  We present a theoretical analysis of a Majorana-based qubit consisting of two
topological superconducting islands connected via a Josephson junction. The
qubit is operated by electrostatic gates which control the coupling of two of
the four Majorana zero modes. At the end of the operation, readout is performed
in the charge basis. Even though the operations are not topologically
protected, the proposed experiment can potentially shed light on the coherence
of the parity degree of freedom in Majorana devices and serve as a first step
towards topological Majorana qubits. We discuss in detail the charge-stability
diagram and its use for characterizing the parameters of the devices, including
the overlap of the Majorana edge states. We describe the multi-level spectral
properties of the system and present a detailed study of its controlled
coherent oscillations, as well as decoherence resulting from coupling to a
non-Markovian environment. In particular, we study a gate-controlled protocol
where conversion between Coulomb-blockade and transmon regimes generates
coherent oscillations of the qubit state due to the overlap of Majorana modes.
We show that, in addition to fluctuations of the Majorana coupling,
considerable measurement errors may be accumulated during the conversion
intervals when electrostatic fluctuations in the superconducting islands are
present. These results are also relevant for several proposed implementations
of topological qubits which rely on readout based on charge detection.
