GKP码的时频框架
A Time-Frequency Framework for GKP Codes
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中文总结 AI 辅助
本文为格GKP码建立时频框架，利用Zak变换与Gabor分析实现逻辑向量的等距表示、稳定重建及位移综合征恢复。
中文摘要 AI 辅助
我们为格GKP码发展了一个时频框架，其中理想码字在调制空间$M^\infty$中实现，并通过向量值Zak变换与连续综合征环面上的有限逻辑纤维识别。多窗Gabor分析随后通过有限块的对偶格系数表示逻辑向量。我们证明了归一化块映射是等距的，获得了显式恢复投影，并推导了稳定的逻辑重建。我们进一步构造了作为格包络Gabor乘子的可归一化GKP近似，并建立了弱-$*$收敛和渐近等距编码。最后，我们从平移系数块之间的相位关系恢复位移综合征，并量化其在加性扰动下的稳定性。
英文摘要
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.
发表机构
- NTNU（挪威科技大学）
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