Qudit cubic code

Qudit cubic code[1–3]

Generalization of the Haah cubic code to modular qudits.

Performance over the erasure and depolarizing channels was studied in Ref. [4].

Fracton stabilizer code — Haah cubic [5] codes 1-4, 7, 8, and 10 do not have string logical operators and are the first examples of Type-II fracton phases. The remaining cubic codes are fractal Type-I fracton codes [6,7]. There is evidence that a qutrit and a \(q=5\) qudit cubic code from Ref. [1] have no string operators and are thus Type-II fracton codes (see [6; Eqs. (D11-D13)]).

Homological rotor code — The qudit cubic code can be generalized to rotors [2,3].
Analog stabilizer code — The qudit cubic code can be generalized to oscillators [3].

[1]: I. H. Kim, “3D local qupit quantum code without string logical operator”, (2012) arXiv:1202.0052
[2]: J. Haah, Two generalizations of the cubic code model, KITP Conference: Frontiers of Quantum Information Physics, UCSB, Santa Barbara, CA.
[3]: V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
[4]: A. J. Moncy and P. Kiran Sarvepalli, “Performance of Nonbinary Cubic Codes”, 2018 International Symposium on Information Theory and Its Applications (ISITA) (2018) DOI
[5]: J. Haah, “Local stabilizer codes in three dimensions without string logical operators”, Physical Review A 83, (2011) arXiv:1101.1962 DOI
[6]: A. Dua, I. H. Kim, M. Cheng, and D. J. Williamson, “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
[7]: M. Pretko, X. Chen, and Y. You, “Fracton phases of matter”, International Journal of Modern Physics A 35, 2030003 (2020) arXiv:2001.01722 DOI

Page edit log

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