  Electron spins in semiconductor quantum dots are promising candidates for the
experimental realization of solid-state qubits. We analyze the dynamics of a
system of three qubits arranged in a linear geometry and a system of four
qubits arranged in a square geometry. Calculations are performed for several
quantum dot confining potentials. In the three-qubit case, three-body effects
are identified that have an important quantitative influence upon quantum
computation. In the four-qubit case, the full Hamiltonian is found to include
both three-body and four-body interactions that significantly influence the
dynamics in physically relevant parameter regimes. We consider the implications
of these results for the encoded universality paradigm applied to the
four-electron qubit code; in particular, we consider what is required to
circumvent the four-body effects in an encoded system (four spins per encoded
qubit) by the appropriate tuning of experimental parameters.
