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PreprintCC BY 4.0arXiv

A Time-Frequency Framework for GKP Codes

  • Franz Luef
  • Eduard Ortega
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Abstract

We develop a time--frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space M∞ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-* convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

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Cite

BibTeXRIS
BibTeX
@misc{luef2026timefrequency,
  title = {{A Time-Frequency Framework for GKP Codes}},
  author = {Franz Luef and Eduard Ortega},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10802},
  eprint = {2609.10802},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10802},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A Time-Frequency Framework for GKP Codes
AU  - Franz Luef
AU  - Eduard Ortega
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.10802
UR  - https://arxiv.org/abs/2609.10802
AB  - We develop a time--frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space M^∞ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-* convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.
ER  -