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Fractal surface code[1–3]

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].

Decoding

Sweep local automaton decoder [3].

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].
  • Linear binary 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].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Homological code
Parents
Homological code
Fractal surface codes are obtained by removing qubits from the 3D surface code on a cubic lattice.
Fractal surface code

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
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Zoo Code ID: fractal_surface

Cite as:
“Fractal surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fractal_surface
BibTeX:
@incollection{eczoo_fractal_surface, title={Fractal surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/fractal_surface} }
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Permanent link:
https://errorcorrectionzoo.org/c/fractal_surface

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/higher_d/fractal_surface.yml.