Charge qubits and limitations of electrostatic quantum gates

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.