A Tighter Distance Upper-Bound for Gottesman-Kitaev-Preskill Codes

Gottesman-Kitaev-Preskill (GKP) codes are stabi-lizer codes that allow one to encode qubits into oscillators, and are known to be hardware efficient. The stabilizer group of a G KP code is isomorphic to a lattice. A particular generator matrix of this lattice can be related to a canonical form via a symplectic matrix. An upper bound to the distance of a G KP code based on the Euler decomposition of this symplectic matrix has been derived in the literature. We derive an upper-bound that is tighter than this bound whenever this symplectic matrix is not orthogonal. This enables us to show that the bound in the prior literature is tight only for the non-interesting case when the symplectic matrix is orthogonal. We then provide some necessary conditions for a class of G KP codes to achieve the improved upper bound. This allows us to upper-bound the largest possible distance of a G KP code in this class.