Generalized quantum partial adiabatic evolution

Due to the intrinsic similarity between partial adiabatic evolution and global adiabatic evolution, we generalize the partial adiabatic evolution proposed recently to its local adiabatic algorithm version. However, unlike that the local adiabatic evolution can speed up the global adiabatic algorithm quadratically, we prove that this new quantum algorithm presented here just has the same time complexity as the original partial adiabatic evolution. This may imply the optimality of the original partial adiabatic evolution or its generalized version. Additionally, a concrete example is given to further support our conclusion.