  We investigate the characteristics of purely electrostatic interactions with
external gates in constructing full single qubit manipulations. The quantum bit
is naturally encoded in the spatial wave function of the electron system.
Single-electron{transistor arrays based on quantum dots or insulating
interfaces typically allow for electrostatic controls where the inter-island
tunneling is considered constant, e.g. determined by the thickness of an
insulating layer. A representative array of 3x3 quantum dots with two mobile
electrons is analyzed using a Hubbard Hamiltonian and a capacitance matrix
formalism. Our study shows that it is easy to realize the first quantum gate
for single qubit operations, but that a second quantum gate only comes at the
cost of compromising the low-energy two-level system needed to encode the
qubit. We use perturbative arguments and the Feshbach formalism to show that
the compromising of the two-level system is a rather general feature for
electrostatically interacting qubits and is not just related to the specific
details of the system chosen. We show further that full implementation requires
tunable tunneling or external magnetic fields.
