# Surface-code-fragment (SCF) holographic code[1]

## Description

Holographic tensor-network code constructed out of a network of encoding isometries of the \([[5,1,2]]\) rotated surface code. The structure of the isometry is similar to that of the HaPPY code since both isometries are rank-six tensors. In the case of the SCF holographic code, the isometry is only a planar-perfect tensor (as opposed to a perfect tensor).

## Rate

Zero-rate version of the code surpasses the hashing bound under certain Pauli noise [2].

## Code Capacity Threshold

\(7.1\%\) and \(8.2\%\) for even and odd raddi reduced-rate codes, respectively, under depolarizing using the integer optimization decoder [1].

## Parents

- Qubit CSS code
- Holographic tensor-network code — The encoding of the heptagon holographic code is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a planar-perfect tensor.

## Child

- \([[5,1,2]]\) rotated surface code — The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code. The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a planar-perfect tensor.

## Cousin

- Planar-perfect-tensor code — The encoding of the heptagon holographic code is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a planar-perfect tensor.

## References

- [1]
- R. J. Harris et al., “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020) arXiv:2008.10206 DOI
- [2]
- J. Fan et al., “Overcoming the Zero-Rate Hashing Bound with Holographic Quantum Error Correction”, (2024) arXiv:2408.06232

## Page edit log

- Victor V. Albert (2024-07-01) — most recent

## Cite as:

“Surface-code-fragment (SCF) holographic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/holographic_5_1_2