  Phase transitions in dissipative quantum systems are intriguing because they
are induced by the interplay between coherent quantum and incoherent classical
fluctuations. Here, we investigate the crossover from a quantum to a classical
absorbing phase transition arising in the quantum contact process (QCP). The
Lindblad equation contains two parameters, $\omega$ and $\kappa$, which adjust
the contributions of the quantum and classical effects, respectively. We find
that in one dimension when the QCP starts from a homogeneous state with all
active sites, there exists a critical line in the region $0 \le \kappa <
\kappa_*$ along which the exponent $\alpha$ (which is associated with the
density of active sites) decreases continuously from a quantum to the classical
directed percolation (DP) value. This behavior suggests that the quantum
coherent effect remains to some extent near $\kappa=0$. However, when the QCP
in one dimension starts from a heterogeneous state with all inactive sites
except for one active site, all the critical exponents have the classical DP
values for $\kappa \ge 0$. In two dimensions, anomalous crossover behavior does
not occur, and classical DP behavior appears in the entire region of $\kappa
\ge 0$ regardless of the initial configuration. Neural network machine learning
is used to identify the critical line and determine the correlation length
exponent. Numerical simulations using the quantum jump Monte Carlo technique
and tensor network method are performed to determine all the other critical
exponents of the QCP.
