  We provide rigorous bounds for the error of the adiabatic approximation of
quantum mechanics under four sources of experimental error: perturbations in
the initial condition, systematic time-dependent perturbations in the
Hamiltonian, coupling to low-energy quantum systems, and decoherent
time-dependent perturbations in the Hamiltonian. For decoherent perturbations,
we find both upper and lower bounds on the evolution time to guarantee the
adiabatic approximation performs within a prescribed tolerance. Our new results
include explicit definitions of constants, and we apply them to the spin-1/2
particle in a rotating magnetic field, and to the superconducting flux qubit.
We compare the theoretical bounds on the superconducting flux qubit to
simulation results.
