  We study the short-imaginary-time quantum critical dynamics (SITQCD) in the
J-Q$_3$ spin chain, which hosts a quasi-long-range-order phase to a valence
bond solid transition. By using the scaling form of the SITQCD with a saturated
ordered phase, we are able to locate the critical point at $q_{\rm
c}=0.170(14)$. We also obtain the critical initial slip exponent
$\theta=-0.507(3)$ and the static exponent $\beta/\nu=0.498(2)$. More
strikingly, we find that the scaling dimension of the initial order parameter
$x_{0}$ is close to zero, which suggests that the initial order parameter is a
marginal operator. As a result, there is no initial increase behavior of the
order parameter in the short-imaginary-time relaxation process for this model,
which is very different from the relaxation dynamics in the Ising-type phase
transitions. Our numerical results are realized by the projector quantum Monte
Carlo algorithm.
