Dynamical GKP codes

Dynamical codes protect quantum information with sequences of measurements, generalizing the stabilizer formalism of static codes. However, the dynamical formalism has been developed exclusively for qubit systems. While Gottesman-Kitaev-Preskill (GKP) codes are promising candidates for quantum computing that encode qubits in the continuous phase space of oscillators, no dynamical formalism has been developed for them. In this work, we bridge this gap and introduce dynamical GKP codes, combining continuous-variable lattice encodings with time-dependent translation measurement dynamics. To obtain the update rules and the distance of a given measurement sequence, we first generalize GKP codes to subsystem GKP codes using scaled symplectic lattices. Building on this structure, we adapt the ancestry, detector, and benign-error machinery originally developed for dynamical qubit codes. From this construction, we obtain a general and efficient method to compute the syndrome and distance of a dynamical GKP code from a persistent stabilizer lattice and a bounded inference window. We illustrate the framework with the example of the two-mode square-hexagonal Floquet GKP code. This code has reduced measurement weight compared to both the square and hexagonal GKP codes, while combining their set of logical gates, resulting in easier implementation of the full single-qubit Clifford group. Finally, we show that measurements with irrational symplectic overlap, which have no analog in discrete qubit codes, reduce to destructive homodyne detection in the single-mode case. These results extend an important qubit-code tool to continuous-variable systems and identify a new bridge between discrete stabilizer measurements and continuous homodyne detection.