  We study the imaginary-time relaxation critical dynamics of a quantum system
with a vanishing initial correlation length and an arbitrary initial order
parameter $M_0$. We find that in quantum critical dynamics, the behavior of
$M_0$ under scale transformations deviates from a simple power-law, which was
proposed for very small $M_0$ previously. A universal characteristic function
is then suggested to describe the rescaled initial magnetization, similar to
classical critical dynamics. This characteristic function is shown to be able
to describe the quantum critical dynamics in both short- and long-time stages
of the evolution. The one-dimensional transverse-field Ising model is employed
to numerically determine the specific form of the characteristic function. We
demonstrate that it is applicable as long as the system is in the vicinity of
the quantum critical point. The universality of the characteristic function is
confirmed by numerical simulations of models belonging to the same universality
class.
