Continuous-variable gate teleportation and bosonic-code error correction

We examine continuous-variable gate teleportation using entangled states made from pure product states sent through a beam splitter. We show that such states are Choi states for a (typically) nonunitary gate, and we derive the associated Kraus operator for teleportation, which can be used to realize non-Gaussian, nonunitary 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 (GKP) 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.