mixtures (BFM) analogous to BCS superconductivity. The possibility to trap fermionic states into localized bosonic components of BFM makes these systems also of potential interest for quantum computing. The basic element of a quantum computer is the so called quantum bit, a special quantum state usually realized with a two-level system, which is manipulated by logical quantum gates (unitary operators) and which must be preserved intact for an extended period of time to allow computation [@Nielsen]. An important role in this context is played by the interaction of the quantum system with the environment (thermal reservoir) which unavoidably produces decoherence. For an ideal q-bit one would like to keep the quantum system well protected from decoherence but not completely isolated from the environment to allow gate manipulations. This condition could be realized in Bose-Fermi mixtures by trapping fermionic states into localized BEC, so to form quantum dots with two (or more) fermionic levels inside, and by using Feshbach resonances to control the boson-fermion interaction, i.e. the coupling of the quantum bit with the environment.

The aim of this paper is twofold. From one side, we investigate the possibility of using ultracold Bose-Fermi mixtures to form matter wave quantum dots with 