  Recently, the quantum contact process, in which branching and coagulation
processes occur both coherently and incoherently, was theoretically and
experimentally investigated in driven open quantum spin systems. In the
semi-classical approach, the quantum coherence effect was regarded as a process
in which two consecutive atoms are involved in the excitation of a neighboring
atom from the inactive (ground) state to the active state (excited $s$ state).
In this case, both second-order and first-order transitions occur. Therefore, a
tricritical point exists at which the transition belongs to the tricritical
directed percolation (TDP) class. On the other hand, when an atom is excited to
the $d$ state, long-range interaction is induced. Here, to account for this
long-range interaction, we extend the TDP model to one with long-range
interaction in the form of $\sim 1/r^{d+\sigma}$ (denoted as LTDP), where $r$
is the separation, $d$ is the spatial dimension, and $\sigma$ is a control
parameter. In particular, we investigate the properties of the LTDP class below
the upper critical dimension $d_c=$ min$(3,\,1.5\sigma)$. We numerically obtain
a set of critical exponents in the LTDP class and determine the interval of
$\sigma$ for the LTDP class. Finally, we construct a diagram of universality
classes in the space ($d$, $\sigma$).
