  We examine continuous-variable gate teleportation using entangled states made
from pure product states sent through a beamsplitter. We show that such states
are Choi states for a (typically) non-unitary gate, and we derive the
associated Kraus operator for teleportation, which can be used to realize
non-Gaussian, non-unitary quantum operations on an input state. With this
result, we show how gate teleportation is used to perform error correction on
bosonic qubits encoded using the Gottesman-Kitaev-Preskill code. This result is
presented in the context of deterministically produced macronode cluster
states, generated by constant-depth linear optical networks, supplemented with
a probabilistic supply of GKP states. The upshot of our technique is that state
injection for both gate teleportation and error correction can be achieved
without active squeezing operations -- an experimental bottleneck for quantum
optical implementations.
