  Long range interaction between distant spins is an important building block
for the realization of large quantum-dot network in which couplings between
pairs of spins can be selectively addressed. Recent experiments on coherent
logical states oscillation between remote spins facilitated by intermediate
electron states has paved the first step for large scale quantum information
processing. Reaching this ultimate goal requires extensive studies on the
superexchange interaction on different quantum-dot spatial arrangements and
electron configurations. Here, we consider a linear triple-quantum-dot with two
anti-parallel spins in the outer dots forming the logical states while various
number of electrons in the middle dot forming a mediator, which facilitates the
superexchange interaction. We show that the superexchange is enhanced when the
number of mediating electrons increases. In addition, we show that forming a
four-electron triplet in the mediator dot further enhance the superexchange
strength. Our work can be a guide to scale up the quantum-dot array with
controllable and dense connectivity.
