Perfect state transfer on Cayley graphs over dihedral groups: A complete and practical characterization

@misc{wang2026perfect,
  title = {{Perfect state transfer on Cayley graphs over dihedral groups: A complete and practical characterization}},
  author = {Shixin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30347},
  eprint = {2608.30347},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30347},
}

TY  - UNPB
TI  - Perfect state transfer on Cayley graphs over dihedral groups: A complete and practical characterization
AU  - Shixin Wang
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30347
UR  - https://arxiv.org/abs/2608.30347
AB  - Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. Explicit characterizations of connection sets admitting perfect state transfer in Cayley graphs are rare and, so far, are known only for a few abelian Cayley graphs. In this paper, we characterize the conjugation-closed connection sets of connected Cayley graphs over dihedral groups that admit perfect state transfer. By applying Ramanujan sums, Möbius inversion, and arguments based on the p-adic exponential valuation of rational numbers, we convert the eigenvalue constraints imposed by perfect state transfer into explicit structural conditions on the connection set. This yields a complete and practical characterization, which gives an effective criterion for recognizing and constructing such Cayley graphs and also determines the exact minimum perfect state transfer time.
ER  -