  Quantum simulations can provide new insights into the physics of strongly
correlated electronic systems. A well studied system, but still open in many
regards, is the Hubbard-Holstein Hamiltonian, where electronic repulsion is in
competition with attraction generated by the electron-phonon coupling. In this
context we study the phase diagram of four quantum dots in a suspended carbon
nanotube and coupled to its flexural degrees of freedom. The system is
described by a Hamiltonian of the Hubbard-Holstein class, where electrons on
different sites interact with the same phonon. We find that the system presents
a transition from the Mott insulating state to a polaronic state, with the
appearance of pairing correlations and the breaking of the translational
symmetry. Our study shows that this system thus constitutes a relevant example
of a correlated system that could be studied by experimental realization.
