Continuous-variable Gate Teleportation and Bosonic-code Error Correction*

Large continuous-variable (CV) cluster states containing 1000s of modes have recently been used to experimentally demonstrate measurement-based single- and two-mode gate operations. A key primitive component of CV cluster states is a chain of two-mode squeezed states (TMSSs) called a macronode wire. Non-local, two-mode homodyne measurements are used to teleport an input state along the wire. Adjusting the measurement bases applies Gaussian gates to the input states via gate teleportation.

We extend this protocol by replacing the TMSSs in the macronode wire with more general two-mode entangled states. Teleporting through these states realizes non-Gaussian and potentially non-unitary CV gates—a powerful supplement to standard gate teleportation. We apply our general result to show how Gottesman-Kitaev-Preskill (GKP) error correction can be performed in a teleportation-based fashion in the macronode wire (and thus other CV cluster-state architectures) even when the GKP-state source is probabilistic.