  Adiabatic quantum algorithms are characterized by their run time and
accuracy. The relation between the two is essential for quantifying adiabatic
algorithmic performance, yet is often poorly understood. We study the dynamics
of a continuous time, adiabatic quantum search algorithm, and find rigorous
results relating the accuracy and the run time. Proceeding with estimates, we
show that under fairly general circumstances the adiabatic algorithmic error
exhibits a behavior with two discernible regimes: the error decreases
exponentially for short times, then decreases polynomially for longer times. We
show that the well known quadratic speedup over classical search is associated
only with the exponential error regime. We illustrate the results through
examples of evolution paths derived by minimization of the adiabatic error. We
also discuss specific strategies for controlling the adiabatic error and run
time.
