Stabiliser codes over prime fields as weighted graphs and as quantum sets of lines
Abstract
We further develop the theory of quantum stabiliser codes and its links with graph theory and finite geometry. We prove that an [[n,k,d]]_p stabiliser code over a finite field with p elements, p prime, gives rise to a graph on n-k vertices together with two disjoint independent sets of size k. Its edges are weighted with elements of a finite field of size p. If k=0, such codes are known as graph states. Second, we view stabiliser codes as so-called quantum sets of lines in projective space, allowing us to determined their minimum distance geometrically. We describe two geometrical operations to obtain new quantum sets of lines from given quantum sets of lines. As an application, we list the number of stabiliser codes up to equivalence, for given small parameters.
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BibTeX
@misc{ball2026stabiliser,
title = {{Stabiliser codes over prime fields as weighted graphs and as quantum sets of lines}},
author = {Simeon Ball and Gülizar Günay and Robin Simoens and Ricard Vilar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05489},
eprint = {2611.05489},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05489},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stabiliser codes over prime fields as weighted graphs and as quantum sets of lines
AU - Simeon Ball
AU - Gülizar Günay
AU - Robin Simoens
AU - Ricard Vilar
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05489
UR - https://arxiv.org/abs/2611.05489
AB - We further develop the theory of quantum stabiliser codes and its links with graph theory and finite geometry. We prove that an [[n,k,d]]_p stabiliser code over a finite field with p elements, p prime, gives rise to a graph on n-k vertices together with two disjoint independent sets of size k. Its edges are weighted with elements of a finite field of size p. If k=0, such codes are known as graph states. Second, we view stabiliser codes as so-called quantum sets of lines in projective space, allowing us to determined their minimum distance geometrically. We describe two geometrical operations to obtain new quantum sets of lines from given quantum sets of lines. As an application, we list the number of stabiliser codes up to equivalence, for given small parameters.
ER -