  A reciprocating quantum refrigerator is studied with the purpose of
determining the limitations of cooling to absolute zero. We find that if the
energy spectrum of the working medium possesses an uncontrollable gap, then
there is a minimum achievable temperature above zero. Such a gap, combined with
a negligible amount of noise, prevents adiabatic following during the
demagnetization stage which is the necessary condition for reaching $T_c \to
0$. The refrigerator is based on an Otto cycle where the working medium is an
interacting spin system with an energy gap. For this system the external
control Hamiltonian does not commute with the internal interaction. As a result
during the demagnetization and magnetization segments of the operating cycle
the system cannot follow adiabatically the temporal change in the energy
levels. We connect the nonadiabatic dynamics to quantum friction. An adiabatic
measure is defined characterizing the rate of change of the Hamiltonian. Closed
form solutions are found for a constant adiabatic measure for all the cycle
segments. We have identified a family of quantized frictionless cycles with
increasing cycle times. These cycles minimize the entropy production. Such
frictionless cycles are able to cool to $T_c=0$. External noise on the controls
eliminates these frictionless cycles. The influence of phase and amplitude
noise on the demagnetization and magnetization segments is explicitly derived.
An extensive numerical study of optimal cooling cycles was carried out which
showed that at sufficiently low temperature the noise always dominates
restricting the minimum temperature.
