  By computing the low-lying energy excitation spectra with the density matrix
renormalization group algorithm we show that boundaries polarized in the
direction of the transverse field lead to scale-invariant conformal towers of
states at the critical point of the quantum four-state Potts model - a special
symmetric case of the Ashkin-Teller model. Furthermore, by direct comparison of
the excitation spectra we phenomenologically establish the duality between the
transverse-polarized and three-state-mixed boundary conditions at the
four-state Potts critical point. Finally, for completeness, we verify that in
the quantum three-state Potts model the "new" boundary conditions dual to the
mixed ones can be realized by polarizing edge spins along the transverse field.
