Dissipative dynamics of a spin system with three-body interaction

In this note we explicitly solve the Lindblad equation for a system of three spins with a three-body interaction, coupled to the environment by bath operators that inject or absorb spin carriers. We exemplify the properties of this solution in the context of a simple instance of Feynman's quantum computer in which a two-qubit program line is executed, applying the $\sqrt{NOT}$ primitive to a one-qubit register.