Checkerboard model code

Checkerboard model code[1]

Description

A foliated type-I fracton code defined on a cubic lattice that admits weight-eight \(X\)- and \(Z\)-type stabilizer generators on the eight vertices of each cube in the lattice. A tetrahedral Ising model can be used to obtain the checkerboard model by gauging [2–4,4] its subsystem symmetry [5].

Variants include the twisted checkerboard model [6].

Decoding

Parallelized matching decoder [7].

Code Capacity Threshold

Independent \(X,Z\) noise: \(\approx 7.5\%\), higher than 3D surface code and color code [8].

Cousins

X-cube model code— The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary [9].
Fracton Floquet code— The ISG of the X-cube Floquet code can be that of the X-cube model code or the checkerboard model code.

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 Qubit Hamiltonian-based QECC Quantum

Parents

Qubit CSS code

Fracton stabilizer codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based QECC Quantum

The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary [9]. Hence, it is a foliated type-I fracton code.

Lifted-product (LP) codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

The checkerboard model code can be formulated directly as an LP code [10].

Checkerboard model code

References

[1]: S. Vijay, J. Haah, and L. Fu, “A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations”, Physical Review B 92, (2015) arXiv:1505.02576 DOI
[2]: M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[3]: L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
[4]: W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[5]: S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[6]: H. Song, A. Prem, S.-J. Huang, and M. A. Martin-Delgado, “Twisted fracton models in three dimensions”, Physical Review B 99, (2019) arXiv:1805.06899 DOI
[7]: B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI
[8]: H. Song, J. Schönmeier-Kromer, K. Liu, O. Viyuela, L. Pollet, and M. A. Martin-Delgado, “Optimal Thresholds for Fracton Codes and Random Spin Models with Subsystem Symmetry”, Physical Review Letters 129, (2022) arXiv:2112.05122 DOI
[9]: W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order in the checkerboard model”, Physical Review B 99, (2019) arXiv:1806.08633 DOI
[10]: Y. Tan, B. Roberts, N. Tantivasadakarn, B. Yoshida, and N. Y. Yao, “Fracton models from product codes”, (2024) arXiv:2312.08462

Page edit log

Victor V. Albert (2024-01-30) — most recent

Cite as:

“Checkerboard model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/checkerboard

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/fracton/checkerboard.yml.