Zak transform as a framework for quantum computation with the Gottesman-Kitaev-Preskill code

The Gottesman-Kitaev-Preskill (GKP) code encodes a qubit into bosonic mode using periodic wave functions. This periodicity makes the GKP natural setting for Zak transform, which is tailor-made to provide simple description We review transform and its connection basis of states in Hilbert space, decompose shift operators that underpin stabilizers correctable errors, find transforms position function appear naturally error correction. construct subsystem decomposition (SSD)---the modular variable SSD---by dividing mode's expressed basis, virtual gauge mode. Tracing over gives logical-qubit state, preceding trace with particular logical-gauge interaction different logical state---that associated