  Continuous-variable (CV) cluster states are a universal resource for
fault-tolerant quantum computation when supplemented with the
Gottesman-Kitaev-Preskill (GKP) bosonic code. We generalize the recently
introduced subsystem decomposition of a bosonic code [Phys. Rev. Lett. 125,
040501 (2020)], and we use it to analyze CV cluster-state quantum computing
with GKP states. Specifically, we decompose squeezed vacuum states and
approximate GKP states to reveal their encoded logical information, and we
decompose several gates crucial to CV cluster-state quantum computing. Then, we
use the subsystem decomposition to quantify damage to the logical information
in approximate GKP states teleported through noisy CV cluster states. Each of
these studies uses the subsystem decomposition to circumvent complications
arising from the full CV nature of the mode in order to focus on the encoded
qubit information.
