  In composite Higgs models, the generation of quark masses requires the
standard model-like quarks to be partially or fully composite states which are
accompanied by composite quark partners. The composite quark partners decay
into a standard model-like quark and an electroweak gauge boson or Higgs boson,
which can be searched for at the LHC. In this article, we study the
phenomenological implications of composite quarks in the minimal composite
Higgs model based on the coset $SO(5)/SO(4)$. We focus on light quark partners
which are embedded in the $SO(4)$ singlet representation. In this case, a
dominant decay mode of the partner quark is into a Higgs boson and a jet, for
which no experimental bounds have been established so far. The presence of
$SO(4)$ singlet partners leads to an enhancement of the di-Higgs production
cross section at the LHC. This will be an interesting experimental signature in
the near future, but, unfortunately, there are no direct bounds available yet
from the experimental analyses. However, we find that the currently available
standard model-like Higgs searches can be used in order to obtain the first
constraints on partially and fully composite quark models with light quark
partners in the $SO(4)$ singlet. We obtain a flavor- and composition parameter
independent bound on the quark partner mass of $M_{U_h} > 310 \mbox{ GeV}$ for
partially composite quark models and $M_{U_h} > 212 \mbox{ GeV}$ for fully
composite quark models.
