## Description

Lattice stabilizer code in two spatial dimensions.

Any prime-qudit code can be converted to several copies of the prime-qudit 2D surface code along with some trivial codes [1].

## Decoding

Tensor-network based decoder for 2D codes subject to correlated noise [2].Standard stabilizer-based error correction can be performed even in the presence of perturbations to the codespace [3].

## Parent

## Children

- Twist-defect surface code
- Abelian TQD stabilizer code — All Abelian TQD codes can be realized as modular-qudit lattice stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and condensing certain bosonic anyons [4]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.
- Galois-qudit topological code

## Cousins

- Kitaev surface code — Translation-invariant 2D qubit lattice stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [5–7]. There exists an algorithm with which one can determine the fusion and braiding rules of a 2D translationally invariant qubit code, and decompose the given code into copies of the surface code [8].
- Abelian quantum-double stabilizer code — Translation-invariant 2D prime-qudit lattice stabilizer codes are equivalent to several copies of the prime-qudit surface code via a local constant-depth Clifford circuit [1].
- Holographic code — 2D lattice stabilizer codes admit a bulk-boundary correspondence similar to that of holographic codes, namely, the boundary Hilbert space of the former cannot be realized via local degrees of freedom [9].

## References

- [1]
- J. Haah, “Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices”, Journal of Mathematical Physics 62, (2021) arXiv:1812.11193 DOI
- [2]
- C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D 8, 269 (2021) arXiv:1809.10704 DOI
- [3]
- W. Zhong, O. Shtanko, and R. Movassagh, “Advantage of Quantum Neural Networks as Quantum Information Decoders”, (2024) arXiv:2401.06300
- [4]
- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
- [5]
- H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
- [6]
- H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014) arXiv:1107.2707 DOI
- [7]
- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [8]
- Z. Liang et al., “Extracting topological orders of generalized Pauli stabilizer codes in two dimensions”, (2023) arXiv:2312.11170
- [9]
- T. Schuster et al., “A holographic view of topological stabilizer codes”, (2023) arXiv:2312.04617

## Page edit log

- Victor V. Albert (2024-01-27) — most recent

## Cite as:

“2D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_stabilizer