  We study theoretically a double quantum dot hydrogen molecule in the GaAs
conduction band as the basic elementary gate for a quantum computer with the
electron spins in the dots serving as qubits. Such a two-dot system provides
the necessary two-qubit entanglement required for quantum computation. We
determine the excitation spectrum of two horizontally coupled quantum dots with
two confined electrons, and study its dependence on an external magnetic field.
In particular, we focus on the splitting of the lowest singlet and triplet
states, the double occupation probability of the lowest states, and the
relative energy scales of these states. We point out that at zero magnetic
field it is difficult to have both a vanishing double occupation probability
for a small error rate and a sizable exchange coupling for fast gating. On the
other hand, finite magnetic fields may provide finite exchange coupling for
quantum computer operations with small errors. We critically discuss the
applicability of the envelope function approach in the current scheme and also
the merits of various quantum chemical approaches in dealing with few-electron
problems in quantum dots, such as the Hartree-Fock self-consistent field
method, the molecular orbital method, the Heisenberg model, and the Hubbard
model. We also discuss a number of relevant issues in quantum dot quantum
computing in the context of our calculations, such as the required design
tolerance, spin decoherence, adiabatic transitions, magnetic field control, and
error correction.
