  The cost and the error of the adiabatic theorem for preparing the final
eigenstate are discussed in terms of path length. Previous studies in terms of
the norm of the Hamiltonian and its derivatives with the spectral gap are
limited to describe the cost of adiabatic state preparation for large systems.
We argue that total time is not a good measure for determining the
computational difficulty of adiabatic quantum computation by developing a no-go
theorem. From the result of time-periodic Hamiltonian cases, we suggest that
there are proxies for computational cost which typically grow as path length
increases when the error is kept fixed and small and consider possible
conjectures on how general the behavior is.
