari, and S. Ouseph, “Real-space RG, error correction and Petz map”, Journal of High Energy Physics 2022, (2022) arXiv:2012.14001 DOI
- [5]
- C. Cao, G. Cheng, and B. Swingle, “Large \(N\) Matrix Quantum Mechanics as a Quantum Memory”, (2022) arXiv:2211.08448
- [6]
- G. Bentsen, P. Nguyen, and B. Swingle, “Approximate Quantum Codes From Long Wormholes”, (2024) arXiv:2310.07770
- [7]
- P. Hayden and G. Penington, “Learning the Alpha-bits of black holes”, Journal of High Energy Physics 2019, (2019) arXiv:1807.06041 DOI
- [8]
- A. Almheiri, X. Dong, and D. Harlow, “Bulk locality and quantum error correction in AdS/CFT”, Journal of High Energy Physics 2015, (2015) arXiv:1411.7041 DOI
- [9]
- F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017) arXiv:1612.00017 DOI
- [10]
- N. Bao and J. Naskar, “Code properties of the holographic Sierpinski triangle”, Physical Review D 106, (2022) arXiv:2203.01379 DOI
- [11]
- T. Schuster et al., “A holographic view of topological stabilizer codes”, (2023) arXiv:2312.04617

## Page edit log

- Victor V. Albert (2022-03-08) — most recent
- Joel Rajakumar (2021-12-20)

## Cite as:

“Holographic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/holographic