Local-dimension-invariant low-density parity-check stabilizer codes

Abstract

The protection of quantum information in systems leveraging more than two basis states (qudits) is an ongoing challenge. A tool for transforming the plethora of qubit stabilizer codes into qudit stabilizer codes is the local-dimension-invariant (LDI) extension to the stabilizer formalism. A drawback of the LDI method is that generally in order to ensure the distance of the resultant code is at least preserved the newly selected local-dimension will need to be any prime bigger than a large cutoff value. In this work we consider the repercussions of this large cutoff value, then by focusing on quantum low-density parity-check (qLDPC) codes we are able to bring down this cutoff value. Notably, we are able to reduce it in some cases to the initial local-dimension and for arbitrary CSS qubit-derived codes to singly exponential in d with no dependence on the number of logicals k. This reduces the size of the caveat in LDI forms in their distance preservation condition.

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BibTeX RIS

BibTeX

@misc{islam2026localdimensioninvariant,
  title = {{Local-dimension-invariant low-density parity-check stabilizer codes}},
  author = {Md. Shahinul Islam and Aaryan Reza and Lane G. Gunderman},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15893},
  eprint = {2610.15893},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15893},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Local-dimension-invariant low-density parity-check stabilizer codes
AU  - Md. Shahinul Islam
AU  - Aaryan Reza
AU  - Lane G. Gunderman
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15893
UR  - https://arxiv.org/abs/2610.15893
AB  - The protection of quantum information in systems leveraging more than two basis states (qudits) is an ongoing challenge. A tool for transforming the plethora of qubit stabilizer codes into qudit stabilizer codes is the local-dimension-invariant (LDI) extension to the stabilizer formalism. A drawback of the LDI method is that generally in order to ensure the distance of the resultant code is at least preserved the newly selected local-dimension will need to be any prime bigger than a large cutoff value. In this work we consider the repercussions of this large cutoff value, then by focusing on quantum low-density parity-check (qLDPC) codes we are able to bring down this cutoff value. Notably, we are able to reduce it in some cases to the initial local-dimension and for arbitrary CSS qubit-derived codes to singly exponential in d with no dependence on the number of logicals k. This reduces the size of the caveat in LDI forms in their distance preservation condition.
ER  -

Abstract

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