Exchange interaction between three and four coupled quantum dots: Theory and applications to quantum computing

Several prominent proposals have suggested that spins of localized electrons could serve as quantum computer qubits. The exchange interaction has been invoked as a means of implementing two qubit gates. In this paper, we analyze the strength and form of the exchange interaction under relevant conditions. We find that, when several spins are engaged in mutual interactions, the quantitative strengths or even qualitative forms of the interactions can change. A variety of interaction forms can arise depending on the symmetry of the system. It is shown that the changes can be dramatic within a Heitler-London model. Hund-M\"ulliken calculations are also presented, and support the qualitative conclusions from the Heitler-London model. The effects need to be considered in spin-based quantum computer designs, either as a source of gate error to be overcome or a new interaction to be exploited.