  The contact process is a paradigmatic classical stochastic system displaying
critical behavior even in one dimension. It features a non-equilibrium phase
transition into an absorbing state that has been widely investigated and shown
to belong to the directed percolation universality class. When the same process
is considered in a quantum setting much less is known. So far mainly
semi-classical studies have been conducted and the nature of the transition in
low dimensions is still a matter of debate. Also from a numerical point of
view, from which the system may look fairly simple --- especially in one
dimension --- results are lacking. In particular the presence of the absorbing
state poses a substantial challenge which appears to affect the reliability of
algorithms targeting directly the steady-state. Here we perform real-time
numerical simulations of the open dynamics of the quantum contact process and
shed light on the existence and on the nature of an absorbing state phase
transition in one dimension. We find evidence for the transition being
continuous and provide first estimates for the critical exponents. Beyond the
conceptual interest, the simplicity of the quantum contact process makes it an
ideal benchmark problem for scrutinizing numerical methods for open quantum
non-equilibrium systems.
