aiqbench analysis
This paper advances fault-tolerant quantum computing by establishing a rigorous time-frequency framework for lattice Gottesman-Kitaev-Preskill (GKP) codes using Gabor analysis and Zak transforms. The authors prove key mathematical properties such as isometry and stable logical reconstruction, which are essential for robust error correction in continuous-variable quantum systems.
A Time-Frequency Framework for GKP Codes
arXiv:2609.10802v1 Announce Type: cross 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.
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.
Mathematical Physics (math-ph); Information Theory (cs.IT); Functional Analysis (math.FA); Operator Algebras (math.OA); Quantum Physics (quant-ph)
arXiv:2609.10802 [math-ph]
(or arXiv:2609.10802v1 [math-ph] for this version)
https://doi.org/10.48550/arXiv.2609.10802 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 215 words
