In this dissertation, we present theoretical studies of coupled seminconductor quantum dots containing up to two electrons. Our focus is the exchange interaction between the two electron spins because this quantity governs the fundamental quantum logic gate operation. To gain critical insight into engineering the exchange interaction, we have performed detailed calculations in two types of quantum dot systems: For two-dimensional GaAs coupled dots, numerical exact diagonalization method is utilized to solve the one- and two-particle Schrodinger equations. We have calculated the stability diagram of the coupled dots for different dot geometries and under various magnetic fields, which reveals the relation between the stability diagram topology and the interdot coupling strength under different conditions. We have also performed detailed analysis on the interdot detuning and dot deformation effects which exemplify the general von Neumann-Wigner theorem on molecular state energies.;For coupled quantum dots formed in a single InAs quantum wire, we use the variational Heitler-London method to obtain the single- and two-particle ground state energies. We show the variational Heitler-London methods yield lower ground state energies than the conventional approach. We find that the exchange energy decreases as the wire diameter becomes smaller due to enhanced coulomb interaction in a quasi-one-dimensional system.
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