  We extend, to the quantum domain, the results obtained in [Nucl. Phys. B 885
(2014) 150] and [Phys. Lett. B 738 (2014) 405] concerning the Niederer's
transformation for the Pais-Uhlenbeck oscillator. Namely, the quantum
counterpart (an unitary operator) of the transformation which maps the free
higher derivatives theory into the Pais-Uhlenbeck oscillator is constructed.
Some consequences of this transformation are discussed.
