  The design and study of hybrid qubits is driven by their ability to get along
the best of charge qubits and of spin qubits, {\em i.e.} the speed of operation
of the former and the very slow decoherence rates of the latter ones. There are
several proposals to implement hybrid qubits, this works focuses on the
spectral properties of an one-electron hybrid qubit. By design, the information
would be stored in the electronic spin and the switching between the qubit
basis states would be achieved using an external ac electric field. The
electron is confined in a two-dimensional quantum dot, whose confining
potential is given by a quartic potential, features that are typical of GaAS
quantum dots. Besides the confining potential that characterizes the quantum
dot there are two static magnetic fields applied to the system, one is a large
constant Zeeman field and the other one has a constant gradient. We study the
spectral properties of the model Hamiltonian, a Scr\"odinger-Pauli Hamiltonian
with realistic parameters, using the Ritz method. In particular, we look for
regions of the parameter space where the lowest eigenenergies and their
eigenfunctions allow to define a qubit which is stable under perturbations to
the design parameters. We put special attention to the constraints that the
design imposes over the magnetic fields, the tuning of the energy gap between
the qubit states and the expectation value of the spin operator where the
information would be stored.
