## Description

Kitaev surface code on a fractal geometry, which is obtained by removing qubits from the surface code on a cubic lattice. A related construction, the fractal product code, is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [1]. The underlying classical codes form classical self-correcting memories [4–6].

## Parent

- Homological code — Fractal surface codes are obtained by removing qubits from the 3D surface code on a cubic lattice.

## Cousins

- 3D surface code — Fractal surface codes are obtained by removing qubits from the 3D surface code on a cubic lattice.
- Hypergraph product (HGP) code — The fractal product code is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [1].
- Self-correcting quantum code — The classical codes underlying the fractal product code form classical self-correcting memories [4–6].

## References

- [1]
- C. G. Brell, “A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)”, New Journal of Physics 18, 013050 (2016) arXiv:1411.7046 DOI
- [2]
- G. Zhu, T. Jochym-O’Connor, and A. Dua, “Topological Order, Quantum Codes, and Quantum Computation on Fractal Geometries”, PRX Quantum 3, (2022) arXiv:2108.00018 DOI
- [3]
- A. Dua, T. Jochym-O'Connor, and G. Zhu, “Quantum error correction with fractal topological codes”, Quantum 7, 1122 (2023) arXiv:2201.03568 DOI
- [4]
- A. Vezzani, “Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result”, Journal of Physics A: Mathematical and General 36, 1593 (2003) arXiv:cond-mat/0212497 DOI
- [5]
- R. Campari and D. Cassi, “Generalization of the Peierls-Griffiths theorem for the Ising model on graphs”, Physical Review E 81, (2010) arXiv:1002.1227 DOI
- [6]
- M. Shinoda, “Existence of phase transition of percolation on Sierpiński carpet lattices”, Journal of Applied Probability 39, 1 (2002) DOI

## Page edit log

- Victor V. Albert (2022-01-12) — most recent

## Cite as:

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