# 4D surface code[1]

Also known as Kitaev tesseract code.

## Description

A generalization of the Kitaev surface code defined on a 4D lattice. The code serves as a self-correcting quantum memory [1,2].

4D toric code often either refers to the construction on the four-dimensional torus or is an alternative name for the general construction.

## Code Capacity Threshold

Independent \(X,Z\) noise: \(2.117\%\) with Hastings decoder [3] and \(7.3\%\) with RG decoder for 4D surface code [4]. It is conjectured via a statistical-mechanical mapping that the optimal ML decoder yields a threshold of \(11.003\%\) [5].

## Threshold

Phenomenological noise model for the 4D toric code: \(4.35\%\) with RG decoder [4], and \(4.3\%\) under improved BP-OSD decoder [6].Gate-based depolarizing noise: \(0.31\%\) with RG decoder for 4D toric code [4].\(1.59\%\) for independent \(X,Z\) noise and faulty syndrome measurements using the Hastings decoder [3].

## Parents

- Homological code
- Lattice stabilizer code
- Self-correcting quantum code — The 4D toric code is a self-correcting quantum memory [1,2].
- Abelian topological code — The 4D Kitaev surface code realizes 4D \(\mathbb{Z}_2\) gauge theory.

## Cousin

- Haah cubic code (CC) — The energy of any partial implementation of CC1 is proportional to the boundary length similar to the 4D toric code, which can potentially surpress the effects of thermal errors, but it is currently an open problem.

## References

- [1]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [2]
- R. Alicki et al., “On thermal stability of topological qubit in Kitaev’s 4D model”, (2008) arXiv:0811.0033
- [3]
- N. P. Breuckmann et al., “Local Decoders for the 2D and 4D Toric Code”, (2016) arXiv:1609.00510
- [4]
- K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, “Renormalization Group Decoder for a Four-Dimensional Toric Code”, IEEE Transactions on Information Theory 65, 2545 (2019) arXiv:1708.09286 DOI
- [5]
- K. Takeda and H. Nishimori, “Self-dual random-plaquette gauge model and the quantum toric code”, Nuclear Physics B 686, 377 (2004) arXiv:hep-th/0310279 DOI
- [6]
- O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI

## Page edit log

- Victor V. Albert (2024-04-26) — most recent

## Cite as:

“4D surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/4d_surface