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We present a voltage gate-based method for controlling the Majorana transmon,using a sinusoidal modulation of the induced offset charge $n_g$. Working inthe transmon regime and in the instantaneous eigenstates basis, we find thetime evolution under this protocol that realises tunable $X$-$Z$ rotations. Weoptimise the parameters of the system for different single-qubit gates in boththe laboratory frame and the qubit rotating frame, obtaining qubit controlerrors $1-\mathcal{F}$ smaller than $\sim2\times 10^{-4}$. In addition to this,we conduct an analysis of the effects of the charge noise, assuming wide-band$1/f$ additive noise in $n_g$, both for the free and the driven evolutions. Forthe free evolution, the relaxation and dephasing rates are calculatedperturbatively, obtaining long dephasing times of the order of milliseconds atthe system's sweet-spots. For the driven case, the average fidelity for the$X$-gate is obtained via a numerical simulation, demonstrating remarkableresilience.

Implementation of single-qubit gates via parametric modulation in the Majorana transmon