  Jets provide one of the primary probes of the quark-gluon plasma produced in
ultrarelativistic heavy ion collisions and the cold nuclear matter explored in
deep inelastic scattering experiments. However, despite important developments
in the last years, a description of the real-time evolution of QCD jets inside
a medium is still far from being complete. In our previous work, we have
explored quantum technologies as a promising alternative theoretical laboratory
to simulate jet evolution in QCD matter, to overcome inherent technical
difficulties in present calculations. Here, we extend our previous
investigation from the single particle $|q\rangle$ to the
$|q\rangle+|qg\rangle$ Fock space, taking into account gluon production. Based
on the light-front Hamiltonian formalism, we construct a digital quantum
circuit that tracks the evolution of a multi-particle jet probe in the presence
of a medium described as a stochastic color field. Studying the momentum
broadening of the jet state, we observe sizable sub-eikonal effects by
comparing to eikonal estimates. We also study the medium-induced modifications
to the gluon emission probability, which exhibit small corrections compared to
the vacuum splitting function. In addition, we study the time evolution of the
von-Neumann entropy associated with the quark component; we find that the
exponential of the entropy grows linearly in time for the bare quark but
super-linearly when taking into account gluon emission.
