To implement fault-tolerant quantum computation with continuous variables,
Gottesman-Kitaev-Preskill (GKP) qubits have been recognized as an important
technological element. However, the analog outcome of GKP qubits, which
includes beneficial information to improve the error tolerance, has been
wasted, because the GKP qubits have been treated as only discrete variables. In
this paper, we propose a hybrid quantum error correction approach that combines
digital information with the analog information of the GKP qubits using the
maximum-likelihood method. As an example, we demonstrate that the three-qubit
bit-flip code can correct double errors, whereas the conventional method based
on majority voting on the binary measurement outcome can correct only a single
error. As another example, a concatenated code known as Knill's C4/C6 code can
achieve the hashing bound for the quantum capacity of the Gaussian quantum
channel. To the best of our knowledge, this approach is the first attempt to
draw both digital and analog information from a single quantum state to improve
quantum error correction performance
