# Holographic code[1]

## Description

Block quantum code whose encoding isometry serves to model aspects of the AdS/CFT holographic duality. Encoding map models radial time evolution for a fixed time slice in Anti de Sitter (AdS) space, mapping operators in the bulk of AdS, represented by logical qudits, onto operators on the boundary of the corresponding Conformal Field Theory (CFT), represented by physical qudits. Encoding can often be represented by a tensor network associated with a tiling of hyperbolic space. See Defn 4.3 of Ref. [2] for a technical formulation.

Typically, the encoding isometry \(U\) obeys the entanglement-wedge reconstruction condition, which states that for any boundary region \(R\), any bulk operator \(O\) localized to the entanglement wedge of \(R\) must be implementable by some boundary operator \(\tilde{O}\) localized to \(R\). Formally, \(UO = \tilde{O}U\) and \([\tilde{O},UU^\dagger] = 0\). The entanglement wedge is the space enclosed within the Ryu–Takayanagi surface in the bulk (minimal surface) with boundary \(R\).

## Protection

## Transversal Gates

## Code Capacity Threshold

## Notes

## Parent

- Quantum Lego code — Quantum Lego codes codes whose encoders are tensor networks discretizing hyperbolic space can be thought of as holographic codes. More generally, tensor-network codes are types of LEGO codes made from stabilizer codes where logical and physical legs are pre-assigned and logical legs are not contracted. In other words, logical legs resulting from the conversion of codes to tensors must remain logical in the final tensor network, and the same for physical. Contracting logical legs is another word for gluing two logical legs together.

## Children

- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code
- Three-qutrit code — Three-qutrit code is a minimal model for holography [3,8].

## Cousins

- Operator-algebra error-correcting code — Properties of holographic codes are often quantified in the Heisenberg picture, i.e., in terms of operator algebras [3,4].
- Renormalization group (RG) cat code — The RG cat code encoder has similar coarse-graining features as that of a holographic code [9].
- Matrix-model code — Matrix-model codes are motivated by the Ads/CFT correspondence because it is manifest in continuous nonabelian gauge theories with large gauge groups [10].
- Concatenated quantum code — A holographic code whose encoding circuit is arranged in a tree geometry reduces to a concatenated code.
- Hyperbolic surface code — Both holographic and hyperbolic surface codes utilize tesselations of hyperbolic surfaces. Encodings for the former are hyperbolically tiled tensor networks, while the latter is defined on hyperbolically tiled physical-qubit lattices.

## 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]
- K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
- [3]
- 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
- [4]
- F. Pastawski and J. Preskill, “Code Properties from Holographic Geometries”, Physical Review X 7, (2017) arXiv:1612.00017 DOI
- [5]
- S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021) arXiv:2103.13404 DOI
- [6]
- N. Bao, C. Cao, and G. Zhu, “Deconfinement and error thresholds in holography”, Physical Review D 106, (2022) arXiv:2202.04710 DOI
- [7]
- D. Harlow and H. Ooguri, “Symmetries in quantum field theory and quantum gravity”, (2019) arXiv:1810.05338
- [8]
- D. Harlow, “The Ryu–Takayanagi Formula from Quantum Error Correction”, Communications in Mathematical Physics 354, 865 (2017) arXiv:1607.03901 DOI
- [9]
- 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
- [10]
- C. Cao, G. Cheng, and B. Swingle, “Large \(N\) Matrix Quantum Mechanics as a Quantum Memory”, (2022) arXiv:2211.08448

## 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