  We explore the imaginary-time relaxation dynamics near quantum critical
points with semi-ordered initial states. Different from the case with
homogeneous ordered initial states, in which the order parameter $M$ decays
homogeneously as $M\propto \tau^{-\beta/\nu z}$, here $M$ depends on the
location $x$, showing rich scaling behaviors. Similar to the classical
relaxation dynamics with an initial domain wall in Model A, which describes the
purely dissipative dynamics, here as the imaginary time evolves, the domain
wall expands into an interfacial region with growing size. In the interfacial
region, the local order parameter decays as $M\propto \tau^{-\beta_1/\nu z}$,
with $\beta_1$ being an additional dynamic critical exponent. Far away from the
interfacial region the local order parameter decays as $M\propto
\tau^{-\beta/\nu z}$ in the short-time stage, then crosses over to the scaling
behavior of $M\propto \tau^{-\beta_1/\nu z}$ when the location $x$ is absorbed
in the interfacial region. A full scaling form characterizing these scaling
properties is developed. The quantum Ising model in both one and two dimensions
are taken as examples to verify the scaling theory. In addition, we find that
for the quantum Ising model the scaling function is an analytical function and
$\beta_1$ is not an independent exponent.
