Galois-qudit topological code[13] 

Description

Abelian topological code, such as a surface [1,3] or color [2] code, constructed on lattices of Galois qudits.

Parents

Child

  • Kitaev surface code — The Galois-qudit surface code for \(q=2\) reduces to the surface code.

Cousins

  • Abelian quantum-double stabilizer code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [4]; see Sec. 5.3 of Ref. [5]. Galois-qudit topological surface and color codes yield Abelian quantum-double codes via this decomposition.
  • Two-dimensional color code — The 2D color code has been extended to Galois qudits.

References

[1]
S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007) arXiv:quant-ph/0609070 DOI
[2]
P. Sarvepalli, “Topological color codes over higher alphabet”, 2010 IEEE Information Theory Workshop (2010) DOI
[3]
I. Andriyanova, D. Maurice, and J.-P. Tillich, “New constructions of CSS codes obtained by moving to higher alphabets”, (2012) arXiv:1202.3338
[4]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[5]
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
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Zoo Code ID: galois_topological

Cite as:
“Galois-qudit topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_topological
BibTeX:
@incollection{eczoo_galois_topological, title={Galois-qudit topological code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/galois_topological} }
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Cite as:

“Galois-qudit topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_topological

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/qldpc/galois_topological.yml.