2D lattice stabilizer code 

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 [24].

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 [5,911]. 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 \(\mathcal{E}(|\psi\rangle\langle\psi|)\) [8,1215].

Parent

Children

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 [1719]. 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
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Zoo Code ID: 2d_stabilizer

Cite as:
“2D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_stabilizer
BibTeX:
@incollection{eczoo_2d_stabilizer, title={2D lattice stabilizer code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/2d_stabilizer} }
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“2D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/stabilizer/lattice/2d_stabilizer.yml.