We consider syndrome-based decoding of continuous-variable (CV) codes encoding qubits
in oscillators, assuming independent Gaussian error model. In the channel setting, given the
measured syndrome, the conditional distribution of errors is described by an integer-variable
Gaussian, with the coupling matrix determined by the syndrome and the code structure. For
the encoded qubits, only the parity of these variables is relevant. As a result, maximum-
likelihood (ML) decoding corresponds to an analog of the Nishimori line in a random-bond …