  Quantum heat engines are commonly believed to achieve their optimal
efficiency when operated under quasi-static conditions. However, when running
at finite power, they suffer effective friction due to the generation of
coherences and transitions between energy eigenstates. It was noted that it is
possible to increase the power of a quantum heat engine using external control
schemes or suitable dephasing noise. Here, we investigate the thermodynamic
cost associated with dephasing noise schemes using both numerical and
analytical methods. Our findings unveil that the observed gain in power is
generally not free of thermodynamic costs, as it involves heat flows from
thermal baths into the dephasing bath. These contributions must be duly
accounted for when determining the engine's overall efficiency. Interestingly,
we identify a particular working regime where these costs become negligible,
demonstrating that quantum heat engines can be operated at any power with an
efficiency per cycle that approaches arbitrarily closely that under quasistatic
operation.
