Time frequency framework improves lattice gkp quantum code analysis
A Time-Frequency Framework for GKP Codes
Information Theory
Summary
Quantum error correction helps protect quantum information, but designing and analyzing these codes can be complex. The authors created a new mathematical framework using time and frequency concepts to better understand certain quantum codes called lattice GKP codes. Their approach allows for stable reconstruction of logical quantum information and helps detect error syndromes more reliably. This could make it easier to work with these codes for future quantum technologies.
What this means in practice
- •For quantum hardware engineers: Design error correction routines with better logical state recovery using time-frequency methods for GKP codes.
- •For signal processing developers: Apply multi-window Gabor techniques inspired by quantum code analysis to improve signal encoding and decoding stability.
A theory result. No direct application yet.
Authors
Franz Luef, Eduard Ortega
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.