## 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 (i.e., \(\mathbb{Z}_2\) topological order) along with some trivial codes [1]. Any 2D topological order requires weight-four Hamiltonian terms, i.e., it cannot be stabilized via weight-two or weight-three terms on 2D lattices of qubits or qutrits [2–4].

## Decoding

Tensor-network based decoder for 2D codes subject to correlated noise [5].Standard stabilizer-based error correction can be performed even in the presence of perturbations to the codespace [6,7].

## Code Capacity Threshold

Noise thresholds can be formulated as anyon condensation transitions in a topological field theory [8], generalizing the mapping of the effect of noise on a code state to a statistical mechanical model [5,9–11]. Namely, the noise threshold for a noise channel \(\cal{E}\) acting on a 2D stabilizer state \(|\psi\rangle\) can be obtained from the properties of the resulting (mixed) state \(\mathcal{E}(|\psi\rangle\langle\psi|)\) [8,12–15].

## Parent

## Children

- Twist-defect color code
- 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 [16]. 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 [17–19]. 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 [20].
- Abelian quantum-double stabilizer code — Translation-invariant 2D prime-qudit lattice stabilizer codes are equivalent to several copies of the prime-qudit surface code and a trivial 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 [21].

## 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]
- S. Bravyi and M. Vyalyi, “Commutative version of the k-local Hamiltonian problem and common eigenspace problem”, (2004) arXiv:quant-ph/0308021
- [3]
- D. Aharonov and L. Eldar, “On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems”, (2011) arXiv:1102.0770
- [4]
- D. Aharonov, O. Kenneth, and I. Vigdorovich, “On the Complexity of Two Dimensional Commuting Local Hamiltonians”, (2018) arXiv:1803.02213 DOI
- [5]
- 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
- [6]
- W. Zhong, O. Shtanko, and R. Movassagh, “Advantage of Quantum Neural Networks as Quantum Information Decoders”, (2024) arXiv:2401.06300
- [7]
- A. Lavasani and S. Vijay, “The Stability of Gapped Quantum Matter and Error-Correction with Adiabatic Noise”, (2024) arXiv:2402.14906
- [8]
- Y. Bao et al., “Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions”, (2023) arXiv:2301.05687
- [9]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [10]
- A. A. Kovalev and L. P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling”, Physical Review A 87, (2013) arXiv:1208.2317 DOI
- [11]
- A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
- [12]
- J. Y. Lee, C.-M. Jian, and C. Xu, “Quantum Criticality Under Decoherence or Weak Measurement”, PRX Quantum 4, (2023) arXiv:2301.05238 DOI
- [13]
- R. Fan et al., “Diagnostics of mixed-state topological order and breakdown of quantum memory”, (2024) arXiv:2301.05689
- [14]
- Y.-H. Chen and T. Grover, “Separability transitions in topological states induced by local decoherence”, (2024) arXiv:2309.11879
- [15]
- K. Su, Z. Yang, and C.-M. Jian, “Tapestry of dualities in decohered quantum error correction codes”, (2024) arXiv:2401.17359
- [16]
- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
- [17]
- 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
- [18]
- H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014) arXiv:1107.2707 DOI
- [19]
- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [20]
- Z. Liang et al., “Extracting topological orders of generalized Pauli stabilizer codes in two dimensions”, (2023) arXiv:2312.11170
- [21]
- 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