  A Majorana-based qubit can form within the fermion parity subspace of a
two-island topological superconductor setup consisting of four Majorana zero
modes. We study the relaxation time of this Majorana qubit induced by the
intrinsic Majorana-phonon interaction in combination with single electron
tunneling between the two islands. The theory is developed for both the
spinless p-wave Kitaev chain model and the spin-orbit-coupled semiconductor
nanowire with induced superconductivity and Zeeman splitting. We systematically
discuss the dependence of the relaxation rate on Majorana charge distribution
(varying with the chemical potential, magnetic field, superconducting paring,
spin-orbit coupling and island length) as well as on phonon wavelength and the
length of the tunnel barrier. Importantly, the accompanied phonon energy can be
tuned by the voltage bias over the tunnel barrier, leading to relatively easy
manipulation and suppression of the relaxation rate.

Mesoscale and Nanoscale Physics;Quantum Physics