  Adiabatic limit is the presumption of the adiabatic geometric quantum
computation and of the adiabatic quantum algorithm. But in reality, the
variation speed of the Hamiltonian is finite. Here we develop a general
formulation of adiabatic quantum computing, which accurately describes the
evolution of the quantum state in a perturbative way, in which the adiabatic
limit is the zeroth-order approximation. As an application of this formulation,
non-adiabatic correction or error is estimated for several physical
implementations of the adiabatic geometric gates. A quantum computing process
consisting of many adiabatic gate operations is considered, for which the total
non-adiabatic error is found to be about the sum of those of all the gates.
This is a useful constraint on the computational power. The formalism is also
briefly applied to the adiabatic quantum algorithm.
