  Numerous sufficient conditions for adiabaticity of the evolution of a driven
quantum system have been known for quite a long time. In contrast, necessary
adiabatic conditions are scarce. A practicable necessary condition well-suited
for many-body systems has been proven recently in [Phys. Rev. Lett. 119, 200401
(2017)]. Here we tailor this condition for estimating run times of quantum
adiabatic algorithms. As an illustration, the condition is applied to the
adiabatic algorithm for searching in an unstructured database (adiabatic Grover
search algorithm). We find that thus obtained lower bound on the run time of
this algorithm reproduces $\sqrt N$ scaling ($N$ being the number of database
entries) of the explicitly known optimal run time. This observation highlights
the merits of the new adiabatic condition and its potential relevance to
adiabatic quantum computing.
