True Galois-qudit stabilizer code[13] 

Description

Also called a linear stabilizer code. A \([[n,k,d]]_q\) stabilizer code whose stabilizer's symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(GF(q)\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only.

The number of generators \(r\) for a true stabilizer code is a multiple of \(m\) (recall that \(q=p^m\) for Galois qudits). As a result, the number \(k=n-r/m\) of logical qudits is an integer.

Each code can be represented by a stabilizer generator matrix \(H=(A|B)\), where each row \((a|b)\) is the Galois symplectic representation of a stabilizer generator.

Protection

Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits.

Notes

See Ref. [4,5] for introductions to various stabilizer code constructions.

Parent

Children

Cousin

  • Linear \(q\)-ary code — A true Galois-qudit stabilizer code is the closest quantum analogue of a linear code over \(GF(q)\) because the \(q\)-ary vectors corresponding to the symplectic representation of the stabilizers form a linear subspace.

References

[1]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[2]
A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
[3]
D. Gottesman. Surviving as a quantum computer in a classical world
[4]
A. Klappenecker, “Algebraic quantum coding theory”, Quantum Error Correction 307 (2013) DOI
[5]
M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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Zoo Code ID: galois_true_stabilizer

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/galois_true_stabilizer.yml.