Stabilizer subsystem decompositions for single- and multi-mode Gottesman-Kitaev-Preskill codes*

The Gottesman-Kitaev-Preskill (GKP) error correcting code encodes a finite dimensional logical space in one or more bosonic modes, and has recently been demostrated in trapped ions and su- perconducting microwave cavities. In this work we introduce a new subsystem decomposition for GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to quantum stabilizer codes. The decomposition has the defining property that a partial trace over the non-logical stabilizer subsystem is equivalent to an ideal decoding of the logical state. We describe how to decompose arbitrary states across the subsystem decomposition using set of transformations that move between the decompositions of different GKP codes. Besides providing a convenient theoretical view on GKP codes, such a decomposition is also of practical use. We use the stabilizer subsystem decomposition to efficiently simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis simulations, we are able to to consider essentially arbitrarily large photon numbers for realistic noise channels such as loss and dephasing.