# Pastawski-Yoshida-Harlow-Preskill (HaPPY) code[1]

## Description

Also known as a hyperbolic pentagon code (HyPeC). Holographic code constructed out of a network of hexagonal perfect tensors that tesselates hyperbolic space. Physical qubits are associated with uncontracted tensor legs at the boundary of the tesselation, while logical qubits are associated with uncontracted legs in the bulk. The code serves as a minimal model for several aspects of the AdS/CFT holographic duality and potentially a dF/CFT duality [2].

The construction below is described for qubits, but straightforward generalizations exist to modular qudits, oscillators, and rotors [3]. Encoding is accomplished using a tensor network of five-qubit encoding isometries, which are six-legged perfect tensors (with five legs corresponding to the physical qubits and one for the encoded logical qubit). A \(2n\)-legged perfect tensor is proportional to an isometry for any bipartition of its indices into a set \(A\) and a complementary set \(A^{\perp}\) such that \(|A|\leq|A^{\perp}|\).

To construct the encoding, one first uniformly tiles the hyperbolic AdS/CFT disc using pentagons and hexagons. Then, one places a 6-legged five-qubit encoding tensor at each hexagon and pentagon, contracting legs between neighboring shapes and leaving one leg uncontracted at each pentagon. This construction forms an encoding isometry from the uncontracted legs in the bulk to the uncontracted legs at the boundary.

The single-qubit HaPPY code has a central pentagon encoding one bulk operator and hexagons tiling all other layers. The pentagon-hexagon HaPPY code has alternating layers of pentagons and hexagons in the tiling. The pentagon HaPPY code consists of a purely pentagonal tiling.

## Protection

## Rate

## Encoding

## Transversal Gates

## Decoding

## Threshold

## Notes

## Parents

- Qubit stabilizer code — The HaPPY code is a stabilizer code because it is defined by a contracted network of stabilizer tensors; see Thm. 6 in Ref. [1].
- Holographic code

## Cousins

- Five-qubit perfect code — The five-qubit encoding isometry tiles various holographic codes because its corresponding tensor is perfect [1].
- Majorana stabilizer code — HaPPY code Hamiltonian can be expressed in terms of mutually commuting two-body Majorana operators [8].

## 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]
- J. Cotler and A. Strominger, “The Universe as a Quantum Encoder”, (2022) arXiv:2201.11658
- [3]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [4]
- S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021) arXiv:2103.13404 DOI
- [5]
- R. J. Harris et al., “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020) arXiv:2008.10206 DOI
- [6]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [7]
- S. Antonini et al., “Holographic measurement and bulk teleportation”, Journal of High Energy Physics 2022, (2022) arXiv:2209.12903 DOI
- [8]
- A. Jahn et al., “Majorana dimers and holographic quantum error-correcting codes”, Physical Review Research 1, (2019) arXiv:1905.03268 DOI

## Page edit log

- Victor V. Albert (2022-07-28) — most recent
- Victor V. Albert (2021-12-29)
- Joel Rajakumar (2021-12-20)

## Cite as:

“Pastawski-Yoshida-Harlow-Preskill (HaPPY) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/happy

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/happy.yml.