A Time-Frequency Framework for GKP Codes

2026-09-09

AI Summary

本文通过构建时频框架，利用多窗口Gabor分析和Zak变换等方法，解决了GKP码的理想码字表示及稳定逻辑重构问题。

Abstract

We develop a time--frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ 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.

Problem

• time-frequency framework
• GKP codes
• logical vector reconstruction
• displacement syndromes

Innovation

• time-frequency framework
• GKP codes
• vector-valued Zak transform
• Gabor analysis
• stable logical reconstruction