  We propose a new scheme of measurement-based quantum computation (MBQC) using
an error-correcting code against photon-loss in circuit quantum
electrodynamics. We describe a specific protocol of logical single-qubit gates
given by sequential cavity measurements for logical MBQC and a generalised
Schr\"odinger cat state is used for a continuous-variable (CV) logical qubit
captured in a microwave cavity. It is assumed that a three CV-qudit entangled
state is initially prepared in three jointed cavities and the microwave qudit
states are individually controlled, operated, and measured through a readout
resonator coupled with an ancillary superconducting qubit. We then examine a
practical approach of how to create the CV-qudit cluster state via a cross-Kerr
interaction induced by intermediary superconducting qubits between neighbouring
cavities under the Jaynes-Cummings Hamiltonian. This approach could be scalable
for building 2D logical cluster states and therefore will pave a new pathway of
logical MBQC in superconducting circuits toward fault-tolerant quantum
computing.
