  The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode
is a promising bosonic code for quantum computation due to its tolerance for
noise and all-Gaussian gate set. We present a toolkit for phase-space
description and manipulation of GKP encodings that includes Wigner functions
for ideal and approximate GKP states, for various types of mixed GKP states,
and for GKP-encoded operators. One advantage of a phase-space approach is that
Gaussian unitaries, required for computation with GKP codes, correspond to
simple transformations on the arguments of Wigner functions. We use this fact
and our toolkit to describe GKP error correction, including magic-state
preparation, entirely in phase space using operations on Wigner functions.
While our focus here is on the square-lattice GKP code, we provide a general
framework for GKP codes defined on any lattice.
