  We investigate the scattering of a quark on a heavy nucleus at high energies
using the time-dependent basis light-front quantization (tBLFQ) formalism,
which is the first application of the tBLFQ formalism in QCD. We present the
real-time evolution of the quark wave function in a strong classical color
field of the relativistic nucleus, described as the Color Glass Condensate. The
quark and the nucleus color field are simulated in the QCD SU(3) color space.
We calculate the total and the differential cross sections, and the quark
distribution in coordinate and color spaces using the tBLFQ approach. We
recover the eikonal cross sections in the eikonal limit. We find that the
differential cross section from the tBLFQ simulation is in agreement with a
perturbative calculation at large $p_\perp$, and it deviates from the
perturbative calculation at small $p_\perp$ due to higher-order contributions.
In particular, we relax the eikonal limit by letting the quark carry realistic
finite longitudinal momenta. We study the sub-eikonal effect on the quark
through the transverse coordinate distribution of the quark with different
longitudinal momentum, and we find the sub-eikonal effect to be sizable. Our
results can significantly reduce the theoretical uncertainties in small
$p_\perp$ region which has important implications to the phenomenology of the
hadron-nucleus and deep inelastic scattering at high energies.
